module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.JordanHolder | {
"line": 324,
"column": 4
} | {
"line": 324,
"column": 28
} | {
"line": 324,
"column": 29
} | [
{
"pp": "case refine_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (last s₁) x₁\nhsat₂ : IsMaximal (last s₂) x₂\nhequiv : s₁.Equivalent s₂\nhlast : Iso (last s₁, x₁) (last s₂, x₂)\ne : Fin s₁.length.succ ≃ Fin s₂.length.succ :=\n Tra... | [
"case refine_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (last s₁) x₁\nhsat₂ : IsMaximal (last s₂) x₂\nhequiv : s₁.Equivalent s₂\nhlast : Iso (last s₁, x₁) (last s₂, x₂)\ne : Fin s₁.length.succ ≃ Fin s₂.length.succ :=\n Trans.trans (Tr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 52
} | {
"line": 331,
"column": 53
} | [
{
"pp": "case refine_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (last s₁) x₁\nhsat₂ : IsMaximal (last s₂) x₂\nhequiv : s₁.Equivalent s₂\nhlast : Iso (last s₁, x₁) (last s₂, x₂)\ne : Fin s₁.length.succ ≃ Fin s₂.length.succ :=\n Tra... | [
"case refine_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (last s₁) x₁\nhsat₂ : IsMaximal (last s₂) x₂\nhequiv : s₁.Equivalent s₂\nhlast : Iso (last s₁, x₁) (last s₂, x₂)\ne : Fin s₁.length.succ ≃ Fin s₂.length.succ :=\n Trans.trans (Tr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 13
} | {
"line": 334,
"column": 14
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nh : s₁.Equivalent s₂\n⊢ s₁.length = s₂.length",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"setOf",
"id",
"RelSeries.length",
"Nat",
"Prod",
"Composit... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nh : s₁.Equivalent s₂\n⊢ s₁.length = s₂.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 46
} | {
"line": 273,
"column": 47
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\nN : Submodule R M\n⊢ sSup {m | IsSimpleModule R ↥m ∧ m ≤ N} = N",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"cong... | [
"R : Type u_2\ninst✝³ : Ring R\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\nN : Submodule R M\n⊢ sSup {m | IsAtom m ∧ m ≤ N} = N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 46
} | {
"line": 278,
"column": 47
} | [
{
"pp": "R : Type u_2\ninst✝⁴ : Ring R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsSemisimpleModule R M\ninst✝ : Nontrivial M\n⊢ ∃ m, IsSimpleModule R ↥m",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"... | [
"R : Type u_2\ninst✝⁴ : Ring R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsSemisimpleModule R M\ninst✝ : Nontrivial M\n⊢ ∃ m, IsAtom m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 46
} | {
"line": 281,
"column": 47
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ sSup {m | IsSimpleModule R ↥m} = ⊤",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"_private.Mathlib.RingTheory.Sim... | [
"R : Type u_2\ninst✝³ : Ring R\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ sSup {m | IsAtom m} = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 59
} | {
"line": 362,
"column": 60
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set ι\np : ι → Submodule R M\nhp' : ⨆ i ∈ s, p i = ⊤\nhp : ∀ i ∈ s, ComplementedLattice (Submodule R ↥(p i))\ni : ι\nhi : i ∈ s\n⊢ ComplementedLattice ↑(Set.Iic (p i))",
"ppTerm": "?m.50",
... | [
"ι : Type u_1\nR : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set ι\np : ι → Submodule R M\nhp' : ⨆ i ∈ s, p i = ⊤\nhp : ∀ i ∈ s, ComplementedLattice (Submodule R ↥(p i))\ni : ι\nhi : i ∈ s\n⊢ ComplementedLattice (Submodule R ↥(p i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 360,
"column": 8
} | {
"line": 363,
"column": 33
} | {
"line": 363,
"column": 33
} | [
{
"pp": "case refine_2.refine_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx₁ x₂ y₁ y₂ : X\nhsat₁ : IsMaximal (last s) x₁\nhsat₂ : IsMaximal (last s) x₂\nhsaty₁ : IsMaximal (snoc s x₁ hsat₁).last y₁\nhsaty₂ : IsMaximal (snoc s x₂ hsat₂).last y₂\nhr₁ : Iso (last s, x... | [] | erw [Equiv.swap_apply_of_ne_of_ne h2 h1, snoc_castSucc, snoc_castSucc,
snoc_castSucc, snoc_castSucc, Fin.succ_castSucc, snoc_castSucc,
Fin.succ_castSucc, snoc_castSucc, snoc_castSucc, snoc_castSucc]
exact (s.step i).iso_refl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.JordanHolder | {
"line": 360,
"column": 8
} | {
"line": 363,
"column": 33
} | {
"line": 363,
"column": 33
} | [
{
"pp": "case refine_2.refine_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx₁ x₂ y₁ y₂ : X\nhsat₁ : IsMaximal (last s) x₁\nhsat₂ : IsMaximal (last s) x₂\nhsaty₁ : IsMaximal (snoc s x₁ hsat₁).last y₁\nhsaty₂ : IsMaximal (snoc s x₂ hsat₂).last y₂\nhr₁ : Iso (last s, x... | [] | erw [Equiv.swap_apply_of_ne_of_ne h2 h1, snoc_castSucc, snoc_castSucc,
snoc_castSucc, snoc_castSucc, Fin.succ_castSucc, snoc_castSucc,
Fin.succ_castSucc, snoc_castSucc, snoc_castSucc, snoc_castSucc]
exact (s.step i).iso_refl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 546,
"column": 11
} | {
"line": 546,
"column": 22
} | {
"line": 546,
"column": 23
} | [
{
"pp": "ι : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁸ : Ring R✝\ninst✝⁷ : Ring S\nM : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R✝ M\nm : Submodule R✝ M\nN : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR : Type u_6\ninst✝² : DivisionRing R\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\nv... | [
"ι : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁸ : Ring R✝\ninst✝⁷ : Ring S\nM : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R✝ M\nm : Submodule R✝ M\nN : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR : Type u_6\ninst✝² : DivisionRing R\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\nv : M\nhv : v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 378,
"column": 6
} | {
"line": 378,
"column": 66
} | {
"line": 379,
"column": 8
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : head s₁ = head s₂\nht : last s₁ = last s₂\n⊢ ¬0 < s₂.length → ¬0 < s₁.length",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : head s₁ = head s₂\nht : last s₁ = last s₂\n⊢ s₂.length = 0 → s₁.length = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 597,
"column": 25
} | {
"line": 597,
"column": 36
} | {
"line": 597,
"column": 37
} | [
{
"pp": "case add\nR : Type u_2\ninst✝⁴ : Ring R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsSemisimpleModule R M\ninst✝ : Module.Finite (End R M) M\ns : Finset M\nhs : span (End R M) ↑s = ⊤\nf : End (End R M) M\nr : R\nhr : ∀ m ∈ s, f m = r • m\nm x✝ y✝ : M\nhx✝ : x✝ ∈ span (End R M... | [
"case add\nR : Type u_2\ninst✝⁴ : Ring R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsSemisimpleModule R M\ninst✝ : Module.Finite (End R M) M\ns : Finset M\nhs : span (End R M) ↑s = ⊤\nf : End (End R M) M\nr : R\nhr : ∀ m ∈ s, f m = r • m\nm x✝ y✝ : M\nhx✝ : x✝ ∈ span (End R M) ↑s\nhy✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 25
