module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 587,
"column": 2
} | {
"line": 587,
"column": 16
} | {
"line": 589,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\nh : n = n\n⊢ X.eqToIso h = Iso.refl (X ^⦋n⦌)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.CosimplicialObject.eqToIso",
"congrArg",
"CategoryTheory.Functor.mapIso_refl",
... | [] | simp [eqToIso] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 587,
"column": 2
} | {
"line": 587,
"column": 16
} | {
"line": 589,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\nh : n = n\n⊢ X.eqToIso h = Iso.refl (X ^⦋n⦌)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.CosimplicialObject.eqToIso",
"congrArg",
"CategoryTheory.Functor.mapIso_refl",
... | [] | simp [eqToIso] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 587,
"column": 2
} | {
"line": 587,
"column": 16
} | {
"line": 589,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\nh : n = n\n⊢ X.eqToIso h = Iso.refl (X ^⦋n⦌)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.CosimplicialObject.eqToIso",
"congrArg",
"CategoryTheory.Functor.mapIso_refl",
... | [] | simp [eqToIso] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 830,
"column": 12
} | {
"line": 830,
"column": 29
} | {
"line": 830,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : C ⥤ D\nX✝ Y✝ : Augmented C\nη : X✝ ⟶ Y✝\nn✝ : SimplexCategory\n⊢ F.map η.left ≫ 𝟙 (F.obj Y✝.left) ≫ F.map (Y✝.hom.app n✝) =\n (𝟙 (F.obj X✝.left) ≫ F.map (X✝.hom.app n✝)) ≫ F.map (η.ri... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : C ⥤ D\nX✝ Y✝ : Augmented C\nη : X✝ ⟶ Y✝\nn✝ : SimplexCategory\n⊢ F.map η.left ≫ F.map (Y✝.hom.app n✝) = (𝟙 (F.obj X✝.left) ≫ F.map (X✝.hom.app n✝)) ≫ F.map (η.right.app n✝)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 830,
"column": 30
} | {
"line": 830,
"column": 47
} | {
"line": 830,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : C ⥤ D\nX✝ Y✝ : Augmented C\nη : X✝ ⟶ Y✝\nn✝ : SimplexCategory\n⊢ F.map η.left ≫ F.map (Y✝.hom.app n✝) = (𝟙 (F.obj X✝.left) ≫ F.map (X✝.hom.app n✝)) ≫ F.map (η.right.app n✝)",
"ppTerm"... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : C ⥤ D\nX✝ Y✝ : Augmented C\nη : X✝ ⟶ Y✝\nn✝ : SimplexCategory\n⊢ F.map η.left ≫ F.map (Y✝.hom.app n✝) = F.map (X✝.hom.app n✝) ≫ F.map (η.right.app n✝)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 845,
"column": 16
} | {
"line": 845,
"column": 33
} | {
"line": 845,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : C ⥤ D\nη : X✝ ⟶ Y✝\nA : Augmented C\nn : SimplexCategory\n⊢ η.app A.left ≫ 𝟙 (Y✝.obj A.left) ≫ Y✝.map (A.hom.app n) =\n (𝟙 (X✝.obj A.left) ≫ X✝.map (A.hom.app n)) ≫ η.app (A.right.ob... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : C ⥤ D\nη : X✝ ⟶ Y✝\nA : Augmented C\nn : SimplexCategory\n⊢ η.app A.left ≫ Y✝.map (A.hom.app n) = (𝟙 (X✝.obj A.left) ≫ X✝.map (A.hom.app n)) ≫ η.app (A.right.obj n)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 845,
"column": 34
} | {
"line": 845,
"column": 51
} | {
"line": 845,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : C ⥤ D\nη : X✝ ⟶ Y✝\nA : Augmented C\nn : SimplexCategory\n⊢ η.app A.left ≫ Y✝.map (A.hom.app n) = (𝟙 (X✝.obj A.left) ≫ X✝.map (A.hom.app n)) ≫ η.app (A.right.obj n)",
"ppTerm": "?m.8... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : C ⥤ D\nη : X✝ ⟶ Y✝\nA : Augmented C\nn : SimplexCategory\n⊢ η.app A.left ≫ Y✝.map (A.hom.app n) = X✝.map (A.hom.app n) ≫ η.app (A.right.obj n)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 878,
"column": 12
} | {
"line": 878,
"column": 29
} | {
"line": 878,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX✝ X : CosimplicialObject C\nX₀ : C\nf : X₀ ⟶ X ^⦋0⦌\nw : ∀ (i : SimplexCategory) (g₁ g₂ : ⦋0⦌ ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂\ni j : SimplexCategory\ng : i ⟶ j\n⊢ 𝟙 X₀ ≫ f ≫ X.map (⦋0⦌.const j 0) = (f ≫ X.map (⦋0⦌.const i 0)) ≫ X.map g",
"ppTerm": "?m.81",... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX✝ X : CosimplicialObject C\nX₀ : C\nf : X₀ ⟶ X ^⦋0⦌\nw : ∀ (i : SimplexCategory) (g₁ g₂ : ⦋0⦌ ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂\ni j : SimplexCategory\ng : i ⟶ j\n⊢ f ≫ X.map (⦋0⦌.const j 0) = (f ≫ X.map (⦋0⦌.const i 0)) ≫ X.map g"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 878,
"column": 8
} | {
"line": 878,
"column": 62
} | {
"line": 878,
"column": 63
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX✝ X : CosimplicialObject C\nX₀ : C\nf : X₀ ⟶ X ^⦋0⦌\nw : ∀ (i : SimplexCategory) (g₁ g₂ : ⦋0⦌ ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂\ni j : SimplexCategory\ng : i ⟶ j\n⊢ 𝟙 X₀ ≫ f ≫ X.map (⦋0⦌.const j 0) = (f ≫ X.map (⦋0⦌.const i 0)) ≫ X.map g",
"ppTerm": "?m.81",... | [] | rw [Category.id_comp, Category.assoc, ← X.map_comp, w] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | {
"line": 163,
"column": 2
} | {
"line": 176,
"column": 7
} | {
"line": 178,
"column": 0
} | [
{
"pp": "i n m : ℕ\nh : n + (i + 1) = m\n⊢ σ₀Iter (i + 1) h = σ₀Iter i ⋯ ≫ σ 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"SimplexCategory.σ₀Iter_succ._proof_2",
"SimplexCategory.instOfNatToTypeOrderHomFinHAddNatLenOfNat._proof_1",
"Eq.mpr",
"instNeZeroNatHAdd_... | [] | refine ConcreteCategory.hom_ext _ _ (fun k ↦ ?_)
ext
rw [dsimp% ConcreteCategory.comp_apply (σ₀Iter i) (σ 0)]
by_cases! hk : k.val ≤ i
· rw [σ₀Iter_coe_eq_of_lt .., coe_σ]
obtain hk | rfl := hk.lt_or_eq
· grind [Fin.predAbove_of_le_castSucc, Fin.coe_castPred, σ₀Iter_coe_eq_of_lt]
· grind [Fin.predAb... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | {
"line": 163,
"column": 2
} | {
"line": 176,
"column": 7
} | {
"line": 178,
"column": 0
} | [
{
"pp": "i n m : ℕ\nh : n + (i + 1) = m\n⊢ σ₀Iter (i + 1) h = σ₀Iter i ⋯ ≫ σ 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"SimplexCategory.σ₀Iter_succ._proof_2",
"SimplexCategory.instOfNatToTypeOrderHomFinHAddNatLenOfNat._proof_1",
"Eq.mpr",
"instNeZeroNatHAdd_... | [] | refine ConcreteCategory.hom_ext _ _ (fun k ↦ ?_)
ext
rw [dsimp% ConcreteCategory.comp_apply (σ₀Iter i) (σ 0)]
by_cases! hk : k.val ≤ i
· rw [σ₀Iter_coe_eq_of_lt .., coe_σ]
obtain hk | rfl := hk.lt_or_eq
· grind [Fin.predAbove_of_le_castSucc, Fin.coe_castPred, σ₀Iter_coe_eq_of_lt]
· grind [Fin.predAb... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.CechNerve | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 40
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case h₁.refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom\nX : Augmented C\nF : Arrow C\nA : X ⟶ F.augmentedCechNerve\nx : SimplexCategoryᵒᵖ\n⊢ ((equivalenceLeftToRight X F (equivalenceRightToLeft X F A)).left.app x... | [] | · simpa using congr_app A.w.symm x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 824,
"column": 4
} | {
"line": 825,
"column": 23
} | {
"line": 825,
"column": 23
} | [
{
