module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv | {
"line": 48,
"column": 13
} | {
"line": 48,
"column": 15
} | {
"line": 48,
"column": 16
} | [
{
"pp": "case left\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type w\ninst✝² : Category.{w', w} A\ninst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A\ninst✝ : IsDenseSub... | [
"case left\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type w\ninst✝² : Category.{w', w} A\ninst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A\ninst✝ : IsDenseSubsite J K G\n... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 491,
"column": 30
} | {
"line": 491,
"column": 32
} | {
"line": 491,
"column": 33
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝² : G.IsCoverDense K\ninst✝¹ : G.IsLocallyFull K\ninst✝ : G.IsLocallyFaithful K\nℱ : Sheaf K (Type u_5)\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.obj) T\n... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝² : G.IsCoverDense K\ninst✝¹ : G.IsLocallyFull K\ninst✝ : G.IsLocallyFaithful K\nℱ : Sheaf K (Type u_5)\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.obj) T\nhx : x.Compa... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 91
} | {
"line": 336,
"column": 4
} | [
{
"pp": "case refine_1.refine_3\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂... | [
"case refine_1.refine_4\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.... | · exact ⟨k.2, a ≫ pullback.snd _ _, by simp [reassoc_of% h₁], by simp [reassoc_of% h₂]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 723,
"column": 6
} | {
"line": 723,
"column": 56
} | {
"line": 723,
"column": 56
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nA : Type u_4\ninst✝³ : Category.{v_4, u_4} A\ninst✝² : IsDenseSubsite J K G\ninst✝¹ : (G.sheafPushforwardContinuous A J K).IsEquivalence\ninst✝ ... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nA : Type u_4\ninst✝³ : Category.{v_4, u_4} A\ninst✝² : IsDenseSubsite J K G\ninst✝¹ : (G.sheafPushforwardContinuous A J K).IsEquivalence\ninst✝ : HasWeakShe... | ← IsCoverDense.restrictHomEquivHom_naturality_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 46,
"column": 49
} | {
"line": 52,
"column": 18
} | {
"line": 54,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nX : C\nhX : IsTerminal X\nI : Type u_1\nY : I → C\n⊢ J.CoversTop Y ↔ Sieve.ofObjects Y X ∈ J X",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.GrothendieckTopology.superset_covering",
"C... | [] | by
constructor
· tauto
· intro h W
apply J.superset_covering _ (J.pullback_stable (hX.from W) h)
rintro T a ⟨i, ⟨b⟩⟩
exact ⟨i, ⟨b⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 121,
"column": 25
} | {
"line": 121,
"column": 27
} | {
"line": 121,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nX Y₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\n⊢ ∀ ⦃f₂ : Y₂ ⟶ X⦄ (h₁ : (Sieve.ofObjects Y X).arrows f₁) (h₂ : (Sieve.ofObjects Y X).arrows f₂),\n g₁ ≫ f₁ =... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nX Y₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\n⊢ ∀ (h₁ : (Sieve.ofObjects Y X).arrows f₁) (h₂ : (Sieve.ofObjects Y X).arrows f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ →\n ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 724,
"column": 2
} | {
"line": 724,
"column": 18
} | {
"line": 726,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni : E.I₀\nhs : (e.hom ≫ e.inv).s₀ = (𝟙 E).s₀\nhh : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (𝟙 E).h₀ i ≫ eqToHom ⋯\nh✝ :\n ∃ (hs₁ : ∀ (i j : E.I₀) (k : E.I₁ i j), (e.hom ≫ e.inv).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 E).s₁ ... | [] | simpa using hh i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 18
} | {
"line": 732,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni : F.I₀\nhs : (e.inv ≫ e.hom).s₀ = (𝟙 F).s₀\nhh : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (𝟙 F).h₀ i ≫ eqToHom ⋯\nh✝ :\n ∃ (hs₁ : ∀ (i j : F.I₀) (k : F.I₁ i j), (e.inv ≫ e.hom).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 F).s₁ ... | [] | simpa using hh i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 810,
"column": 28
} | {
"line": 811,
"column": 67
} | {
"line": 813,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS : C\nE✝¹ E✝ : PreOneHypercover S\nF✝ : PreOneHypercover S\nG : PreOneHypercover S\nE F : PreOneHypercover S\nf : E ≅ F\nc : WalkingMulticospan E.multicospanShape\n⊢ (Hom.mapMulticospan f.hom).map ((eqToIso ⋯).hom.app... | [] | by
cases c <;> rw [eqToIso.hom, eqToHom_app, eqToHom_map] <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 64
} | {
"line": 111,
"column": 64
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF : C ⥤ D\nS : Sheaf J RingCat\nR : Sheaf K RingCat\ninst✝² : F.IsContinuous J K\nφ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R\ninst✝¹ : (pushforward... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF : C ⥤ D\nS : Sheaf J RingCat\nR : Sheaf K RingCat\ninst✝² : F.IsContinuous J K\nφ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R\ninst✝¹ : (pushforward φ).IsRightA... | pullbackPushforwardAdjunction_homEquiv_pullbackObjUnitToUnit | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Simple | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 32
} | {
"line": 91,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasZeroMorphisms C\nD : Type u_1\ninst✝³ : Category.{v_1, u_1} D\ninst✝² : HasZeroMorphisms D\nF : C ⥤ D\ninst✝¹ : F.IsEquivalence\nX : C\ninst✝ : Simple X\n⊢ Simple (F.obj X)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"E... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasZeroMorphisms C\nD : Type u_1\ninst✝³ : Category.{v_1, u_1} D\ninst✝² : HasZeroMorphisms D\nF : C ⥤ D\ninst✝¹ : F.IsEquivalence\nX : C\ninst✝ : Simple X\n⊢ Simple (F.asEquivalence.functor.obj X)"
] | rw [← F.asEquivalence_functor] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Simple | {
"line": 29,
"column": 4
} | {
"line": 29,
"column": 22
} | {
"line": 29,
"column": 22
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ IsSimpleOrder (Submodule R ↑(of R M)) ↔ IsSimpleModule R M",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"AddCommGroup.toAddComm... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ IsSimpleOrder (Submodule R ↑(of R M)) ↔ IsSimpleOrder (Submodule R M)"
