module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 167,
"column": 8
} | {
"line": 167,
"column": 29
} | [
{
"pp": "case h\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ModelCategory.Injective | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Abelian C\nA : CochainComplex C ℤ\nhA : CochainComplex.plus C A\nB : CochainComplex C ℤ\nhB : CochainComplex.plus C B\nX : CochainComplex C ℤ\nhX : CochainComplex.plus C X\nY : CochainComplex C ℤ\nhY : CochainComplex.plus C Y\ni : A ⟶ B\ninst✝¹ : M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Basic | {
"line": 75,
"column": 10
} | {
"line": 75,
"column": 32
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_4, u_1} C\ninst✝¹ : Category.{u_3, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k : ι\nf : i ⟶ j\ng : j ⟶ k\ni' j' k' : ι\nf' : i' ⟶ j'\ng' : j' ⟶ k'\nα : mk₁ f ⟶ mk₁ f'\nβ : mk₁ g ⟶ mk₁ g'\nn₀ n₁ : ℤ\nhαβ : α.app 1 = β.app 0\nhn₁ : n₀ + 1 = n₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Monoidal | {
"line": 250,
"column": 42
} | {
"line": 258,
"column": 53
} | [
{
"pp": "C : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : MonoidalCategory C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : (curriedTensor C).Additive\ninst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝³ : AddMonoid I\nc : ComplexShape I\ninst✝² : c.TensorSigns\nK :... | by
by_cases hij : c.Rel i j
· simp only [rightUnitor'_inv, assoc, mapBifunctor.d_eq,
Preadditive.comp_add, mapBifunctor.ι_D₁, mapBifunctor.ι_D₂,
tensor_unit_d₂, comp_zero, add_zero]
rw [mapBifunctor.d₁_eq _ _ _ _ hij _ _ (by simp)]
dsimp
simp only [one_smul, whisker_exchange_assoc, whiskerRi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 80
} | [
{
"pp": "case h.h\nI : Type u\ninst✝⁶ : AddMonoid I\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : MonoidalCategory C\nX₁ X₂ X₃ X₄ : GradedObject I C\ninst✝³ : X₃.HasTensor X₄\ninst✝² : X₂.HasTensor (tensorObj X₃ X₄)\ninst✝¹ : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))\nj : I\nA : C\nf g : tensorObj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 606,
"column": 2
} | {
"line": 606,
"column": 13
} | [
{
"pp": "I : Type u\ninst✝² : AddMonoid I\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : MonoidalCategory C\nn fst✝¹ snd✝¹ : ℕ\nproperty✝¹ : (fst✝¹, snd✝¹) ∈ (fun i ↦ i.1 + i.2) ⁻¹' {n}\nfst✝ snd✝ : ℕ\nproperty✝ : (fst✝, snd✝) ∈ (fun i ↦ i.1 + i.2) ⁻¹' {n}\nh :\n (fun x ↦\n match x with\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 613,
"column": 2
} | {
"line": 613,
"column": 13
} | [
{
"pp": "I : Type u\ninst✝² : AddMonoid I\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : MonoidalCategory C\nn fst✝³ fst✝² snd✝¹ : ℕ\nproperty✝¹ : (fst✝³, fst✝², snd✝¹) ∈ {i | i.1 + i.2.1 + i.2.2 = n}\nfst✝¹ fst✝ snd✝ : ℕ\nproperty✝ : (fst✝¹, fst✝, snd✝) ∈ {i | i.1 + i.2.1 + i.2.2 = n}\nh :\n (fun x ↦\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Differentials | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 47
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ i₂ i₃ i₄ i₅ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₄ : i₃ ⟶ i₄\nf₅ : i₄ ⟶ i₅\nn₀ n₁ n₂ n₃ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nhn₃ : n₂ + 1 = n₃\n⊢... | ← cancel_epi (X.toCycles f₃ f₄ _ rfl n₁), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.SpectralObject.FirstPage | {
"line": 119,
"column": 14
} | {
"line": 120,
"column": 17
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝⁴ : Category.{?u.9037, u_1} C\ninst✝³ : Abelian C\ninst✝² : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝¹ : data.HasFirstPageComputation\ninst✝ : X.HasSpectralSequence data\npq pq' : κ\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.FirstPage | {