"pp": "n : ℕ\nθ : ⦋n⦌ ⟶ ⦋n + 1⦌\ninst✝ : Mono θ\n⊢ ¬Function.Surjective ⇑(Hom.toOrderHom θ)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Epi",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instOfNatNat",
"_private.Mat... | [] | rw [← epi_iff_surjective]
grind [→ le_of_epi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 824,
"column": 4
} | {
"line": 825,
"column": 23
} | {
"line": 825,
"column": 23
} | [
{
"pp": "n : ℕ\nθ : ⦋n⦌ ⟶ ⦋n + 1⦌\ninst✝ : Mono θ\n⊢ ¬Function.Surjective ⇑(Hom.toOrderHom θ)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Epi",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instOfNatNat",
"_private.Mat... | [] | rw [← epi_iff_surjective]
grind [→ le_of_epi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Basic | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 68
} | {
"line": 260,
"column": 2
} | [
{
"pp": "S T : SSet\nf : S ⟶ T\nn : ℕ\ni : Fin (n + 1)\nx : S _⦋n⦌\n⊢ (ConcreteCategory.hom (f.app (op ⦋n + 1⦌))) ((ConcreteCategory.hom (SimplicialObject.σ S i)) x) =\n (ConcreteCategory.hom (SimplicialObject.σ T i)) ((ConcreteCategory.hom (f.app (op ⦋n⦌))) x)",
"ppTerm": "?m.47",
"assigned": true,
... | [
"S T : SSet\nf : S ⟶ T\nn : ℕ\ni : Fin (n + 1)\nx : S _⦋n⦌\n⊢ (ConcreteCategory.hom (SimplicialObject.σ S i ≫ f.app (op ⦋n + 1⦌))) x =\n (ConcreteCategory.hom (f.app (op ⦋n⦌) ≫ SimplicialObject.σ T i)) x"
] | change (S.σ i ≫ f.app (op ⦋n + 1⦌)) x = (f.app (op ⦋n⦌) ≫ T.σ i) x | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicTopology.SimplicialSet.Degenerate | {
"line": 127,
"column": 61
} | {
"line": 127,
"column": 73
} | {
"line": 127,
"column": 73
} | [
{
"pp": "X : SSet\nn✝ n : ℕ\nhn : ∀ (x : X _⦋n⦌), ∃ m f, ∃ (_ : Epi f), ∃ y, x = (ConcreteCategory.hom (X.map f.op)) ↑y\ni : Fin (n + 1)\nm : ℕ\nf : ⦋n⦌ ⟶ ⦋m⦌\nhf : Epi f\nz : ↑(X.nonDegenerate m)\n⊢ (ConcreteCategory.hom (SimplicialObject.σ X i)) ((ConcreteCategory.hom (X.map f.op)) ↑z) =\n (ConcreteCategor... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex | {
"line": 337,
"column": 33
} | {
"line": 337,
"column": 43
} | {
"line": 337,
"column": 43
} | [
{
"pp": "X Y : SSet\nB : X.Subcomplex\nf : Y ⟶ X\n⊢ B.preimage f = ⊤ ↔ ⊤ ≤ B.preimage f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"congrArg",
"PartialOrder.toPreorder",
... | [
"X Y : SSet\nB : X.Subcomplex\nf : Y ⟶ X\n⊢ B.preimage f = ⊤ ↔ B.preimage f = ⊤"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 183,
"column": 40
} | {
"line": 183,
"column": 67
} | {
"line": 183,
"column": 67
} | [
{
"pp": "X : SSet\nx y : X.op.N\nf : ⦋(mk (opObjEquiv x.simplex) ⋯).dim⦌ ⟶ ⦋(mk (opObjEquiv y.simplex) ⋯).dim⦌\nhf : (ConcreteCategory.hom (X.map f.op)) (mk (opObjEquiv y.simplex) ⋯).simplex = (mk (opObjEquiv x.simplex) ⋯).simplex\n⊢ (ConcreteCategory.hom (X.op.map (SimplexCategory.rev.map f).op)) y.simplex = x... | [] | by simp [op_map, dsimp% hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Fin.Finset | {
"line": 56,
"column": 8
} | {
"line": 56,
"column": 22
} | {
"line": 56,
"column": 22
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\na x : α\nhab : a < x\nx✝ : ↥{a, x}\nhx : x ∈ {a, x}\n⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨x, ⋯⟩] a_1 = ⟨x, hx⟩",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Fin.orderIsoPair._proof_4",
"Finset",
"Membership.mem... | [] | exact ⟨1, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Fin.Finset | {
"line": 56,
"column": 8
} | {
"line": 56,
"column": 22
} | {
"line": 56,
"column": 22
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\na x : α\nhab : a < x\nx✝ : ↥{a, x}\nhx : x ∈ {a, x}\n⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨x, ⋯⟩] a_1 = ⟨x, hx⟩",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Fin.orderIsoPair._proof_4",
"Finset",
"Membership.mem... | [] | exact ⟨1, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Fin.Finset | {
"line": 56,
"column": 8
} | {
"line": 56,
"column": 22
} | {
"line": 56,
"column": 22
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\na x : α\nhab : a < x\nx✝ : ↥{a, x}\nhx : x ∈ {a, x}\n⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨x, ⋯⟩] a_1 = ⟨x, hx⟩",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Fin.orderIsoPair._proof_4",
"Finset",
"Membership.mem... | [] | exact ⟨1, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Fin.Finset | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 22
} | {
"line": 77,
"column": 6
} | [
{
"pp": "case inr.inl\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\na c x : α\nhab : a < x\nhbc : x < c\nx✝ : ↥{a, x, c}\nhx : x ∈ {a, x, c}\n⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨x, ⋯⟩, ⟨c, ⋯⟩] a_1 = ⟨x, hx⟩",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.m... | [] | exact ⟨1, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Fin.Finset | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 22
} | {
"line": 77,
"column": 6
} | [
{
"pp": "case inr.inl\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\na c x : α\nhab : a < x\nhbc : x < c\nx✝ : ↥{a, x, c}\nhx : x ∈ {a, x, c}\n⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨x, ⋯⟩, ⟨c, ⋯⟩] a_1 = ⟨x, hx⟩",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.m... | [] | exact ⟨1, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Fin.Finset | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 22
} | {
"line": 77,
"column": 6
} | [
{
"pp": "case inr.inl\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\na c x : α\nhab : a < x\nhbc : x < c\nx✝ : ↥{a, x, c}\nhx : x ∈ {a, x, c}\n⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨x, ⋯⟩, ⟨c, ⋯⟩] a_1 = ⟨x, hx⟩",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.m... | [] | exact ⟨1, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Nerve | {
"line": 108,
"column": 33
} | {
"line": 108,
"column": 45
} | {
"line": 108,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nx : C\n⊢ ComposableArrows.hom ((ConcreteCategory.hom (SimplicialObject.σ (nerve C) 0)) (ComposableArrows.mk₀ x)) =\n eqToHom ⋯ ≫ (ComposableArrows.mk₁ (𝟙 x)).hom ≫ eqToHom ⋯",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Nerve | {
"line": 116,
"column": 33
} | {
"line": 116,
"column": 45
} | {
"line": 116,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₀ X₁ X₂ : C\nf : X₀ ⟶ X₁\ng : X₁ ⟶ X₂\n⊢ ComposableArrows.hom ((ConcreteCategory.hom (SimplicialObject.δ (nerve C) 2)) (ComposableArrows.mk₂ f g)) =\n eqToHom ⋯ ≫ (ComposableArrows.mk₁ f).hom ≫ eqToHom ⋯",
"ppTerm": "?m.30",
"assigned": true,
"used... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Nerve | {
"line": 120,
"column": 33
} | {
"line": 120,
"column": 45
} | {
"line": 120,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₀ X₁ X₂ : C\nf : X₀ ⟶ X₁\ng : X₁ ⟶ X₂\n⊢ ComposableArrows.hom ((ConcreteCategory.hom (SimplicialObject.δ (nerve C) 0)) (ComposableArrows.mk₂ f g)) =\n eqToHom ⋯ ≫ (ComposableArrows.mk₁ g).hom ≫ eqToHom ⋯",
"ppTerm": "?m.30",
"assigned": true,
