module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 635,
"column": 6
} | {
"line": 635,
"column": 27
} | {
"line": 636,
"column": 4
} | [
{
"pp": "case of\nC : Type u\ninst✝ : Category.{v, u} C\na✝¹ b✝ : Cat\nf✝ : a✝¹ ⟶ b✝\na✝ : ↑a✝¹\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app (of a✝) =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Ho... | [] | simpa using! (refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 711,
"column": 8
} | {
"line": 711,
"column": 12
} | {
"line": 712,
"column": 8
} | [
{
"pp": "case star.of.unit\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nZ : D\nF : C ⥤ D\nM : (x : C) → Z ⟶ F.obj x\nhM : ∀ (x y : C) (f : x ⟶ y), M x ≫ F.map f = M y\nG : WithInitial C ⥤ D\nh : incl ⋙ G ≅ F\nhG : G.obj star ≅ Z\nhh : ∀ (x : C), hG.symm.hom ≫ G.map (star... | [
"case star.of.unit\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nZ : D\nF : C ⥤ D\nM : (x : C) → Z ⟶ F.obj x\nhM : ∀ (x y : C) (f : x ⟶ y), M x ≫ F.map f = M y\nG : WithInitial C ⥤ D\nh : incl ⋙ G ≅ F\nhG : G.obj star ≅ Z\nhh : ∀ (x : C), hG.symm.hom ≫ G.map (starInitial.to (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Final | {
"line": 1014,
"column": 71
} | {
"line": 1014,
"column": 94
} | {
"line": 1014,
"column": 94
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : F.Final\ninst✝ : IsFilteredOrEmpty C\nX Y : D\nf g : X ⟶ Y\nP : StructuredArrow X F → Prop := fun h ↦ ∃ Z q₁ q₂, h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂\nh₀ : P (Struct... | [
"case refine_1\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : F.Final\ninst✝ : IsFilteredOrEmpty C\nX Y : D\nf g : X ⟶ Y\nP : StructuredArrow X F → Prop := fun h ↦ ∃ Z q₁ q₂, h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂\nh₀ : P (StructuredArrow.mk... | StructuredArrow.w_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Final | {
"line": 1158,
"column": 2
} | {
"line": 1158,
"column": 26
} | {
"line": 1159,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : SmallCategory C\nF G : C ⥤ Cat\nα : F ⟶ G\nhα : ∀ (X : C), (α.app X).toFunctor.Final\nH : Grothendieck G ⥤ Type u₁\ni : colimit (map α ⋙ H) ≅ colimit H :=\n (colimitFiberwiseColimitIso (map α ⋙ H)).symm ≪≫\n HasColimit.isoOfNatIso (fiberwiseColimitMapCompEquivalence α H) ≪≫ col... | [
"C : Type u₁\ninst✝ : SmallCategory C\nF G : C ⥤ Cat\nα : F ⟶ G\nhα : ∀ (X : C), (α.app X).toFunctor.Final\nH : Grothendieck G ⥤ Type u₁\ni : colimit (map α ⋙ H) ≅ colimit H :=\n (colimitFiberwiseColimitIso (map α ⋙ H)).symm ≪≫\n HasColimit.isoOfNatIso (fiberwiseColimitMapCompEquivalence α H) ≪≫ colimitFiberwis... | convert! Iso.isIso_hom i | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Limits.Shapes.KernelPair | {
"line": 223,
"column": 12
} | {
"line": 223,
"column": 20
} | {
"line": 225,
"column": 0
} | [
{
"pp": "case e'_8\nC : Type u\ninst✝² : Category.{v, u} C\nR X Y : C\nf : X ⟶ Y\na : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ a = a ≫ 𝟙 X",
"ppTerm": "?e'_8",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Categor... | [] | { simp } | Lean.Elab.Tactic.evalTacticSeqBracketed | Lean.Parser.Tactic.tacticSeqBracketed |
Mathlib.CategoryTheory.Limits.Shapes.KernelPair | {
"line": 223,
"column": 12
} | {
"line": 223,
"column": 20
} | {
"line": 225,
"column": 0
} | [
{
"pp": "case e'_8\nC : Type u\ninst✝² : Category.{v, u} C\nR X Y : C\nf : X ⟶ Y\na : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ a = a ≫ 𝟙 X",
"ppTerm": "?e'_8",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Categor... | [] | { simp } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.KernelPair | {
"line": 223,
"column": 12
} | {
"line": 223,
"column": 20
} | {
"line": 225,
"column": 0
} | [
{
"pp": "case e'_9\nC : Type u\ninst✝² : Category.{v, u} C\nR X Y : C\nf : X ⟶ Y\na : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ f = 𝟙 X ≫ f",
"ppTerm": "?e'_9",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Categor... | [] | { simp } | Lean.Elab.Tactic.evalTacticSeqBracketed | Lean.Parser.Tactic.tacticSeqBracketed |
Mathlib.CategoryTheory.Limits.Shapes.KernelPair | {
"line": 223,
"column": 12
} | {
"line": 223,
"column": 20
} | {
"line": 225,
"column": 0
} | [
{
"pp": "case e'_9\nC : Type u\ninst✝² : Category.{v, u} C\nR X Y : C\nf : X ⟶ Y\na : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ f = 𝟙 X ≫ f",
"ppTerm": "?e'_9",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Categor... | [] | { simp } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHomProperties | {
"line": 194,
"column": 58
} | {
"line": 205,
"column": 50
} | {
"line": 207,
"column": 0
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\n⊢ (RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P) ↔\n (toMorphismProperty fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Categ... | [] | by
refine ⟨fun h ↦ MorphismProperty.RespectsIso.mk _ ?_ ?_, fun h ↦ ⟨?_, ?_⟩⟩
· intro X Y Z e f hf
exact h.right f.hom e.commRingCatIsoToRingEquiv hf
· intro X Y Z e f hf
exact h.left f.hom e.commRingCatIsoToRingEquiv hf
· intro X Y Z _ _ _ f e hf
exact MorphismProperty.RespectsIso.postcomp (toMorph... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 100
} | {
"line": 250,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙... | [] | rw [← Category.comp_id (pullback.snd ..), ← condition, Category.assoc, IsIso.inv_hom_id_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 100
} | {
"line": 250,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙... | [] | rw [← Category.comp_id (pullback.snd ..), ← condition, Category.assoc, IsIso.inv_hom_id_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 100
} | {
"line": 250,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙... | [] | rw [← Category.comp_id (pullback.snd ..), ← condition, Category.assoc, IsIso.inv_hom_id_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 100
} | {
"line": 251,
"column": 4
} | [
{