] | isSimpleModule_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 41
} | {
"line": 406,
"column": 2
} | [
{
"pp": "case property.refine_1\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns✝ : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X :=\n fun x ↦\n match x with\n | { down := j } => WalkingPair.casesOn j ... | [
"case property.refine_2\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns✝ : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X :=\n fun x ↦\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nhι : U ... | · apply interUnionPullbackConeLift_left | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Category.ModuleCat.Stalk | {
"line": 101,
"column": 48
} | {
"line": 112,
"column": 87
} | {
"line": 113,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : SmallCategory C\ninst✝¹ : IsFiltered C\nR : C ⥤ RingCat\nM : C ⥤ Ab\ninst✝ : (i : C) → Module ↑(R.obj i) ↑(M.obj i)\nH :\n ∀ {i j : C} (f : i ⟶ j) (r : ↑(R.obj i)) (m : ↑(M.obj i)),\n (ConcreteCategory.hom (M.map f)) (r • m) = (ConcreteCategory.hom (R.map f)) r • (ConcreteCat... | [] | by
rintro ⟨U₁, a₁⟩ ⟨U₂, a₂⟩ ⟨V, b⟩
obtain ⟨s, α, β, h₁, h₂, h₃, h₄⟩ := crown₄
(rightToMax U₁ V ≫ leftToMax (max U₁ V) (max U₂ V)) (rightToMax (max U₁ U₂) V)
(rightToMax U₂ V ≫ rightToMax (max U₁ V) (max U₂ V)) (rightToMax (max U₁ U₂) V)
(leftToMax U₁ V ≫ leftToMax (max U₁ V) (max U₂ V))
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Topology.Homology | {
"line": 91,
"column": 36
} | {
"line": 91,
"column": 60
} | {
"line": 91,
"column": 61
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nM N : TopModuleCat R\nφ : M ⟶ N\ns : CokernelCofork φ\nm : (CokernelCofork.ofπ (cokerπ φ) ⋯).pt ⟶ s.pt\nh : cokerπ φ ≫ m = Cofork.π s\n⊢ m = ofHom { toLinearMap := (↑(Hom.hom φ)).range.liftQ ↑(Hom.hom (Cofork.π s)) ⋯, cont := ⋯ }",
"ppTerm": ... | [
"R : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nM N : TopModuleCat R\nφ : M ⟶ N\ns : CokernelCofork φ\nm : (CokernelCofork.ofπ (cokerπ φ) ⋯).pt ⟶ s.pt\nh : cokerπ φ ≫ m = Cofork.π s\n⊢ cokerπ φ ≫ m = cokerπ φ ≫ ofHom { toLinearMap := (↑(Hom.hom φ)).range.liftQ ↑(Hom.hom (Cofork.π s)) ⋯, cont := ⋯ }"
] | ← cancel_epi (cokerπ φ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1422,
"column": 7
} | {
"line": 1422,
"column": 48
} | {
"line": 1424,
"column": 0
} | [
{
"pp": "S : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nG : Type u_5\ninst✝²² : Semiring R\ninst✝²¹ : Semiring S\ninst✝²⁰ : AddCommMonoid V\ninst✝¹⁹ : Module R V\ninst✝¹⁸ : TopologicalSpace V\ninst✝¹⁷ : Module S V\ninst✝¹⁶ : ContinuousConstSMul S V\ninst✝¹⁵ : AddCommMonoid W\ninst✝¹⁴ : Module R W\ninst... | [] | simp [LinearMapClass.map_smul_of_tower f] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.Ring.Epi | {
"line": 27,
"column": 68
} | {
"line": 41,
"column": 84
} | {
"line": 43,
"column": 0
} | [
{
"pp": "R S : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ Epi (ofHom (algebraMap R S)) ↔ Algebra.IsEpi R S",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NonAssocSemirin... | [] | by
simp_rw [Algebra.isEpi_iff_forall_one_tmul_eq, eq_comm]
constructor
· intro H
have := H.1 (CommRingCat.ofHom <| Algebra.TensorProduct.includeLeftRingHom)
(CommRingCat.ofHom <| (Algebra.TensorProduct.includeRight (R := R) (A := S)).toRingHom)
(by ext r; change algebraMap R S r ⊗ₜ 1 = 1 ⊗ₜ algebr... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.Over | {
"line": 42,
"column": 71
} | {
"line": 53,
"column": 50
} | {
"line": 55,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\n⊢ PreservesFilteredColimitsOfSize.{u_2, u_3, v_1, v_1, max u_1 v_1, u_1} (Under.forget X)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.IsColimit.fac",
"CategoryTheory.instCateg... | [] | by
refine ⟨fun J hJ hJ' ↦ ⟨fun {F} ↦ ⟨fun {c} hc ↦ ⟨.ofExistsUnique fun s ↦ ?_⟩⟩⟩⟩
obtain i := Nonempty.some ((inferInstance : Nonempty J))
let s' : Cocone F := ⟨Under.mk ((F.obj i).hom ≫ s.ι.app i), fun j ↦ Under.homMk (s.ι.app j) (by
obtain ⟨k, hik, hjk, -⟩ := IsFilteredOrEmpty.cocone_objs i j
simp only... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Comma.CardinalArrow | {
"line": 30,
"column": 2
} | {
"line": 43,
"column": 18
} | {
"line": 45,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : SmallCategory C\n⊢ Finite (Arrow C) ↔ Nonempty (FinCategory C)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Fintype.ofFinite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"HEq.refl",
"CategoryTheory.FinCategory.finty... | [] | constructor
· intro
refine ⟨?_, fun a b ↦ ?_⟩
· have := Finite.of_injective (fun (a : C) ↦ Arrow.mk (𝟙 a))
(fun _ _ ↦ congr_arg Comma.left)
apply Fintype.ofFinite
· have := Finite.of_injective (fun (f : a ⟶ b) ↦ Arrow.mk f)
(fun f g h ↦ by
change (Arrow.mk f).hom = (Arrow.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Comma.CardinalArrow | {
"line": 30,
"column": 2
} | {
"line": 43,
"column": 18
} | {
"line": 45,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : SmallCategory C\n⊢ Finite (Arrow C) ↔ Nonempty (FinCategory C)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Fintype.ofFinite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"HEq.refl",
"CategoryTheory.FinCategory.finty... | [] | constructor
· intro
refine ⟨?_, fun a b ↦ ?_⟩
· have := Finite.of_injective (fun (a : C) ↦ Arrow.mk (𝟙 a))
(fun _ _ ↦ congr_arg Comma.left)
apply Fintype.ofFinite
· have := Finite.of_injective (fun (f : a ⟶ b) ↦ Arrow.mk f)
(fun f g h ↦ by
change (Arrow.mk f).hom = (Arrow.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Comma.CardinalArrow | {
"line": 96,
"column": 38
} | {
"line": 96,
"column": 50
} | {
"line": 96,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Small.{w, u} C\nx✝ : Arrow (Shrink.{w, u} C)\n⊢ ((Shrink.equivalence C).functor.mapArrow.obj ((Shrink.equivalence C).inverse.mapArrow.obj x✝)).hom =\n eqToHom ⋯ ≫ x✝.hom ≫ eqToHom ⋯",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant | {
"line": 127,
"column": 20
} | {
"line": 127,
"column": 85
} | {
"line": 129,
"column": 0