"used... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Nerve | {
"line": 124,
"column": 33
} | {
"line": 124,
"column": 45
} | {
"line": 124,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₀ X₁ X₂ : C\nf : X₀ ⟶ X₁\ng : X₁ ⟶ X₂\n⊢ ComposableArrows.hom ((ConcreteCategory.hom (SimplicialObject.δ (nerve C) 1)) (ComposableArrows.mk₂ f g)) =\n eqToHom ⋯ ≫ (ComposableArrows.mk₁ (f ≫ g)).hom ≫ eqToHom ⋯",
"ppTerm": "?m.35",
"assigned": true,
... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 410,
"column": 6
} | {
"line": 410,
"column": 35
} | {
"line": 410,
"column": 35
} | [
{
"pp": "n : ℕ\ni : Fin (n + 2)\n⊢ Subcomplex.range (stdSimplex.δ i) = face {i}ᶜ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"SSet.yonedaEquiv",
"Eq.mpr",
"SSet.Subcomplex.range",
"SSet.Subcomplex.ofSimplex",
"SSet.Subcomplex.range_eq_ofSimplex",
"Op... | [
"n : ℕ\ni : Fin (n + 2)\n⊢ Subcomplex.ofSimplex (yonedaEquiv (stdSimplex.δ i)) = face {i}ᶜ"
] | Subcomplex.range_eq_ofSimplex | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 610,
"column": 8
} | {
"line": 610,
"column": 37
} | {
"line": 610,
"column": 37
} | [
{
"pp": "case left\nn m : ℕ\nf₁ : unop (op ⦋m⦌) ⟶ ⦋n⦌\nh₁✝ : objEquiv.symm f₁ ∈ Δ[n].nonDegenerate m\nf₂ : unop (op ⦋m⦌) ⟶ ⦋n⦌\nh₂✝ : objEquiv.symm f₂ ∈ Δ[n].nonDegenerate m\nh₁ : Function.Injective ⇑(Hom.toOrderHom f₁)\nh₂ : Function.Injective ⇑(Hom.toOrderHom f₂)\nh₃ : Finset.image (⇑(Hom.toOrderHom f₁)) univ... | [
"case left\nn m : ℕ\nf₁ : unop (op ⦋m⦌) ⟶ ⦋n⦌\nh₁✝ : objEquiv.symm f₁ ∈ Δ[n].nonDegenerate m\nf₂ : unop (op ⦋m⦌) ⟶ ⦋n⦌\nh₂✝ : objEquiv.symm f₂ ∈ Δ[n].nonDegenerate m\nh₁ : Function.Injective ⇑(Hom.toOrderHom f₁)\nh₂ : Function.Injective ⇑(Hom.toOrderHom f₂)\nh₃ : Finset.image (⇑(Hom.toOrderHom f₁)) univ = Finset.im... | ← OrderHom.range_eq_iff h₁ h₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ComplexShapeSigns | {
"line": 272,
"column": 24
} | {
"line": 272,
"column": 41
} | {
"line": 274,
"column": 0
} | [
{
"pp": "I₁ : Type u_1\nI₂ : Type u_2\nI₃ : Type u_3\nI₁₂ : Type u_4\nI₂₃ : Type u_5\nJ : Type u_6\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₃ : ComplexShape I₃\nc₁₂ : ComplexShape I₁₂\nc₂₃ : ComplexShape I₂₃\nc✝ : ComplexShape J\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\nI : Type u_7\ninst✝¹ : AddMonoid I\nc :... | [] | dsimp; rw [ε_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ComplexShapeSigns | {
"line": 272,
"column": 24
} | {
"line": 272,
"column": 41
} | {
"line": 274,
"column": 0
} | [
{
"pp": "I₁ : Type u_1\nI₂ : Type u_2\nI₃ : Type u_3\nI₁₂ : Type u_4\nI₂₃ : Type u_5\nJ : Type u_6\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₃ : ComplexShape I₃\nc₁₂ : ComplexShape I₁₂\nc₂₃ : ComplexShape I₂₃\nc✝ : ComplexShape J\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\nI : Type u_7\ninst✝¹ : AddMonoid I\nc :... | [] | dsimp; rw [ε_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Bifunctor | {
"line": 54,
"column": 8
} | {
"line": 55,
"column": 56
} | {
"line": 55,
"column": 56
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D\ninst✝⁴ : HasZeroMorphisms C₁\ninst✝³ : HasZeroMorphisms C₂\ninst✝² : HasZeroMorphisms D\nF : C₁ ⥤ C₂ ⥤ D\nI₁ : Type u_4\nI₂ : Type u_5\nJ : Type u_6\nc₁ : Comple... | [] | dsimp
rw [K₁.shape _ _ h₁, Functor.map_zero, zero_app] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Bifunctor | {
"line": 54,
"column": 8
} | {
"line": 55,
"column": 56
} | {
"line": 55,
"column": 56
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D\ninst✝⁴ : HasZeroMorphisms C₁\ninst✝³ : HasZeroMorphisms C₂\ninst✝² : HasZeroMorphisms D\nF : C₁ ⥤ C₂ ⥤ D\nI₁ : Type u_4\nI₂ : Type u_5\nJ : Type u_6\nc₁ : Comple... | [] | dsimp
rw [K₁.shape _ _ h₁, Functor.map_zero, zero_app] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 80
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : DecidableEq I₁₂\ninst✝ : K.Ha... | [
"case neg\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : DecidableEq I₁₂\ninst✝ : K.HasTotal c₁₂\n... | · rw [K.d₁_eq_zero' c₁₂ h₁ i₂ i₁₂']
intro h₂
exact h₁₂ (by simpa only [← h, ← h₂] using ComplexShape.rel_π₁ c₂ c₁₂ h₁ i₂) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 58
} | {
"line": 81,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nd : ℤ\nX : ℕ → Set (Cochain K L d)\nφ : (n : ℕ) → ↑(X n) → ↑(X (n + 1))\np₀ : ℤ\nhφ : ∀ (n : ℕ) (x : ↑(X n)), (↑(φ n x)).EqUpTo (↑x) (p₀ + ↑n)\nx₀ : ↑(X 0)\nn : ℕ\np q : ℤ\nhpq : p + d = q\nhp : p ≤ p₀ + ↑n\n⊢ (lim... | [] | exact sequence_eqUpTo φ hφ _ _ _ (by lia) _ _ _ (by lia) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 389,
"column": 16
} | {
"line": 389,
"column": 51
} | {
"line": 389,
"column": 51
} | [
{
"pp": "case neg\nC₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²⁰ : Category.{v_1, u_1} C₁\ninst✝¹⁹ : Category.{v_2, u_2} C₂\ninst✝¹⁸ : Category.{v_3, u_5} C₃\ninst✝¹⁷ : Category.{v_4, u_6} C₄\ninst✝¹⁶ : Category.{v_5, u_3} C₁₂\ninst✝¹⁵ : HasZeroMorphisms C₁\ninst✝¹⁴ : HasZe... | [
"case neg\nC₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²⁰ : Category.{v_1, u_1} C₁\ninst✝¹⁹ : Category.{v_2, u_2} C₂\ninst✝¹⁸ : Category.{v_3, u_5} C₃\ninst✝¹⁷ : Category.{v_4, u_6} C₄\ninst✝¹⁶ : Category.{v_5, u_3} C₁₂\ninst✝¹⁵ : HasZeroMorphisms C₁\ninst✝¹⁴ : HasZeroMorphisms ... | d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 383,
"column": 10
} | {
"line": 389,
"column": 52
} | {
"line": 390,
"column": 8
} | [
{
"pp": "case e_a\nC₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²⁰ : Category.{v_1, u_1} C₁\ninst✝¹⁹ : Category.{v_2, u_2} C₂\ninst✝¹⁸ : Category.{v_3, u_5} C₃\ninst✝¹⁷ : Category.{v_4, u_6} C₄\ninst✝¹⁶ : Category.{v_5, u_3} C₁₂\ninst✝¹⁵ : HasZeroMorphisms C₁\ninst✝¹⁴ : HasZe... | [] | by_cases h₃ : c₁.Rel i₁ (c₁.next i₁)
· rw [d₁_eq _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
dsimp [ComplexShape.r]
intro h₄
apply h₂
rw [← h₄, ComplexShape.next_π₁ c₂ c₁₂ h₃ i₂]
· rw [d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 383,
"column": 10
} | {
"line": 389,
"column": 52
} | {
"line": 390,
"column": 8
} | [
{
"pp": "case e_a\nC₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²⁰ : Category.{v_1, u_1} C₁\ninst✝¹⁹ : Category.{v_2, u_2} C₂\ninst✝¹⁸ : Category.{v_3, u_5} C₃\ninst✝¹⁷ : Category.{v_4, u_6} C₄\ninst✝¹⁶ : Category.{v_5, u_3} C₁₂\ninst✝¹⁵ : HasZeroMorphisms C₁\ninst✝¹⁴ : HasZe... | [] | by_cases h₃ : c₁.Rel i₁ (c₁.next i₁)
· rw [d₁_eq _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
dsimp [ComplexShape.r]
intro h₄
apply h₂
rw [← h₄, ComplexShape.next_π₁ c₂ c₁₂ h₃ i₂]
· rw [d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 84,
"column": 2
} | {
"line": 87,
"column": 22
} | {
"line": 88,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nA : C\nn₀ n₁ : ℤ\nh✝ : n₀ + 1 = n₁\nX Y : C\nhX : t.le 0 X\nhY : t.ge 1 Y\nf : X ⟶ (shiftF... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nA : C\nn₀ n₁ : ℤ\nh✝ : n₀ + 1 = n₁\nX Y : C\nhX : t.le 0 X\nhY : t.ge 1 Y\nf : X ⟶ (shiftFunctor C n₀)... | have hT' : Triangle.mk (T.mor₁ ≫ e.hom) (e.inv ≫ T.mor₂) T.mor₃ ∈ distTriang C := by