} | {
"line": 118,
"column": 26
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ (R ∙ x).annihilator = torsionOf R M x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"Set.instSingletonSet",
"id",
"Ideal... | [
"R : Type u_3\nM : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ (R ∙ x).annihilator = (LinearMap.toSpanSingleton R M x).ker"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 424,
"column": 14
} | {
"line": 424,
"column": 25
} | {
"line": 424,
"column": 26
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih :\n ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),\n head s ≤ x →\n s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)\ns : CompositionSeries ... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih :\n ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),\n head s ≤ x →\n s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)\ns : CompositionSeries X\nx : X\nhm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 13
} | {
"line": 141,
"column": 14
} | [
{
"pp": "ι : Type u_3\nR : Type u_4\nM : Type u_5\nv : ι → M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh_ne_zero : ∀ (i : ι), torsionOf R M (v i) = ⊥\ni : ι\nr : R\nhi : r • v i ∈ Submodule.span R (v '' (Set.univ \\ {i}))\nhv : R ∙ v i ⊓ Submodule.span R (Set.range fun i_1 ↦ v ↑i_1) = ⊥\nth... | [
"ι : Type u_3\nR : Type u_4\nM : Type u_5\nv : ι → M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh_ne_zero : ∀ (i : ι), torsionOf R M (v i) = ⊥\ni : ι\nr : R\nhi : r • v i ∈ Submodule.span R (v '' (Set.univ \\ {i}))\nhv : R ∙ v i ⊓ Submodule.span R (Set.range fun i_1 ↦ v ↑i_1) = ⊥\nthis : r • v i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 426,
"column": 40
} | {
"line": 426,
"column": 51
} | {
"line": 426,
"column": 52
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih :\n ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),\n head s ≤ x →\n s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)\ns : CompositionSeries ... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih :\n ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),\n head s ≤ x →\n s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)\ns : CompositionSeries X\nx : X\nhm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 166,
"column": 76
} | {
"line": 166,
"column": 87
} | {
"line": 166,
"column": 88
} | [
{
"pp": "R : Type ?u.5\nA : Type ?u.7\nM : Type ?u.13\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\na : A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\np : Submodule R M\nhp : p ∈ ((Algebra.lsmul R R M) a).invtSubmodule\nx : AEval R M a\... | [
"R : Type ?u.5\nA : Type ?u.7\nM : Type ?u.13\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\na : A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\np : Submodule R M\nhp : p ∈ ((Algebra.lsmul R R M) a).invtSubmodule\nx : AEval R M a\nhx : x ∈ (m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 452,
"column": 35
} | {
"line": 452,
"column": 52
} | {
"line": 452,
"column": 53
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih : ∀ (s₁ s₂ : CompositionSeries X), head s₁ = head s₂ → last s₁ = last s₂ → s₁.length = n → s₁.Equivalent s₂\ns₁ s₂ : CompositionSeries X\nhb : head s₁ = head s₂\nht : last s₁ = last s₂\nhle : s₁.length = n + 1\nh0s₂ : 0 < s₂.lengt... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih : ∀ (s₁ s₂ : CompositionSeries X), head s₁ = head s₂ → last s₁ = last s₂ → s₁.length = n → s₁.Equivalent s₂\ns₁ s₂ : CompositionSeries X\nhb : head s₁ = head s₂\nht : last s₁ = last s₂\nhle : s₁.length = n + 1\nh0s₂ : 0 < s₂.length\nt : Compo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 73
} | {
"line": 289,
"column": 74
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nM' : Type u_7\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nf g : PolynomialModule R M →ₗ[R] M'\nh : ∀ (a : ℕ), f ∘ₗ lsingle R a = g ∘ₗ lsingle R a\n⊢ f = g",
"ppTerm": "?m.100",
"assigned": true,
... | [
"R : Type u_2\nM : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nM' : Type u_7\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nf g : PolynomialModule R M →ₗ[R] M'\nh : ∀ (a : ℕ), f ∘ₗ lsingle R a = g ∘ₗ lsingle R a\n⊢ ∀ (q : ℕ →₀ M), f { coeff := q } = g { coeff := q }"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ReesAlgebra | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 21
} | {
"line": 87,
"column": 6
} | [
{
"pp": "case succ.refine_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\nhn : ∀ {r : R}, r ∈ I ^ n → (monomial n) r ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)\nr : R\nhr : r ∈ I * I ^ n\n⊢ ∀ r ∈ I, ∀ n_1 ∈ I ^ n, (monomial (n + 1)) (r • n_1) ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
... | [
"case succ.refine_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\nhn : ∀ {r : R}, r ∈ I ^ n → (monomial n) r ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)\nr✝ : R\nhr✝ : r✝ ∈ I * I ^ n\nr : R\nhr : r ∈ I\ns : R\nhs : s ∈ I ^ n\n⊢ (monomial (n + 1)) (r • s) ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1)... | intro r hr s hs | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Finiteness.Nakayama | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 59
} | {
"line": 71,
"column": 60
} | [
{
"pp": "case insert.refine_1\nR : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\ns✝ : Set M\ni : M\ns : Set M\na✝ : i ∉ s\nhs✝ : s.Finite\nih :\n (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r -... | [
"case insert.refine_1\nR : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\ns✝ : Set M\ni : M\ns : Set M\na✝ : i ∉ s\nhs✝ : s.Finite\nih :\n (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Nakayama | {
"line": 87,
"column": 48
} | {
"line": 87,
"column": 70
} | {
"line": 87,
"column": 71
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\nhn✝ : N.FG\nhin : N ≤ I • N\nr : R\nhr : r - 1 ∈ I\nhr' : ∀ n ∈ N, r • n = 0\nn : M\nhn : n ∈ N\n⊢ -(r - 1) • n = n",
"ppTerm": "?m.75",
"assigned": true,
"usedConst... | [
"R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\nhn✝ : N.FG\nhin : N ≤ I • N\nr : R\nhr : r - 1 ∈ I\nhr' : ∀ n ∈ N, r • n = 0\nn : M\nhn : n ∈ N\n⊢ r • n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 459,
"column": 4
} | {
"line": 459,
"column": 80
} | {
"line": 459,
"column": 81
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nP Q : Ideal R\nhc : P ⊔ Q = ⊤\nmap : Fin 2 → Ideal R :=\n fun x ↦\n match x with\n | 0 => P\n | 1 => Q\nheq : ⨆ i ∈ ⊤, torsionBySet R M ↑(map i) = torsionBySet R M ↑(⨅ i ∈ ⊤, map ... | [
