module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 37
} | {
"line": 139,
"column": 2
} | [
{
"pp": "R : Type uR\nM : Type uM\nι : Type uι\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : Finite ι\ni j : ι\n⊢ (b.dualBasis i) (b j) = if j = i then 1 else 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"case e'_3.h₁\nR : Type uR\nM : Type uM\nι : Type uι\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : Finite ι\ni j : ι\n⊢ j = i ↔ i = j"
] | convert! b.toDual_apply i j using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 182,
"column": 2
} | {
"line": 184,
"column": 89
} | {
"line": 186,
"column": 0
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\n⊢ (Dual.eval R M).range = ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjectiv... | [] | classical
cases nonempty_fintype ι
rw [← b.toDual_toDual, range_comp, b.toDual_range, Submodule.map_top, toDual_range _] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 182,
"column": 2
} | {
"line": 184,
"column": 89
} | {
"line": 186,
"column": 0
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\n⊢ (Dual.eval R M).range = ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjectiv... | [] | classical
cases nonempty_fintype ι
rw [← b.toDual_toDual, range_comp, b.toDual_range, Submodule.map_top, toDual_range _] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 182,
"column": 2
} | {
"line": 184,
"column": 89
} | {
"line": 186,
"column": 0
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\n⊢ (Dual.eval R M).range = ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjectiv... | [] | classical
cases nonempty_fintype ι
rw [← b.toDual_toDual, range_comp, b.toDual_range, Submodule.map_top, toDual_range _] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 101,
"column": 47
} | {
"line": 101,
"column": 52
} | {
"line": 101,
"column": 52
} | [
{
"pp": "R : Type u_1\nM : Type u\nM' : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : HasRankNullity.{u, u_1} R\nf : M →ₗ[R] M'\np : Submodule R M'\n⊢ ∀ x ∈ comap f p, f x ∈ p",
"ppTerm": "?m.96",
"assigned": true,
"use... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 101,
"column": 47
} | {
"line": 101,
"column": 52
} | {
"line": 101,
"column": 52
} | [
{
"pp": "R : Type u_1\nM : Type u\nM' : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : HasRankNullity.{u, u_1} R\nf : M →ₗ[R] M'\np : Submodule R M'\n⊢ ∀ x ∈ comap f p, f x ∈ p",
"ppTerm": "?m.96",
"assigned": true,
"use... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 101,
"column": 47
} | {
"line": 101,
"column": 52
} | {
"line": 101,
"column": 52
} | [
{
"pp": "R : Type u_1\nM : Type u\nM' : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : HasRankNullity.{u, u_1} R\nf : M →ₗ[R] M'\np : Submodule R M'\n⊢ ∀ x ∈ comap f p, f x ∈ p",
"ppTerm": "?m.96",
"assigned": true,
"use... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 93,
"column": 29
} | {
"line": 93,
"column": 34
} | {
"line": 94,
"column": 6
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nW : Submodule K V\nv : V\nhv : v ∉ W\nhW : W ⊔ K ∙ v = ⊤\nthis : W ⊓ K ∙ v = ⊥\n⊢ v ≠ 0",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Submodule",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 11
} | {
"line": 106,
"column": 4
} | [
{
"pp": "case refine_2.hdim\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nW : Submodule K V\nv : V\nhv : v ∉ W\nhW : finrank K (V ⧸ W) = 1\n⊢ 1 ≤ finrank K ↥(K ∙ v)",
"ppTerm": "?refine_2.hdim",
"assigned": true,
"usedConst... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 111,
"column": 56
} | {
"line": 111,
"column": 61
} | {
"line": 111,
"column": 61
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\np : Submodule K V\nv : V\nhv : v ∉ p\n⊢ v ≠ 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Submodule",
"False",
"Submodule.addSubmonoidCl... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 111,
"column": 56
} | {
"line": 111,
"column": 61
} | {
"line": 111,
"column": 61
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\np : Submodule K V\nv : V\nhv : v ∉ p\n⊢ v ≠ 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Submodule",
"False",
"Submodule.addSubmonoidCl... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 111,
"column": 56
} | {
"line": 111,
"column": 61
} | {
"line": 111,
"column": 61
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\np : Submodule K V\nv : V\nhv : v ∉ p\n⊢ v ≠ 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Submodule",
"False",
"Submodule.addSubmonoidCl... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Contraction | {
"line": 162,
"column": 43
} | {
"line": 162,
"column": 71
} | {
"line": 162,
"column": 71
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb : Basis ι R M\nx : M →ₗ[R] N\n⊢ (dualTensorHomEquivOfBasis b) ((dualTensorHomEquivO... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb : Basis ι R M\nx : M →ₗ[R] N\n⊢ x = x"
] | LinearEquiv.apply_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.Finite.Matrix | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 51
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type u'\nM : Type v\ninst✝⁹ : Ring R\ninst✝⁸ : Ring S\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : StrongRankCondition R\ninst✝² : StrongRankCondition S\ninst✝¹ : Module R S\ninst✝ : SMulCommClass R S S\n⊢ Module.rank S (M →ₗ[R] ... | [] | rw [rank_linearMap, rank_self, lift_one, mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.FreeModule.Finite.Matrix | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 51
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type u'\nM : Type v\ninst✝⁹ : Ring R\ninst✝⁸ : Ring S\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : StrongRankCondition R\ninst✝² : StrongRankCondition S\ninst✝¹ : Module R S\ninst✝ : SMulCommClass R S S\n⊢ Module.rank S (M →ₗ[R] ... | [] | rw [rank_linearMap, rank_self, lift_one, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FreeModule.Finite.Matrix | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 51