refine isomorphic_distinguished _ (Triangle.shift_distinguished _ mem (-n₀)) _ ?_
refine Triangle.isoMk _ _ (Iso.refl _) e.symm (Iso.refl _) ?_ ?_ ?_
all_goals simp [T] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 134,
"column": 6
} | {
"line": 140,
"column": 38
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case e_a\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\ne : c.Embedding c'\ninst✝ : e.IsRelIff\nh : Homotopy (extendMap f e) (extendMap g e... | [] | by_cases hi : c.Rel (c.prev i) i
· have hi' : c'.Rel (e.f (c.prev i)) (e.f i) := by rwa [e.rel_iff]
simp [prevD_eq _ hi, prevD_eq _ hi', L.extend_d_eq _ rfl rfl]
· rw [prevD_eq_zero _ _ hi]
by_cases hi' : c'.Rel (c'.prev (e.f i)) (e.f i)
· simp [prevD_eq _ hi', extend_d_to_eq_zero _ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 134,
"column": 6
} | {
"line": 140,
"column": 38
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case e_a\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\ne : c.Embedding c'\ninst✝ : e.IsRelIff\nh : Homotopy (extendMap f e) (extendMap g e... | [] | by_cases hi : c.Rel (c.prev i) i
· have hi' : c'.Rel (e.f (c.prev i)) (e.f i) := by rwa [e.rel_iff]
simp [prevD_eq _ hi, prevD_eq _ hi', L.extend_d_eq _ rfl rfl]
· rw [prevD_eq_zero _ _ hi]
by_cases hi' : c'.Rel (c'.prev (e.f i)) (e.f i)
· simp [prevD_eq _ hi', extend_d_to_eq_zero _ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 131,
"column": 12
} | {
"line": 131,
"column": 24
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)\nH_zero : H 0\nH_one ... | [] | exact H_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 131,
"column": 12
} | {
"line": 131,
"column": 24
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)\nH_zero : H 0\nH_one ... | [] | exact H_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 131,
"column": 12
} | {
"line": 131,
"column": 24
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)\nH_zero : H 0\nH_one ... | [] | exact H_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 24
} | {
"line": 155,
"column": 2
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n\nH_zero : H 0\nH_one ... | [] | exact H_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 24
} | {
"line": 155,
"column": 2
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n\nH_zero : H 0\nH_one ... | [] | exact H_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 24
} | {
"line": 155,
"column": 2
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n\nH_zero : H 0\nH_one ... | [] | exact H_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.TStructure | {
"line": 68,
"column": 4
} | {
"line": 72,
"column": 81
} | {
"line": 73,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nX : DerivedCategory C\nK : CochainComplex C ℤ\ne₂ : Q.obj K ≅ X\nh : (K.shortComplexTruncLE 0).ShortExact\n⊢ ∃ X_1 Y,\n ∃ (_ : ∃ K x, K.IsStrictlyLE 0) (_ : ∃ K x, K.IsStrictlyGE 1), ∃ f g h, Triangle.mk f g h ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nX : DerivedCategory C\nK : CochainComplex C ℤ\ne₂ : Q.obj K ≅ X\nh : (K.shortComplexTruncLE 0).ShortExact\n⊢ triangleOfSES h ≅\n Triangle.mk (Q.map (K.ιTruncLE 0) ≫ e₂.hom) (e₂.inv ≫ Q.map (K.πTruncGE 1))\n (inv (Q.ma... | refine ⟨Q.obj (K.truncLE 0), Q.obj (K.truncGE 1),
⟨_, Iso.refl _, inferInstance⟩, ⟨_, Iso.refl _, inferInstance⟩,
Q.map (K.ιTruncLE 0) ≫ e₂.hom, e₂.inv ≫ Q.map (K.πTruncGE 1),
inv (Q.map (K.shortComplexTruncLEX₃ToTruncGE 0 1 (by lia))) ≫ (triangleOfSES h).mor₃,
isomorphic_distinguished _ (triang... | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.LiftingProperties.Limits | {
"line": 33,
"column": 4
} | {
"line": 36,
"column": 94
} | {
"line": 38,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX Y Z W : C\nf : X ⟶ Y\ns : X ⟶ Z\ng : Z ⟶ W\nt : Y ⟶ W\nh : IsPushout s f g t\nZ' W' : C\ng' : Z' ⟶ W'\ninst✝ : HasLiftingProperty f g'\nu : Z ⟶ Z'\nv : W ⟶ W'\nsq : CommSq u g g' v\n⊢ sq.HasLift",
"ppTerm": "?m.40",
"assigned": true,
"usedCons... | [] | have w : (s ≫ u) ≫ g' = f ≫ (t ≫ v) := by
rw [← Category.assoc, ← h.w, Category.assoc, Category.assoc, sq.w]
exact ⟨h.desc u (CommSq.mk w).lift (by rw [CommSq.fac_left]), h.inl_desc ..,
h.hom_ext (by rw [h.inl_desc_assoc, sq.w]) (by rw [h.inr_desc_assoc, CommSq.fac_right])⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.LiftingProperties.Limits | {
"line": 33,
"column": 4
} | {
"line": 36,
"column": 94
} | {
"line": 38,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX Y Z W : C\nf : X ⟶ Y\ns : X ⟶ Z\ng : Z ⟶ W\nt : Y ⟶ W\nh : IsPushout s f g t\nZ' W' : C\ng' : Z' ⟶ W'\ninst✝ : HasLiftingProperty f g'\nu : Z ⟶ Z'\nv : W ⟶ W'\nsq : CommSq u g g' v\n⊢ sq.HasLift",
"ppTerm": "?m.40",
"assigned": true,
"usedCons... | [] | have w : (s ≫ u) ≫ g' = f ≫ (t ≫ v) := by
rw [← Category.assoc, ← h.w, Category.assoc, Category.assoc, sq.w]
exact ⟨h.desc u (CommSq.mk w).lift (by rw [CommSq.fac_left]), h.inl_desc ..,
h.hom_ext (by rw [h.inl_desc_assoc, sq.w]) (by rw [h.inr_desc_assoc, CommSq.fac_right])⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.KInjective | {
"line": 183,
"column": 2
} | {
"line": 184,
"column": 47
} | {
"line": 185,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nn : ℤ\nz : Cocycle K L n\ninst✝ : L.IsKInjective\nhK : HomologicalComplex.Acyclic K\nm : ℤ\nhm : m + 1 = n\nφ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L\nhφ : Cochain.ofHom φ = (↑z).rightSh... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nn : ℤ\nz : Cocycle K L n\ninst✝ : L.IsKInjective\nhK : HomologicalComplex.Acyclic K\nm : ℤ\nhm : m + 1 = n\nφ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L\nhφ : Cochain.ofHom φ = (↑z).rightShift n 0 ⋯\nh... | rw [← hφ, hf, δ_units_smul, Cochain.rightShift_units_smul,
Cochain.δ_rightUnshift _ _ _ _ 0 (by simp)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory | {
"line": 70,
"column": 4
} | {
"line": 71,
"column": 60
} | {
"line": 73,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} D₁\ninst✝¹ : Category.{v_4, u_4} D₂\nA B : ObjectProperty Cᵒᵖ\ninst✝ : A.IsVerdierLeftLocalizing B\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nhX : B.unop X✝\nhY : A.unop ... | [] | obtain ⟨Z, a, b, h₁, h₂, fac⟩ := IsVerdierLeftLocalizing.fac f.op hY hX
exact ⟨_, b.unop, a.unop, h₁, h₂, Quiver.Hom.op_inj fac⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory | {
"line": 70,
"column": 4
} | {
"line": 71,
"column": 60
} | {
"line": 73,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} D₁\ninst✝¹ : Category.{v_4, u_4} D₂\nA B : ObjectProperty Cᵒᵖ\ninst✝ : A.IsVerdierLeftLocalizing B\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nhX : B.unop X✝\nhY : A.unop ... | [] | obtain ⟨Z, a, b, h₁, h₂, fac⟩ := IsVerdierLeftLocalizing.fac f.op hY hX