"pp": "case refine_1.refine_2\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙... | [] | rw [← Category.comp_id (pullback.snd ..), ← condition, Category.assoc, IsIso.inv_hom_id_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 100
} | {
"line": 251,
"column": 4
} | [
{
"pp": "case refine_1.refine_2\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙... | [] | rw [← Category.comp_id (pullback.snd ..), ← condition, Category.assoc, IsIso.inv_hom_id_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 100
} | {
"line": 251,
"column": 4
} | [
{
"pp": "case refine_1.refine_2\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙... | [] | rw [← Category.comp_id (pullback.snd ..), ← condition, Category.assoc, IsIso.inv_hom_id_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 330,
"column": 2
} | {
"line": 331,
"column": 73
} | {
"line": 332,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : CartesianMonoidalCategory C\nA B : C\ninst✝² : GrpObj A\ninst✝¹ : GrpObj B\nf : A ⟶ B\ninst✝ : IsMonHom f\n⊢ ι ≫ f = f ≫ ι",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.GrpObj.inv",
"CategoryTheory.C... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : CartesianMonoidalCategory C\nA B : C\ninst✝² : GrpObj A\ninst✝¹ : GrpObj B\nf : A ⟶ B\ninst✝ : IsMonHom f\n⊢ lift (lift f (ι ≫ f)) f = lift (lift f (f ≫ ι)) f"
] | suffices lift (lift f (ι ≫ f)) f =
lift (lift f (f ≫ ι)) f by simpa using (this =≫ fst _ _) =≫ snd _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Data.W.Basic | {
"line": 105,
"column": 6
} | {
"line": 108,
"column": 29
} | {
"line": 108,
"column": 29
} | [
{
"pp": "case succ\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nn : ℕ\nih :\n ∀ ⦃m : ℕ⦄,\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n n =\n (fun n ↦\... | [] | rcases m with - | m
· simp_all
· refine congr_arg Nat.succ (ih ?_)
simp_all [funext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.W.Basic | {
"line": 105,
"column": 6
} | {
"line": 108,
"column": 29
} | {
"line": 108,
"column": 29
} | [
{
"pp": "case succ\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nn : ℕ\nih :\n ∀ ⦃m : ℕ⦄,\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n n =\n (fun n ↦\... | [] | rcases m with - | m
· simp_all
· refine congr_arg Nat.succ (ih ?_)
simp_all [funext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Subfield | {
"line": 74,
"column": 66
} | {
"line": 84,
"column": 18
} | {
"line": 86,
"column": 0
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : DivisionRing α\n⊢ #(WType (Operands s)) ≤ max #↑s ℵ₀",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"_private.Mathlib.SetTheory.Cardinal.Subfield.0.Subfield.Operands",
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSe... | [] | by
convert! WType.cardinalMk_le_max_aleph0_of_finite' using 1
· rw [lift_uzero, mk_sum, lift_uzero]
have : lift.{u, 0} #(Fin 6) < ℵ₀ := lift_lt_aleph0.mpr (lt_aleph0_of_finite _)
obtain h | h := lt_or_ge #s ℵ₀
· rw [max_eq_right h.le, max_eq_right]
exact (add_lt_aleph0 this h).le
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.ErdosKaplansky | {
"line": 82,
"column": 7
} | {
"line": 82,
"column": 11
} | {
"line": 82,
"column": 12
} | [
{
"pp": "case inr\nK : Type u\ninst✝ : DivisionRing K\naleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)\ncard_K : ℵ₀ < #K\nthis✝² : Module.rank K (ℕ → K) < #K\nιK : Type u\nbK : Module.Basis ιK K (ℕ → K)\nL : Subfield K := Subfield.closure (Set.range fun i ↦ bK i.1 i.2)\nhLK : #↥L < #K\nthis✝¹ : Module (↥L)ᵐᵒᵖ K := Modul... | [
"case inr\nK : Type u\ninst✝ : DivisionRing K\naleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)\ncard_K : ℵ₀ < #K\nthis✝² : Module.rank K (ℕ → K) < #K\nιK : Type u\nbK : Module.Basis ιK K (ℕ → K)\nL : Subfield K := Subfield.closure (Set.range fun i ↦ bK i.1 i.2)\nhLK : #↥L < #K\nthis✝¹ : Module (↥L)ᵐᵒᵖ K := Module.compHom K ... | eq0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 22
} | {
"line": 298,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ne : ι → M\nε : ι → Dual R M\nh : DualBases e ε\ni : ι\n⊢ (h.basis.repr (e i)) i = (Finsupp.single i 1) i",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
... | [] | simp [h.eval_same] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 22
} | {
"line": 298,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ne : ι → M\nε : ι → Dual R M\nh : DualBases e ε\ni : ι\n⊢ (h.basis.repr (e i)) i = (Finsupp.single i 1) i",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
... | [] | simp [h.eval_same] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 22
} | {
"line": 298,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ne : ι → M\nε : ι → Dual R M\nh : DualBases e ε\ni : ι\n⊢ (h.basis.repr (e i)) i = (Finsupp.single i 1) i",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
... | [] | simp [h.eval_same] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 6
} | {
"line": 344,
"column": 2
} | [
{
"pp": "case e_a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nX Y Z : C\ninst✝² : HasRightDual X\ninst✝¹ : HasRightDual Y\ninst✝ : HasRightDual Z\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ η_ X Xᘁ ≫ f ▷ Xᘁ ≫ g ▷ Xᘁ =\n η_ Y Yᘁ ≫ (g ⊗ₘ (ρ_ Yᘁ).inv ≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε... | [
"case e_a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nX Y Z : C\ninst✝² : HasRightDual X\ninst✝¹ : HasRightDual Y\ninst✝ : HasRightDual Z\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ η_ Y Yᘁ ≫ (g ⊗ₘ (ρ_ Yᘁ).inv ≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom) =\n η_ X Xᘁ ≫ f... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Category.CommHopfAlgCat | {
"line": 183,
"column": 15
} | {
"line": 191,
"column": 53
} | {
"line": 193,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nA : Type u\ninst✝¹ : CommRing A\ninst✝ : HopfAlgebra R A\n⊢ CartesianMonoidalCategory.lift (𝟙 (Opposite.op (of R A))) (ofHom (antipodeAlgHom R A)).op ≫ MonObj.mul =\n SemiCartesianMonoidalCategory.toUnit (Opposite.op (of R A)) ≫ MonObj.one",
"ppTerm": "?m.64",
... | [] | by
ext x
-- TODO: Add more simp lemmas to make this `simpa ... using ...` again.