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type u'\nM : Type v\ninst✝⁹ : Ring R\ninst✝⁸ : Ring S\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Free R M\ninst✝⁴ : Module.Finite R M\ninst✝³ : StrongRankCondition R\ninst✝² : StrongRankCondition S\ninst✝¹ : Module R S\ninst✝ : SMulCommClass R S S\n⊢ Module.rank S (M →ₗ[R] ... | [] | rw [rank_linearMap, rank_self, lift_one, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 65
} | {
"line": 135,
"column": 0
} | [
{
"pp": "K : Type u_13\nK₁ : Type u_14\nV : Type u_16\nV₁ : Type u_17\nn : Type u_19\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nI₁ I₁' : K₁ →+* K\nB : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V\nv : n → V₁\nhv₁ : B.IsOrthoᵢ v\nhv₂ : ∀ (i : n... | [] | exact (smul_eq_zero.mp this).elim _root_.id (hv₂ i · |>.elim) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 694,
"column": 7
} | {
"line": 694,
"column": 38
} | {
"line": 694,
"column": 38
} | [
{
"pp": "R : Type u_1\nM : Type u_5\nM₁ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB : M →ₗ[R] M →ₗ[R] M₁\nhB : B.IsRefl\nhB' : B.SeparatingLeft\n⊢ B.ker = ⊥",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nM : Type u_5\nM₁ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB : M →ₗ[R] M →ₗ[R] M₁\nhB : B.IsRefl\nhB' : B.SeparatingLeft\n⊢ B.SeparatingLeft"
] | ← separatingLeft_iff_ker_eq_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 725,
"column": 68
} | {
"line": 726,
"column": 70
} | {
"line": 727,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : M →ₗ[R] M →ₗ[R] R\nhB : B.IsSymm\nW : Submodule R M\nhW : IsCompl W B.ker\nhB' : (B.domRestrict₁₂ W W).IsRefl\nx : M\nhx : x ∈ W\nhx' : ∀ y ∈ W, (B x) y = 0\ny : M\n⊢ ∃ u ∈ W, ∃ v ∈ B.ker, u + v = y",
... | [] | by
rw [← Submodule.mem_sup, hW.sup_eq_top]; exact Submodule.mem_top | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 877,
"column": 33
} | {
"line": 877,
"column": 38
} | {
"line": 879,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nx : M\n⊢ B x = 0 → x = 0 ↔ (B x = 0 ↔ x = 0)",
"ppTerm": "?m.57",
"assig... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 877,
"column": 33
} | {
"line": 877,
"column": 38
} | {
"line": 879,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nx : M\n⊢ B x = 0 → x = 0 ↔ (B x = 0 ↔ x = 0)",
"ppTerm": "?m.57",
"assig... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 877,
"column": 33
} | {
"line": 877,
"column": 38
} | {
"line": 879,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nx : M\n⊢ B x = 0 → x = 0 ↔ (B x = 0 ↔ x = 0)",
"ppTerm": "?m.57",
"assig... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 885,
"column": 33
} | {
"line": 885,
"column": 38
} | {
"line": 885,
"column": 38
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nx : M\n⊢ (B x) x = 0 ∧ x ≠ 0 ∨ (B x) x ≠ 0 ∧ x = 0 → x ≠ 0 ∧ (B x) x ≤ 0",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 885,
"column": 33
} | {
"line": 885,
"column": 38
} | {
"line": 885,
"column": 38
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nx : M\n⊢ (B x) x = 0 ∧ x ≠ 0 ∨ (B x) x ≠ 0 ∧ x = 0 → x ≠ 0 ∧ (B x) x ≤ 0",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 885,
"column": 33
} | {
"line": 885,
"column": 38
} | {
"line": 885,
"column": 38
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nx : M\n⊢ (B x) x = 0 ∧ x ≠ 0 ∨ (B x) x ≠ 0 ∧ x = 0 → x ≠ 0 ∧ (B x) x ≤ 0",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Kernels | {
"line": 34,
"column": 61
} | {
"line": 34,
"column": 66
} | {
"line": 36,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM N P : ModuleCat R\nf : M ⟶ N\n⊢ ↟(Hom.hom f).ker.subtype ≫ f = 0",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Submodule",
"LinearMap.comp.congr_simp",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"ModuleCat... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.ModuleCat.Kernels | {
"line": 34,
"column": 61
} | {
"line": 34,
"column": 66
} | {
"line": 36,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM N P : ModuleCat R\nf : M ⟶ N\n⊢ ↟(Hom.hom f).ker.subtype ≫ f = 0",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Submodule",
"LinearMap.comp.congr_simp",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"ModuleCat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Kernels | {
"line": 34,
"column": 61
} | {
"line": 34,
"column": 66
} | {
"line": 36,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM N P : ModuleCat R\nf : M ⟶ N\n⊢ ↟(Hom.hom f).ker.subtype ≫ f = 0",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Submodule",
"LinearMap.comp.congr_simp",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"ModuleCat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 334,
"column": 15
} | {
"line": 334,
"column": 20
} | {
"line": 334,
"column": 20
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\nhp : p < ⊤\nhp' : Projective R (M ⧸ p)\nx : M\nhx : x ∉ p\nf : Dual R M\nhf : f x ≠ 0\nhf' : map f p = ⊥\n⊢ f ≠ 0",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Su... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 334,
"column": 15
} | {
"line": 334,
"column": 20
} | {
"line": 334,
"column": 20
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\nhp : p < ⊤\nhp' : Projective R (M ⧸ p)\nx : M\nhx : x ∉ p\nf : Dual R M\nhf : f x ≠ 0\nhf' : map f p = ⊥\n⊢ f ≠ 0",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Su... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 334,
"column": 15
} | {
"line": 334,
"column": 20
} | {
"line": 334,
"column": 20
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\nhp : p < ⊤\nhp' : Projective R (M ⧸ p)\nx : M\nhx : x ∉ p\nf : Dual R M\nhf : f x ≠ 0\nhf' : map f p = ⊥\n⊢ f ≠ 0",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Su... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers | {
"line": 354,
"column": 4
} | {
"line": 355,