"case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nP Q : Ideal R\nhc : P ⊔ Q = ⊤\nmap : Fin 2 → Ideal R :=\n fun x ↦\n match x with\n | 0 => P\n | 1 => Q\nheq : ⨆ i ∈ ⊤, torsionBySet R M ↑(map i) = torsionBySet R M ↑(⨅ i ∈ ⊤, map i)\nthis : ⨆... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 478,
"column": 4
} | {
"line": 478,
"column": 54
} | {
"line": 478,
"column": 55
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nP Q : Ideal R\nhc : P ⊔ Q = ⊤\nmap : Fin 2 → Ideal R :=\n fun x ↦\n match x with\n | 0 => P\n | 1 => Q\nheq : ⊤.SupIndep fun i ↦ torsionBySet R M ↑(map i)\n⊢ Disjoint (torsionBySe... | [
"case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nP Q : Ideal R\nhc : P ⊔ Q = ⊤\nmap : Fin 2 → Ideal R :=\n fun x ↦\n match x with\n | 0 => P\n | 1 => Q\nheq : ⊤.SupIndep fun i ↦ torsionBySet R M ↑(map i)\n⊢ Disjoint (torsionBySet R M ↑P) (t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nakayama | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 73
} | {
"line": 130,
"column": 74
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : N'.FG\nhIJ : I ≤ J.jacobson\nhNN : N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap N.mkQ)\n⊢ map N.mkQ (I • N') = map N.mkQ N'",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : N'.FG\nhIJ : I ≤ J.jacobson\nhNN : N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap N.mkQ)\n⊢ N ⊔ N' = N ⊔ I • N'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nakayama | {
"line": 165,
"column": 47
} | {
"line": 165,
"column": 58
} | {
"line": 165,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN N' P : Submodule R M\nhN' : N'.FG\nhN'le : N' ≤ P\nhNN' : P ≤ N ⊔ I • N'\n⊢ N ⊔ I • N' ≤ N ⊔ N'",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submo... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN N' P : Submodule R M\nhN' : N'.FG\nhN'le : N' ≤ P\nhNN' : P ≤ N ⊔ I • N'\n⊢ I • N' ≤ N ⊔ N'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 1029,
"column": 2
} | {
"line": 1029,
"column": 13
} | {
"line": 1029,
"column": 14
} | [
{
"pp": "M : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : IsAddTorsionFree M\nx y : M\nh : y ≠ 0\nr s : ℕ\nhrs : ↑r • y = ↑s • y\n⊢ r = s",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : IsAddTorsionFree M\nx y : M\nh : y ≠ 0\nr s : ℕ\nhrs : ↑r • y = ↑s • y\n⊢ r = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nakayama | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 15
} | {
"line": 168,
"column": 16
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN N' P : Submodule R M\nhN' : N'.FG\nhN'le : N' ≤ P\nhNN' : P ≤ N ⊔ I • N'\nhNN'' : P ≤ N ⊔ N'\n⊢ map N.mkQ P ≤ map N.mkQ N'",
"ppTerm": "?m.123",
"assigned": false,
"usedConstants": [... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN N' P : Submodule R M\nhN' : N'.FG\nhN'le : N' ≤ P\nhNN' : P ≤ N ⊔ I • N'\nhNN'' : P ≤ N ⊔ N'\n⊢ map N.mkQ P ≤ map N.mkQ N'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nakayama | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 17
} | {
"line": 171,
"column": 18
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN N' P : Submodule R M\nhN' : N'.FG\nhN'le : N' ≤ P\nhNN' : P ≤ N ⊔ I • N'\nhNN'' : P ≤ N ⊔ N'\nh1 : map N.mkQ P = map N.mkQ N'\n⊢ map N.mkQ P ≤ map N.mkQ (I • N')",
"ppTerm": "?a✝",
... | [
"case a\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN N' P : Submodule R M\nhN' : N'.FG\nhN'le : N' ≤ P\nhNN' : P ≤ N ⊔ I • N'\nhNN'' : P ≤ N ⊔ N'\nh1 : map N.mkQ P = map N.mkQ N'\n⊢ map N.mkQ P ≤ I • map N.mkQ N'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Differentials.Basic | {
"line": 57,
"column": 41
} | {
"line": 57,
"column": 47
} | {
"line": 57,
"column": 48
} | [
{
"pp": "A B : CommRingCat\nM : ModuleCat ↑B\nf : A ⟶ B\nd : ↑B → ↑M\nd_add : ∀ (b b' : ↑B), d (b + b') = d b + d b'\nd_mul : ∀ (b b' : ↑B), d (b * b') = b • d b' + b' • d b\nd_map : ∀ (a : ↑A), d ((ConcreteCategory.hom f) a) = 0\nthis✝ : Algebra ↑A ↑B := (CommRingCat.Hom.hom f).toAlgebra\nthis : Module ↑A ↑M :... | [
"A B : CommRingCat\nM : ModuleCat ↑B\nf : A ⟶ B\nd : ↑B → ↑M\nd_add : ∀ (b b' : ↑B), d (b + b') = d b + d b'\nd_mul : ∀ (b b' : ↑B), d (b * b') = b • d b' + b' • d b\nd_map : ∀ (a : ↑A), d ((ConcreteCategory.hom f) a) = 0\nthis✝ : Algebra ↑A ↑B := (CommRingCat.Hom.hom f).toAlgebra\nthis : Module ↑A ↑M := Module.com... | d_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | {
"line": 129,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\nx : R\nx✝ : (Quotient.mk (I ^ 2)) x ∈ I.cotangentIdeal\ny : R\nhy : y ∈ ↑I\ne : (Quotient.mk (I ^ 2)).toSemilinearMap y = (Quotient.mk (I ^ 2)) x\n⊢ x ∈ I",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"R : Type u\ninst✝ : CommRing R\nI : Ideal R\nx : R\nx✝ : (Quotient.mk (I ^ 2)) x ∈ I.cotangentIdeal\ny : R\nhy : y ∈ ↑I\ne : (Quotient.mk (I ^ 2)).toSemilinearMap y = (Quotient.mk (I ^ 2)) x\n⊢ x ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 13
} | {
"line": 370,
"column": 14
} | [
{
"pp": "case h\nA : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nsurj : Function.Surjective ⇑(algebraMap A B)\nI : Ideal B\nJ : Ideal A\neq : comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J\nx' : ↥I\ny' : A\nmem : y' ∈ J\nhy' : (algebraMap A B) y' = ↑x'\n⊢ ... | [
"case h\nA : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nsurj : Function.Surjective ⇑(algebraMap A B)\nI : Ideal B\nJ : Ideal A\neq : comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J\nx' : ↥I\ny' : A\nmem : y' ∈ J\nhy' : (algebraMap A B) y' = ↑x'\n⊢ I.toCotangen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Filtration | {
"line": 420,
"column": 59
} | {
"line": 420,
"column": 70
} | {
"line": 420,
"column": 71
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\nh : I ≤ ⊥.jacobson\nx : M\nhx : x ∈ ⨅ i, I ^ i • ⊤\nr : ↥I\nhr : ↑r • x = x\n⊢ 1 - ↑r - 1 ∈ ⊥.jacobson",
"ppTerm": "?m.82",
"assign... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\nh : I ≤ ⊥.jacobson\nx : M\nhx : x ∈ ⨅ i, I ^ i • ⊤\nr : ↥I\nhr : ↑r • x = x\n⊢ ↑r ∈ ⊥.jacobson"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Differentials.Presheaf | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | {
"line": 79,
"column": 14
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nS : Cᵒᵖ ⥤ CommRingCat\nF : C ⥤ D\nR : Dᵒᵖ ⥤ CommRingCat\nM : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nφ : S ⟶ F.op ⋙ R\nd : M.Derivation φ\nX : Dᵒᵖ\n⊢ d.d 1 = 0",
"ppTerm": "?m.87",
"assigned": fals... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nS : Cᵒᵖ ⥤ CommRingCat\nF : C ⥤ D\nR : Dᵒᵖ ⥤ CommRingCat\nM : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nφ : S ⟶ F.op ⋙ R\nd : M.Derivation φ\nX : Dᵒᵖ\n⊢ d.d 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Differentials.Presheaf | {
"line": 87,
"column": 24
} | {
"line": 87,
"column": 35
} | {
"line": 87,
"column": 36