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\nF : J ⥤ C\ni₀ : J\nh : F.IsEventuallyConstantTo i₀\ninst✝ : IsCofiltered J\ns : Cone F\nm : s.pt ⟶ h.cone.pt\nhm : ∀ (j : J), m ≫ h.cone.π.app j = s.π.app j\n⊢ m = s.π.app i₀",
"ppTerm": "?m.95",
"assign... | [] | simp only [← hm i₀, cone_π_app, coneπApp_eq_id, cone_pt, comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant | {
"line": 127,
"column": 20
} | {
"line": 127,
"column": 85
} | {
"line": 129,
"column": 0
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\nF : J ⥤ C\ni₀ : J\nh : F.IsEventuallyConstantTo i₀\ninst✝ : IsCofiltered J\ns : Cone F\nm : s.pt ⟶ h.cone.pt\nhm : ∀ (j : J), m ≫ h.cone.π.app j = s.π.app j\n⊢ m = s.π.app i₀",
"ppTerm": "?m.95",
"assign... | [] | simp only [← hm i₀, cone_π_app, coneπApp_eq_id, cone_pt, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant | {
"line": 127,
"column": 20
} | {
"line": 127,
"column": 85
} | {
"line": 129,
"column": 0
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\nF : J ⥤ C\ni₀ : J\nh : F.IsEventuallyConstantTo i₀\ninst✝ : IsCofiltered J\ns : Cone F\nm : s.pt ⟶ h.cone.pt\nhm : ∀ (j : J), m ≫ h.cone.π.app j = s.π.app j\n⊢ m = s.π.app i₀",
"ppTerm": "?m.95",
"assign... | [] | simp only [← hm i₀, cone_π_app, coneπApp_eq_id, cone_pt, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Ring.FinitePresentation | {
"line": 65,
"column": 4
} | {
"line": 66,
"column": 25
} | {
"line": 67,
"column": 2
} | [
{
"pp": "case h₁\nJ : Type uJ\ninst✝² : Category.{vJ, uJ} J\ninst✝¹ : IsFiltered J\nR : CommRingCat\nF : J ⥤ CommRingCat\nα : (Functor.const J).obj R ⟶ F\nS : CommRingCat\nf : R ⟶ S\nc : Cocone F\nhc : IsColimit c\ninst✝ : PreservesColimit F (forget CommRingCat)\nhf : (CommRingCat.Hom.hom f).EssFiniteType\ni : ... | [] | rw [← CommRingCat.hom_comp, ← CommRingCat.hom_comp, reassoc_of% ha, reassoc_of% hb]
simp [← α.naturality] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Ring.FinitePresentation | {
"line": 65,
"column": 4
} | {
"line": 66,
"column": 25
} | {
"line": 67,
"column": 2
} | [
{
"pp": "case h₁\nJ : Type uJ\ninst✝² : Category.{vJ, uJ} J\ninst✝¹ : IsFiltered J\nR : CommRingCat\nF : J ⥤ CommRingCat\nα : (Functor.const J).obj R ⟶ F\nS : CommRingCat\nf : R ⟶ S\nc : Cocone F\nhc : IsColimit c\ninst✝ : PreservesColimit F (forget CommRingCat)\nhf : (CommRingCat.Hom.hom f).EssFiniteType\ni : ... | [] | rw [← CommRingCat.hom_comp, ← CommRingCat.hom_comp, reassoc_of% ha, reassoc_of% hb]
simp [← α.naturality] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TensorProduct.Pi | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 47
} | {
"line": 116,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\ninst✝⁵ : Fintype ι\ninst✝⁴ : DecidableEq ι\nN : Type u_4\ninst✝³ : Semiring N\ninst✝² : Algebra R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : ι → R\n⊢ (piScalarRight R S N ι).s... | [] | simp [Algebra.algebraMap_eq_smul_one, Pi.smul_def', LinearEquiv.symm_apply_eq,
piScalarRight_apply, piScalarRightHom_tmul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.TensorProduct.Pi | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 47
} | {
"line": 116,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\ninst✝⁵ : Fintype ι\ninst✝⁴ : DecidableEq ι\nN : Type u_4\ninst✝³ : Semiring N\ninst✝² : Algebra R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : ι → R\n⊢ (piScalarRight R S N ι).s... | [] | simp [Algebra.algebraMap_eq_smul_one, Pi.smul_def', LinearEquiv.symm_apply_eq,
piScalarRight_apply, piScalarRightHom_tmul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Pi | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 47
} | {
"line": 116,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\ninst✝⁵ : Fintype ι\ninst✝⁴ : DecidableEq ι\nN : Type u_4\ninst✝³ : Semiring N\ninst✝² : Algebra R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : ι → R\n⊢ (piScalarRight R S N ι).s... | [] | simp [Algebra.algebraMap_eq_smul_one, Pi.smul_def', LinearEquiv.symm_apply_eq,
piScalarRight_apply, piScalarRightHom_tmul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Ring.FinitePresentation | {
"line": 156,
"column": 17
} | {
"line": 156,
"column": 19
} | {
"line": 156,
"column": 20
} | [
{
"pp": "case refine_2\nJ : Type uJ\ninst✝² : Category.{vJ, uJ} J\ninst✝¹ : IsFiltered J\nR : CommRingCat\nS : Under R\nhS : (Hom.hom S.hom).FinitePresentation\nF : J ⥤ Under R\ninst✝ : PreservesColimit (F ⋙ Under.forget R) (forget CommRingCat)\nc : Cocone F\nhc : IsColimit c\ni j : J\nf₁ : (F ⋙ coyoneda.obj (O... | [
"case refine_2\nJ : Type uJ\ninst✝² : Category.{vJ, uJ} J\ninst✝¹ : IsFiltered J\nR : CommRingCat\nS : Under R\nhS : (Hom.hom S.hom).FinitePresentation\nF : J ⥤ Under R\ninst✝ : PreservesColimit (F ⋙ Under.forget R) (forget CommRingCat)\nc : Cocone F\nhc : IsColimit c\ni j : J\nf₁ : (F ⋙ coyoneda.obj (Opposite.op S... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 122,
"column": 4
} | {
"line": 125,
"column": 10
} | {
"line": 126,
"column": 4
} | [
{
"pp": "T : Type u_1\ninst✝⁵ : Category.{v_1, u_1} T\nA : Type u_2\ninst✝⁴ : Category.{v_2, u_2} A\nL : A ⥤ T\nJ : Type u_3\ninst✝³ : Category.{v_3, u_3} J\nP : MorphismProperty T\ninst✝² : P.RespectsIso\ninst✝¹ : PreservesColimitsOfShape J L\ninst✝ : HasColimitsOfShape J A\nc : (D : J ⥤ T) → [HasColimit D] → ... | [
"T : Type u_1\ninst✝⁵ : Category.{v_1, u_1} T\nA : Type u_2\ninst✝⁴ : Category.{v_2, u_2} A\nL : A ⥤ T\nJ : Type u_3\ninst✝³ : Category.{v_3, u_3} J\nP : MorphismProperty T\ninst✝² : P.RespectsIso\ninst✝¹ : PreservesColimitsOfShape J L\ninst✝ : HasColimitsOfShape J A\nc : (D : J ⥤ T) → [HasColimit D] → Cocone D\nhc... | have heq : Y.hom = hd.desc { pt := X, ι := { app j := (d.diag.obj j).hom } } := by
refine hd.hom_ext fun j ↦ ?_
simp only [IsColimit.fac]
simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 316,
"column": 4
} | {
"line": 319,
"column": 58
} | {
"line": 320,
"column": 4
} | [
{
"pp": "T : Type u_1\ninst✝⁴ : Category.{v_1, u_1} T\nP : MorphismProperty T\nX✝ : T\ninst✝³ : HasPullbacks T\ninst✝² : P.IsMultiplicative\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.HasOfPostcompProperty P\nX Y : T\nf : X ⟶ Y\nJ : Type\nx✝¹ : SmallCategory J\nx✝ : FinCategory J\n⊢ PreservesLimitsOfShape J ... | [