"line": 121,
"column": 21
} | {
"line": 121,
"column": 46
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝⁴ : Category.{?u.9037, u_1} C\ninst✝³ : Abelian C\ninst✝² : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝¹ : data.HasFirstPageComputation\ninst✝ : X.HasSpectralSequence data\npq pq' : κ\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.FirstPage | {
"line": 125,
"column": 2
} | {
"line": 128,
"column": 9
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Abelian C\ninst✝² : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝¹ : data.HasFirstPageComputation\ninst✝ : X.HasSpectralSequence data\npq pq' : κ\nhpq :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 29
} | [
{
"pp": "ι : Type u_2\nκ : Type u_3\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr : ℤ\nhr : r₀ ≤ r\npq' : κ\ni₀ i₁ : ι\nhi₀ : i₀ = data.i₀ r pq' ⋯\nhi₁ : i₁ = data.i₁ pq'\nthis : data.i₀ r pq' ⋯ ≤ data.i₁ pq'\n⊢ i₀ ≤ i₁",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 29
} | [
{
"pp": "ι : Type u_2\nκ : Type u_3\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\npq' : κ\ni₁ i₂ : ι\nhi₁ : i₁ = data.i₁ pq'\nhi₂ : i₂ = data.i₂ pq'\n⊢ i₁ ≤ i₂",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Preorder.toLE",
"id",
"LE.l... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 29
} | [
{
"pp": "ι : Type u_2\nκ : Type u_3\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr : ℤ\nhr : r₀ ≤ r\npq' : κ\ni₂ i₃ : ι\nhi₂ : i₂ = data.i₂ pq'\nhi₃ : i₃ = data.i₃ r pq' ⋯\n⊢ i₂ ≤ i₃",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Abelian.SpectralOb... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 30
} | [
{
"pp": "ι : Type u_2\nκ : Type u_3\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' : κ\ni₃ i₃' : ι\nhi₃ : i₃ = data.i₃ r pq' ⋯\nhi₃' : i₃' = data.i₃ r' pq' ⋯\n⊢ i₃ ≤ i₃'",
"usedConstants": [
"Eq.mpr",
"Ca... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Square | {
"line": 334,
"column": 12
} | {
"line": 334,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nsq : Square C\nF : C ⥤ D\n⊢ F.map sq.f₁₂ ≫ F.map sq.f₂₄ = F.map sq.f₁₃ ≫ F.map sq.f₃₄",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Square | {
"line": 351,
"column": 19
} | {
"line": 351,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\nX✝ Y✝ : Square C\nφ : X✝ ⟶ Y✝\n⊢ (X✝.map F).f₁₂ ≫ F.map φ.τ₂ = F.map φ.τ₁ ≫ (Y✝.map F).f₁₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Square | {
"line": 352,
"column": 19
} | {
"line": 352,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\nX✝ Y✝ : Square C\nφ : X✝ ⟶ Y✝\n⊢ (X✝.map F).f₁₃ ≫ F.map φ.τ₃ = F.map φ.τ₁ ≫ (Y✝.map F).f₁₃",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Square | {
"line": 353,
"column": 19
} | {
"line": 353,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\nX✝ Y✝ : Square C\nφ : X✝ ⟶ Y✝\n⊢ (X✝.map F).f₂₄ ≫ F.map φ.τ₄ = F.map φ.τ₂ ≫ (Y✝.map F).f₂₄",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Square | {
"line": 354,
"column": 19
} | {
"line": 354,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\nX✝ Y✝ : Square C\nφ : X✝ ⟶ Y✝\n⊢ (X✝.map F).f₃₄ ≫ F.map φ.τ₄ = F.map φ.τ₃ ≫ (Y✝.map F).f₃₄",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 56
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k : ι\nf : i ⟶ j\ng : j ⟶ k\nhf : IsIso f\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.δ f g n₀ n₁ hn₁ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 57
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k : ι\nf : i ⟶ j\ng : j ⟶ k\nhg : IsIso g\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.δ f g n₀ n₁ hn₁ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 189,
"column": 32
} | {
"line": 189,