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nS : Cᵒᵖ ⥤ CommRingCat\nF : C ⥤ D\nS' R : Dᵒᵖ ⥤ CommRingCat\nM N : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nφ : S ⟶ F.op ⋙ R\nφ' : S' ⟶ R\nd : M.Derivation φ\nf : M ⟶ N\nX Y : Dᵒᵖ\ng : X ⟶ Y\nx : ↑(R.obj X)\... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nS : Cᵒᵖ ⥤ CommRingCat\nF : C ⥤ D\nS' R : Dᵒᵖ ⥤ CommRingCat\nM N : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nφ : S ⟶ F.op ⋙ R\nφ' : S' ⟶ R\nd : M.Derivation φ\nf : M ⟶ N\nX Y : Dᵒᵖ\ng : X ⟶ Y\nx : ↑(R.obj X)\n⊢ (ModuleCa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 386,
"column": 50
} | {
"line": 386,
"column": 94
} | {
"line": 386,
"column": 95
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nsurj : Function.Surjective ⇑(algebraMap A B)\nI : Ideal B\nJ : Ideal A\neq : comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J\neqmap : I = map (algebraMap A B) J\nx : J.Cotangent\nx' : ↥J\nhx' : J.toCo... | [
"A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nsurj : Function.Surjective ⇑(algebraMap A B)\nI : Ideal B\nJ : Ideal A\neq : comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J\neqmap : I = map (algebraMap A B) J\nx : J.Cotangent\nx' : ↥J\nhx' : J.toCotangent x' =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 225,
"column": 21
} | {
"line": 225,
"column": 36
} | {
"line": 226,
"column": 6
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM M₁ M₂ : PresheafOfModules R\nφ : M₁.presheaf ⟶ M₂.presheaf\nhφ :\n ∀ (X : Cᵒᵖ) (r : ↑(R.obj X)) (m : ↑(M₁.obj X)),\n (ConcreteCategory.hom (φ.app X)) (r • m) = r • (ConcreteCategory.hom (φ.app X)) m\nX : Cᵒᵖ\n⊢ ∀ (x y : ↑(M₁.1 X)),\n ... | [] | simp +instances | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 225,
"column": 21
} | {
"line": 225,
"column": 36
} | {
"line": 226,
"column": 6
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM M₁ M₂ : PresheafOfModules R\nφ : M₁.presheaf ⟶ M₂.presheaf\nhφ :\n ∀ (X : Cᵒᵖ) (r : ↑(R.obj X)) (m : ↑(M₁.obj X)),\n (ConcreteCategory.hom (φ.app X)) (r • m) = r • (ConcreteCategory.hom (φ.app X)) m\nX : Cᵒᵖ\n⊢ ∀ (x y : ↑(M₁.1 X)),\n ... | [] | simp +instances | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 225,
"column": 21
} | {
"line": 225,
"column": 36
} | {
"line": 226,
"column": 6
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM M₁ M₂ : PresheafOfModules R\nφ : M₁.presheaf ⟶ M₂.presheaf\nhφ :\n ∀ (X : Cᵒᵖ) (r : ↑(R.obj X)) (m : ↑(M₁.obj X)),\n (ConcreteCategory.hom (φ.app X)) (r • m) = r • (ConcreteCategory.hom (φ.app X)) m\nX : Cᵒᵖ\n⊢ ∀ (x y : ↑(M₁.1 X)),\n ... | [] | simp +instances | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ComplexShape | {
"line": 157,
"column": 14
} | {
"line": 157,
"column": 25
} | {
"line": 157,
"column": 26
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ∀ (k : ι), ¬c.Rel j k\n⊢ ¬∃ j_1, c.Rel j j_1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Exists",
"id",
"Eq",
"Not",
"ComplexShape.Rel"
],
"usedFVars... | [
"ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ∀ (k : ι), ¬c.Rel j k\n⊢ ∀ (x : ι), ¬c.Rel j x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ComplexShape | {
"line": 162,
"column": 41
} | {
"line": 162,
"column": 74
} | {
"line": 162,
"column": 75
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ¬c.Rel j (c.next j)\nk : ι\nhk' : c.Rel j k\n⊢ c.Rel j (c.next j)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"ComplexShape.next",
"Eq",
"ComplexShape.Rel",
"Comp... | [
"ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ¬c.Rel j (c.next j)\nk : ι\nhk' : c.Rel j k\n⊢ c.Rel j k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.End | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 13
} | {
"line": 178,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nm n m' n' : M\nf : m ⟶ m'\ng : n ⟶ n'\nX : C\ninst✝ : F.LaxMonoidal\nthis :\n ((F.map g).app ((F.obj m).obj X) ≫ (F.obj n').map ((F.map f).app X)) ≫ (μ F m' n').app X =\n ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nm n m' n' : M\nf : m ⟶ m'\ng : n ⟶ n'\nX : C\ninst✝ : F.LaxMonoidal\nthis :\n ((F.map g).app ((F.obj m).obj X) ≫ (F.obj n').map ((F.map f).app X)) ≫ (μ F m' n').app X =\n (μ F m n).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.End | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 13
} | {
"line": 257,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nm₁ m₂ m₃ : M\nX : C\ninst✝ : F.LaxMonoidal\nthis :\n (F.obj m₃).map ((μ F m₁ m₂).app X) ≫ (μ F (m₁ ⊗ m₂) m₃).app X ≫ (F.map (α_ m₁ m₂ m₃).hom).app X =\n 𝟙 ((F.obj m₃).o... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nm₁ m₂ m₃ : M\nX : C\ninst✝ : F.LaxMonoidal\nthis :\n (F.obj m₃).map ((μ F m₁ m₂).app X) ≫ (μ F (m₁ ⊗ m₂) m₃).app X ≫ (F.map (α_ m₁ m₂ m₃).hom).app X =\n 𝟙 ((F.obj m₃).obj ((F.obj m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GradedObject | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 53
} | {
"line": 94,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nβ : Type u_1\nX Y : GradedObject β C\nf : X ⟶ Y\nhf : ∀ (i : β), IsIso (f i)\n⊢ IsIso f",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsIso",
"id",
"CategoryTheory.GradedObject.categoryOfGradedObjects",
... | [
"C : Type u\ninst✝ : Category.{v, u} C\nβ : Type u_1\nX Y : GradedObject β C\nf : X ⟶ Y\nhf : ∀ (i : β), IsIso (f i)\n⊢ IsIso (X.isoMk Y fun i ↦ asIso (f i)).hom"
] | change IsIso (isoMk X Y (fun i => asIso (f i))).hom | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 53
} | {
"line": 91,
"column": 54
} | [
{
"pp": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nX₁ : ι → V\nd₁ : (i j : ι) → X₁ i ⟶ X₁ j\ns₁ : ∀ (i j : ι), ¬c.Rel i j → d₁ i j = 0\nh₁ : ∀ (i j k : ι), c.Rel i j → c.Rel j k → d₁ i j ≫ d₁ j k = 0\nd₂ : (i j : ι) → X₁ i ⟶ X₁ j\ns₂ : ∀ (i j... | [
"case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nX₁ : ι → V\nd₁ : (i j : ι) → X₁ i ⟶ X₁ j\ns₁ : ∀ (i j : ι), ¬c.Rel i j → d₁ i j = 0\nh₁ : ∀ (i j k : ι), c.Rel i j → c.Rel j k → d₁ i j ≫ d₁ j k = 0\nd₂ : (i j : ι) → X₁ i ⟶ X₁ j\ns₂ : ∀ (i j : ι), ¬c.Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 25
} | {
"line": 228,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nA B : HomologicalComplex V c\nf : A.Hom B\ni j : ι\nhij : c.Rel i j\n⊢ f.f i ≫ B.d i j = A.d i j ≫ f.f j",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Homologic... | [] | exact f.comm' i j hij | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 25
} | {
"line": 228,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nA B : HomologicalComplex V c\nf : A.Hom B\ni j : ι\nhij : c.Rel i j\n⊢ f.f i ≫ B.d i j = A.d i j ≫ f.f j",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Homologic... | [] | exact f.comm' i j hij | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 25
} | {