simp only [unop_comp, unop_tensorObj, hom_comp, coe_tensorObj, lift_unop_hom, unop_id, hom_id,
Quiver.Hom.unop_op, ConcreteCategory.hom_ofHom, ← Algebra.TensorProduct.lmul'_comp_map,
mul_op_of_unop_hom, AlgHom.coe_c... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 338,
"column": 52
} | {
"line": 354,
"column": 46
} | {
"line": 356,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nX Y Z : C\ninst✝² : HasRightDual X\ninst✝¹ : HasRightDual Y\ninst✝ : HasRightDual Z\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ (f ≫ g)ᘁ = gᘁ ≫ fᘁ",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCoh... | [] | by
rw [rightAdjointMate_comp]
simp only [rightAdjointMate, comp_whiskerRight]
simp only [← Category.assoc]; congr 3; simp only [Category.assoc]
simp only [← MonoidalCategory.whiskerLeft_comp]; congr 2
symm
calc
_ = 𝟙 _ ⊗≫ (η_ Y Yᘁ ▷ 𝟙_ C ≫ (Y ⊗ Yᘁ) ◁ η_ X Xᘁ) ⊗≫ Y ◁ Yᘁ ◁ f ▷ Xᘁ ⊗≫
Y ◁ ε_ Y Yᘁ ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.DivisionRing | {
"line": 95,
"column": 24
} | {
"line": 95,
"column": 35
} | {
"line": 95,
"column": 36
} | [
{
"pp": "case e_a.refine_2\nK : Type u\nV V₁ V₂ V₃ : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ndb : V₂ →ₗ[K] V\neb : V₃ →ₗ[K] V\ncd ... | [
"case e_a.refine_2\nK : Type u\nV V₁ V₂ V₃ : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ndb : V₂ →ₗ[K] V\neb : V₃ →ₗ[K] V\ncd : V₁ →ₗ[K] V... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 6
} | {
"line": 365,
"column": 2
} | [
{
"pp": "case e_a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nX Y Z : C\ninst✝² : HasLeftDual X\ninst✝¹ : HasLeftDual Y\ninst✝ : HasLeftDual Z\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ η_ (ᘁX) X ≫ ᘁX ◁ f ≫ ᘁX ◁ g =\n η_ (ᘁY) Y ≫ ((λ_ ᘁY).inv ≫ η_ (ᘁX) X ▷ ᘁY ≫ (ᘁX ◁ f) ▷ ᘁY ≫ (α_ (ᘁX) Y ᘁY).hom ≫... | [
"case e_a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nX Y Z : C\ninst✝² : HasLeftDual X\ninst✝¹ : HasLeftDual Y\ninst✝ : HasLeftDual Z\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ η_ (ᘁY) Y ≫ ((λ_ ᘁY).inv ≫ η_ (ᘁX) X ▷ ᘁY ≫ (ᘁX ◁ f) ▷ ᘁY ≫ (α_ (ᘁX) Y ᘁY).hom ≫ ᘁX ◁ ε_ (ᘁY) Y ≫ (ρ_ ᘁX).hom ⊗ₘ g) =\n η_ (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 143,
"column": 2
} | {
"line": 148,
"column": 35
} | {
"line": 149,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : HasRankNullity.{u, u_1} R\ninst✝ : StrongRankCondition R\ns : Set M\nhs : LinearIndepOn R _root_.id s\nt : Set (M ⧸ span R s)\nht : #↑t = Module.rank R (M ⧸ span R s)\nht' : LinearIndependent (ι := ↑t) R S... | [
"R : Type u_1\nM : Type u\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : HasRankNullity.{u, u_1} R\ninst✝ : StrongRankCondition R\ns : Set M\nhs : LinearIndepOn R _root_.id s\nt : Set (M ⧸ span R s)\nht : #↑t = Module.rank R (M ⧸ span R s)\nht' : LinearIndependent (ι := ↑t) R Subtype.val\n... | have hst : Disjoint s (sec '' t) := by
rw [Set.disjoint_iff]
rintro _ ⟨hxs, ⟨x, hxt, rfl⟩⟩
apply ht'.ne_zero ⟨x, hxt⟩
rw [Subtype.coe_mk, ← hsec x, mkQ_apply, Quotient.mk_eq_zero]
exact Submodule.subset_span hxs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 38
} | {
"line": 118,
"column": 2
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\np : Submodule K V\nv : V\nhv : v ∉ p\nx : V\na : K\nhx : a • v ∈ p\nhx' : a • v = x\n⊢ x ∈ ⊥",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"Submodule"... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\np : Submodule K V\nv : V\nhv : v ∉ p\nx : V\na : K\nhx : a • v ∈ p\nhx' : a • v = x\n⊢ a = 0"
] | suffices a = 0 by simp [← hx', this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.FreeModule.Finite.Matrix | {
"line": 91,
"column": 71
} | {
"line": 94,
"column": 83
} | {
"line": 96,
"column": 0
} | [
{
"pp": "K : Type u_1\nM : Type u_2\nL : Type v\ninst✝⁷ : CommRing K\ninst✝⁶ : Ring M\ninst✝⁵ : Algebra K M\ninst✝⁴ : Module.Free K M\ninst✝³ : Module.Finite K M\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ninst✝ : Algebra K L\n⊢ Nat.card (M →ₐ[K] L) ≤ finrank K M",
"ppTerm": "?m.22",
"assigned": true,
... | [] | by
convert! toNat_le_toNat (cardinalMk_algHom_le_rank K M L) ?_
· rw [toNat_lift, finrank]
· rw [lift_lt_aleph0]; have := Module.nontrivial K L; apply Module.rank_lt_aleph0 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer | {
"line": 93,
"column": 23
} | {
"line": 93,
"column": 68
} | {
"line": 94,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\nf g : X ⟶ Y\nZ : C\nπ : Y ⟶ Z\nq : IsSplitCoequalizer f g π\nF : C ⥤ D\n⊢ F.map q.rightSection ≫ F.map π = 𝟙 (F.obj Z)",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Eq.... | [] | rw [← F.map_comp, q.rightSection_π, F.map_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer | {
"line": 93,
"column": 23
} | {
"line": 93,
"column": 68
} | {
"line": 94,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\nf g : X ⟶ Y\nZ : C\nπ : Y ⟶ Z\nq : IsSplitCoequalizer f g π\nF : C ⥤ D\n⊢ F.map q.rightSection ≫ F.map π = 𝟙 (F.obj Z)",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Eq.... | [] | rw [← F.map_comp, q.rightSection_π, F.map_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer | {
"line": 93,
"column": 23
} | {
"line": 93,
"column": 68
} | {
"line": 94,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\nf g : X ⟶ Y\nZ : C\nπ : Y ⟶ Z\nq : IsSplitCoequalizer f g π\nF : C ⥤ D\n⊢ F.map q.rightSection ≫ F.map π = 𝟙 (F.obj Z)",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Eq.... | [] | rw [← F.map_comp, q.rightSection_π, F.map_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 793,
"column": 2
} | {
"line": 793,
"column": 26
} | {
"line": 794,
"column": 2
} | [
{
"pp": "n : Type u_19\nR : Type u_20\nM : Type u_21\nM₁ : Type u_22\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R M\ninst✝² : Module R M₁\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M₁\nB : M →ₗ[R] M →ₗ[R] M₁\nv : Basis n R M\nhO : B.IsOrthoᵢ ⇑v\nh : ∀ (i : ... | [
"n : Type u_19\nR : Type u_20\nM : Type u_21\nM₁ : Type u_22\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R M\ninst✝² : Module R M₁\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M₁\nB : M →ₗ[R] M →ₗ[R] M₁\nv : Basis n R M\nhO : B.flip.IsOrthoᵢ ⇑v\nh : ∀ (i : n), (B ... | rw [isOrthoᵢ_flip] at hO | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 840,
"column": 6
} | {
"line": 840,