"column": 21
} | {
"line": 356,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nJ : Type w\ninst✝¹ : SmallCategory J\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : J ⥤ C\nc₁ : Cofan fun f ↦ F.obj f.fst.1\nc₂ : Cofan F.obj\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), c₁.ι.app { as := f } ≫ s = F.map f.snd ≫ c₂.... | [
"case refine_2\nC : Type u\ninst✝² : Category.{v, u} C\nJ : Type w\ninst✝¹ : SmallCategory J\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : J ⥤ C\nc₁ : Cofan fun f ↦ F.obj f.fst.1\nc₂ : Cofan F.obj\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), c₁.ι.app { as := f } ≫ s = F.map f.snd ≫ c₂.ι.app { as :... | · refine t₂.desc (Cofan.mk _ fun j => ?_)
apply q.ι.app j | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 762,
"column": 24
} | {
"line": 762,
"column": 29
} | {
"line": 763,
"column": 2
} | [
{
"pp": "K : Type u_1\nV₁ : Type u_2\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\ninst✝ : FiniteDimensional K V₁\nf g : Dual K V₁\nx : V₁\nh : LinearMap.ker f = LinearMap.ker g\nh' : f x = g x\nhx : f x ≠ 0\np : Submodule K V₁ := K ∙ x\n⊢ p ≠ ⊥",
"ppTerm": "?m.62",
"assigned... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 769,
"column": 24
} | {
"line": 769,
"column": 29
} | {
"line": 770,
"column": 2
} | [
{
"pp": "K : Type u_1\nV₁ : Type u_2\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\ninst✝ : FiniteDimensional K V₁\nf g : Dual K V₁\nx : V₁\nh : LinearMap.ker f = LinearMap.ker g\nh' : f x = g x\nhx : f x ≠ 0\np : Submodule K V₁ := K ∙ x\nhp : p ≠ ⊥\nhpf : Disjoint (LinearMap.ker f) p... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 760,
"column": 13
} | {
"line": 777,
"column": 20
} | {
"line": 779,
"column": 0
} | [
{
"pp": "K : Type u_1\nV₁ : Type u_2\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\ninst✝ : FiniteDimensional K V₁\nf g : Dual K V₁\nx : V₁\nh : LinearMap.ker f = LinearMap.ker g\nh' : f x = g x\nhx : f x ≠ 0\n⊢ f = g",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants":... | [] | by
let p := K ∙ x
have hp : p ≠ ⊥ := by aesop
have hpf : Disjoint (LinearMap.ker f) p := by
rw [disjoint_iff, Submodule.eq_bot_iff]
rintro y ⟨hfy : f y = 0, hpy : y ∈ p⟩
obtain ⟨t, rfl⟩ := Submodule.mem_span_singleton.mp hpy
have ht : t = 0 := by simpa [hx] using hfy
simp [ht]
have hf : f ≠ ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 256,
"column": 75
} | {
"line": 256,
"column": 100
} | {
"line": 257,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (𝟙 A) 0 ≫ prod.snd = 0\nhp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) (fun s ↦ s.ι ≫ prod.fst) ⋯ ⋯\nhp2 : IsColimit (CokernelCofork.... | [] | rw [← Category.assoc, hz] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 218,
"column": 53
} | {
"line": 218,
"column": 84
} | {
"line": 219,
"column": 18
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteCoproducts C\ninst✝³ : HasCokernels C\ninst✝² : IsNormalEpiCategory C\nX Y Z : C\na : X ⟶ Y\nb : X ⟶ Z\ninst✝¹ : Epi a\ninst✝ : Epi b\nP : C\nf : P ⟶ X\nhfa : f ≫ a = 0\ni : IsColimit (CokernelCofork.ofπ a hfa)... | [] | by rw [PushoutCocone.condition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 1059,
"column": 2
} | {
"line": 1067,
"column": 15
} | {
"line": 1069,
"column": 0
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set ι\nv : ι → V\nhli : LinearIndepOn K v s\n⊢ ∃ f, ∀ i ∈ s, f (v i) = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"LinearIndepOn.extend",
"Eq.mpr",
... | [] | replace hli : LinearIndepOn K id (v '' s) := LinearIndepOn.id_image hli
let b : Basis _ K V := .mk (hli.linearIndepOn_extend (Set.subset_univ _)) <| by
simpa using hli.span_extend_eq_span <| Set.subset_univ _
refine ⟨b.constr K 1, fun i hi ↦ ?_⟩
replace hi : v i ∈ hli.extend (Set.subset_univ _) :=
hli.sub... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 1059,
"column": 2
} | {
"line": 1067,
"column": 15
} | {
"line": 1069,
"column": 0
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set ι\nv : ι → V\nhli : LinearIndepOn K v s\n⊢ ∃ f, ∀ i ∈ s, f (v i) = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"LinearIndepOn.extend",
"Eq.mpr",
... | [] | replace hli : LinearIndepOn K id (v '' s) := LinearIndepOn.id_image hli
let b : Basis _ K V := .mk (hli.linearIndepOn_extend (Set.subset_univ _)) <| by
simpa using hli.span_extend_eq_span <| Set.subset_univ _
refine ⟨b.constr K 1, fun i hi ↦ ?_⟩
replace hi : v i ∈ hli.extend (Set.subset_univ _) :=
hli.sub... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 631,
"column": 4
} | {
"line": 631,
"column": 70
} | {
"line": 633,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\n⊢ biprod.lift f (-g) ≫ biproductToPushout f g = 0",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
... | [] | rw [biprod.lift_desc, neg_comp, pushout.condition, add_neg_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 631,
"column": 4
} | {
"line": 631,
"column": 70
} | {
"line": 633,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\n⊢ biprod.lift f (-g) ≫ biproductToPushout f g = 0",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
... | [] | rw [biprod.lift_desc, neg_comp, pushout.condition, add_neg_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 631,
"column": 4
} | {
"line": 631,
"column": 70
} | {
"line": 633,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\n⊢ biprod.lift f (-g) ≫ biproductToPushout f g = 0",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
... | [] | rw [biprod.lift_desc, neg_comp, pushout.condition, add_neg_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 767,
"column": 48
} | {
"line": 769,
"column": 93
} | {
"line": 771,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nφ : S₁ ⟶ S₂\n⊢ S₁.leftHomologyIso.inv ≫ leftHomologyMap φ = homologyMap φ ≫ S₂.leftHomologyIso.inv",
"ppTerm": "?m.69",
"assigned": true,
"usedConsta... | [] | by
simpa only [LeftHomologyData.homologyIso_leftHomologyData, Iso.symm_inv] using!