"T : Type u_1\ninst✝⁴ : Category.{v_1, u_1} T\nP : MorphismProperty T\nX✝ : T\ninst✝³ : HasPullbacks T\ninst✝² : P.IsMultiplicative\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.HasOfPostcompProperty P\nX Y : T\nf : X ⟶ Y\nJ : Type\nx✝¹ : SmallCategory J\nx✝ : FinCategory J\nthis : PreservesLimitsOfShape J (pullba... | have : PreservesLimitsOfShape J
(MorphismProperty.Over.pullback P ⊤ f ⋙ MorphismProperty.Over.forget _ _ _) :=
inferInstanceAs <| PreservesLimitsOfShape J <|
Over.forget _ _ _ ⋙ CategoryTheory.Over.pullback f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Constructions.Over.Products | {
"line": 88,
"column": 81
} | {
"line": 89,
"column": 26
} | {
"line": 89,
"column": 26
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf : Y ⟶ X\ng : Z ⟶ X\nc : PullbackCone f g\nhc : IsLimit c\ns : BinaryFan (Over.mk f) (Over.mk g)\n⊢ hc.lift (pullbackConeEquivBinaryFan.inverse.obj s) ≫ (Over.mk (c.fst ≫ f)).hom = s.pt.hom",
"ppTerm": "?m.109",
"assigned": true,
... | [] | by
simpa using! s.fst.w | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Constructions.Over.Products | {
"line": 254,
"column": 14
} | {
"line": 254,
"column": 31
} | {
"line": 254,
"column": 32
} | [
{
"pp": "case none.none.id\nJ✝ : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z B : C\nJ : Type w\nF : Discrete J ⥤ Over B\nc : Cone F\n⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) none =\n Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) none ≫\n ... | [
"case none.none.id\nJ✝ : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z B : C\nJ : Type w\nF : Discrete J ⥤ Over B\nc : Cone F\n⊢ Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) none =\n Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) none ≫\n WidePullbackShape.Hom.re... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Constructions.Over.Products | {
"line": 256,
"column": 14
} | {
"line": 256,
"column": 31
} | {
"line": 256,
"column": 32
} | [
{
"pp": "case some.some.id\nJ✝ : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z B : C\nJ : Type w\nF : Discrete J ⥤ Over B\nc : Cone F\nval✝ : J\n⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) (some val✝) =\n Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := va... | [
"case some.some.id\nJ✝ : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z B : C\nJ : Type w\nF : Discrete J ⥤ Over B\nc : Cone F\nval✝ : J\n⊢ Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) (some val✝) =\n Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) (some val✝) ≫\n ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.Algebra | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 62
} | {
"line": 49,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : NonAssocSemiring R\ninst✝³ : NoZeroDivisors R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Nontrivial A\nf : R →+* A\np : ℕ\nhp : p ≠ 0\ninst✝ : CharP R p\nthis : Nontrivial R\n⊢ CharP A p",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Nat.Pr... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : NonAssocSemiring R\ninst✝³ : NoZeroDivisors R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Nontrivial A\nf : R →+* A\np : ℕ\nhp : p ≠ 0\ninst✝ : CharP R p\nthis : Nontrivial R\nH : Nat.Prime p\n⊢ CharP A p"
] | have H := (CharP.char_is_prime_or_zero R p).resolve_right hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Category.Ring.Under.Property | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 50
} | {
"line": 115,
"column": 0
} | [
{
"pp": "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhQi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQe : HasEqualizers fun {R S} [CommRing R] [CommRing S] ↦ Q\nR : CommRingCat\nD : WalkingParallelPair ⥤ (toMorphismProperty fun {R S} [CommRing R] [CommRing S]... | [] | exact hasLimit_of_created D (Under.forget _ _ R) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.CharP.Invertible | {
"line": 63,
"column": 20
} | {
"line": 63,
"column": 82
} | {
"line": 64,
"column": 2
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝¹ : Ring R\np : ℕ\ninst✝ : CharP R p\nn : ℕ\nh : n.Coprime p\n⊢ ↑(n.gcdA p) * ↑n = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAd... | [] | by rw [CharP.natCast_gcdA_mul_intCast_eq_gcd, h, Nat.cast_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 128,
"column": 40
} | {
"line": 128,
"column": 80
} | {
"line": 128,
"column": 80
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\nN : Type u_4\nP : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R N\... | [] | by simp [Module.Flat.eqLocus_lTensor_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CharP.Pi | {
"line": 30,
"column": 71
} | {
"line": 30,
"column": 83
} | {
"line": 30,
"column": 84
} | [
{
"pp": "ι : Type u\nhi : Nonempty ι\nR : Type v\ninst✝¹ : Semiring R\np : ℕ\ninst✝ : CharP R p\nx : ℕ\ni : ι\nh : ↑x = 0\nj : ι\n⊢ (Pi.evalRingHom (fun x ↦ R) j) ↑x = 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"... | [
"ι : Type u\nhi : Nonempty ι\nR : Type v\ninst✝¹ : Semiring R\np : ℕ\ninst✝ : CharP R p\nx : ℕ\ni : ι\nh : ↑x = 0\nj : ι\n⊢ ↑x = 0"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 96,
"column": 2
} | {
"line": 111,
"column": 50
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommRing R\nP : Prop\n⊢ (∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P) → ∀ p > 0, MixedCharZero R p → P",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"ringChar.of_eq",
"NonAssocSemiring.toAddCommMonoidWithO... | [] | · intro h q q_pos q_mixedChar
rcases q_mixedChar.charP_quotient with ⟨I, hI_ne_top, _⟩
-- Krull's Thm: There exists a prime ideal `P` such that `I ≤ P`
rcases Ideal.exists_le_maximal I hI_ne_top with ⟨M, hM_max, h_IM⟩
let r := ringChar (R ⧸ M)
have r_pos : r ≠ 0 := by
have q_zero :=
co... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.CharP.Subring | {
"line": 26,
"column": 61
} | {
"line": 26,
"column": 73
} | {
"line": 26,
"column": 74
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\np : ℕ\ninst✝ : CharP R p\nS : Subsemiring R\nx : ℕ\nh : ↑x = 0\n⊢ S.subtype ↑x = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Subsemiring... | [