"column": 39
} | {
"line": 842,
"column": 0
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx :... | [] | · simpa [hx, hy] using hs (x + y) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 898,
"column": 2
} | {
"line": 898,
"column": 29
} | {
"line": 900,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nW : Submodule R M\nhW : Disjoint W (ker B)\nhB' : (domRestrict₁₂ B W W).IsRefl\n... | [] | simpa [hW.eq_bot] using key | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic | {
"line": 286,
"column": 8
} | {
"line": 286,
"column": 33
} | {
"line": 286,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ : C\ninst✝ : HasZeroMorphisms C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nm : NormalMono f.unop\nZ' : Cᵒᵖ\nm' : Y ⟶ Z'\nw' : NormalMono.g.op ≫ f ≫ m' = 0\n⊢ ∀ (j : WalkingParallelPair),\n m'.unop ≫ (KernelFork.ofι f.unop ⋯).π.app j = (KernelFork.ofι (m'.unop ≫ f.unop) ⋯... | [] | rintro (⟨⟩ | ⟨⟩) <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels | {
"line": 144,
"column": 90
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : HasZeroMorphisms C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : HasZeroMorphisms D\nG : C ⥤ D\ninst✝³ : G.PreservesZeroMorphisms\nX Y : C\nf : X ⟶ Y\ninst✝² : HasKernel f\ninst✝¹ : HasKernel (G.map f)\ninst✝ : PreservesLimit (parallelPair f 0)... | [] | by
rw [← cancel_mono (kernel.ι _)]
simp [PreservesKernel.iso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic | {
"line": 306,
"column": 8
} | {
"line": 306,
"column": 33
} | {
"line": 306,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ : C\ninst✝ : HasZeroMorphisms C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nm : NormalEpi f.unop\nZ' : Cᵒᵖ\nm' : Z' ⟶ X\nw' : (m' ≫ f) ≫ NormalEpi.g.op = 0\n⊢ ∀ (j : WalkingParallelPair),\n (CokernelCofork.ofπ f.unop ⋯).ι.app j ≫ m'.unop = (CokernelCofork.ofπ (f.unop ≫ m'... | [] | rintro (⟨⟩ | ⟨⟩) <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 89
} | {
"line": 416,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nv : V\n⊢ (∀ φ ∈ dualAnnihilator W, φ v = 0) ↔ v ∈ W",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"Sub... | [] | rw [← SetLike.ext_iff.mp dualAnnihilator_dualCoannihilator_eq v, mem_dualCoannihilator] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 89
} | {
"line": 416,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nv : V\n⊢ (∀ φ ∈ dualAnnihilator W, φ v = 0) ↔ v ∈ W",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"Sub... | [] | rw [← SetLike.ext_iff.mp dualAnnihilator_dualCoannihilator_eq v, mem_dualCoannihilator] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 89
} | {
"line": 416,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nv : V\n⊢ (∀ φ ∈ dualAnnihilator W, φ v = 0) ↔ v ∈ W",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"Sub... | [] | rw [← SetLike.ext_iff.mp dualAnnihilator_dualCoannihilator_eq v, mem_dualCoannihilator] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 48,
"column": 81
} | {
"line": 48,
"column": 94
} | {
"line": 48,
"column": 94
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteProducts C\ninst✝³ : HasKernels C\ninst✝² : IsNormalMonoCategory C\nX Y Z : C\na : X ⟶ Z\nb : Y ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nP : C\nf : Z ⟶ P\nhaf : a ≫ f = 0\ni : IsLimit (KernelFork.ofι a haf)\nQ : C... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteProducts C\ninst✝³ : HasKernels C\ninst✝² : IsNormalMonoCategory C\nX Y Z : C\na : X ⟶ Z\nb : Y ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nP : C\nf : Z ⟶ P\nhaf : a ≫ f = 0\ni : IsLimit (KernelFork.ofι a haf)\nQ : C\ng : Z ⟶ Q\... | prod.lift_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 113,
"column": 73
} | {
"line": 113,
"column": 86
} | {
"line": 113,
"column": 86
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : IsNormalMonoCategory C\nX Y : C\nf g : X ⟶ Y\nhuv : pullback.fst (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g) = pullback.snd (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g)\n⊢ ... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : IsNormalMonoCategory C\nX Y : C\nf g : X ⟶ Y\nhuv : pullback.fst (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g) = pullback.snd (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g)\n⊢ pullback.fst... | prod.lift_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 115,
"column": 41
} | {
"line": 115,
"column": 54
} | {
"line": 115,
"column": 54
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : IsNormalMonoCategory C\nX Y : C\nf g : X ⟶ Y\nhuv : pullback.fst (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g) = pullback.snd (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g)\n⊢ ... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : IsNormalMonoCategory C\nX Y : C\nf g : X ⟶ Y\nhuv : pullback.fst (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g) = pullback.snd (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g)\n⊢ pullback.snd... | prod.lift_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 352,
"column": 83
} | {
"line": 352,
"column": 96
} | {
"line": 352,
"column": 96
} | [
{
"pp": "case h₂\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c d : X ⟶ Y\n⊢ prod.lift b d ≫ σ = prod.lift (prod.lift a c ≫ σ) (prod.lift b d ≫ σ) ≫ prod.snd",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Catego... | [
"case h₂\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c d : X ⟶ Y\n⊢ prod.lift b d ≫ σ = prod.lift b d ≫ σ"
] | prod.lift_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 27
} | {
"line": 374,
"column": 2
} | [
{
"pp": "case a\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n| 0 - a - b",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HSub.hSub",
"Category... | [
"case a\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n| -b - a"
] | rw [← neg_def, neg_sub] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Algebra.Category.Grp.Biproducts | {
"line": 89,
"column": 17
} | {
"line": 91,
"column": 9
} | {
"line": 92,
"column": 4
} | [
{
"pp": "J : Type w\nf : J → AddCommGrpCat\ns : Fan f\n⊢ (fun j ↦ (ConcreteCategory.hom (s.π.app { as := j })) 0) = 0",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"AddMonoidHom.instAddMonoidHomCl... | [] | by