"column": 43
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₁₂ : i ⟶ k\nh₁₂ : f₁ ≫ f₂ = f₁₂\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 301,
"column": 6
} | {
"line": 301,
"column": 34
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\nX : SpectralObject C ι\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq : κ\nhpq : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 317,
"column": 6
} | {
"line": 317,
"column": 34
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\nX : SpectralObject C ι\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq : κ\nhpq : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 210,
"column": 36
} | {
"line": 210,
"column": 47
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nhi : IsLimit (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 289,
"column": 36
} | {
"line": 289,
"column": 47
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₁₂ : i ⟶ k\nh₁₂ : f₁ ≫ f₂ = f₁₂\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 311,
"column": 40
} | {
"line": 311,
"column": 51
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nhp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 410,
"column": 45
} | {
"line": 410,
"column": 56
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₁₂ : i ⟶ k\nh₁₂ : f₁ ≫ f₂ = f₁₂\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nA : C\nx₂ : A ⟶ (X.H n₁).obj (mk₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 415,
"column": 6
} | {
"line": 415,
"column": 62
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₁₂ : i ⟶ k\nh₁₂ : f₁ ≫ f₂ = f₁₂\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nA : C\nx₂ : A ⟶ (X.H n₁).obj (mk₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 434,
"column": 62
} | {
"line": 437,
"column": 68
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₁₂ : i ⟶ k\nh₁₂ : f₁ ≫ f₂ = f₁₂\nn₀ n₁ n₂ : ℤ\nA : C\nx : (X.H n₁).obj (mk₁ f₁₂) ⟶ A\nh : (X.H n₁).map (twoδ₂Toδ₁ f₁ f... | by
dsimp only [descE]
rw [← Category.assoc]
apply (X.cokernelSequenceE_exact f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂).g_desc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 464,
"column": 43
} | {
"line": 464,
"column": 54
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nA : C\nx₂ : A ⟶ (X.H n₁).obj (mk₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 468,
"column": 8
} | {
"line": 468,
"column": 19
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nA : C\nx₂ : A ⟶ (X.H n₁).obj (mk₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 144,
"column": 11
} | {
"line": 144,
"column": 45
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\ni₀ i₁ i₂ i₃ i₄ i₅ : ι\nf₁ : i₀ ⟶ i₁\n... | by rw [h₃, data.hc₁₃ r pq pq' hpq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 145,
"column": 14
} | {
"line": 145,
"column": 44
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\ni₀ i₁ i₂ i₃ i₄ i₅ : ι\nf₁ : i₀ ⟶ i₁\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.EventuallyConst | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\ninst✝ : One β\ns : Set α\nc : β\nh : EventuallyConst (s.mulIndicator fun x ↦ c) l\nhc : c ≠ 1\n⊢ EventuallyConst s l",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 52
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ i₂ i₃ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₁₂ : i₀ ⟶ i₂\nh₁₂ : f₁ ≫ f₂ = f₁₂\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\n⊢ X.opcyclesToE f₁ f₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 609,
"column": 10
} | {
"line": 610,
"column": 13
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ i₂ i₃ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₁₂ : i₀ ⟶ i₂\nh₁₂ : f₁ ≫ f₂ = f₁₂\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nA : C\nx₂ : A ⟶ X.opc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 51