"line": 228,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nA B : HomologicalComplex V c\nf : A.Hom B\ni j : ι\nhij : c.Rel i j\n⊢ f.f i ≫ B.d i j = A.d i j ≫ f.f j",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Homologic... | [] | exact f.comm' i j hij | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 277,
"column": 2
} | {
"line": 278,
"column": 9
} | {
"line": 278,
"column": 10
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ : A\nX : C\n⊢ (shiftFunctorAdd C (a₁ + a₂) a₃ ≪≫\n isoWhiskerRight (shiftFunctorAdd C a₁ a₂) (shiftFunctor C a₃) ≪≫\n (shiftFunctor C a₁).associator (shiftFunctor C a₂) (sh... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ : A\nX : C\n⊢ (shiftFunctorAdd C (a₁ + a₂) a₃).hom.app X ≫\n (shiftFunctor C a₃).map ((shiftFunctorAdd C a₁ a₂).hom.app X) ≫\n 𝟙 (((shiftFunctor C a₁ ⋙ shiftFunctor C a₂) ⋙ shiftFunctor C a₃).obj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 13
} | {
"line": 287,
"column": 14
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nX : C\n⊢ (shiftFunctorAdd' C 0 a a ⋯).hom.app X = (shiftFunctor C a).map ((shiftFunctorZero C A).inv.app X)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nX : C\n⊢ (shiftFunctorAdd' C 0 a a ⋯).hom.app X = (shiftFunctor C a).map ((shiftFunctorZero C A).inv.app X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 13
} | {
"line": 299,
"column": 14
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nX : C\n⊢ (shiftFunctorAdd' C 0 a a ⋯).inv.app X = (shiftFunctor C a).map ((shiftFunctorZero C A).hom.app X)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nX : C\n⊢ (shiftFunctorAdd' C 0 a a ⋯).inv.app X = (shiftFunctor C a).map ((shiftFunctorZero C A).hom.app X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 13
} | {
"line": 310,
"column": 14
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nX : C\n⊢ (shiftFunctorAdd' C a 0 a ⋯).hom.app X = (shiftFunctorZero C A).inv.app ((shiftFunctor C a).obj X)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nX : C\n⊢ (shiftFunctorAdd' C a 0 a ⋯).hom.app X = (shiftFunctorZero C A).inv.app ((shiftFunctor C a).obj X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 13
} | {
"line": 321,
"column": 14
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nX : C\n⊢ (shiftFunctorAdd' C a 0 a ⋯).inv.app X = (shiftFunctorZero C A).hom.app ((shiftFunctor C a).obj X)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nX : C\n⊢ (shiftFunctorAdd' C a 0 a ⋯).inv.app X = (shiftFunctorZero C A).hom.app ((shiftFunctor C a).obj X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 683,
"column": 4
} | {
"line": 683,
"column": 15
} | {
"line": 683,
"column": 16
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : HasZeroMorphisms V\nα : Type u_2\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX✝ : α → V\nd_X : (n : α) → X✝ (n + 1) ⟶ X✝ n\nsq_X : ∀ (n : α), d_X (n + 1) ≫ d_X n = 0\nY✝ : α → V\nd_Y : (n : α) → Y✝ (n + 1) ⟶ ... | [
"ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : HasZeroMorphisms V\nα : Type u_2\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX✝ : α → V\nd_X : (n : α) → X✝ (n + 1) ⟶ X✝ n\nsq_X : ∀ (n : α), d_X (n + 1) ≫ d_X n = 0\nY✝ : α → V\nd_Y : (n : α) → Y✝ (n + 1) ⟶ Y✝ n\nsq_Y :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 13
} | {
"line": 336,
"column": 14
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A\nh₁₂ : a₁ + a₂ = a₁₂\nh₂₃ : a₂ + a₃ = a₂₃\nh₁₂₃ : a₁ + a₂ + a₃ = a₁₂₃\nX : C\n⊢ (shiftFunctorAdd' C a₁₂ a₃ a₁₂₃ ⋯).hom.app X ≫\n (shiftFunctor C a₃).map ((shiftFunctorAdd' C ... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A\nh₁₂ : a₁ + a₂ = a₁₂\nh₂₃ : a₂ + a₃ = a₂₃\nh₁₂₃ : a₁ + a₂ + a₃ = a₁₂₃\nX : C\n⊢ (shiftFunctorAdd' C a₁₂ a₃ a₁₂₃ ⋯).hom.app X ≫\n (shiftFunctor C a₃).map ((shiftFunctorAdd' C a₁ a₂ a₁₂ h₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 13
} | {
"line": 347,
"column": 14
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A\nh₁₂ : a₁ + a₂ = a₁₂\nh₂₃ : a₂ + a₃ = a₂₃\nh₁₂₃ : a₁ + a₂ + a₃ = a₁₂₃\nX : C\n⊢ (shiftFunctor C a₃).map ((shiftFunctorAdd' C a₁ a₂ a₁₂ h₁₂).inv.app X) ≫\n (shiftFunctorAdd' C... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A\nh₁₂ : a₁ + a₂ = a₁₂\nh₂₃ : a₂ + a₃ = a₂₃\nh₁₂₃ : a₁ + a₂ + a₃ = a₁₂₃\nX : C\n⊢ (shiftFunctor C a₃).map ((shiftFunctorAdd' C a₁ a₂ a₁₂ h₁₂).inv.app X) ≫\n (shiftFunctorAdd' C a₁₂ a₃ a₁₂₃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 13
} | {
"line": 357,
"column": 14
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ : A\nX : C\n⊢ (shiftFunctorAdd C (a₁ + a₂) a₃).hom.app X ≫ (shiftFunctor C a₃).map ((shiftFunctorAdd C a₁ a₂).hom.app X) =\n (shiftFunctorAdd' C a₁ (a₂ + a₃) (a₁ + a₂ + a₃) ⋯).hom.app X ≫\n ... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ : A\nX : C\n⊢ (shiftFunctorAdd C (a₁ + a₂) a₃).hom.app X ≫ (shiftFunctor C a₃).map ((shiftFunctorAdd C a₁ a₂).hom.app X) =\n (shiftFunctorAdd' C a₁ (a₂ + a₃) (a₁ + a₂ + a₃) ⋯).hom.app X ≫\n (shiftFunc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 13
} | {
"line": 366,
"column": 14
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ : A\nX : C\n⊢ (shiftFunctor C a₃).map ((shiftFunctorAdd C a₁ a₂).inv.app X) ≫ (shiftFunctorAdd C (a₁ + a₂) a₃).inv.app X =\n (shiftFunctorAdd C a₂ a₃).inv.app ((shiftFunctor C a₁).obj X) ≫\n ... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ : A\nX : C\n⊢ (shiftFunctor C a₃).map ((shiftFunctorAdd C a₁ a₂).inv.app X) ≫ (shiftFunctorAdd C (a₁ + a₂) a₃).inv.app X =\n (shiftFunctorAdd C a₂ a₃).inv.app ((shiftFunctor C a₁).obj X) ≫\n (shiftFun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Single | {
"line": 230,
"column": 29
} | {
"line": 230,
"column": 40
} | {
"line": 230,
"column": 41
} | [
{
"pp": "V : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasZeroObject V\nC : ChainComplex V ℕ\nX : V\nf : { f // C.d 1 0 ≫ f = 0 }\ni : ℕ\nhi : (ComplexShape.down ℕ).Rel i 0\n⊢ i = 1",
"ppTerm": "?m.121",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"V : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasZeroObject V\nC : ChainComplex V ℕ\nX : V\nf : { f // C.d 1 0 ≫ f = 0 }\ni : ℕ\nhi : (ComplexShape.down ℕ).Rel i 0\n⊢ i = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 486,
"column": 2
} | {
"line": 488,
"column": 9
} | {
"line": 488,
"column": 10
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nn m : A\nh : n + m = 0\nX : C\n⊢ (shiftFunctor C n).map ((shiftFunctorAdd' C n m 0 h).inv.app X ≫ (shiftFunctorZero C A).hom.app X) =\n (shiftFunctorAdd' C m n 0 ⋯).inv.app ((shiftFunctor C n).obj X) ≫\n... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nn m : A\nh : n + m = 0\nX : C\n⊢ (shiftFunctor C n).map ((shiftFunctorAdd' C n m 0 h).inv.app X) ≫ (shiftFunctorAdd' C 0 n n ⋯).inv.app X =\n (shiftFunctorAdd' C m n 0 ⋯).inv.app ((shiftFunctor C n).obj X) ≫ (shiftF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Single | {