simp only [Functor.const_obj_obj, map_zero]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Basic | {
"line": 207,
"column": 16
} | {
"line": 209,
"column": 42
} | {
"line": 210,
"column": 6
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : Category.{v_3, u_3} E\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroMorphisms D\ninst✝¹ : HasZeroMorphisms E\nS S₁ S₂ S₃ : ShortComplex C\nF : C ⥤ D\ninst✝ : F.PreservesZeroMorphisms\nX✝ Y... | [] | by
dsimp
simp only [← F.map_comp, φ.comm₁₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 43
} | {
"line": 62,
"column": 0
} | [
{
"pp": "A : Type u\nB : Type v\ninst✝¹ : CommGroup A\ninst✝ : CommGroup B\nf : A →* B\nh : (QuotientGroup.mk' f.range).ker = ker 1\n⊢ f.range = ⊤",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"MonoidHom.range",
"CommMonoid.toCommSemigroup",
"MonoidHom",
"Monoid.... | [] | rwa [ker_one, QuotientGroup.ker_mk'] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 637,
"column": 2
} | {
"line": 638,
"column": 43
} | {
"line": 640,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ leftRightHomologyComparison' h.left h.right = h.iso.hom",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.ShortC... | [] | simp only [← cancel_epi h.left.π, ← cancel_mono h.right.ι, assoc,
π_leftRightHomologyComparison'_ι, comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 637,
"column": 2
} | {
"line": 638,
"column": 43
} | {
"line": 640,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ leftRightHomologyComparison' h.left h.right = h.iso.hom",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.ShortC... | [] | simp only [← cancel_epi h.left.π, ← cancel_mono h.right.ι, assoc,
π_leftRightHomologyComparison'_ι, comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 637,
"column": 2
} | {
"line": 638,
"column": 43
} | {
"line": 640,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ leftRightHomologyComparison' h.left h.right = h.iso.hom",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.ShortC... | [] | simp only [← cancel_epi h.left.π, ← cancel_mono h.right.ι, assoc,
π_leftRightHomologyComparison'_ι, comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 339,
"column": 42
} | {
"line": 339,
"column": 51
} | {
"line": 339,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\ncommf' : h₁.f' ≫ φK = φ.τ₁ ≫ h₂.f'\nφH : h₁.H ⟶ h₂.H := h₁.descH (φK ≫ h₂.π) ⋯\n⊢ h₁.π ≫ ... | [] | simp [φH] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 339,
"column": 42
} | {
"line": 339,
"column": 51
} | {
"line": 339,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\ncommf' : h₁.f' ≫ φK = φ.τ₁ ≫ h₂.f'\nφH : h₁.H ⟶ h₂.H := h₁.descH (φK ≫ h₂.π) ⋯\n⊢ h₁.π ≫ ... | [] | simp [φH] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 339,
"column": 42
} | {
"line": 339,
"column": 51
} | {
"line": 339,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\ncommf' : h₁.f' ≫ φK = φ.τ₁ ≫ h₂.f'\nφH : h₁.H ⟶ h₂.H := h₁.descH (φK ≫ h₂.π) ⋯\n⊢ h₁.π ≫ ... | [] | simp [φH] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 432,
"column": 42
} | {
"line": 432,
"column": 51
} | {
"line": 432,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nφQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯\ncommg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃\nφH : h₁.H ⟶ h₂.H := h₂.liftH (h₁.ι ≫ φQ) ⋯\n⊢ φH ≫ ... | [] | simp [φH] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 432,
"column": 42
} | {
"line": 432,
"column": 51
} | {
"line": 432,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nφQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯\ncommg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃\nφH : h₁.H ⟶ h₂.H := h₂.liftH (h₁.ι ≫ φQ) ⋯\n⊢ φH ≫ ... | [] | simp [φH] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 432,
"column": 42
} | {
"line": 432,
"column": 51
} | {
"line": 432,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nφQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯\ncommg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃\nφH : h₁.H ⟶ h₂.H := h₂.liftH (h₁.ι ≫ φQ) ⋯\n⊢ φH ≫ ... | [] | simp [φH] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1183,
"column": 82
} | {
"line": 1184,
"column": 35
} | {
"line": 1186,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.RightHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\n⊢ (ofEpiOfIsIsoOfMono' φ h).p = φ.τ₂ ≫ h.p",
"ppTerm": "?m.69",
"assigned": true,
"usedConstan... | [] | by
simp [ofEpiOfIsIsoOfMono', opMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 233,
"column": 11
} | {
"line": 233,
"column": 64
} | {
"line": 234,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nψ : RightHomologyMapData φ h₁ h₂\nF : C ⥤ D\ninst✝² : F.Pres... | [] | by simpa only [F.map_comp] using! F.congr_map ψ.commp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 453,
"column": 18
} | {
"line": 453,
"column": 25
} | {
"line": 453,
"column": 26
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ + φ₃).τ₂ = S₁.g ≫ (h.h₂ + h'.h₂) + (h.h₁ + h'.h₁) ≫ S₂.f + (φ₂ + φ₄).τ₂",
"ppTerm": "?m.168",
"assigned": true,
"usedConstant... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ φ₁.τ₂ + φ₃.τ₂ = S₁.g ≫ (h.h₂ + h'.h₂) + (h.h₁ + h'.h₁) ≫ S₂.f + (φ₂ + φ₄).τ₂"
] | add_τ₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 453,
"column": 26
} | {
"line": 453,
"column": 33
} | {
"line": 453,
"column": 34
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ φ₁.τ₂ + φ₃.τ₂ = S₁.g ≫ (h.h₂ + h'.h₂) + (h.h₁ + h'.h₁) ≫ S₂.f + (φ₂ + φ₄).τ₂",
"ppTerm": "?m.176",
"assigned": true,
"usedConstan... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ φ₁.τ₂ + φ₃.τ₂ = S₁.g ≫ (h.h₂ + h'.h₂) + (h.h₁ + h'.h₁) ≫ S₂.f + (φ₂.τ₂ + φ₄.τ₂)"
] | add_τ₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Preadditive.Injective.Basic | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 92
} | {
"line": 149,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nP Q : C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasBinaryBiproduct P Q\ninst✝¹ : Injective P\ninst✝ : Injective Q\nX✝ Y✝ : C\ng : X✝ ⟶ P ⊞ Q\nf : X✝ ⟶ Y✝\nmono : Mono f\n⊢ ∃ h, f ≫ h = g",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nP Q : C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasBinaryBiproduct P Q\ninst✝¹ : Injective P\ninst✝ : Injective Q\nX✝ Y✝ : C\ng : X✝ ⟶ P ⊞ Q\nf : X✝ ⟶ Y✝\nmono : Mono f\n⊢ f ≫ biprod.lift (factorThru (g ≫ biprod.fst) f) (factorThru (g ≫ biprod.snd) f) = g"