LeftHomologyData.leftHomologyIso_hom_naturality φ S₁.leftHomologyData S₂.leftHomologyData | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 139,
"column": 2
} | {
"line": 140,
"column": 60
} | {
"line": 141,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.RightHomologyData\nhg : S.g = 0\n⊢ IsIso h.ι",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.RightHomologyData.ι_g'",
"Eq.mpr",
"_priv... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.RightHomologyData\nhg : S.g = 0\nφ : h.Q ⟶ (KernelFork.ofι h.ι ⋯).pt\nhφ : φ ≫ Fork.ι (KernelFork.ofι h.ι ⋯) = 𝟙 h.Q\n⊢ IsIso h.ι"
] | have ⟨φ, hφ⟩ := KernelFork.IsLimit.lift' h.hι' (𝟙 _)
(by rw [← cancel_epi h.p, id_comp, p_g', comp_zero, hg]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 335,
"column": 4
} | {
"line": 336,
"column": 67
} | {
"line": 337,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\n⊢ h₁.f' ≫ φK = φ.τ₁ ≫ h₂.f'",
"ppTerm": "?m.113",
"assigned": true,
"usedCons... | [] | rw [← cancel_mono h₂.i, assoc, assoc, LeftHomologyData.liftK_i,
LeftHomologyData.f'_i_assoc, LeftHomologyData.f'_i, φ.comm₁₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 29
} | {
"line": 221,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\nhg : S.g = 0\n⊢ 𝟙 S.X₂ ≫ S.g = 0",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | simp only [hg, comp_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 695,
"column": 16
} | {
"line": 695,
"column": 70
} | {
"line": 696,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ne : S₁ ≅ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\n⊢ cyclesMap' e.hom h₁ h₂ ≫ cyclesMap' e.inv h₂ h₁ = 𝟙 h₁.K",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
... | [] | by rw [← cyclesMap'_comp, e.hom_inv_id, cyclesMap'_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.Injective.Basic | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 36
} | {
"line": 297,
"column": 36
} | [
{
"pp": "case h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u_1\ninst✝³ : Category.{v_1, u_1} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝² : G.Full\ninst✝¹ : G.Faithful\nI : D\nhI : Injective (G.obj I)\nX Y : D\nf : X ⟶ I\ng : X ⟶ Y\ninst✝ : Mono g\nthis : PreservesLimitsOfSize.{0, 0, v_1, v₁, u_1, u₁}... | [] | exact G.map_injective (by simpa) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.ShortComplex.Ab | {
"line": 51,
"column": 63
} | {
"line": 51,
"column": 68
} | {
"line": 51,
"column": 68
} | [
{
"pp": "S : ShortComplex Ab\n⊢ ∀ (a b : ↑S.X₁),\n ⟨(ConcreteCategory.hom S.f) (a + b), ⋯⟩ = ⟨(ConcreteCategory.hom S.f) a, ⋯⟩ + ⟨(ConcreteCategory.hom S.f) b, ⋯⟩",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"AddMonoidHom.instAddMo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.ShortComplex.Ab | {
"line": 51,
"column": 63
} | {
"line": 51,
"column": 68
} | {
"line": 51,
"column": 68
} | [
{
"pp": "S : ShortComplex Ab\n⊢ ∀ (a b : ↑S.X₁),\n ⟨(ConcreteCategory.hom S.f) (a + b), ⋯⟩ = ⟨(ConcreteCategory.hom S.f) a, ⋯⟩ + ⟨(ConcreteCategory.hom S.f) b, ⋯⟩",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"AddMonoidHom.instAddMo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Ab | {
"line": 51,
"column": 63
} | {
"line": 51,
"column": 68
} | {
"line": 51,
"column": 68
} | [
{
"pp": "S : ShortComplex Ab\n⊢ ∀ (a b : ↑S.X₁),\n ⟨(ConcreteCategory.hom S.f) (a + b), ⋯⟩ = ⟨(ConcreteCategory.hom S.f) a, ⋯⟩ + ⟨(ConcreteCategory.hom S.f) b, ⋯⟩",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"AddMonoidHom.instAddMo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Restrict | {
"line": 54,
"column": 8
} | {
"line": 54,
"column": 31
} | {
"line": 55,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nC' : Type u₃\ninst✝¹ : Category.{v₃, u₃} C'\nD' : Type u₄\ninst✝ : Category.{v₄, u₄} D'\niC : C ⥤ C'\niD : D ⥤ D'\nL' : C' ⥤ D'\nR' : D' ⥤ C'\nadj : L' ⊣ R'\nhiC : iC.FullyFaithful\nhiD : iD.FullyFaithful\nL : C ⥤ D\n... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nC' : Type u₃\ninst✝¹ : Category.{v₃, u₃} C'\nD' : Type u₄\ninst✝ : Category.{v₄, u₄} D'\niC : C ⥤ C'\niD : D ⥤ D'\nL' : C' ⥤ D'\nR' : D' ⥤ C'\nadj : L' ⊣ R'\nhiC : iC.FullyFaithful\nhiD : iD.FullyFaithful\nL : C ⥤ D\nR : D ⥤ C\nc... | apply hiD.map_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 41
} | {
"line": 98,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\nthis : S.HasHomology\n⊢ S.Exact ↔ IsZero h.left.H",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.HomologyData.left",
"Categor... | [] | exact LeftHomologyData.exact_iff h.left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 240,
"column": 74
} | {
"line": 242,
"column": 39
} | {
"line": 242,
"column": 39
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\nh : mk f = ⊤\n⊢ IsIso f",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg"... | [] | by
rw [← ofMkLEMk_comp h.le, Category.comp_id]
exact (isoOfMkEqMk _ _ h).isIso_hom | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 355,
"column": 95
} | {
"line": 356,
"column": 63
} | {
"line": 358,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA B : C\ninst✝ : HasBinaryProduct A B\n⊢ (opProdIsoCoprod A B).inv.unop ≫ coprod.inl.unop = prod.fst",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.opProdIsoCoprod",
"Opposite",
"Q... | [] | by