"line": 298,
"column": 29
} | {
"line": 298,
"column": 40
} | {
"line": 298,
"column": 41
} | [
{
"pp": "V : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasZeroObject V\nC : CochainComplex V ℕ\nX : V\nf : { f // f ≫ C.d 0 1 = 0 }\ni : ℕ\nhi : (ComplexShape.up ℕ).Rel 0 i\n⊢ i = 1",
"ppTerm": "?m.119",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"V : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasZeroObject V\nC : CochainComplex V ℕ\nX : V\nf : { f // f ≫ C.d 0 1 = 0 }\ni : ℕ\nhi : (ComplexShape.up ℕ).Rel 0 i\n⊢ i = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 944,
"column": 4
} | {
"line": 944,
"column": 15
} | {
"line": 944,
"column": 16
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : HasZeroMorphisms V\nα : Type u_2\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX✝ : α → V\nd_X : (n : α) → X✝ n ⟶ X✝ (n + 1)\nsq_X : ∀ (n : α), d_X n ≫ d_X (n + 1) = 0\nY✝ : α → V\nd_Y : (n : α) → Y✝ n ⟶ Y✝ (n ... | [
"ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : HasZeroMorphisms V\nα : Type u_2\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX✝ : α → V\nd_X : (n : α) → X✝ n ⟶ X✝ (n + 1)\nsq_X : ∀ (n : α), d_X n ≫ d_X (n + 1) = 0\nY✝ : α → V\nd_Y : (n : α) → Y✝ n ⟶ Y✝ (n + 1)\nsq_Y :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Additive | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 55
} | {
"line": 293,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nV : Type u\ninst✝¹¹ : Category.{v, u} V\ninst✝¹⁰ : Preadditive V\nW : Type u_2\ninst✝⁹ : Category.{v_1, u_2} W\ninst✝⁸ : Preadditive W\nW₁ : Type u_3\nW₂ : Type u_4\ninst✝⁷ : Category.{v_2, u_3} W₁\ninst✝⁶ : Category.{v_3, u_4} W₂\ninst✝⁵ : HasZeroMorphisms W₁\ninst✝⁴ : HasZeroM... | [] | · apply (isZero_single_obj_X c j _ _ h).eq_of_tgt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 53
} | {
"line": 431,
"column": 54
} | [
{
"pp": "case a\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nx : S\nI : Ideal (S ⊗[R] S) := Ideal.span ↑(Finset.image (fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) (EssFiniteType.finset R S))\nthis :\n (IsScalarTower.toAlgHom R (S ⊗[R] S) (S ⊗[R] S ⧸ I))... | [
"case a\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nx : S\nI : Ideal (S ⊗[R] S) := Ideal.span ↑(Finset.image (fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) (EssFiniteType.finset R S))\nthis :\n (IsScalarTower.toAlgHom R (S ⊗[R] S) (S ⊗[R] S ⧸ I)).comp Tensor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 628,
"column": 11
} | {
"line": 628,
"column": 50
} | {
"line": 628,
"column": 51
} | [
{
"pp": "case e'_4.e'_4\nR : Type u\nS : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nA : Type u_2\nB : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R A B\ninst✝ : Is... | [
"case e'_4.e'_4\nR : Type u\nS : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nA : Type u_2\nB : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsScalarTower ... | ← IsScalarTower.algebraMap_apply R A B, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 806,
"column": 75
} | {
"line": 806,
"column": 86
} | {
"line": 806,
"column": 87
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : Ω[A⁄R]\nhx :\n (LinearMap.rTensor Ω[A⁄R] (Algebra.linearM... | [
"R : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : Ω[A⁄R]\nhx :\n (LinearMap.rTensor Ω[A⁄R] (Algebra.linearMap A B)) ((T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 820,
"column": 6
} | {
"line": 821,
"column": 56
} | {
"line": 821,
"column": 57
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : A →₀ A\nhx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A... | [
"R : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : A →₀ A\nhx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 834,
"column": 8
} | {
"line": 834,
"column": 23
} | {
"line": 835,
"column": 8
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : A →₀ A\nhx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A... | [
"R : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : A →₀ A\nhx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapD... | convert! i.prop | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 833,
"column": 4
} | {
"line": 836,
"column": 40
} | {
"line": 838,
"column": 0
} | [
{
"pp": "case w\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : A →₀ A\nhx : x ∈ ↑(mapRange.linearMap (Algebra.lin... | [] | · have : x i ≠ 0 ∧ algebraMap A B i = c := by
convert! i.prop
simp_rw [Finset.mem_filter, Finsupp.mem_support_iff]
simp [RingHom.mem_ker, ha, this.2] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 722,
"column": 2
} | {
"line": 724,
"column": 56
} | {
"line": 726,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nK L : ChainComplex C ℕ\nφ : K ⟶ L\ninst✝¹ : HomologicalComplex.HasHomology K 0\ninst✝ : HomologicalComplex.HasHomology L 0\n⊢ K.isoHomologyι₀.inv ≫ HomologicalComplex.homologyMap φ 0 = HomologicalComplex.opcyclesMap φ 0 ≫ L.isoH... | [] | simp only [assoc, ← cancel_mono (L.homologyι 0),
HomologicalComplex.homologyι_naturality, HomologicalComplex.isoHomologyι_inv_hom_id_assoc,
HomologicalComplex.isoHomologyι_inv_hom_id, comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 722,
"column": 2
} | {
"line": 724,
"column": 56
} | {
"line": 726,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nK L : ChainComplex C ℕ\nφ : K ⟶ L\ninst✝¹ : HomologicalComplex.HasHomology K 0\ninst✝ : HomologicalComplex.HasHomology L 0\n⊢ K.isoHomologyι₀.inv ≫ HomologicalComplex.homologyMap φ 0 = HomologicalComplex.opcyclesMap φ 0 ≫ L.isoH... | [] | simp only [assoc, ← cancel_mono (L.homologyι 0),
HomologicalComplex.homologyι_naturality, HomologicalComplex.isoHomologyι_inv_hom_id_assoc,
HomologicalComplex.isoHomologyι_inv_hom_id, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 722,
"column": 2
} | {
"line": 724,
"column": 56
} | {
"line": 726,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nK L : ChainComplex C ℕ\nφ : K ⟶ L\ninst✝¹ : HomologicalComplex.HasHomology K 0\ninst✝ : HomologicalComplex.HasHomology L 0\n⊢ K.isoHomologyι₀.inv ≫ HomologicalComplex.homologyMap φ 0 = HomologicalComplex.opcyclesMap φ 0 ≫ L.isoH... | [] | simp only [assoc, ← cancel_mono (L.homologyι 0),
HomologicalComplex.homologyι_naturality, HomologicalComplex.isoHomologyι_inv_hom_id_assoc,
HomologicalComplex.isoHomologyι_inv_hom_id, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 37
} | {
"line": 197,
"column": 38
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\ni : ι\nhi : ¬c.Rel i (c.next i)\nA : C\nf g : A ⟶ X φ i\nh : f ≫ sndX φ i = g ≫ sndX φ i\n⊢ f ≫ (XI... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\ni : ι\nhi : ¬c.Rel i (c.next i)\nA : C\nf g : A ⟶ X φ i\nh : f ≫ sndX φ i = g ≫ sndX φ i\n⊢ f ≫ (XIso φ i hi).h... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 83