exact ⟨_, b.unop, a.unop, h₁, h₂, Quiver.Hom.op_inj fac⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 687,
"column": 9
} | {
"line": 687,
"column": 17
} | {
"line": 687,
"column": 17
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nY Z : C\ng : Y ⟶ Z\nn₀ n₁ : ℤ\nhn : n₀ + 1 = n₁\nhf : IsIso ((t.truncGE n₁).map g)\n⊢ t.IsLE ((t... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nY Z : C\ng : Y ⟶ Z\nn₀ : ℤ\nhf : IsIso ((t.truncGE (n₀ + 1)).map g)\n⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj Y) n... | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 162,
"column": 8
} | {
"line": 163,
"column": 48
} | {
"line": 164,
"column": 8
} | [
{
"pp": "C : 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 : CochainComplex C... | [
"C : 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 : CochainComplex C ℤ\nι : K ⟶ ... | have h₁ : i.f n ≫ π.f n = 0 := by
simp [← HomologicalComplex.comp_f, hπ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 249,
"column": 8
} | {
"line": 249,
"column": 38
} | {
"line": 249,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ModelCategory C\nX Y Z : C\ninst✝¹ : IsCofibrant X\ninst✝ : IsFibrant Y\nx✝¹ x✝ : X ⟶ Y\nh : RightHomotopyRel x✝¹ x✝\n⊢ LeftHomotopyClass.mk x✝¹ = LeftHomotopyClass.mk x✝",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ModelCategory C\nX Y Z : C\ninst✝¹ : IsCofibrant X\ninst✝ : IsFibrant Y\nx✝¹ x✝ : X ⟶ Y\nh : RightHomotopyRel x✝¹ x✝\n⊢ LeftHomotopyRel x✝¹ x✝"
] | LeftHomotopyClass.mk_eq_mk_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 94
} | {
"line": 322,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝ : EnoughInjectives C\nn : ℤ\nthis✝¹ : IsIso ((S f n).homologyπ n)\nthis✝ : Mono (homologyπ ((cokernel f).truncGE n) n ≫ homologyMap (p f n) n)\nthis : IsIso (homologyπ ((cokernel f).truncGE n) n... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝ : EnoughInjectives C\nn : ℤ\nthis✝¹ : IsIso ((S f n).homologyπ n)\nthis✝ : Mono (homologyπ ((cokernel f).truncGE n) n ≫ homologyMap (p f n) n)\nthis : IsIso (homologyπ ((cokernel f).truncGE n) n)\n⊢ Mono (i... | rw [← IsIso.inv_hom_id_assoc ((truncGE (cokernel f) n).homologyπ n) (homologyMap (p f n) n)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 339,
"column": 12
} | {
"line": 339,
"column": 29
} | {
"line": 339,
"column": 30
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝¹ : EnoughInjectives C\nn : ℤ\ninst✝ : Mono f\nT : ShortComplex C :=\n { X₁ := homology K n, X₂ := homology L n, X₃ := homology (cokernel f) n, f := homologyMap f n,\n g := homologyMap (coker... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝¹ : EnoughInjectives C\nn : ℤ\ninst✝ : Mono f\nT : ShortComplex C :=\n { X₁ := homology K n, X₂ := homology L n, X₃ := homology (cokernel f) n, f := homologyMap f n,\n g := homologyMap (cokernel.π f) n, ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.GuitartExact.VerticalComposition | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 31
} | {
"line": 52,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_4\nD₂ : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C₁\ninst✝³ : Category.{v_2, u_2} C₂\ninst✝² : Category.{v_4, u_4} D₁\ninst✝¹ : Category.{v_5, u_5} D₂\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : ... | [
"C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_4\nD₂ : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C₁\ninst✝³ : Category.{v_2, u_2} C₂\ninst✝² : Category.{v_4, u_4} D₁\ninst✝¹ : Category.{v_5, u_5} D₂\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExa... | rw [guitartExact_iff_initial] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GuitartExact.VerticalComposition | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 31
} | {
"line": 128,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁸ : Category.{v_1, u_1} C₁\ninst✝⁷ : Category.{v_2, u_2} C₂\ninst✝⁶ : Category.{v_3, u_3} C₃\ninst✝⁵ : Category.{v_4, u_4} D₁\ninst✝⁴ : Category.{v_5, u_5} D₂\ninst✝³ : Category.{v_6, u_6} D₃\nH₁ : C₁ ⥤ D₁\nL... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁸ : Category.{v_1, u_1} C₁\ninst✝⁷ : Category.{v_2, u_2} C₂\ninst✝⁶ : Category.{v_3, u_3} C₃\ninst✝⁵ : Category.{v_4, u_4} D₁\ninst✝⁴ : Category.{v_5, u_5} D₂\ninst✝³ : Category.{v_6, u_6} D₃\nH₁ : C₁ ⥤ D₁\nL₁ : C₁ ⥤ C₂\... | rw [guitartExact_iff_initial] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GuitartExact.Basic | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 31
} | {
"line": 289,
"column": 2
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} C₃\ninst✝³ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\ninst✝² : L.IsEquivalence\ninst✝¹ : R.IsEquivalen... | [
"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} C₃\ninst✝³ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\ninst✝² : L.IsEquivalence\ninst✝¹ : R.IsEquivalence\ninst✝ : ... | rw [guitartExact_iff_initial] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 95,
"column": 22
} | {
"line": 95,
"column": 39
} | {
"line": 96,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\ninst✝⁴ : HasDerivedCategory C\ninst✝³ : HasDerivedCategory D\nS : ShortComplex C\nhS : S.ShortExact\nF : C ⥤ D\ninst✝² : F.Additive\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : Prese... | [
"C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\ninst✝⁴ : HasDerivedCategory C\ninst✝³ : HasDerivedCategory D\nS : ShortComplex C\nhS : S.ShortExact\nF : C ⥤ D\ninst✝² : F.Additive\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : PreservesFiniteCo... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 97,
"column": 62
} | {
"line": 97,
"column": 79
} | {
"line": 98,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\ninst✝⁴ : HasDerivedCategory C\ninst✝³ : HasDerivedCategory D\nS : ShortComplex C\nhS : S.ShortExact\nF : C ⥤ D\ninst✝² : F.Additive\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : Prese... | [
"C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\ninst✝⁴ : HasDerivedCategory C\ninst✝³ : HasDerivedCategory D\nS : ShortComplex C\nhS : S.ShortExact\nF : C ⥤ D\ninst✝² : F.Additive\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : PreservesFiniteCo... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Preadditive.Projective.Resolution | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 29
} | {
"line": 136,
"column": 6
} | [
{
"pp": "case succ\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nZ : C\nP : ProjectiveResolution Z\ninst✝ : Projective Z\nn✝ : ℕ\n⊢ Projective (((single₀ C).obj Z).X (n✝ + 1))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"ChainComp... | [
"case succ\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nZ : C\nP : ProjectiveResolution Z\ninst✝ : Projective Z\nn✝ : ℕ\n⊢ IsZero (((single₀ C).obj Z).X (n✝ + 1))"
] | apply IsZero.projective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Abelian.LeftDerived | {