] | refine ⟨biprod.lift (factorThru (g ≫ biprod.fst) f) (factorThru (g ≫ biprod.snd) f), ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Preadditive.Injective.Basic | {
"line": 210,
"column": 2
} | {
"line": 213,
"column": 60
} | {
"line": 215,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : EnoughInjectives C\nX : C\n⊢ ∃ p, IsZero X → IsZero p.J",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.Injective",
"CategoryTheory.Mono",
"CategoryTheory.EnoughInjectives.presentation",
"Fa... | [] | by_cases h : IsZero X
· have := h.injective
exact ⟨{ J := X, f := 𝟙 X}, by tauto⟩
· exact ⟨(EnoughInjectives.presentation X).some, by tauto⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.Injective.Basic | {
"line": 210,
"column": 2
} | {
"line": 213,
"column": 60
} | {
"line": 215,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : EnoughInjectives C\nX : C\n⊢ ∃ p, IsZero X → IsZero p.J",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.Injective",
"CategoryTheory.Mono",
"CategoryTheory.EnoughInjectives.presentation",
"Fa... | [] | by_cases h : IsZero X
· have := h.injective
exact ⟨{ J := X, f := 𝟙 X}, by tauto⟩
· exact ⟨(EnoughInjectives.presentation X).some, by tauto⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.ShortExact | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 13
} | {
"line": 130,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex C\nhS : S.ShortExact\ninst✝ : Balanced C\nthis✝¹ : HasZeroObject C\nthis✝ : Mono S.f\nthis : Epi S.g\n⊢ IsZero S.X₃ → IsIso S.f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Ca... | [
"case mpr\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex C\nhS : S.ShortExact\ninst✝ : Balanced C\nthis✝¹ : HasZeroObject C\nthis✝ : Mono S.f\nthis : Epi S.g\nhX₃ : IsZero S.X₃\n⊢ IsIso S.f"
] | intro hX₃ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Algebra.Homology.ShortComplex.ShortExact | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 13
} | {
"line": 130,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex C\nhS : S.ShortExact\ninst✝ : Balanced C\nthis✝¹ : HasZeroObject C\nthis✝ : Mono S.f\nthis : Epi S.g\n⊢ IsZero S.X₃ → IsIso S.f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Ca... | [
"case mpr\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex C\nhS : S.ShortExact\ninst✝ : Balanced C\nthis✝¹ : HasZeroObject C\nthis✝ : Mono S.f\nthis : Epi S.g\nhX₃ : IsZero S.X₃\n⊢ IsIso S.f"
] | intro hX₃ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 207,
"column": 60
} | {
"line": 208,
"column": 66
} | {
"line": 210,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nX Y : C\ninst✝² : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝¹ : HasKernel f\nX' Y' : C\nf' : X' ⟶ Y'\ninst✝ : HasKernel f'\nsq : Arrow.mk f ⟶ Arrow.mk f'\n⊢ (kernelSubobjectIso f).hom ≫ kernel.map f f' sq.left sq.right ⋯ = kernelSubobjectMap sq ≫ (kernelSubobjectIso f'... | [] | by
simp [← Iso.comp_inv_eq, kernel_map_comp_kernelSubobjectIso_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 704,
"column": 2
} | {
"line": 704,
"column": 10
} | {
"line": 705,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : HasPullbacks C\nX Y Z W : C\nf : X ⟶ Y\ng : X ⟶ Z\nh : Y ⟶ W\nk : Z ⟶ W\ninst✝¹ : Mono h\ninst✝ : Mono g\ncomm : f ≫ h = g ≫ k\nt : IsLimit (PullbackCone.mk f g comm)\np : Subobject Y\n⊢ (map g).obj ((pullback f).obj p) = (pullback k).obj ((map h).obj... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : HasPullbacks C\nX Y Z W : C\nf : X ⟶ Y\ng : X ⟶ Z\nh : Y ⟶ W\nk : Z ⟶ W\ninst✝¹ : Mono h\ninst✝ : Mono g\ncomm : f ≫ h = g ≫ k\nt : IsLimit (PullbackCone.mk f g comm)\n⊢ ∀ (p : Subobject Y), (map g).obj ((pullback f).obj p) = (pullback k).obj ((map h).obj p)"
] | revert p | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 57,
"column": 4
} | {
"line": 58,
"column": 32
} | {
"line": 59,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\n⊢ factorThruImage S.f ≫ imageToKernel' S.f S.g ⋯ = kernel.lift S.g S.f ⋯",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"imageToKernel'._proof_1",
"CategoryTheory.Abelian.toPreadditive",... | [] | simp only [← cancel_mono (kernel.ι _), kernel.lift_ι, imageToKernel',
Category.assoc, image.fac] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 60,
"column": 2
} | {
"line": 61,
"column": 29
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\nthis : factorThruImage S.f ≫ imageToKernel' S.f S.g ⋯ = kernel.lift S.g S.f ⋯\n⊢ Epi (kernel.lift S.g S.f ⋯) → Epi (imageToKernel' S.f S.g ⋯)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
... | [
"case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\nthis : factorThruImage S.f ≫ imageToKernel' S.f S.g ⋯ = kernel.lift S.g S.f ⋯\n⊢ Epi (imageToKernel' S.f S.g ⋯) → Epi (kernel.lift S.g S.f ⋯)"
] | · intro
exact epi_of_epi_fac this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 6
} | {
"line": 178,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\n⊢ S.Exact ↔ { X₁ := S.X₁, X₂ := S.X₂, X₃ := Abelian.coimage S.g, f := S.f, g := Abelian.coimage.π S.g, zero := ⋯ }.Exact",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\n⊢ { X₁ := S.X₁, X₂ := S.X₂, X₃ := Abelian.coimage S.g, f := S.f, g := Abelian.coimage.π S.g, zero := ⋯ }.Exact ↔ S.Exact"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | {
"line": 203,
"column": 2
} | {
"line": 205,
"column": 57
} | {
"line": 206,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\ntfae_1_to_2 :\n (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →\n ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g... | [
"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\ntfae_1_to_2 :\n (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →\n ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F)... | tfae_have 3 → 4
| hF => by
exact preservesFiniteColimits_of_preservesCokernels F | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic | {
"line": 145,
"column": 35
} | {
"line": 150,
"column": 92
} | {
"line": 152,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : Type u_1\ninst✝³ : Category.{v_1, u_1} J\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\ne : C ≌ D\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\n⊢ HasExactColimitsOfShape J D",