rw [← unop_comp, inl_opProdIsoCoprod_inv, Quiver.Hom.unop_op] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 282,
"column": 16
} | {
"line": 282,
"column": 31
} | {
"line": 283,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE : L.LeftExtension F\nY Y' : D\ne : Y ≅ Y'\nh : E.IsPointwiseLeftKanExtensionAt Y\... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 283,
"column": 17
} | {
"line": 283,
"column": 32
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE : L.LeftExtension F\nY Y' : D\ne : Y ≅ Y'\nh : E.IsPointwiseLeftKanExtensionAt Y'... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 471,
"column": 16
} | {
"line": 471,
"column": 31
} | {
"line": 472,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE E' : L.RightExtension F\nY Y' : D\ne : Y ≅ Y'\nh : E.IsPointwiseRightKanExtension... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 472,
"column": 17
} | {
"line": 472,
"column": 32
} | {
"line": 474,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE E' : L.RightExtension F\nY Y' : D\ne : Y ≅ Y'\nh : E.IsPointwiseRightKanExtension... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 675,
"column": 4
} | {
"line": 675,
"column": 9
} | {
"line": 676,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\ninst✝ : L.HasPointwiseLeftKanExtension F\nG : D ⥤ H\nα : F ⟶ L ⋙ G\nY : D\nβ : L.pointwiseLeftKanExtension F ⟶ G := ⋯\n⊢ L.pointwiseLeftKanExte... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 9
} | {
"line": 785,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\ninst✝ : L.HasPointwiseRightKanExtension F\nG : D ⥤ H\nα : L ⋙ G ⟶ F\nY : D\nβ : G ⟶ L.pointwiseRightKanExtension F := ⋯\n⊢ L.whiskerLeft\n ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 504,
"column": 2
} | {
"line": 507,
"column": 66
} | {
"line": 508,
"column": 2
} | [
{
"pp": "C : Type u_1\nC' : Type u_2\nH : Type u_3\nD : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} C'\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\nL : C ⥤ D\nF : C ⥤ H\nF' : D ⥤ H\nG : C' ⥤ C\ninst✝ : G.IsEquivalence\nα : L ⋙ F' ⟶ F\n⊢ F'.IsRightKanExtension α ↔ F... | [
"C : Type u_1\nC' : Type u_2\nH : Type u_3\nD : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} C'\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\nL : C ⥤ D\nF : C ⥤ H\nF' : D ⥤ H\nG : C' ⥤ C\ninst✝ : G.IsEquivalence\nα : L ⋙ F' ⟶ F\neq : CostructuredArrow.IsUniversal (RightE... | let eq : (RightExtension.mk _ α).IsUniversal ≃ (RightExtension.mk _
((associator _ _ _).hom ≫ whiskerLeft G α)).IsUniversal :=
(RightExtension.isUniversalPrecompEquiv L F G _).trans
(IsTerminal.equivOfIso (CostructuredArrow.isoMk (Iso.refl _))) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 715,
"column": 16
} | {
"line": 715,
"column": 31
} | {
"line": 716,
"column": 2
} | [
{
"pp": "C : Type u_1\nC' : Type u_2\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} C'\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\ninst✝ : Category.{v_5, u_5} D'\nL : C ⥤ D\nL' : D ⥤ D'\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nF₂ : D' ⥤ H\nα : F₀ ⟶ ... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 716,
"column": 17
} | {
"line": 716,
"column": 32
} | {
"line": 719,
"column": 0
} | [
{
"pp": "C : Type u_1\nC' : Type u_2\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} C'\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\ninst✝ : Category.{v_5, u_5} D'\nL : C ⥤ D\nL' : D ⥤ D'\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nF₂ : D' ⥤ H\nα : F₀ ⟶ ... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic | {
"line": 577,
"column": 4
} | {
"line": 577,
"column": 74
} | {
"line": 578,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nJ : Type u'\ninst✝² : Category.{v', u'} J\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : lim.PreservesEpimorphisms\n⊢ PreservesFiniteColimits lim",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPr... | [
"case inst\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nJ : Type u'\ninst✝² : Category.{v', u'} J\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : lim.PreservesEpimorphisms\n⊢ lim.PreservesHomology",
"case inst\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nJ : Type u'\ninst✝² : Category.{v',... | apply +allowSynthFailures preservesFiniteColimits_of_preservesHomology | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyWith___elabRules_Mathlib_Tactic_applyWith_1 | Mathlib.Tactic.applyWith |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 301,
"column": 12
} | {
"line": 301,
"column": 35
} | {
"line": 301,
"column": 36
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝ : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type (max w v₁)\ns : Cocone (functorToRepresentables P)\nX Y : Cᵒᵖ\nf : X ⟶ Y\nx : P.obj X\nthis :\n uliftYoneda.{w, v₁, u₁}.map f.unop ≫ s.ι.app (op (P.elementsMk X x)) =\n s.ι.app (op (P.elem... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝ : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type (max w v₁)\ns : Cocone (functorToRepresentables P)\nX Y : Cᵒᵖ\nf : X ⟶ Y\nx : P.obj X\nthis :\n uliftYoneda.{w, v₁, u₁}.map f.unop ≫ s.ι.app (op (P.elementsMk X x)) =\n s.ι.app (op (P.elementsMk Y ((h... | uliftYonedaEquiv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 301,
"column": 36
} | {
"line": 301,
"column": 59
} | {
"line": 302,
"column": 10