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ i₂ i₃ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₂₃ : i₁ ⟶ i₃\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\n⊢ X.πE f₁ f₂ f₃ n₀ n₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 201,
"column": 21
} | {
"line": 202,
"column": 15
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr : ℤ\nhr : r₀ ≤ r\npq pq' pq'' : κ\nhpq : (c r).Rel pq pq'\nhpq' : (c r).Rel pq' pq''\nn₀ n₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 735,
"column": 2
} | {
"line": 735,
"column": 13
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ : ι\nf : i₀ ⟶ i₁\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ (X.cyclesIsoH f n₀ n₁ hn₁).hom ≫ X.toCycles (𝟙 i₀) f f ⋯ n₀ = 𝟙 (X.cycles (𝟙 i₀) f n₀)",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 741,
"column": 2
} | {
"line": 741,
"column": 13
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ : ι\nf : i₀ ⟶ i₁\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.toCycles (𝟙 i₀) f f ⋯ n₀ ≫ (X.cyclesIsoH f n₀ n₁ hn₁).hom = 𝟙 ((X.H n₀).obj (mk₁ f))",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 764,
"column": 2
} | {
"line": 764,
"column": 13
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ : ι\nf : i₀ ⟶ i₁\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.fromOpcycles f (𝟙 i₁) f ⋯ n₁ ≫ (X.opcyclesIsoH f n₀ n₁ hn₁).inv = 𝟙 (X.opcycles f (𝟙 i₁) n₁)",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 770,
"column": 2
} | {
"line": 770,
"column": 13
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ : ι\nf : i₀ ⟶ i₁\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ (X.opcyclesIsoH f n₀ n₁ hn₁).inv ≫ X.fromOpcycles f (𝟙 i₁) f ⋯ n₁ = 𝟙 ((X.H n₁).obj (mk₁ f))",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 214,
"column": 2
} | {
"line": 215,
"column": 72
} | [
{
"pp": "case refine_2\nC : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr : ℤ\nhr : r₀ ≤ r\npq pq' pq'' : κ\nhpq : (c r).Rel pq pq'\nhpq' : (c r).Rel ... | · simp only [← Iso.comp_inv_eq, Category.assoc]
exact (pageD_eq X data r hr pq' pq'' hpq' _ _ _ _ _ rfl rfl ..).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 243,
"column": 19
} | {
"line": 243,
"column": 67
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' pq'' : κ\ni₀' i₀ i₁ i₂ i₃ : ι\nhi₀' : i₀' = dat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 292,
"column": 10
} | {
"line": 292,
"column": 58
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' pq'' : κ\nhpq' : (c r).next pq' = pq''\ni₀' i₀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 862,
"column": 2
} | {
"line": 862,
"column": 88
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ i₂ i₃ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₁₂ : i₀ ⟶ i₂\nf₂₃ : i₁ ⟶ i₃\nh₁₂ : f₁ ≫ f₂ = f₁₂\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ ... | rw [← cancel_mono (X.fromOpcycles f₁ f₂₃ (f₁₂ ≫ f₃) (by cat_disch) n₁), hx, zero_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 333,
"column": 19
} | {
"line": 333,
"column": 67
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\ni₀ i₁ i₂ i₃ i₃' : ι\nhi₀ : i₀ = data.i₀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 911,
"column": 4
} | {
"line": 911,
"column": 20
} | [
{
"pp": "case e_a.e_a.h₀\nC : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ i₂ i₃ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₁₂ : i₀ ⟶ i₂\nh₁₂ : f₁ ≫ f₂ = f₁₂\ni₀' i₁' i₂' i₃' : ι\nf₁' : i₀' ⟶ i₁'\nf₂' : i₁' ⟶ i₂'\nf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 387,
"column": 8
} | {
"line": 387,
"column": 56