} | {
"line": 207,
"column": 84
} | [
{
"pp": "case h₁\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\nj : ι\nA : C\nf g : X φ j ⟶ A\nh₂ : inrX φ j ≫ f = inrX φ j ≫ g\nhij : c.Rel j (c.next j)\... | [
"case h₁\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\nj : ι\nA : C\nf g : X φ j ⟶ A\nh₂ : inrX φ j ≫ f = inrX φ j ≫ g\nhij : c.Rel j (c.next j)\nh₁ : inlX φ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 37
} | {
"line": 212,
"column": 38
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\ni : ι\nhi : ¬c.Rel i (c.next i)\nA : C\nf g : X φ i ⟶ A\nh : inrX φ i ≫ f = inrX φ i ≫ g\n⊢ (XIso φ... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\ni : ι\nhi : ¬c.Rel i (c.next i)\nA : C\nf g : X φ i ⟶ A\nh : inrX φ i ≫ f = inrX φ i ≫ g\n⊢ (XIso φ i hi).inv ≫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 145,
"column": 26
} | {
"line": 145,
"column": 57
} | {
"line": 145,
"column": 58
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni✝ : ι\nh : Homotopy (f - g) 0\ni : ι\n⊢ f.f i = (((dNext i) fun i j ↦ h.hom i j) + (prevD i) fun i j ↦ h.hom i j) + g.f i",
"ppTerm": "?m.122"... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni✝ : ι\nh : Homotopy (f - g) 0\ni : ι\n⊢ f.f i = (C.dFrom i ≫ (fromNext i) fun i j ↦ h.hom i j) + ((toPrev i) fun i j ↦ h.hom i j) ≫ D.dTo i + g.f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 78
} | {
"line": 212,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nn : ℤ\nγ : Cochain F G n\np q : ℤ\nhpq : p + n = q\n⊢ γ.v p q hpq ≫ (HomologicalComplex.XIsoOfEq G ⋯).hom = γ.v p q ⋯",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"CategoryTheory.Cate... | [] | simp only [HomologicalComplex.XIsoOfEq, eqToIso_refl, Iso.refl_hom, comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Homotopy | {
"line": 498,
"column": 42
} | {
"line": 498,
"column": 53
} | {
"line": 498,
"column": 54
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 0 ⟶ Q.X 1\ncomm_zero : e.f 0 = zero ≫ Q.d 1 0\none : P.X 1 ⟶ Q.X 2\ncomm_one : e.f 1 = P.d 1 0 ... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 0 ⟶ Q.X 1\ncomm_zero : e.f 0 = zero ≫ Q.d 1 0\none : P.X 1 ⟶ Q.X 2\ncomm_one : e.f 1 = P.d 1 0 ≫ zero + one... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 35
} | {
"line": 580,
"column": 35
} | [
{
"pp": "case a\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ ¬(ComplexShape.up ℕ).Rel ((ComplexShape.up ℕ).prev 0) 0",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"congr... | [
"case a\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ ¬(ComplexShape.up ℕ).Rel 0 0"
] | rw [CochainComplex.prev_nat_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 627,
"column": 42
} | {
"line": 627,
"column": 53
} | {
"line": 627,
"column": 54
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 1 ⟶ Q.X 0\ncomm_zero : e.f 0 = P.d 0 1 ≫ zero\none : P.X 2 ⟶ Q.X 1\ncomm_one : e.f 1 = zero ≫... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 1 ⟶ Q.X 0\ncomm_zero : e.f 0 = P.d 0 1 ≫ zero\none : P.X 2 ⟶ Q.X 1\ncomm_one : e.f 1 = zero ≫ Q.d 0 1 + P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 647,
"column": 9
} | {
"line": 647,
"column": 45
} | {
"line": 647,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m✝ : ℤ\nz : Cochain F G n\nm : ℤ\nhnm : n + 1 = m\nh : δ n m z = 0\n⊢ z ∈ cocycle F G n",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m✝ : ℤ\nz : Cochain F G n\nm : ℤ\nhnm : n + 1 = m\nh : δ n m z = 0\n⊢ δ n m z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 669,
"column": 4
} | {
"line": 670,
"column": 11
} | {
"line": 670,
"column": 12
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m i : ℤ\nz : Cochain F G 0\nhz : δ 0 1 z = 0\n⊢ z.v i i ⋯ ≫ G.d i (i + 1) = F.d i (i + 1) ≫ z.v (i + 1) (i + 1) ⋯",
"ppTerm": "?m.92",
"assigned": f... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m i : ℤ\nz : Cochain F G 0\nhz : δ 0 1 z = 0\n⊢ z.v i i ⋯ ≫ G.d i (i + 1) = F.d i (i + 1) ≫ z.v (i + 1) (i + 1) ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 668,
"column": 8
} | {
"line": 668,
"column": 65
} | {
"line": 669,
"column": 8
} | [
{
"pp": "case e_a.e_a.succ\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 1 ⟶ Q.X 0\ncomm_zero : e.f 0 = P.d 0 1 ≫ zero\none : P.X 2 ⟶ Q.X 1\ncomm_o... | [
"case e_a.e_a.succ\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 1 ⟶ Q.X 0\ncomm_zero : e.f 0 = P.d 0 1 ≫ zero\none : P.X 2 ⟶ Q.X 1\ncomm_one : e.f 1 =... | rw [mkCoinductiveAux₃ e zero comm_zero one comm_one succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 254,
"column": 2
} | {
"line": 256,
"column": 9
} | {
"line": 256,
"column": 10
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\np q : ℤ\nhpq : p + 1 = q\n⊢ (↑(fst φ)).v p q hpq ≫ (inl φ).v q p ⋯ + (snd φ).v p p ⋯ ≫ (inr φ).f p = 𝟙 ((mappingCone φ).X p)",
"ppTerm": "?m.111",
"assigned": ... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\np q : ℤ\nhpq : p + 1 = q\n⊢ (↑(fst φ)).v p q hpq ≫ (inl φ).v q p ⋯ + (snd φ).v p p ⋯ ≫ (inr φ).f p = 𝟙 ((mappingCone φ).X p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 732,
"column": 25
} | {
"line": 732,
"column": 36
} | {
"line": 733,
"column": 4
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC✝ D✝ E✝ : HomologicalComplex V c\nf✝ g✝ : C✝ ⟶ D✝\nh k : D✝ ⟶ E✝\ni : ι\nC D E : HomologicalComplex V c\nf : HomotopyEquiv C D\ng : HomotopyEquiv D E\n⊢ Homotopy ((f.hom ≫ g.hom) ≫ g.inv ≫ f.inv) (𝟙 C)",
... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC✝ D✝ E✝ : HomologicalComplex V c\nf✝ g✝ : C✝ ⟶ D✝\nh k : D✝ ⟶ E✝\ni : ι\nC D E : HomologicalComplex V c\nf : HomotopyEquiv C D\ng : HomotopyEquiv D E\n⊢ Homotopy (f.hom ≫ g.hom ≫ g.inv ≫ f.inv) (𝟙 C)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 734,
"column": 25
} | {
"line": 734,
"column": 36
} | {
"line": 735,
"column": 4
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC✝ D✝ E✝ : HomologicalComplex V c\nf✝ g✝ : C✝ ⟶ D✝\nh k : D✝ ⟶ E✝\ni : ι\nC D E : HomologicalComplex V c\nf : HomotopyEquiv C D\ng : HomotopyEquiv D E\n⊢ Homotopy ((g.inv ≫ f.inv) ≫ f.hom ≫ g.hom) (𝟙 E)",
... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC✝ D✝ E✝ : HomologicalComplex V c\nf✝ g✝ : C✝ ⟶ D✝\nh k : D✝ ⟶ E✝\ni : ι\nC D E : HomologicalComplex V c\nf : HomotopyEquiv C D\ng : HomotopyEquiv D E\n⊢ Homotopy (g.inv ≫ f.inv ≫ f.hom ≫ g.hom) (𝟙 E)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 28
} | {
"line": 340,
"column": 6