"ppTerm": "?m.25",
"assigned": true,
"usedCons... | [] | by
haveI : HasColimitsOfShape J D := Adjunction.hasColimitsOfShape_of_equivalence e.inverse
refine ⟨⟨fun _ _ _ => ⟨@fun K => ?_⟩⟩⟩
refine preservesLimit_of_natIso K (?_ : e.congrRight.inverse ⋙ colim ⋙ e.functor ≅ colim)
apply e.symm.congrRight.fullyFaithfulFunctor.preimageIso
exact isoWhiskerLeft (_ ⋙ colim)... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic | {
"line": 351,
"column": 2
} | {
"line": 352,
"column": 53
} | {
"line": 353,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : HasCofilteredLimitsOfSize.{w₂, w₂', v, u} C\ninst✝⁴ : UnivLE.{w, w₂}\ninst✝³ : UnivLE.{w', w₂'}\ninst✝² : AB5StarOfSize.{w₂, w₂', v, u} C\nthis : HasCofilteredLimitsOfSize.{w, w', v, u} C\nJ : Type w'\ninst✝¹ : Category.{w, w'} J\ninst✝ : IsCofiltered J\... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : HasCofilteredLimitsOfSize.{w₂, w₂', v, u} C\ninst✝⁴ : UnivLE.{w, w₂}\ninst✝³ : UnivLE.{w', w₂'}\ninst✝² : AB5StarOfSize.{w₂, w₂', v, u} C\nthis✝ : HasCofilteredLimitsOfSize.{w, w', v, u} C\nJ : Type w'\ninst✝¹ : Category.{w, w'} J\ninst✝ : IsCofiltered J\nthis : IsC... | haveI := IsCofiltered.of_equivalence ((ShrinkHoms.equivalence.{w₂} J).trans <|
Shrink.equivalence.{w₂', w₂} (ShrinkHoms.{w'} J)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 60
} | {
"line": 319,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝ : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type (max w v₁)\ns : Cocone (functorToRepresentables P)\nm : (coconeOfRepresentable P).pt ⟶ s.pt\nhm : ∀ (j : P.Elementsᵒᵖ), uliftYonedaEquiv.symm (unop j).snd ≫ m = s.ι.app j\nX : Cᵒᵖ\nx : (cocone... | [] | rw [← hm, uliftYonedaEquiv_comp, Equiv.apply_symm_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 197,
"column": 21
} | {
"line": 197,
"column": 51
} | {
"line": 197,
"column": 51
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nx✝⁴ : Mono f\nx✝³ : C\nx✝² : P x✝³\nx✝¹ x✝ : x✝³ ⟶ X\nh : x✝¹ ≫ f = x✝ ≫ f\n⊢ x✝¹ = x✝",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQu... | [] | by simpa [cancel_mono] using h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 241,
"column": 6
} | {
"line": 242,
"column": 33
} | {
"line": 244,
"column": 0
} | [
{
"pp": "case mpr.refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCodetecting\nX Y : C\nf : X ⟶ Y\ninst✝ : Epi f\nhf : ∀ (G : C), P G → Function.Surjective ⇑(ConcreteCategory.hom ((yoneda.obj G).map f.op))\nA : C\nhA : P A\ng : X ⟶ A\n⊢ ∀ (y₁ y₂ : Y ⟶ A), f ≫ y₁ = g → f ≫ y₂ ... | [] | intro l₁ l₂ h₁ h₂
rw [← cancel_epi f, h₁, h₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 241,
"column": 6
} | {
"line": 242,
"column": 33
} | {
"line": 244,
"column": 0
} | [
{
"pp": "case mpr.refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCodetecting\nX Y : C\nf : X ⟶ Y\ninst✝ : Epi f\nhf : ∀ (G : C), P G → Function.Surjective ⇑(ConcreteCategory.hom ((yoneda.obj G).map f.op))\nA : C\nhA : P A\ng : X ⟶ A\n⊢ ∀ (y₁ y₂ : Y ⟶ A), f ≫ y₁ = g → f ≫ y₂ ... | [] | intro l₁ l₂ h₁ h₂
rw [← cancel_epi f, h₁, h₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 622,
"column": 2
} | {
"line": 623,
"column": 41
} | {
"line": 624,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ Type (max w v₁ v₂)), F.op.HasLeftKanExtension P\nΦ : uliftYoneda.{max w v₂, v₁, u₁}.LeftExtension (F ⋙ uliftYoneda.{max w v₁, v₂, u₂})\nf g : Functor.LeftExtension.mk F.op.lan (compULif... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ Type (max w v₁ v₂)), F.op.HasLeftKanExtension P\nΦ : uliftYoneda.{max w v₂, v₁, u₁}.LeftExtension (F ⋙ uliftYoneda.{max w v₁, v₂, u₂})\nf g : Functor.LeftExtension.mk F.op.lan (compULiftYonedaIsoUL... | have eq₂ := congr_hom (CC := fun X ↦ X) (congr_app (congr_app (StructuredArrow.w g) x.unop.1.unop)
(F.op.obj x.unop.1)) (ULift.up (𝟙 _)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 6
} | {
"line": 333,
"column": 2
} | [
{
"pp": "C : Type u\nF : C → Type v\nX Y : Type u\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ IsCompl (Set.range ⇑(ConcreteCategory.hom (BinaryCofan.mk f (↾Subtype.val)).inl))\n (Set.range ⇑(ConcreteCategory.hom (BinaryCofan.mk f (↾Subtype.val)).inr))",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\nF : C → Type v\nX Y : Type u\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ IsCompl (Set.range ⇑(ConcreteCategory.hom (BinaryCofan.mk f (↾Subtype.val)).inr))\n (Set.range ⇑(ConcreteCategory.hom (BinaryCofan.mk f (↾Subtype.val)).inl))"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Ring.Periodic | {
"line": 83,
"column": 59
} | {
"line": 83,
"column": 76
} | {
"line": 83,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nc : α\ninst✝¹ : Add α\ninst✝ : CommMonoid β\nι : Type u_4\nf : ι → α → β\ns : Finset ι\nhs : ∀ i ∈ s, Periodic (f i) c\n⊢ ∀ f_1 ∈ List.map f s.toList, Periodic f_1 c",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | simpa [-Periodic] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Ring.Periodic | {
"line": 83,
"column": 59
} | {
"line": 83,
"column": 76
} | {
"line": 83,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nc : α\ninst✝¹ : Add α\ninst✝ : CommMonoid β\nι : Type u_4\nf : ι → α → β\ns : Finset ι\nhs : ∀ i ∈ s, Periodic (f i) c\n⊢ ∀ f_1 ∈ List.map f s.toList, Periodic f_1 c",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | simpa [-Periodic] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Periodic | {
"line": 83,
"column": 59
} | {
"line": 83,
"column": 76
} | {
"line": 83,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nc : α\ninst✝¹ : Add α\ninst✝ : CommMonoid β\nι : Type u_4\nf : ι → α → β\ns : Finset ι\nhs : ∀ i ∈ s, Periodic (f i) c\n⊢ ∀ f_1 ∈ List.map f s.toList, Periodic f_1 c",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | simpa [-Periodic] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Periodic | {
"line": 324,
"column": 34
} | {
"line": 324,
"column": 64
} | {
"line": 324,
"column": 65
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : AddGroup α\ninst✝ : SubtractionMonoid β\nh : Antiperiodic f c\nk : ℤ\n⊢ -f x = ↑(2 * k + 1).negOnePow • f x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"SubtractionMonoid... | [