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 508,
"column": 52
} | {
"line": 508,
"column": 86
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\n... | by rw [h₃, data.hc₁₃ r pq pq' hpq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 509,
"column": 18
} | {
"line": 509,
"column": 48
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 528,
"column": 24
} | {
"line": 528,
"column": 49
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq : κ\nn : ℤ\nhn : n = data.deg pq\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 555,
"column": 21
} | {
"line": 556,
"column": 15
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' pq'' : κ\nhpq : (c r).Rel pq ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 168,
"column": 21
} | {
"line": 168,
"column": 44
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\ns : Injective ⇑f\nh : ¬Surjective ⇑f\nn : ℕ\n⊢ (f ^ n * f).range < (f ^ n).range",
"usedConstants": [
"Submodule",
"RingHomSurjective.ids",
"Preor... | Module.End.mul_eq_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 40
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : ℕ → Submodule R M\nh : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))\nn : ℕ\nw : ∀ (m : ℕ), n ≤ m → (partialSups (⇑OrderDual.toDual ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 278,
"column": 22
} | {
"line": 278,
"column": 38
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup P\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R P\ninst✝³ : Module R N\nι : Type u_5\ninst✝² : Finite ι\nα✝ : Type u_5\ninst✝¹ : Fintype α✝\nih : ∀ {M : α✝ → Submod... | rw [iSup_option] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 359,
"column": 15
} | {
"line": 359,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nr : R\nx : M\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → ∀ (x : M), x ∈ (r ^ n • LinearMap.id).range ↔ x ∈ (r ^ m • LinearMap.id).range\n⊢ ∃ y, r ^ n.succ • y = r ^ n • x",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 446,
"column": 2
} | {
"line": 446,
"column": 70
} | [
{
"pp": "ι : Type u_2\ninst✝ : Finite ι\n⊢ ∀ {α : Type u_2} [Fintype α],\n (∀ {R : α → Type u_1} [inst : (i : α) → Semiring (R i)] [∀ (i : α), IsArtinianRing (R i)],\n IsArtinianRing ((i : α) → R i)) →\n ∀ {R : Option α → Type u_1} [inst : (i : Option α) → Semiring (R i)] [∀ (i : Option α), IsArt... | · exact fun ih ↦ RingEquiv.isArtinianRing (.symm .piOptionEquivProd) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Artinian.Module | {
"line": 589,
"column": 2
} | {
"line": 589,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\n⊢ nilradical R = iInf MaximalSpectrum.asIdeal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\n⊢ {I | I.IsPrime}.Finite",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"setOf",
"Set.Finite",
"id",
"Ideal",
"funext",
"CommRing.toCommSemiring",
"Ideal.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 649,
"column": 40
} | {
"line": 649,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\nJac : Ideal R := Ring.jacobson R\nn : ℕ\nhn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m\nhn : Jac * Jac ^ n = Jac ^ n\nne✝ : ¬Ring.jacobson R ^ n = 0\nN : Ideal R\neq : Jac... | rw [Jac.pow_zero, N.one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 649,
"column": 40
} | {
"line": 649,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\nJac : Ideal R := Ring.jacobson R\nn : ℕ\nhn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m\nhn : Jac * Jac ^ n = Jac ^ n\nne✝ : ¬Ring.jacobson R ^ n = 0\nN : Ideal R\neq : Jac... | rw [Jac.pow_zero, N.one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Artinian.Module | {
"line": 649,
"column": 40
} | {