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain F K m\nβ : Cochain G K n\nh : m + 1 = n\np₁ p₂ p₃ : ℤ\nh₁₂ : p₁ + -1 = p₂\nh₂₃ : p₂ + n = p₃\n⊢ (inl φ).v p₁ p₂ h₁₂ ≫ (desc... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain F K m\nβ : Cochain G K n\nh : m + 1 = n\np₁ p₂ p₃ : ℤ\nh₁₂ : p₁ + -1 = p₂\nh₂₃ : p₂ + n = p₃\n⊢ (inl φ).v p₁ p₂ h₁₂ ≫ (descCochain φ α ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 345,
"column": 2
} | {
"line": 346,
"column": 9
} | {
"line": 346,
"column": 10
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain F K m\nβ : Cochain G K n\nh : m + 1 = n\np₁ p₂ : ℤ\nh₁₂ : p₁ + n = p₂\n⊢ (inr φ).f p₁ ≫ (descCochain φ α β h).v p₁ p₂ h₁₂ =... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain F K m\nβ : Cochain G K n\nh : m + 1 = n\np₁ p₂ : ℤ\nh₁₂ : p₁ + n = p₂\n⊢ (inr φ).f p₁ ≫ (descCochain φ α β h).v p₁ p₂ h₁₂ = β.v p₁ p₂ h... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 455,
"column": 2
} | {
"line": 456,
"column": 9
} | {
"line": 456,
"column": 10
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\np₁ p₂ p₃ : ℤ\nh₁₂ : p₁ + n = p₂\nh₂₃ : p₂ + 1 = p₃\n⊢ (liftCochain φ α β h).v p₁ p... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\np₁ p₂ p₃ : ℤ\nh₁₂ : p₁ + n = p₂\nh₂₃ : p₂ + 1 = p₃\n⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (↑(f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 461,
"column": 2
} | {
"line": 462,
"column": 9
} | {
"line": 462,
"column": 10
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\np₁ p₂ : ℤ\nh₁₂ : p₁ + n = p₂\n⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (snd φ).v p₂ p... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\np₁ p₂ : ℤ\nh₁₂ : p₁ + n = p₂\n⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (snd φ).v p₂ p₂ ⋯ = β.v p₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Quotient | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 13
} | {
"line": 69,
"column": 14
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\na b : C\nm₁ m₂ : a ⟶ b\nh : r m₁ m₂\n⊢ CompClosure r m₁ m₂",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\na b : C\nm₁ m₂ : a ⟶ b\nh : r m₁ m₂\n⊢ CompClosure r m₁ m₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Quotient | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 15
} | {
"line": 74,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\na b e : C\nf : a ⟶ b\nX Y c d : C\nh₁ h₂ : c ⟶ d\nh : r h₁ h₂\ng : b ⟶ c\ni : d ⟶ e\n⊢ CompClosure r (f ≫ g ≫ h₁ ≫ i) (f ≫ g ≫ h₂ ≫ i)",
"ppTerm": "?m.98",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\na b e : C\nf : a ⟶ b\nX Y c d : C\nh₁ h₂ : c ⟶ d\nh : r h₁ h₂\ng : b ⟶ c\ni : d ⟶ e\n⊢ CompClosure r (f ≫ g ≫ h₁ ≫ i) (f ≫ g ≫ h₂ ≫ i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Quotient | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 15
} | {
"line": 79,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\na d e : C\ng : d ⟶ e\nX Y b c : C\ng₁ g₂ : b ⟶ c\nh : r g₁ g₂\nf : a ⟶ b\ni : c ⟶ d\n⊢ CompClosure r ((f ≫ g₁ ≫ i) ≫ g) ((f ≫ g₂ ≫ i) ≫ g)",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\na d e : C\ng : d ⟶ e\nX Y b c : C\ng₁ g₂ : b ⟶ c\nh : r g₁ g₂\nf : a ⟶ b\ni : c ⟶ d\n⊢ CompClosure r (f ≫ g₁ ≫ i ≫ g) (f ≫ g₂ ≫ i ≫ g)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Quotient | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 49
} | {
"line": 232,
"column": 50
} | [
{
"pp": "case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ _root_.Equivalence (HomRel.CompClosure r)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [
"case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ _root_.Equivalence r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 561,
"column": 2
} | {
"line": 562,
"column": 66
} | {
"line": 562,
"column": 67
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK L : CochainComplex C ℤ\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nn' m' : ℤ\nα' : Cochain F L m'\nβ' : Cochain G L n'\nh : n + 1 = m\nh' : m' + 1 = n'\np : ℤ\nhp... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK L : CochainComplex C ℤ\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nn' m' : ℤ\nα' : Cochain F L m'\nβ' : Cochain G L n'\nh : n + 1 = m\nh' : m' + 1 = n'\np : ℤ\nhp : n + n' = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Quotient | {
"line": 310,
"column": 16
} | {
"line": 310,
"column": 57
} | {
"line": 310,
"column": 57
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nr : HomRel C\nD : Type u_2\ninst✝ : Category.{v_2, u_2} D\nF G : Quotient r ⥤ D\nτ₁ τ₂ : F ⟶ G\nh : (functor r).whiskerLeft τ₁ = (functor r).whiskerLeft τ₂\n⊢ τ₁.app = τ₂.app",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Categ... | [] | by ext1 ⟨X⟩; exact NatTrans.congr_app h X | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory | {
"line": 123,
"column": 33
} | {
"line": 123,
"column": 44
} | {
"line": 123,
"column": 45
} | [
{
"pp": "ι : Type u_2\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nf : C ⟶ D\nh : (quotient V c).map f = 0\n⊢ (quotient V c).map f = (quotient V c).map 0",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"ι : Type u_2\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nf : C ⟶ D\nh : (quotient V c).map f = 0\n⊢ (quotient V c).map f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory | {
"line": 124,
"column": 17
} | {
"line": 124,
"column": 28
} | {
"line": 124,
"column": 29
} | [
{
"pp": "ι : Type u_2\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nf : C ⟶ D\nx✝ : Nonempty (Homotopy f 0)\nh : Homotopy f 0\n⊢ (quotient V c).map f = 0",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"ι : Type u_2\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nf : C ⟶ D\nx✝ : Nonempty (Homotopy f 0)\nh : Homotopy f 0\n⊢ (quotient V c).map f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 15
} | {
"line": 194,
"column": 16
} | [
{
"pp": "case mpr\nι : Type u_2\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC : HomologicalComplex V c\nh : Homotopy (𝟙 C) 0\n⊢ 𝟙 ((quotient V c).obj C) = 0",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"case mpr\nι : Type u_2\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC : HomologicalComplex V c\nh : Homotopy (𝟙 C) 0\n⊢ 𝟙 ((quotient V c).obj C) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Shift.Induced | {
"line": 143,
"column": 10
} | {
"line": 143,
"column": 21
} | {
"line": 143,
"column": 22
} | [
{
"pp": "C : Type ?u.2\nD : Type ?u.4\ninst✝⁵ : Category.{v_1, ?u.2} C\ninst✝⁴ : Category.{v_2, ?u.4} D\nF : C ⥤ D\nA : Type ?u.15\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D... | [
"C : Type ?u.2\nD : Type ?u.4\ninst✝⁵ : Category.{v_1, ?u.2} C\ninst✝⁴ : Category.{v_2, ?u.4} D\nF : C ⥤ D\nA : Type ?u.15\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D D).obj F).F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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