"line": 131,
"column": 2
} | {
"line": 136,
"column": 5
} | {
"line": 138,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : Abelian C\ninst✝² : HasProjectiveResolutions C\ninst✝¹ : Abelian D\nX Y : C\nf : X ⟶ Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\nφ : P.complex ⟶ Q.complex\ncomm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f\... | [] | dsimp [isoLeftDerivedObj, Functor.leftDerived]
rw [assoc, ← Functor.map_comp_assoc,
ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_hom_naturality f P Q φ comm F,
Functor.map_comp, assoc]
erw [(HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.naturality]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.LeftDerived | {
"line": 131,
"column": 2
} | {
"line": 136,
"column": 5
} | {
"line": 138,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : Abelian C\ninst✝² : HasProjectiveResolutions C\ninst✝¹ : Abelian D\nX Y : C\nf : X ⟶ Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\nφ : P.complex ⟶ Q.complex\ncomm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f\... | [] | dsimp [isoLeftDerivedObj, Functor.leftDerived]
rw [assoc, ← Functor.map_comp_assoc,
ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_hom_naturality f P Q φ comm F,
Functor.map_comp, assoc]
erw [(HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.naturality]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Basic | {
"line": 192,
"column": 37
} | {
"line": 192,
"column": 54
} | {
"line": 192,
"column": 55
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝³ : Category.{u_4, u_1} C\ninst✝² : Category.{u_3, u_2} ι\ninst✝¹ : Abelian C\nX : SpectralObject C ι\nn : ℤ\ni₀ i₁ : ι\nf : i₀ ⟶ i₁\ninst✝ : IsIso f\nφ : mk₁ f ⟶ mk₁ (inv f) := twoδ₂Toδ₁ f (inv f) (𝟙 i₀) ⋯ ≫ twoδ₁Toδ₀ f (inv f) (𝟙 i₀) ⋯\nthis : IsIso φ\n⊢ 𝟙 ((X.H n)... | [
"C : Type u_1\nι : Type u_2\ninst✝³ : Category.{u_4, u_1} C\ninst✝² : Category.{u_3, u_2} ι\ninst✝¹ : Abelian C\nX : SpectralObject C ι\nn : ℤ\ni₀ i₁ : ι\nf : i₀ ⟶ i₁\ninst✝ : IsIso f\nφ : mk₁ f ⟶ mk₁ (inv f) := twoδ₂Toδ₁ f (inv f) (𝟙 i₀) ⋯ ≫ twoδ₁Toδ₀ f (inv f) (𝟙 i₀) ⋯\nthis : IsIso φ\n⊢ (X.H n).map φ = 0 ≫ (X.... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 455,
"column": 71
} | {
"line": 455,
"column": 90
} | {
"line": 455,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : IsArtinianRing R\nx : R\n⊢ ((Injective fun x_1 ↦ x_1 * x) ∧ Surjective fun x_1 ↦ x_1 * x) ↔ Injective fun x_1 ↦ x_1 * x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"and_iff_left_of_imp",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : IsArtinianRing R\nx : R\n⊢ (Injective fun x_1 ↦ x_1 * x) ↔ Injective fun x_1 ↦ x_1 * x",
"R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : IsArtinianRing R\nx : R\n⊢ (Injective fun x_1 ↦ x_1 * x) → Surjective fun x_1 ↦ x_1 * x"
] | and_iff_left_of_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 346,
"column": 37
} | {
"line": 346,
"column": 47
} | {
"line": 346,
"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\nS : Set (LieSubmodule R L M)\n⊢ sInf (toSubmodule '' S) = ⨅ N ∈ S, ↑N",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"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 (LieSubmodule R L M)\n⊢ ⨅ a ∈ S, ↑a = ⨅ N ∈ S, ↑N"
] | sInf_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 95,
"column": 8
} | {
"line": 95,
"column": 48
} | {
"line": 96,
"column": 6
} | [
{
"pp": "case refine_1.refine_2\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nN : LieSubmodule R L M\ninst✝¹ : LieAlgebra R L\nI : LieIdeal R L\ninst✝ : LieModule R L M\ns : Set M := ⋯\ny : L\nm' : M\nhm' :... | [] | · use x, ⟨⁅y, ↑n⁆, N.lie_mem n.property⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 102,
"column": 32
} | {
"line": 102,
"column": 63
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case a\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nN : LieSubmodule R L M\ninst✝¹ : LieAlgebra R L\nI : LieIdeal R L\ninst✝ : LieModule R L M\n⊢ Submodule.span R {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ≤ ↑... | [] | apply submodule_span_le_lieSpan | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Lie.Submodule | {
"line": 972,
"column": 2
} | {
"line": 972,
"column": 36
} | {
"line": 974,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\nN : LieSubmodule R L M\nM' : Type u_1\ninst✝² : AddCommGroup M'\ninst✝¹ : Module R M'\ninst✝ : LieRingModule L M'\nf : M →ₗ⁅R,L⁆ M'\n⊢ map f N ≤ map f ⊤... | [] | exact LieSubmodule.map_mono le_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Submodule | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 36
} | {
"line": 986,
"column": 0
} | [
{
"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\nN' : LieSubmodule R L ↥N\n⊢ map N.incl N' ≤ map N.incl ⊤",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
... | [] | exact LieSubmodule.map_mono le_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 42
} | {
"line": 95,
"column": 0
} | [
{
"pp": "A : Type u_3\nK : Type u_4\nR : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : Field K\ninst✝⁴ : Ring R\ninst✝³ : Algebra A K\ninst✝² : Algebra A R\ninst✝¹ : Algebra K R\ninst✝ : IsScalarTower A K R\nx : R\n⊢ minpoly K x ∣ map (algebraMap A K) (minpoly A x)",
"ppTerm": "?m.29",
"assigned": true,
"... | [] | refine minpoly.dvd K x ?_
rw [aeval_map_algebraMap, minpoly.aeval] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 42
} | {
"line": 95,
"column": 0
} | [
{
"pp": "A : Type u_3\nK : Type u_4\nR : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : Field K\ninst✝⁴ : Ring R\ninst✝³ : Algebra A K\ninst✝² : Algebra A R\ninst✝¹ : Algebra K R\ninst✝ : IsScalarTower A K R\nx : R\n⊢ minpoly K x ∣ map (algebraMap A K) (minpoly A x)",
"ppTerm": "?m.29",
"assigned": true,
"... | [] | refine minpoly.dvd K x ?_
rw [aeval_map_algebraMap, minpoly.aeval] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 277,
"column": 2
} | {
"line": 278,
"column": 21
} | {
"line": 280,
"column": 0
} | [
{
"pp": "case inr\nA : Type u_1\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Algebra A B\nx : B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : f.Monic\nhg : g.Monic\nhe : f * g = minpoly A x\nh : ¬IsUnit f ∧ ¬IsUnit g\nheval : (Polynomial.aeval x) g = 0\n⊢ False... | [] | · refine aeval_ne_zero_of_dvdNotUnit_minpoly hx hg ⟨hg.ne_zero, f, h.1, ?_⟩ heval
rw [mul_comm, he] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 79,
"column": 2
} | {
"line": 92,
"column": 61
} | {
"line": 94,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nm : ℕ\nhm : m ∣ n\nhm' : m ≠ 1\nb : K\n⊢ b ^ m ≠ a",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.addMonoidWithOne",
"Polynomial.C",
"Nat.instCanonicallyOrderedAd... | [] | have hn : n ≠ 0 := fun e ↦ not_irreducible_C
(1 - a) (by simpa only [e, pow_zero, ← C.map_one, ← map_sub] using H)
obtain ⟨k, rfl⟩ := hm
rintro rfl