"line": 649,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\nJac : Ideal R := Ring.jacobson R\nn : ℕ\nhn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m\nhn : Jac * Jac ^ n = Jac ^ n\nne✝ : ¬Ring.jacobson R ^ n = 0\nN : Ideal R\neq : Jac... | rw [Jac.pow_zero, N.one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 243,
"column": 48
} | {
"line": 243,
"column": 59
} | [
{
"pp": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : IsFaithful R L M\nx : L\nhx : ∀ (m : M), ⁅x, m⁆ = 0\nm : M\n⊢ ((toEnd R L M) x) m = 0 m",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 15
} | [
{
"pp": "case refine_1\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : IsFaithful R L M\nx : L\nhx : (toEnd R L M) x = 0\n⊢ x = 0",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 245,
"column": 4
} | {
"line": 247,
"column": 80
} | [
{
"pp": "case refine_2\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : ∀ (x : L), (∀ (m : M), ⁅x, m⁆ = 0) → x = 0\nx y : L\nhxy : (toEnd R L M) x = (toEnd... | rw [← sub_eq_zero]
refine h _ fun m ↦ ?_
rw [sub_lie, sub_eq_zero, ← toEnd_apply_apply R, ← toEnd_apply_apply R, hxy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 245,
"column": 4
} | {
"line": 247,
"column": 80
} | [
{
"pp": "case refine_2\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : ∀ (x : L), (∀ (m : M), ⁅x, m⁆ = 0) → x = 0\nx y : L\nhxy : (toEnd R L M) x = (toEnd... | rw [← sub_eq_zero]
refine h _ fun m ↦ ?_
rw [sub_lie, sub_eq_zero, ← toEnd_apply_apply R, ← toEnd_apply_apply R, hxy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 255,
"column": 24
} | {
"line": 255,
"column": 35
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\n⊢ Injective ⇑(toEnd R (Module.End R M) M)",
"usedConstants": [
"LieHom",
"Module.End.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 97,
"column": 4
} | {
"line": 99,
"column": 21
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nL' : LieSubalgebra R L\n⊢ ∀ (x y z : ↥L'), ⟨⁅↑x, ↑⟨⁅↑y, ↑z⁆, ⋯⟩⁆, ⋯⟩ = ⟨⁅↑⟨⁅↑x, ↑y⁆, ⋯⟩, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑⟨⁅↑x, ↑z⁆, ⋯⟩⁆, ⋯⟩",
"usedConstants": [
"instIsLieTower",
"leibniz_lie",
"AddMonoid.to... | intros
apply SetCoe.ext
apply leibniz_lie | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 97,
"column": 4
} | {
"line": 99,
"column": 21
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nL' : LieSubalgebra R L\n⊢ ∀ (x y z : ↥L'), ⟨⁅↑x, ↑⟨⁅↑y, ↑z⁆, ⋯⟩⁆, ⋯⟩ = ⟨⁅↑⟨⁅↑x, ↑y⁆, ⋯⟩, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑⟨⁅↑x, ↑z⁆, ⋯⟩⁆, ⋯⟩",
"usedConstants": [
"instIsLieTower",
"leibniz_lie",
"AddMonoid.to... | intros
apply SetCoe.ext
apply leibniz_lie | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 13
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nm : M\nhm : m ∈ ↑N\n⊢ ((toEnd R L M) x ∘ₗ (↑N).subtype) ⟨m, hm⟩ ∈ ↑N",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 385,
"column": 47
} | {
"line": 385,
"column": 66
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nA : Type v\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nA' : Subalgebra R A\nx y : A\nhy : y ∈ (Subalgebra.toSubmodule A').carrier\nhx : x ∈ A'\n⊢ ⁅x, y⁆ ∈ A'",
"usedConstants": [
"Subalgebra.instSetLike",
"Submodule",
"LieRing.toAddCommGroup",
"Su... | change y ∈ A' at hy | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 358,
"column": 6
} | {
"line": 358,
"column": 17
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nK K' : LieSubalgebra R L\nK₂ : LieSubalgebra R L₂\nx' : L\nhx' : x' ∈ ↑K.toSubmodule\ny' : L\nhy' : y' ∈ ↑K.toSubmodule\n⊢ ⁅↑f x', ↑f y'⁆ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Basic | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 38
} | [
{