"R : Type u\ninst✝¹ : Semiring R\np : ℕ\ninst✝ : CharP R p\nS : Subsemiring R\nx : ℕ\nh : ↑x = 0\n⊢ ↑x = 0"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.Subring | {
"line": 33,
"column": 61
} | {
"line": 33,
"column": 73
} | {
"line": 33,
"column": 74
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\np : ℕ\ninst✝ : CharP R p\nS : Subring R\nx : ℕ\nh : ↑x = 0\n⊢ S.subtype ↑x = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Subring.instSetLike... | [
"R : Type u\ninst✝¹ : Ring R\np : ℕ\ninst✝ : CharP R p\nS : Subring R\nx : ℕ\nh : ↑x = 0\n⊢ ↑x = 0"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Stream.Init | {
"line": 416,
"column": 8
} | {
"line": 416,
"column": 73
} | {
"line": 416,
"column": 73
} | [
{
"pp": "case right\nα : Type u\ns₁✝ s₂✝ s₁' s₁ : Stream' α\nx✝ : (fun s₁' s₁ ↦ ∃ s₂, s₁' = (s₁ ⋈ s₂).even) s₁' s₁\ns₂ : Stream' α\nh₁ : s₁' = (s₁ ⋈ s₂).even\n⊢ (fun s₁' s₁ ↦ ∃ s₂, s₁' = (s₁ ⋈ s₂).even) (s₁ ⋈ s₂).even.tail s₁.tail",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Eq.... | [] | exact ⟨tail s₂, by rw [interleave_eq, even_cons_cons, tail_cons]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Stream.Init | {
"line": 416,
"column": 8
} | {
"line": 416,
"column": 73
} | {
"line": 416,
"column": 73
} | [
{
"pp": "case right\nα : Type u\ns₁✝ s₂✝ s₁' s₁ : Stream' α\nx✝ : (fun s₁' s₁ ↦ ∃ s₂, s₁' = (s₁ ⋈ s₂).even) s₁' s₁\ns₂ : Stream' α\nh₁ : s₁' = (s₁ ⋈ s₂).even\n⊢ (fun s₁' s₁ ↦ ∃ s₂, s₁' = (s₁ ⋈ s₂).even) (s₁ ⋈ s₂).even.tail s₁.tail",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Eq.... | [] | exact ⟨tail s₂, by rw [interleave_eq, even_cons_cons, tail_cons]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Stream.Init | {
"line": 416,
"column": 8
} | {
"line": 416,
"column": 73
} | {
"line": 416,
"column": 73
} | [
{
"pp": "case right\nα : Type u\ns₁✝ s₂✝ s₁' s₁ : Stream' α\nx✝ : (fun s₁' s₁ ↦ ∃ s₂, s₁' = (s₁ ⋈ s₂).even) s₁' s₁\ns₂ : Stream' α\nh₁ : s₁' = (s₁ ⋈ s₂).even\n⊢ (fun s₁' s₁ ↦ ∃ s₂, s₁' = (s₁ ⋈ s₂).even) (s₁ ⋈ s₂).even.tail s₁.tail",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Eq.... | [] | exact ⟨tail s₂, by rw [interleave_eq, even_cons_cons, tail_cons]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Colimit.Ring | {
"line": 88,
"column": 52
} | {
"line": 88,
"column": 64
} | {
"line": 88,
"column": 64
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\n⊢ (of G f (Classical.arbitrary ι)) (-1) =\n (Ideal.Quotient.mk\n (Ideal.span\n ... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Colimit.Ring | {
"line": 90,
"column": 44
} | {
"line": 90,
"column": 76
} | {
"line": 90,
"column": 76
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\nx' y' : FreeCommRing ((i : ι) × G i)\ni : ι\nx : G i\nhx :\n (of G f i) x =\n (Ideal.Quotient.... | [
"ι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\nx' y' : FreeCommRing ((i : ι) × G i)\ni : ι\nx : G i\nhx :\n (of G f i) x =\n (Ideal.Quotient.mk\n ... | rw [map_add, of_f, of_f, hx, hy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Stream.Init | {
"line": 644,
"column": 4
} | {
"line": 644,
"column": 16
} | {
"line": 645,
"column": 4
} | [
{
"pp": "α : Type u\ns : Stream' α\n⊢ s.inits.tail = initsCore [s.head, s.tail.head] s.tail.tail",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Stream'.inits",
"id",
"List.cons",
"Stream'",
"List",
"Stream'.tail",
"Eq",
"Stream'.head",
... | [
"α : Type u\ns : Stream' α\n⊢ (initsCore [s.head] s.tail).tail = initsCore [s.head, s.tail.head] s.tail.tail"
] | unfold inits | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Data.Seq.Basic | {
"line": 299,
"column": 8
} | {
"line": 299,
"column": 21
} | {
"line": 299,
"column": 22
} | [
{
"pp": "case bisim.nil.nil.cons\nα : Type u\ns t u✝ : Seq α\nx✝ : α\nu : Seq α\n⊢ BisimO (fun s1 s2 ↦ ∃ s t u, s1 = (s.append t).append u ∧ s2 = s.append (t.append u))\n ((nil.append nil).append (cons x✝ u)).destruct (nil.append (nil.append (cons x✝ u))).destruct",
"ppTerm": "?bisim.nil.nil.cons",
"... | [] | | cons _ u => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Algebra.Colimit.Ring | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 87
} | {
"line": 143,
"column": 88
} | [
{
"pp": "ι : Type u_1\ninst✝² : Preorder ι\nG : ι → Type u_2\ninst✝¹ : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nP : Type u_3\ninst✝ : CommRing P\ng : (i : ι) → G i →+* P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x\nx : FreeCommRing ((i : ι) × G i)\nhx :\n x ∈\n ... | [
"case inl\nι : Type u_1\ninst✝² : Preorder ι\nG : ι → Type u_2\ninst✝¹ : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nP : Type u_3\ninst✝ : CommRing P\ng : (i : ι) → G i →+* P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x\ni j : ι\nhij : i ≤ j\nx : G i\n⊢ (FreeCommRing.lif... | rcases hx with (⟨i, j, hij, x, rfl⟩ | ⟨i, rfl⟩ | ⟨i, x, y, rfl⟩ | ⟨i, x, y, rfl⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Colimit.Ring | {
"line": 254,
"column": 50
} | {
"line": 255,
"column": 85
} | {
"line": 255,
"column": 86
} | [
{
"pp": "ι : Type u_1\ninst✝⁴ : Preorder ι\nG : ι → Type u_2\ninst✝³ : (i : ι) → CommRing (G i)\nf✝ : (i j : ι) → i ≤ j → G i → G j\nP : Type u_3\ninst✝² : CommRing P\ng : (i : ι) → G i →+* P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f✝ i j hij x) = (g i) x\nf'✝ f : (i j : ι) → i ≤ j → G i →+* G j\nG' :... | [] | by
rw [RingHom.comp_assoc, hg₁ i, ← RingHom.comp_assoc, hg₂ i, RingHom.comp_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Basic | {
"line": 821,
"column": 2
} | {
"line": 821,
"column": 41
} | {
"line": 823,
"column": 0
} | [
{
"pp": "α : Type u\nR : α → α → Prop\nhd : α\ntl : Seq α\nh : ∀ (i j : ℕ), i < j → ∀ x ∈ (Seq.cons hd tl).get? i, ∀ y ∈ (Seq.cons hd tl).get? j, R x y\nx : α\nn : ℕ\nhx : some x = tl.get? n\n⊢ R hd x",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"congrArg",
"Option.instMembe... | [] | simpa [← hx] using h 0 (n + 1) (by lia) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Seq.Basic | {
"line": 1052,
"column": 2
} | {
"line": 1063,
"column": 30
} | {
"line": 1065,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq1 α\nf : α → Seq1 β\ng : β → Seq1 γ\n⊢ (s.bind f).bind g = s.bind fun x ↦ (f x).bind g",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"congrArg",