obtain ⟨q, hq⟩ := sub_dvd_pow_sub_pow (X ^ k) (C b) m
rw [mul_comm, pow_mul, map_pow, hq] at H
have : degree q = 0 := by
simpa [isUnit_iff_degree_eq_zero,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 79,
"column": 2
} | {
"line": 92,
"column": 61
} | {
"line": 94,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nm : ℕ\nhm : m ∣ n\nhm' : m ≠ 1\nb : K\n⊢ b ^ m ≠ a",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.addMonoidWithOne",
"Polynomial.C",
"Nat.instCanonicallyOrderedAd... | [] | have hn : n ≠ 0 := fun e ↦ not_irreducible_C
(1 - a) (by simpa only [e, pow_zero, ← C.map_one, ← map_sub] using H)
obtain ⟨k, rfl⟩ := hm
rintro rfl
obtain ⟨q, hq⟩ := sub_dvd_pow_sub_pow (X ^ k) (C b) m
rw [mul_comm, pow_mul, map_pow, hq] at H
have : degree q = 0 := by
simpa [isUnit_iff_degree_eq_zero,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Squarefree.Basic | {
"line": 137,
"column": 96
} | {
"line": 145,
"column": 31
} | {
"line": 147,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nr : R\n⊢ (∀ (x : R), Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ (x : R), ¬Irreducible x) ∨ Squarefree r",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
"Dvd.dvd"... | [] | by
refine ⟨fun h ↦ ?_, ?_⟩
· rcases eq_or_ne r 0 with (rfl | hr)
· exact .inl (by simpa using h)
· exact .inr ((squarefree_iff_no_irreducibles hr).mpr h)
· rintro (⟨rfl, h⟩ | h)
· simpa using h
intro x hx t
exact hx.not_isUnit (h x t) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Perfect | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 32
} | {
"line": 242,
"column": 2
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring S\np : ℕ\ninst✝³ : ExpChar R p\ninst✝² : PerfectRing R p\ninst✝¹ : ExpChar S p\ninst✝ : PerfectRing S p\nf : R →* S\nx : R\n⊢ f ((frobeniusEquiv R p).symm x) = (frobeniusEquiv S p).symm (f x)",
"ppTerm": "?m.32",
"assign... | [
"R : Type u_2\nS : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring S\np : ℕ\ninst✝³ : ExpChar R p\ninst✝² : PerfectRing R p\ninst✝¹ : ExpChar S p\ninst✝ : PerfectRing S p\nf : R →* S\nx : R\n⊢ (frobeniusEquiv S p) (f ((frobeniusEquiv R p).symm x)) = (frobeniusEquiv S p) ((frobeniusEquiv S p).symm (f x))"
] | apply_fun (frobeniusEquiv S p) | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.FieldTheory.Perfect | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 32
} | {
"line": 247,
"column": 2
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring S\np : ℕ\ninst✝³ : ExpChar R p\ninst✝² : PerfectRing R p\ninst✝¹ : ExpChar S p\ninst✝ : PerfectRing S p\nf : R →+* S\nx : R\n⊢ f ((frobeniusEquiv R p).symm x) = (frobeniusEquiv S p).symm (f x)",
"ppTerm": "?m.32",
"assig... | [
"R : Type u_2\nS : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring S\np : ℕ\ninst✝³ : ExpChar R p\ninst✝² : PerfectRing R p\ninst✝¹ : ExpChar S p\ninst✝ : PerfectRing S p\nf : R →+* S\nx : R\n⊢ (frobeniusEquiv S p) (f ((frobeniusEquiv R p).symm x)) = (frobeniusEquiv S p) ((frobeniusEquiv S p).symm (f x))"
... | apply_fun (frobeniusEquiv S p) | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.FieldTheory.Separable | {
"line": 379,
"column": 33
} | {
"line": 379,
"column": 38
} | {
"line": 379,
"column": 38
} | [
{
"pp": "F : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nhp : Nat.Prime p\nf : F[X]\nhf : Irreducible f\nh1 : ¬f.Separable\ng : F[X]\nhg : Irreducible g\nhgf : (expand F p) g = f\nN : ℕ\nih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f\nhn : f.n... | [
"F : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nhp : Nat.Prime p\nf : F[X]\nhf : Irreducible f\nh1 : ¬f.Separable\ng : F[X]\nhg : Irreducible g\nhgf : (expand F p) g = f\nN : ℕ\nih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f\nhn : f.natDegree = N... | ← hg1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Perfect | {
"line": 408,
"column": 80
} | {
"line": 410,
"column": 79
} | {
"line": 412,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\np n : ℕ\ninst✝¹ : ExpChar R p\ninst✝ : PerfectRing R p\ny : R\n⊢ (X ^ p ^ n - C y).roots = p ^ n • {(iterateFrobeniusEquiv R p n).symm y}",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"iterateFrobeniusEquiv",
"P... | [] | by
have H := roots_expand_pow (p := p) (n := n) (f := X - C y)
rwa [roots_X_sub_C, Multiset.map_singleton, map_sub, expand_X, expand_C] at H | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Determinant | {
"line": 764,
"column": 37
} | {
"line": 764,
"column": 50
} | {
"line": 764,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (Set.range v)\ni k : ι\nhik : k ≠ i\n⊢ update v i (v k) k = update v i (v k) ... | [] | by simp [hik] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 52
} | {
"line": 74,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : Ring A\ninst✝³ : Module ℚ A\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Module ℚ M\na : A\nm : M\nk : ℕ\nh : a ^ k • m = 0\nhn : IsNilpotent a\n⊢ exp a • m = ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i • m",
"ppTerm": "?m.48",
"assigned": true,
"usedConst... | [
"case inl\nA : Type u_1\ninst✝⁴ : Ring A\ninst✝³ : Module ℚ A\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Module ℚ M\na : A\nm : M\nk : ℕ\nh : a ^ k • m = 0\nhn : IsNilpotent a\nh₀ : nilpotencyClass a ≤ k\n⊢ exp a • m = ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i • m",
"case inr\nA : Type u_1\ninst✝⁴... | rcases le_or_gt (nilpotencyClass a) k with h₀ | h₀ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.Semisimple | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 80
} | {
"line": 156,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhs : f.IsFinitelySemisimple\nthis : ∀ (p : Submodule R M) (hp₁ : p ∈ f.invtSubmodule), Module.Finite R ↥p → LinearMap.restrict f hp₁ = 0\nx : M\nk : ℕ\nhk : f ^ k = 0\np : Submodule R M := Submodu... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhs : f.IsFinitelySemisimple\nthis : ∀ (p : Submodule R M) (hp₁ : p ∈ f.invtSubmodule), Module.Finite R ↥p → LinearMap.restrict f hp₁ = 0\nx : M\nk : ℕ\nhk : f ^ k = 0\np : Submodule R M := Submodule.span R {x... | have hg : {(f ^ i) x | (i : ℕ) (_ : i ≤ k)} = g '' Iic k := by ext; simp [g] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AdjoinRoot | {
"line": 695,
"column": 4
} | {
"line": 695,
"column": 14
} | {
"line": 696,
"column": 4
} | [
{
"pp": "case hsp\nR : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝¹ : CommRing R\ng : R[X]\ninst✝ : Field K\nf : K[X]\nhf : f ≠ 0\nf' : K[X] := ⋯\ndeg_f' : f'.natDegree = f.natDegree\nminpoly_eq : minpoly K (root f) = f'\n⊢ ⊤ ≤ Submodule.span K (Set.range fun i ↦ root f ^ ↑i)",
"... | [
"case hsp\nR : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝¹ : CommRing R\ng : R[X]\ninst✝ : Field K\nf : K[X]\nhf : f ≠ 0\nf' : K[X] := f * C f.leadingCoeff⁻¹\ndeg_f' : f'.natDegree = f.natDegree\nminpoly_eq : minpoly K (root f) = f'\ny : AdjoinRoot f\n⊢ y ∈ Submodule.span K (Set.range f... | rintro y - | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 9