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝ : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type (max w v₁)\ns : Cocone (functorToRepresentables P)\nX Y : Cᵒᵖ\nf : X ⟶ Y\nx : P.obj X\nthis :\n uliftYoneda.{w, v₁, u₁}.map f.unop ≫ s.ι.app (op (P.elementsMk X x)) =\n s.ι.app (op (P.elem... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝ : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type (max w v₁)\ns : Cocone (functorToRepresentables P)\nX Y : Cᵒᵖ\nf : X ⟶ Y\nx : P.obj X\nthis :\n uliftYoneda.{w, v₁, u₁}.map f.unop ≫ s.ι.app (op (P.elementsMk X x)) =\n s.ι.app (op (P.elementsMk Y ((h... | uliftYonedaEquiv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Comma.LocallySmall | {
"line": 35,
"column": 24
} | {
"line": 35,
"column": 29
} | {
"line": 35,
"column": 29
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nT : Type u₃\ninst✝⁴ : Category.{v₁, u₁} A\ninst✝³ : Category.{v₂, u₂} B\ninst✝² : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\ninst✝¹ : LocallySmall.{w, v₁, u₁} A\ninst✝ : LocallySmall.{w, v₂, u₂} B\nX Y : Comma L R\nx✝² x✝¹ : X ⟶ Y\nx✝ : (fun g ↦ (g.left, g.right)) x✝² = (fun g... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Comma.LocallySmall | {
"line": 35,
"column": 24
} | {
"line": 35,
"column": 29
} | {
"line": 35,
"column": 29
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nT : Type u₃\ninst✝⁴ : Category.{v₁, u₁} A\ninst✝³ : Category.{v₂, u₂} B\ninst✝² : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\ninst✝¹ : LocallySmall.{w, v₁, u₁} A\ninst✝ : LocallySmall.{w, v₂, u₂} B\nX Y : Comma L R\nx✝² x✝¹ : X ⟶ Y\nx✝ : (fun g ↦ (g.left, g.right)) x✝² = (fun g... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Comma.LocallySmall | {
"line": 35,
"column": 24
} | {
"line": 35,
"column": 29
} | {
"line": 35,
"column": 29
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nT : Type u₃\ninst✝⁴ : Category.{v₁, u₁} A\ninst✝³ : Category.{v₂, u₂} B\ninst✝² : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\ninst✝¹ : LocallySmall.{w, v₁, u₁} A\ninst✝ : LocallySmall.{w, v₂, u₂} B\nX Y : Comma L R\nx✝² x✝¹ : X ⟶ Y\nx✝ : (fun g ↦ (g.left, g.right)) x✝² = (fun g... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 474,
"column": 6
} | {
"line": 474,
"column": 13
} | {
"line": 475,
"column": 4
} | [
{
"pp": "case zero\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Trident f\nk : s.pt ⟶ t.pt\nw : k ≫ t.ι = s.ι\n⊢ k ≫ t.π.app zero = s.π.app zero",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"w"
],... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 474,
"column": 6
} | {
"line": 474,
"column": 13
} | {
"line": 475,
"column": 4
} | [
{
"pp": "case zero\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Trident f\nk : s.pt ⟶ t.pt\nw : k ≫ t.ι = s.ι\n⊢ k ≫ t.π.app zero = s.π.app zero",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"w"
],... | [] | exact w | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 474,
"column": 6
} | {
"line": 474,
"column": 13
} | {
"line": 475,
"column": 4
} | [
{
"pp": "case zero\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Trident f\nk : s.pt ⟶ t.pt\nw : k ≫ t.ι = s.ι\n⊢ k ≫ t.π.app zero = s.π.app zero",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"w"
],... | [] | exact w | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 497,
"column": 6
} | {
"line": 497,
"column": 13
} | {
"line": 499,
"column": 0
} | [
{
"pp": "case one\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Cotrident f\nk : s.pt ⟶ t.pt\nw : s.π ≫ k = t.π\n⊢ s.ι.app one ≫ k = t.ι.app one",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"w"
],
... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 497,
"column": 6
} | {
"line": 497,
"column": 13
} | {
"line": 499,
"column": 0
} | [
{
"pp": "case one\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Cotrident f\nk : s.pt ⟶ t.pt\nw : s.π ≫ k = t.π\n⊢ s.ι.app one ≫ k = t.ι.app one",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"w"
],
... | [] | exact w | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 497,
"column": 6
} | {
"line": 497,
"column": 13
} | {
"line": 499,
"column": 0
} | [
{
"pp": "case one\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Cotrident f\nk : s.pt ⟶ t.pt\nw : s.π ≫ k = t.π\n⊢ s.ι.app one ≫ k = t.ι.app one",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"w"
],
... | [] | exact w | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 705,
"column": 69
} | {
"line": 712,
"column": 35
} | {
"line": 714,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : HasZeroMorphisms C\nG H : C\ninst✝ : HasBinaryCoproduct G H\n⊢ IsSeparator (G ⨿ H) ↔ (ObjectProperty.pair G H).IsSeparating",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.Ob... | [] | by
refine (isSeparator_iff_of_isColimit_cofan (coprodIsCoprod G H)).trans ?_
convert! Iff.rfl
ext X
simp only [ObjectProperty.pair_iff, ObjectProperty.ofObj_iff]
constructor
· rintro (rfl | rfl); exacts [⟨.left, rfl⟩, ⟨.right, rfl⟩]
· rintro ⟨⟨_ | _⟩, rfl⟩ <;> tauto | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 715,
"column": 4
} | {
"line": 720,
"column": 90
} | {
"line": 721,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nI : Type v₁\ninst✝ : SmallCategory I\nF : I ⥤ C\nc : Cocone (F ⋙ yoneda)\nhc : IsColimit c\nthis :\n IsTerminal\n (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))\n⊢ IsTerminal (colimit ((c.to... | [] | apply IsTerminal.isTerminalOfObj (overEquivPresheafCostructuredArrow c.pt).inverse