"Function.comp",
"Stream'.Seq.map",
... | [] | obtain ⟨a, s⟩ := s
simp only [bind, map_pair, map_join]
rw [← map_comp]
simp only [show (fun x => join (map g (f x))) = join ∘ (map g ∘ f) from rfl]
rw [map_comp _ join]
generalize Seq.map (map g ∘ f) s = SS
rcases map g (f a) with ⟨⟨a, s⟩, S⟩
induction s using recOn with | nil => ?_ | cons x s_1 => ?_ <;... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 1052,
"column": 2
} | {
"line": 1063,
"column": 30
} | {
"line": 1065,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq1 α\nf : α → Seq1 β\ng : β → Seq1 γ\n⊢ (s.bind f).bind g = s.bind fun x ↦ (f x).bind g",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"congrArg",
"Function.comp",
"Stream'.Seq.map",
... | [] | obtain ⟨a, s⟩ := s
simp only [bind, map_pair, map_join]
rw [← map_comp]
simp only [show (fun x => join (map g (f x))) = join ∘ (map g ∘ f) from rfl]
rw [map_comp _ join]
generalize Seq.map (map g ∘ f) s = SS
rcases map g (f a) with ⟨⟨a, s⟩, S⟩
induction s using recOn with | nil => ?_ | cons x s_1 => ?_ <;... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat | {
"line": 99,
"column": 10
} | {
"line": 99,
"column": 44
} | {
"line": 99,
"column": 44
} | [
{
"pp": "case h.mk.mk\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : FloorRing K\nv : K\ng : GenContFract K := of v\nn : ℕ\nIH : ∀ m < n + 1 + 1, ∃ conts, (of v).contsAux m = Pair.map Rat.cast conts\ngp_n : Pair K\ns_ppred_nth_eq : g.s.get? n = some gp_n\na_eq_one : gp_n.a = 1\nz : ℤ\nb_eq_z :... | [] | simp [nextConts, nextNum, nextDen] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 56
} | {
"line": 112,
"column": 0
} | [
{
"pp": "K : Type u_1\nn : ℕ\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\nterminatedAt_succ_n : s.get? (n + 1) = none\n⊢ squashSeq s n = s",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Stream'.Seq.zipWith",
"GenContFract.Pair.b",
"instHDiv",
... | [] | cases s_nth_eq : s.get? n <;> simp only [*, squashSeq] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 124,
"column": 4
} | {
"line": 126,
"column": 34
} | {
"line": 128,
"column": 0
} | [
{
"pp": "case some\nK : Type u_1\nn : ℕ\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\nm : ℕ\nm_lt_n : m < n\nval✝ : Pair K\ns_succ_nth_eq : s.get? (n + 1) = some val✝\n⊢ (squashSeq s n).get? m = s.get? m",
"ppTerm": "?some",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Strea... | [] | obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n :=
s.ge_stable n.le_succ s_succ_nth_eq
simp [*, squashSeq, m_lt_n.ne] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 124,
"column": 4
} | {
"line": 126,
"column": 34
} | {
"line": 128,
"column": 0
} | [
{
"pp": "case some\nK : Type u_1\nn : ℕ\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\nm : ℕ\nm_lt_n : m < n\nval✝ : Pair K\ns_succ_nth_eq : s.get? (n + 1) = some val✝\n⊢ (squashSeq s n).get? m = s.get? m",
"ppTerm": "?some",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Strea... | [] | obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n :=
s.ge_stable n.le_succ s_succ_nth_eq
simp [*, squashSeq, m_lt_n.ne] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 19
} | {
"line": 81,
"column": 2
} | [
{
"pp": "K : Type u_1\nv : K\ninst✝⁴ : Field K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Archimedean K\nε : K\nε_pos : ε > 0\nN' : ℕ\none_div_ε_lt_N' : 1 / ε < ↑N'\n⊢ ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε",
"ppTerm": "?m.64",
"assigned": true,
"usedConst... | [
"K : Type u_1\nv : K\ninst✝⁴ : Field K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Archimedean K\nε : K\nε_pos : ε > 0\nN' : ℕ\none_div_ε_lt_N' : 1 / ε < ↑N'\nN : ℕ := max N' 5\n⊢ ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε"
] | let N := max N' 5 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 19
} | {
"line": 176,
"column": 6
} | [
{
"pp": "case succ.some\nK : Type u_1\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : FloorRing K\ninst✝ : IsStrictOrderedRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\nIH : Option.map (mapFr Rat.cast) (IntFractPair.stream q n) = IntFractPair.stream (↑q) n\nifp_n : IntFractPair ℚ\nstream_q_nth_eq : IntFractP... | [] | | some ifp_n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat | {
"line": 291,
"column": 4
} | {
"line": 294,
"column": 68
} | {
"line": 295,
"column": 4
} | [
{
"pp": "case some\nq : ℚ\nfract_q_num : ℤ := (Int.fract q).num\nn : ℕ := fract_q_num.natAbs + 1\nifp : IntFractPair ℚ\nstream_nth_eq : IntFractPair.stream q n = some ifp\nifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n\n⊢ ∃ n, IntFractPair.stream q n = none",
"ppTerm": "?some",
"assigned":... | [
"case some\nq : ℚ\nfract_q_num : ℤ := (Int.fract q).num\nn : ℕ := fract_q_num.natAbs + 1\nifp : IntFractPair ℚ\nstream_nth_eq : IntFractPair.stream q n = some ifp\nifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n\nthis : fract_q_num - ↑n = -1\n⊢ ∃ n, IntFractPair.stream q n = none"
] | have : fract_q_num - n = -1 := by
have : 0 ≤ fract_q_num := Rat.num_nonneg.mpr (Int.fract_nonneg q)
simp only [n, Nat.cast_add, Int.natAbs_of_nonneg this, Nat.cast_one,
sub_add_eq_sub_sub_swap, sub_right_comm, sub_self, zero_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 265,
"column": 6
} | {
"line": 309,
"column": 11
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case inr.succ\nK : Type u_1\ng : GenContFract K\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0\nnot_terminatedAt_n : ¬g.TerminatedAt (n' + 1)\ns_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }\n⊢ g.convs (n' + 1 + 1)... | [] | obtain ⟨⟨pa, pb⟩, s_n'th_eq⟩ : ∃ gp_n', g.s.get? n' = some gp_n' :=
g.s.ge_stable n'.le_succ s_nth_eq
-- Notations
let g' := squashGCF g (n' + 1)
set pred_conts := g.contsAux (n' + 1) with succ_n'th_contsAux_eq
set ppred_conts := g.contsAux n' with n'th_contsAux_eq
let pA := pred_c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 265,
"column": 6
} | {
"line": 309,
"column": 11
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case inr.succ\nK : Type u_1\ng : GenContFract K\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0\nnot_terminatedAt_n : ¬g.TerminatedAt (n' + 1)\ns_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }\n⊢ g.convs (n' + 1 + 1)... | [] | obtain ⟨⟨pa, pb⟩, s_n'th_eq⟩ : ∃ gp_n', g.s.get? n' = some gp_n' :=