"pp": "L : Type v\ninst✝ : LieRing L\nx y : L\nh : ⁅x + y, x⁆ + ⁅x + y, y⁆ = 0\n⊢ -⁅y, x⁆ = ⁅x, y⁆",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"LieRing.toAddCommGroup",
"AddMonoid.toAddZeroClass",
"Bracket.bracket",
"AddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 576,
"column": 4
} | {
"line": 576,
"column": 20
} | [
{
"pp": "case mp\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK K' : LieSubalgebra R L\nh : K ≤ K'\ny : ↥K\n⊢ ↑⟨↑y, ⋯⟩ ∈ K",
"usedConstants": [
"LieSubalgebra.instSetLike",
"Membership.mem",
"LieSubalgebra",
"SetLike.instMembership",
... | exact y.property | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 624,
"column": 23
} | {
"line": 624,
"column": 34
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nm : L\nhm : m ∈ s\n⊢ m ∈ lieSpan R L s",
"usedConstants": [
"Eq.mpr",
"LieSubalgebra.instSetLike",
"congrArg",
"Membership.mem",
"id",
"HasSubset.Subset",
"L... | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 636,
"column": 8
} | {
"line": 636,
"column": 19
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nK : LieSubalgebra R L\nhs : s ⊆ ↑K\nm : L\nhm : m ∈ lieSpan R L s\n⊢ m ∈ K",
"usedConstants": [
"LieSubalgebra.instSetLike",
"congrArg",
"Membership.mem",
"Eq.mp",
... | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 719,
"column": 31
} | {
"line": 719,
"column": 42
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nthis : ∀ (s : Set L), lieSpan R L (-s) ≤ lieSpan R L s\n⊢ lieSpan R L s ≤ lieSpan R L (-s)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 12
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ map f I ≤ J ↔ I ≤ comap f J",
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
... | map_le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 163,
"column": 29
} | {
"line": 163,
"column": 40
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nh₂ : N ≤ N'\nm : M\nh : m ∈ lieSpan R L {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x}... | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 56
} | [
{
"pp": "case a\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ⇑f '' ↑I = ↑J\n⊢ map f I ≤ J",
"usedConstants": [
"LieAlgebra.toModule",
... | rw [map, LieSubmodule.lieSpan_le, Submodule.map_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 163,
"column": 75
} | {
"line": 163,
"column": 86
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nh₂ : N ≤ N'\nm : M\nh : ∀ (N_1 : LieSubmodule R L M), {x | ∃ x_1 n, ⁅... | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 389,
"column": 36
} | {
"line": 389,
"column": 47
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\nS : Set (LieSubmodule R L M)\nx : L\nm : M\nhs : ↑∅ ⊆ {x | ∃ p ∈ S, ↑p = x}\nhsm : m ∈ ⨆ i ∈ ∅, i\n⊢ m = 0",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 13
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\n⊢ map f I ⊔ map f f.ker = map f I",
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"Lattice.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 554,
"column": 4
} | {
"line": 554,
"column": 15
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\nh : ∀ (a b : ↥N), a = b\nm : M\nhm : m ∈ N\n⊢ m = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 482,
"column": 2
} | {
"line": 484,
"column": 43
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI I₂ : LieIdeal R L\n⊢ comap I.incl I₂ = ⊥ ↔ Disjoint I I₂",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
"Eq.mpr",
"Submodule",
"LieSubmodule.bot_toSubmod... | rw [disjoint_iff, ← LieSubmodule.toSubmodule_inj, LieIdeal.comap_toSubmodule,
LieSubmodule.bot_toSubmodule, ← LieSubmodule.toSubmodule_inj, LieSubmodule.inf_toSubmodule,
LieSubmodule.bot_toSubmodule, incl_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Submodule | {