apply IsTerminal.ofIso this
refine ?_ ≪≫ (preservesColimitIso (overEquivPresheafCostructuredArrow c.pt).inverse _).symm
apply HasColimit.isoOfNatIso
exact Functor.isoWhiskerLeft _
(CostructuredArrow.toOverCompOverEq... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 715,
"column": 4
} | {
"line": 720,
"column": 90
} | {
"line": 721,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nI : Type v₁\ninst✝ : SmallCategory I\nF : I ⥤ C\nc : Cocone (F ⋙ yoneda)\nhc : IsColimit c\nthis :\n IsTerminal\n (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))\n⊢ IsTerminal (colimit ((c.to... | [] | apply IsTerminal.isTerminalOfObj (overEquivPresheafCostructuredArrow c.pt).inverse
apply IsTerminal.ofIso this
refine ?_ ≪≫ (preservesColimitIso (overEquivPresheafCostructuredArrow c.pt).inverse _).symm
apply HasColimit.isoOfNatIso
exact Functor.isoWhiskerLeft _
(CostructuredArrow.toOverCompOverEq... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Grp.AB | {
"line": 107,
"column": 79
} | {
"line": 107,
"column": 84
} | {
"line": 107,
"column": 84
} | [
{
"pp": "case h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nA B : AddCommGrpCat\nf g : A ⟶ B\nx : ↑A\nh :\n (AddCommGrpCat.Hom.hom f)\n ((AddCommGrpCat.Hom.hom (AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun y ↦ y • x) ⋯))) { down := 1 }) =\n (AddCommGrpCat.Hom.hom g)\n ((AddCommGrpCa... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 79,
"column": 21
} | {
"line": 79,
"column": 26
} | {
"line": 79,
"column": 26
} | [
{
"pp": "C : Type u\nF : C → Type v\n⊢ ∀ (x : (sigma F).pt), ∃ i y, (sigma F).inj i y = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.Discrete.functor",
"Exists",
"id",
"Sigma.fst",
"CategoryTheory.Functor.CoconeTypes.pt",
"CategoryThe... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 79,
"column": 21
} | {
"line": 79,
"column": 26
} | {
"line": 79,
"column": 26
} | [
{
"pp": "C : Type u\nF : C → Type v\n⊢ ∀ (x : (sigma F).pt), ∃ i y, (sigma F).inj i y = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.Discrete.functor",
"Exists",
"id",
"Sigma.fst",
"CategoryTheory.Functor.CoconeTypes.pt",
"CategoryThe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 79,
"column": 21
} | {
"line": 79,
"column": 26
} | {
"line": 79,
"column": 26
} | [
{
"pp": "C : Type u\nF : C → Type v\n⊢ ∀ (x : (sigma F).pt), ∃ i y, (sigma F).inj i y = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.Discrete.functor",
"Exists",
"id",
"Sigma.fst",
"CategoryTheory.Functor.CoconeTypes.pt",
"CategoryThe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 7
} | {
"line": 96,
"column": 0
} | [
{
"pp": "case hf\nC : Type u\nF : C → Type v\nc : CofanTypes F\n⊢ ∀ (j : Discrete C) (x : (Discrete.functor F).obj j), c.ι j x = fromSigma F c ((sigma F).ι j x)",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CategoryTheory.Limits.CofanTypes.sigma_ι_s... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.Periodic | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 37
} | {
"line": 310,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh : Antiperiodic f c\n⊢ f (-c) = -f 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Function.Antiperiodic.neg",
... | [] | simpa only [zero_add] using h.neg 0 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Ring.Periodic | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 37
} | {
"line": 310,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh : Antiperiodic f c\n⊢ f (-c) = -f 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Function.Antiperiodic.neg",
... | [] | simpa only [zero_add] using h.neg 0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Periodic | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 37
} | {
"line": 310,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh : Antiperiodic f c\n⊢ f (-c) = -f 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Function.Antiperiodic.neg",
... | [] | simpa only [zero_add] using h.neg 0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Set.Group | {
"line": 175,
"column": 2
} | {
"line": 185,
"column": 25
} | {
"line": 187,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b : α\n⊢ Pairwise (Disjoint on fun n ↦ Ico (a * b ^ n) (a * b ^ (n + 1)))",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"zpow_lt_zpow_iff_right",
"Eq.mpr",
"MulOne.toOne",... | [] | simp +unfoldPartialApp only [Function.onFun]
simp_rw [Set.disjoint_iff]
intro m n hmn x hx
apply hmn
have hb : 1 < b := by
have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.trans_lt hx.1.2
rwa [mul_lt_mul_iff_left, ← mul_one (b ^ m), zpow_add_one, mul_lt_mul_iff_left] at this
have i1 := hx.1.1.trans_lt hx.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Interval.Set.Group | {
"line": 175,
"column": 2
} | {
"line": 185,
"column": 25
} | {
"line": 187,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b : α\n⊢ Pairwise (Disjoint on fun n ↦ Ico (a * b ^ n) (a * b ^ (n + 1)))",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"zpow_lt_zpow_iff_right",
"Eq.mpr",
"MulOne.toOne",... | [] | simp +unfoldPartialApp only [Function.onFun]
simp_rw [Set.disjoint_iff]
intro m n hmn x hx
apply hmn
have hb : 1 < b := by