} | {
"line": 52,
"column": 4
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\n⊢ IsNilpotent ((aeval ff) P)",
"ppTerm": "?m.91",
"assigned": true,
"usedCo... | [
"case h\nK : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⋯\nP' : K[X] := ⋯\n⊢ (aeval ff) P ^ k = 0"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Solvable | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 33
} | {
"line": 124,
"column": 33
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nk l : ℕ\nD₁ : LieIdeal R L →o LieIdeal R L := { toFun := fun I ↦ ⁅I, I⁆, monotone' := ⋯ }\nh₁ : ∀ (I J : LieIdeal R L), D₁ (I ⊔ J) ≤ D₁ I ⊔ J\n⊢ D (k + l) (I + J) ≤ D k I + D l J",
"ppTerm":... | [
"R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nk l : ℕ\nD₁ : LieIdeal R L →o LieIdeal R L := { toFun := fun I ↦ ⁅I, I⁆, monotone' := ⋯ }\nh₁ : ∀ (n₁ n₂ : ℕ) (a₁ a₂ : LieIdeal R L), (⇑D₁)^[n₁ + n₂] (a₁ ⊔ a₂) ≤ (⇑D₁)^[n₁] a₁ ⊔ (⇑D₁)^[n₂] a₂\n⊢ D (k + l) (... | ← D₁.iterate_sup_le_sup_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Solvable | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 36
} | {
"line": 174,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\n⊢ derivedSeriesOfIdeal R L k I ≤ I",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"LieAlgebra.derivedSeriesOfIdeal_le_self"
],
"usedFVars": [
"R",... | [] | apply derivedSeriesOfIdeal_le_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Lie.TensorProduct | {
"line": 202,
"column": 6
} | {
"line": 202,
"column": 26
} | {
"line": 202,
"column": 26
} | [
{
"pp": "case h.left\nR : Type u\ninst✝⁶ : CommRing R\nL : Type v\nM : Type w\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\nI : LieIdeal R L\nN : LieSubmodule R L M\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\n⊢ x ⊗ₜ[R]... | [
"case h\nR : Type u\ninst✝⁶ : CommRing R\nL : Type v\nM : Type w\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\nI : LieIdeal R L\nN : LieSubmodule R L M\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\n⊢ ↑(incl I) ⟨x, hx⟩ ⊗ₜ[R] ... | use ⟨x, hx⟩, ⟨n, hn⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.TensorProduct | {
"line": 204,
"column": 54
} | {
"line": 204,
"column": 74
} | {
"line": 204,
"column": 74
} | [
{
"pp": "case mpr\nR : Type u\ninst✝⁶ : CommRing R\nL : Type v\nM : Type w\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\nI : LieIdeal R L\nN : LieSubmodule R L M\nm : M\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\nh : ↑... | [
"case h\nR : Type u\ninst✝⁶ : CommRing R\nL : Type v\nM : Type w\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\nI : LieIdeal R L\nN : LieSubmodule R L M\nm : M\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\nh : ↑(toModuleHom R... | use ⟨x, hx⟩, ⟨n, hn⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Solvable | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 45
} | {
"line": 201,
"column": 2
} | [
{
"pp": "case zero\nR : Type u\nL : Type v\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L' →ₗ⁅R⁆ L\nh : Function.Surjective ⇑f\n⊢ f.idealRange = ⊤",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Li... | [] | exact f.idealRange_eq_top_of_surjective h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Solvable | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 21
} | {
"line": 241,
"column": 21
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nk : ℕ\n⊢ ↑↑(derivedSeriesOfIdeal R L k ⊤) = ↑↑(derivedSeries ℤ L k)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"Submodule",
"LieRi... | [
"R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nk : ℕ\n⊢ ↑↑(derivedSeriesOfIdeal R L k ⊤) = ↑↑(derivedSeriesOfIdeal ℤ L k ⊤)"
] | derivedSeries_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Solvable | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 43
} | {
"line": 280,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : LieRing L'\ninst✝² : LieAlgebra R L'\nI✝ J✝ : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI J : LieIdeal R L\ninst✝¹ : IsSolvable ↥I\ninst✝ : IsSolvable ↥J\n⊢ IsSolvable ↥(I + J)",
"ppTerm... | [
"R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : LieRing L'\ninst✝² : LieAlgebra R L'\nI✝ J✝ : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI J : LieIdeal R L\ninst✝¹ : IsSolvable ↥I\ninst✝ : IsSolvable ↥J\nk : ℕ\nhk : derivedSeries R (↥I) k = ⊥\n⊢ IsSol... | obtain ⟨k, hk⟩ := IsSolvable.solvable R I | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.Solvable | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 7
} | {
"line": 296,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : LieRing L'\ninst✝³ : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : IsSolvable L\nk : ℕ\nhk : derivedSeries... | [
"case h\nR : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : LieRing L'\ninst✝³ : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : IsSolvable L\nk : ℕ\nhk : derivedSeries R L... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Solvable | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 41
} | {
"line": 303,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Module.FaithfullyFlat R A\n⊢ IsSolvable (A ⊗[R] L) → IsSolvable L",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"LieAl... | [
"R : Type u\nL : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Module.FaithfullyFlat R A\n⊢ (∃ k, derivedSeries A (A ⊗[R] L) k = ⊥) → ∃ k, derivedSeries R L k = ⊥"
] | rw [isSolvable_iff A, isSolvable_iff R] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Solvable | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 7
} | {
"line": 305,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Module.FaithfullyFlat R A\nk : ℕ\nh : derivedSeries A (A ⊗[R] L) k = ⊥\n⊢ ∃ k, derivedSeries R L k = ⊥",
"ppTerm": "?m.52",
"assigned": true... | [
"case h\nR : Type u\nL : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Module.FaithfullyFlat R A\nk : ℕ\nh : derivedSeries A (A ⊗[R] L) k = ⊥\n⊢ derivedSeries R L k = ⊥"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 387,
"column": 26
} | {
"line": 387,
"column": 36
} | {
"line": 387,
"column": 36
} | [
{
"pp": "V : Type u_5\nK : Type u_6\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nι : Type u_9\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nB : BilinForm K V\nhB : B.Nondegenerate\nb : Basis ι K V\nx : V\ni : ι\nthis : FiniteDimensional K V\n⊢ ((B.toDual hB) x) (b i) = (B x) (b i)",
"ppTerm... | [
"V : Type u_5\nK : Type u_6\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nι : Type u_9\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nB : BilinForm K V\nhB : B.Nondegenerate\nb : Basis ι K V\nx : V\ni : ι\nthis : FiniteDimensional K V\n⊢ (B x) (b i) = (B x) (b i)"
] | toDual_def | Lean.Elab.Tactic.evalRewriteSeq | null |
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