"case inr\nα : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : AddGroup α\ninst✝ : SubtractionMonoid β\nh : Antiperiodic f c\nk : ℤ\n⊢ -f x = ↑(-1) • f x"
] | Int.negOnePow_two_mul_add_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Periodic | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 18
} | {
"line": 356,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : AddMonoid α\ninst✝ : SubtractionMonoid β\nh : Antiperiodic f c\nk : ℕ\n⊢ -f x = (-1) ^ (2 * k + 1) • f x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"one_pow",
"NegZeroClass.toNeg",
"MulOne.toO... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Ring.Periodic | {
"line": 422,
"column": 62
} | {
"line": 423,
"column": 13
} | {
"line": 425,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝² : Add α\ninst✝¹ : IsRightCancelAdd α\ninst✝ : SubtractionCommMonoid β\nhf : Antiperiodic f c\ns : Finset α\n⊢ ∑ k ∈ Finset.map (addRightEmbedding c) s, f k = -∑ k ∈ s, f k",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"N... | [] | by
simp [hf _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Injective | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 10
} | {
"line": 171,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\nc : Set (ExtensionOf i f)\nhchain : IsChain (fun x1 x2 ↦ x1 ≤ x2) c... | [
"R : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\nc : Set (ExtensionOf i f)\nhchain : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nhnonempty ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Nat.Cast.Field | {
"line": 33,
"column": 2
} | {
"line": 36,
"column": 41
} | {
"line": 38,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionSemiring K\nm n : ℕ\nhnm : n ∣ m\nhn : ↑n ≠ 0\n⊢ ↑(m / n) = ↑m / ↑n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Dvd.dvd",
"instHDiv",
"HMul.hMul",
... | [] | obtain ⟨k, rfl⟩ := hnm
have : n ≠ 0 := by rintro rfl; simp at hn
rw [Nat.mul_div_cancel_left _ <| zero_lt_of_ne_zero this, mul_comm n,
cast_mul, mul_div_cancel_right₀ _ hn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Cast.Field | {
"line": 33,
"column": 2
} | {
"line": 36,
"column": 41
} | {
"line": 38,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionSemiring K\nm n : ℕ\nhnm : n ∣ m\nhn : ↑n ≠ 0\n⊢ ↑(m / n) = ↑m / ↑n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Dvd.dvd",
"instHDiv",
"HMul.hMul",
... | [] | obtain ⟨k, rfl⟩ := hnm
have : n ≠ 0 := by rintro rfl; simp at hn
rw [Nat.mul_div_cancel_left _ <| zero_lt_of_ne_zero this, mul_comm n,
cast_mul, mul_div_cancel_right₀ _ hn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Injective | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 8
} | {
"line": 371,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nx : ↥(e... | [
"R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nx : ↥(extensionOfMa... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Nat.Factorization.Induction | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 68
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case succ.succ\nmotive : ℕ → Prop\nzero : motive 0\none : motive 1\nprime : ∀ (p : ℕ), Prime p → motive p\ncomposite : ∀ (a : ℕ), 2 ≤ a → motive a → ∀ (b : ℕ), 2 ≤ b → motive b → motive (a * b)\nn p a : ℕ\nhp : Prime p\nha : motive (a + 1 + 1)\n⊢ motive (p * (a + 1 + 1))",
"ppTerm": "?succ.succ",
... | [] | · exact composite _ hp.two_le (prime _ hp) _ a.one_lt_succ_succ ha | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Totient | {
"line": 69,
"column": 12
} | {
"line": 69,
"column": 18
} | {
"line": 70,
"column": 2
} | [
{
"pp": "⊢ φ 0 = 0 ↔ 0 = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.totient",
"Bool.true",
"Iff",
"Nat",
"Bool",
"Eq.refl",
"instDecidableEqNat",
"OfNat.ofNat",
"D... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Totient | {
"line": 69,
"column": 12
} | {
"line": 69,
"column": 18
} | {
"line": 70,
"column": 2
} | [
{
"pp": "⊢ φ 0 = 0 ↔ 0 = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.totient",
"Bool.true",
"Iff",
"Nat",
"Bool",
"Eq.refl",
"instDecidableEqNat",
"OfNat.ofNat",
"D... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Totient | {
"line": 69,
"column": 12
} | {
"line": 69,
"column": 18
} | {
"line": 70,
"column": 2
} | [
{
"pp": "⊢ φ 0 = 0 ↔ 0 = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.totient",
"Bool.true",
"Iff",
"Nat",
"Bool",
"Eq.refl",
"instDecidableEqNat",
"OfNat.ofNat",
"D... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Totient | {
"line": 103,
"column": 24
} | {
"line": 103,
"column": 77
} | {
"line": 103,
"column": 77
} | [
{
"pp": "a k n : ℕ\na_ne_zero : a ≠ 0\ni : ℕ\nih : #({x ∈ Ico k (k + n % a + a * i) | a.Coprime x}) ≤ φ a * (i + 1)\n⊢ #({x ∈ Ico k (k + n % a + a * i) | a.Coprime x} ∪\n {x ∈ Ico (k + n % a + a * i) (k + n % a + a * i + a) | a.Coprime x}) ≤\n #({x ∈ Ico k (k + n % a + a * i) | a.Coprime x}) + φ a",
... | [
"a k n : ℕ\na_ne_zero : a ≠ 0\ni : ℕ\nih : #({x ∈ Ico k (k + n % a + a * i) | a.Coprime x}) ≤ φ a * (i + 1)\n⊢ #({x ∈ Ico k (k + n % a + a * i) | a.Coprime x} ∪\n {x ∈ Ico (k + n % a + a * i) (k + n % a + a * i + a) | a.Coprime x}) ≤\n #({x ∈ Ico k (k + n % a + a * i) | a.Coprime x}) +\n #({x ∈ Ico (... | ← filter_coprime_Ico_eq_totient a (k + n % a + a * i) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Totient | {
"line": 203,
"column": 12
} | {
"line": 203,
"column": 24
} | {
"line": 203,
"column": 24
} | [
{
"pp": "p : ℕ\nhp : Prime p\nn : ℕ\nh1 : Function.Injective fun x ↦ x * p\nb : ℕ\nh : b < p ^ n\n⊢ b * p < p ^ (n + 1)",
"ppTerm": "?m.255",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Nat.pow_succ",
... | [
"p : ℕ\nhp : Prime p\nn : ℕ\nh1 : Function.Injective fun x ↦ x * p\nb : ℕ\nh : b < p ^ n\n⊢ b * p < p ^ n * p"
] | Nat.pow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Totient | {
"line": 225,
"column": 31
} | {
"line": 225,
"column": 44
} | {
"line": 225,
"column": 45
} | [
{
"pp": "p : ℕ\nh : #({a ∈ range p | p.Coprime a}) = p - 1\nhp : 1 < p\n⊢ Prime p",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"congrArg",
"Finset",
"Nat.instDecidableCoprime",
"HSub.hSub",
"Nat.instLocallyFiniteOrder",
"Eq.mp",
... | [
"p : ℕ\nh : #({a ∈ Ico 0 p | p.Coprime a}) = p - 1\nhp : 1 < p\n⊢ Prime p"
] | range_eq_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
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