have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.trans_lt hx.1.2
rwa [mul_lt_mul_iff_left, ← mul_one (b ^ m), zpow_add_one, mul_lt_mul_iff_left] at this
have i1 := hx.1.1.trans_lt hx.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Constructions | {
"line": 89,
"column": 28
} | {
"line": 89,
"column": 33
} | {
"line": 91,
"column": 0
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nh : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S\ns : Set α\nhs : s ∈ insert univ S\nt : Set α\nht : t ∈ insert univ S\n⊢ s ∩ t ∈ insert univ S",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_o... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Constructions | {
"line": 89,
"column": 28
} | {
"line": 89,
"column": 33
} | {
"line": 91,
"column": 0
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nh : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S\ns : Set α\nhs : s ∈ insert univ S\nt : Set α\nht : t ∈ insert univ S\n⊢ s ∩ t ∈ insert univ S",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_o... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Constructions | {
"line": 89,
"column": 28
} | {
"line": 89,
"column": 33
} | {
"line": 91,
"column": 0
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nh : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S\ns : Set α\nhs : s ∈ insert univ S\nt : Set α\nht : t ∈ insert univ S\n⊢ s ∩ t ∈ insert univ S",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_o... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Lift | {
"line": 102,
"column": 22
} | {
"line": 103,
"column": 82
} | {
"line": 105,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : Filter α\ng : Set α → Filter β\nm : β → γ\nhg : Monotone g\nthis : Monotone (map m ∘ g)\ns : Set γ\n⊢ s ∈ map m (f.lift g) ↔ s ∈ f.lift (map m ∘ g)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"c... | [] | by
simp only [mem_lift_sets hg, mem_lift_sets this, mem_map, Function.comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Neighborhoods | {
"line": 242,
"column": 23
} | {
"line": 242,
"column": 48
} | {
"line": 242,
"column": 48
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns V : Set X\n⊢ (∀ x ∈ s, x ∈ interior V) ↔ ∀ x ∈ s, V ∈ 𝓝 x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Membership.mem",
"nhds",
"iff_self",
"Iff",
"implies... | [] | mem_interior_iff_mem_nhds | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Closure | {
"line": 279,
"column": 6
} | {
"line": 279,
"column": 15
} | {
"line": 279,
"column": 16
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ closure s = (interior sᶜ)ᶜ",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"setOf",
"Set.sUnion",
"id",
"LE.le",
"Set.instCompl",
"And",
... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ closure s = (⋃₀ {t | IsOpen t ∧ t ⊆ sᶜ})ᶜ"
] | interior, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Closure | {
"line": 279,
"column": 16
} | {
"line": 279,
"column": 24
} | {
"line": 279,
"column": 25
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ closure s = (⋃₀ {t | IsOpen t ∧ t ⊆ sᶜ})ᶜ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"setOf",
"Set.sUnion",
"closure.eq_1",
"id",
"LE.le",... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ ⋂₀ {t | IsClosed t ∧ s ⊆ t} = (⋃₀ {t | IsOpen t ∧ t ⊆ sᶜ})ᶜ"
] | closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.NhdsSet | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 45
} | {
"line": 211,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Sort u_4\np : ι → Prop\ns : ι → Set X\nP : X → Prop\n⊢ (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, ⋃ (_ : p i), s i), P x) ↔ ∀ (i : ι), p i → ∀ᶠ (x : X) in 𝓝ˢ (s i), P x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Filter.instSupSet",
"n... | [] | simp only [nhdsSet_iUnion, eventually_iSup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.NhdsSet | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 45
} | {
"line": 211,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Sort u_4\np : ι → Prop\ns : ι → Set X\nP : X → Prop\n⊢ (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, ⋃ (_ : p i), s i), P x) ↔ ∀ (i : ι), p i → ∀ᶠ (x : X) in 𝓝ˢ (s i), P x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Filter.instSupSet",
"n... | [] | simp only [nhdsSet_iUnion, eventually_iSup] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.NhdsSet | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 45
} | {
"line": 211,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Sort u_4\np : ι → Prop\ns : ι → Set X\nP : X → Prop\n⊢ (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, ⋃ (_ : p i), s i), P x) ↔ ∀ (i : ι), p i → ∀ᶠ (x : X) in 𝓝ˢ (s i), P x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Filter.instSupSet",
"n... | [] | simp only [nhdsSet_iUnion, eventually_iSup] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.NhdsSet | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 45
} | {
"line": 214,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Sort u_4\ns : ι → Set X\nP : X → Prop\n⊢ (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, s i), P x) ↔ ∀ (i : ι), ∀ᶠ (x : X) in 𝓝ˢ (s i), P x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instSupSet",
"nhdsSet_iUnion",
"congrArg",... | [] | simp only [nhdsSet_iUnion, eventually_iSup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.