g.s.ge_stable n'.le_succ s_nth_eq
-- Notations
let g' := squashGCF g (n' + 1)
set pred_conts := g.contsAux (n' + 1) with succ_n'th_contsAux_eq
set ppred_conts := g.contsAux n' with n'th_contsAux_eq
let pA := pred_c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 343,
"column": 8
} | {
"line": 343,
"column": 42
} | {
"line": 344,
"column": 8
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nn : ℕ\nIH :\n ∀ {g : GenContFract K},\n (∀ {gp : Pair K} {m : ℕ}, m < n → g.s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b) → g.convs n = g.convs' n\ng : GenContFract K\ns_pos : ∀ {gp : Pair K} {m : ℕ}, m < n + 1 → g.s... | [
"case refl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\ng : GenContFract K\ngp' : Pair K\nm : ℕ\nIH :\n ∀ {g : GenContFract K},\n (∀ {gp : Pair K} {m_1 : ℕ}, m_1 < m.succ → g.s.get? m_1 = some gp → 0 < gp.a ∧ 0 < gp.b) →\n g.convs m.succ = g.convs' m.succ\ns_pos :... | rcases m_lt_n with n | succ_m_lt_n | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.ContinuedFractions.Computation.Approximations | {
"line": 370,
"column": 6
} | {
"line": 376,
"column": 83
} | {
"line": 379,
"column": 4
} | [
{
"pp": "K : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nifp : IntFractPair K\nstream_nth_eq : IntFractPair.stream v n = some ifp\ng : GenContFract K := of v\nconts : Pair K := g.contsAux (n + 1)\npred_conts : Pair K := g.contsAux n\nifp... | [] | match hn : n with
| 0 => subst n; simp [pA, pB, A, B, pred_conts, conts]
| n' + 1 =>
subst n
simp only [succ_ne_zero, false_or] at n_eq_zero_or_not_terminatedAt_pred_n
rw [add_tsub_cancel_right] at n_eq_zero_or_not_terminatedAt_pred_n
exact (SimpContFract.of v).determinant n_... | Lean.Elab.Tactic.evalMatch | Lean.Parser.Tactic.match |
Mathlib.Algebra.CubicDiscriminant | {
"line": 482,
"column": 30
} | {
"line": 482,
"column": 39
} | {
"line": 482,
"column": 40
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nha : P.a ≠ 0\nhP : (Polynomial.map φ P.toPoly).Splits\nx y z : K\nh3 : (map φ P).roots = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ x ∉ y ::ₘ {z} ∧ y ∉ {z} ∧ {z}.Nodup",
"ppTerm": "?m.99",
"assigned": true,
"... | [
"F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nha : P.a ≠ 0\nhP : (Polynomial.map φ P.toPoly).Splits\nx y z : K\nh3 : (map φ P).roots = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ ¬(x = y ∨ x ∈ {z}) ∧ y ∉ {z} ∧ {z}.Nodup"
] | mem_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Congruence.BigOperators | {
"line": 125,
"column": 55
} | {
"line": 125,
"column": 70
} | {
"line": 125,
"column": 70
} | [
{
"pp": "ι : Type u_1\nβ : ι → Type u_2\nM : Type u_3\ninst✝² : DecidableEq ι\ninst✝¹ : AddCommMonoid M\ninst✝ : (i : ι) → AddCommMonoid (β i)\nc : AddCon M\nh : (i : ι) → β i →+ M\nf : Π₀ (i : ι), β i\n⊢ c.mk' (f.sum fun x ↦ ⇑(h x)) = f.sum fun x ↦ ⇑(c.mk'.comp (h x))",
"ppTerm": "?m.37",
"assigned": t... | [
"ι : Type u_1\nβ : ι → Type u_2\nM : Type u_3\ninst✝² : DecidableEq ι\ninst✝¹ : AddCommMonoid M\ninst✝ : (i : ι) → AddCommMonoid (β i)\nc : AddCon M\nh : (i : ι) → β i →+ M\nf : Π₀ (i : ι), β i\n⊢ (f.sum fun a b ↦ c.mk' ((h a) b)) = f.sum fun x ↦ ⇑(c.mk'.comp (h x))"
] | map_dfinsuppSum | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 99,
"column": 4
} | {
"line": 101,
"column": 49
} | {
"line": 102,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : Ring R\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nI : Ideal R\nx : R\nxIJ : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I\np q : n\nM : Matrix n n R\nz : R\nzMx : z * M q p * x + z - 1 ∈ I\nN : Matrix n n R := 1 - ∑ i, single i q (if i = q then 1 - z else M i p *... | [] | · convert! I.mul_mem_left (-M i p * x) zMx
simp [iq, N, single, mul_apply, sum_apply, ite_and, sub_mul]
simp [sub_add, mul_add, mul_sub, mul_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 97,
"column": 2
} | {
"line": 101,
"column": 49
} | {
"line": 102,
"column": 2
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝² : Ring R\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nI : Ideal R\nx : R\nxIJ : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I\np q : n\nM : Matrix n n R\nz : R\nzMx : z * M q p * x + z - 1 ∈ I\nN : Matrix n n R := 1 - ∑ i, single i q (if i = q then 1 - z else M i p... | [
"case h.inr\nR : Type u_1\ninst✝² : Ring R\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nI : Ideal R\nx : R\nxIJ : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I\np q : n\nM : Matrix n n R\nz : R\nzMx : z * M q p * x + z - 1 ∈ I\nN : Matrix n n R := 1 - ∑ i, single i q (if i = q then 1 - z else M i p * x * z)\ni... | · by_cases iq : i = q
· simp [iq, N, zMx, single, mul_apply, sum_apply, ite_and, sub_mul]
· convert! I.mul_mem_left (-M i p * x) zMx
simp [iq, N, single, mul_apply, sum_apply, ite_and, sub_mul]
simp [sub_add, mul_add, mul_sub, mul_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 194,
"column": 4
} | {
"line": 196,
"column": 49
} | {
"line": 197,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nn : Type u_2\ninst✝³ : NonUnitalNonAssocSemiring R\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\nc : RingCon R\nx y : R\n⊢ (matrix n c).ofMatrix x y → c x y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"Ri... | [] | intro h
inhabit n
simpa using h default default default default | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 194,
"column": 4
} | {
"line": 196,
"column": 49
} | {
"line": 197,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nn : Type u_2\ninst✝³ : NonUnitalNonAssocSemiring R\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\nc : RingCon R\nx y : R\n⊢ (matrix n c).ofMatrix x y → c x y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"Ri... | [] | intro h
inhabit n
simpa using h default default default default | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Idempotents | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 53
} | {