"line": 614,
"column": 23
} | {
"line": 614,
"column": 34
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\ns : Set M\nm : M\nhm : m ∈ s\n⊢ m ∈ lieSpan R L s",
"usedConstants": [
"LieSubmodule.instSetLike",
"Eq.mpr",
"congrArg",
"Lie... | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 626,
"column": 23
} | {
"line": 626,
"column": 34
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\ns : Set M\nN : LieSubmodule R L M\nhs : s ⊆ ↑N\nm : M\nhm : m ∈ lieSpan R L s\n⊢ m ∈ N",
"usedConstants": [
"LieSubmodule.instSetLike... | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 705,
"column": 52
} | {
"line": 705,
"column": 63
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\nm : M\nhm : m ∈ ↑N\nN' : Submodule R M\nhN' : ∀ p ∈ {x | ∃ s, (∃ x ∈ N, lieSpan R L {x} = s) ∧ ↑s = x}, p ≤ N'\n⊢ ∀ m ∈ N, ↑(lieS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 13
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nh : minpoly A x = 1\n⊢ 1 = 0",
"usedConstants": [
"Eq.mpr",
"False",
"NeZero.one",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"one_ne_zero._simp_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 99,
"column": 29
} | {
"line": 99,
"column": 66
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nh : IsLieAbelian ↥(lieSpan R L s)\nx : L\nhx : x ∈ s\ny : L\nhy : y ∈ s\nx' : ↥(lieSpan R L s) := ⟨x, ⋯⟩\ny' : ↥(lieSpan R L s) := ⟨y, ⋯⟩\nthis : ⁅x', y'⁆ = 0\n⊢ ⁅x, y⁆ = 0",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 104,
"column": 20
} | {
"line": 104,
"column": 49
} | [
{
"pp": "case refine_2.mem.mem\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nh : ∀ x ∈ s, ∀ y ∈ s, ⁅x, y⁆ = 0\nx✝¹ x✝ : ↥(lieSpan R L s)\nx y w : L\nhw : w ∈ s\nu : L\nhu : u ∈ s\n⊢ ⁅⟨w, ⋯⟩, ⟨u, ⋯⟩⁆ = 0",
"usedConstants": [
"Eq.mpr",
"Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 15
} | [
{
"pp": "case refine_2\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nh : I ≤ LieModule.ker R L ↥I\nx✝¹ x✝ : ↥I\nx : L\nhx : x ∈ I\ny : L\nhy : y ∈ I\n⊢ ⁅⟨x, hx⟩, ⟨y, hy⟩⁆ = 0",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 17
} | [
{
"pp": "case inl\nA : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nmonic : p.Monic\nhp0 : (Polynomial.aeval x) p = 0\nn : ℕ\nhpn : p.degree = ↑n\nind : LinearIndependent A fun i ↦ x ^ ↑i\nq : A[X]\nlt✝ : q.degree < p.degree\nne : ¬q = 0\nhq : (Polynomial.a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 176,
"column": 12
} | {
"line": 176,
"column": 60
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nmonic : p.Monic\nhp0 : (Polynomial.aeval x) p = 0\nn : ℕ\nhpn : p.degree = ↑n\nind : LinearIndependent A fun i ↦ x ^ ↑i\nq : A[X]\nlt✝ : q.degree < p.degree\nne : ¬q = 0\nhq : (Polynomial.aeval x) (∑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 40
} | [
{
"pp": "case h.e'_2.h.e'_6\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : (minpoly A x).natDegree = 0\n⊢ (minpoly A x).coeff 0 = 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 49
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : (minpoly A x).natDegree = 0\neq_one : minpoly A x = 1\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 186,
"column": 2
} | {
"line": 191,
"column": 52
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\n⊢ N ≤ maxTrivSubmodule R L M ↔ ⁅⊤, N⁆ = ⊥",
"usedConstants": [
"le_b... | refine ⟨fun h => ?_, fun h m hm => ?_⟩
· rw [← le_bot_iff, ← ideal_oper_maxTrivSubmodule_eq_bot R L M ⊤]
exact LieSubmodule.mono_lie_right ⊤ h
· rw [mem_maxTrivSubmodule]
rw [LieSubmodule.lie_eq_bot_iff] at h
exact fun x => h x (LieSubmodule.mem_top x) m hm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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