"line": 390,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Type u_3\ninst✝ : Finite I\ne : I → R\nhe : OrthogonalIdempotents e\nthis : Pairwise fun i j ↦ IsCoprime (Ideal.span {1 - e i}) (Ideal.span {1 - e j})\nx : R ⧸ ⨅ i, Ideal.span {1 - e i}\n⊢ ∃ a,\n (RingHom.pi fun i ↦ Ideal.Quotient.mk (Ideal.span {1 - e i})) a =... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Type u_3\ninst✝ : Finite I\ne : I → R\nhe : OrthogonalIdempotents e\nthis : Pairwise fun i j ↦ IsCoprime (Ideal.span {1 - e i}) (Ideal.span {1 - e j})\nx : R\n⊢ ∃ a,\n (RingHom.pi fun i ↦ Ideal.Quotient.mk (Ideal.span {1 - e i})) a =\n (Ideal.quotientInfToPiQuotient f... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Idempotents | {
"line": 456,
"column": 48
} | {
"line": 469,
"column": 9
} | {
"line": 471,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ne f : R\nhe : IsIdempotentElem e\nhf : IsIdempotentElem f\nhef₁ : e + f = 1\nhef₂ : e * f = 0\n⊢ Function.Bijective ⇑((Ideal.Quotient.mk (Ideal.span {e})).prod (Ideal.Quotient.mk (Ideal.span {f})))",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
... | [] | by
let o (i : Fin 2) : R := match i with
| 0 => e
| 1 => f
change Function.Bijective
(piFinTwoEquiv _ ∘ RingHom.pi (fun i : Fin 2 ↦ Ideal.Quotient.mk (Ideal.span {o i})))
rw [(Equiv.bijective _).of_comp_iff']
apply pi_bijective_of_isIdempotentElem
· intro i
fin_cases i <;> simpa [o]
· intro ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 43,
"column": 2
} | {
"line": 47,
"column": 82
} | {
"line": 49,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝⁹ : DecidableEq ι\nR : Type u_4\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\nN₁ : ι → Submodule R M₁\nh₁ : IsInternal N₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nN₂ : ι → Submodule R M₂\nh₂ : IsInternal N₂\nκ₁ : ι →... | [] | ext ⟨i, _⟩ ⟨j, _⟩
simp only [toMatrix_apply, Matrix.blockDiagonal'_apply]
rcases eq_or_ne i j with rfl | hij
· simp [h₂.collectedBasis_repr_of_mem _ (hf _ (Subtype.mem _)), restrict_apply]
· simp [hij, h₂.collectedBasis_repr_of_mem_ne _ hij.symm (hf _ (Subtype.mem _))] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Idempotents | {
"line": 536,
"column": 27
} | {
"line": 536,
"column": 46
} | {
"line": 536,
"column": 46
} | [
{
"pp": "R : Type u_1\ne : R\ninst✝ : Semigroup R\nidem : IsIdempotentElem e\nhc : IsMulCentral e\nr : R\n⊢ e * r = r ∧ r * e = r ↔ e * r = r",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"and_iff_left_of_imp",
"HMul.hMul",
"cong... | [
"R : Type u_1\ne : R\ninst✝ : Semigroup R\nidem : IsIdempotentElem e\nhc : IsMulCentral e\nr : R\n⊢ e * r = r ↔ e * r = r",
"R : Type u_1\ne : R\ninst✝ : Semigroup R\nidem : IsIdempotentElem e\nhc : IsMulCentral e\nr : R\n⊢ e * r = r → r * e = r"
] | and_iff_left_of_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 43,
"column": 2
} | {
"line": 47,
"column": 82
} | {
"line": 49,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝⁹ : DecidableEq ι\nR : Type u_4\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\nN₁ : ι → Submodule R M₁\nh₁ : IsInternal N₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nN₂ : ι → Submodule R M₂\nh₂ : IsInternal N₂\nκ₁ : ι →... | [] | ext ⟨i, _⟩ ⟨j, _⟩
simp only [toMatrix_apply, Matrix.blockDiagonal'_apply]
rcases eq_or_ne i j with rfl | hij
· simp [h₂.collectedBasis_repr_of_mem _ (hf _ (Subtype.mem _)), restrict_apply]
· simp [hij, h₂.collectedBasis_repr_of_mem_ne _ hij.symm (hf _ (Subtype.mem _))] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 7
} | {
"line": 382,
"column": 2
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk : ℕ\nh : MapsTo ⇑(f - μ • 1) ↑((f.genEigenspace μ) ↑k) ↑((f.genEigenspace μ) ↑k)\n⊢ IsNilpotent (LinearMap.restrict (f - μ • 1) h)",
"ppTerm": "?m.74",
"assigned": true,
"usedCons... | [
"case h\nR : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk : ℕ\nh : MapsTo ⇑(f - μ • 1) ↑((f.genEigenspace μ) ↑k) ↑((f.genEigenspace μ) ↑k)\n⊢ LinearMap.restrict (f - μ • 1) h ^ k = 0"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 683,
"column": 2
} | {
"line": 683,
"column": 19
} | {
"line": 684,
"column": 2
} | [
{
"pp": "case insert\nR : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nf : End R M\nk : ℕ∞\nμ₁ μ₂ : R\ns : Finset R\na✝ : μ₂ ∉ s\nih : μ₁ ∉ s → Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ ↦ (f.genEigenspace μ) k)\nhμ₁₂✝... | [
"case insert\nR : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nf : End R M\nk : ℕ∞\nμ₁ μ₂ : R\ns : Finset R\na✝ : μ₂ ∉ s\nih : Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ ↦ (f.genEigenspace μ) k)\nhμ₁₂✝ : μ₁ ∉ insert μ₂ s\n... | specialize ih hμ₁ | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.LinearAlgebra.Trace | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 69
} | {
"line": 111,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\ninst✝² : CommSemiring R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : M →ₗ[R] M\nH : ¬∃ s, Nonempty (Basis (↥s) R M)\n⊢ (trace R M) (f * g) = (trace R M) (g * f)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"LinearMap.trace",
... | [] | · rw [trace, dif_neg H, LinearMap.zero_apply, LinearMap.zero_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 13
} | {
"line": 75,
"column": 4
} | [
{
"pp": "case cons\nα : Type u_1\nL : List (α × Bool)\ninst✝ : DecidableEq α\nhd1 : α × Bool\ntl1 : List (α × Bool)\nih : Red tl1 (reduce tl1)\n⊢ Red (hd1 :: tl1)\n (List.rec [hd1] (fun head tail tail_ih ↦ if hd1.1 = head.1 ∧ hd1.2 = !head.2 then tail else hd1 :: head :: tail)\n (reduce tl1))",
"ppT... | [
"case cons\nα : Type u_1\nL : List (α × Bool)\ninst✝ : DecidableEq α\nhd1 : α × Bool\ntl1 : List (α × Bool)\n⊢ Red tl1 (reduce tl1) →\n Red (hd1 :: tl1)\n (List.rec [hd1] (fun head tail tail_ih ↦ if hd1.1 = head.1 ∧ hd1.2 = !head.2 then tail else hd1 :: head :: tail)\n (reduce tl1))"
] | revert ih | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 19
} | {
"line": 108,
"column": 8
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\np : Prop\nx : α\nb : Bool\nL1 L2 L3 : List (α × Bool)\nx' : α\nb' : Bool\ntail : List (α × Bool)\ny : α\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : x = y ∧ b = !c\nH : tail = L2 ++ (x', b') :: (x', !b') :: L3\n⊢ p",
"ppTerm": "?pos✝",
"assigned"... | [
"case pos\nα : Type u_1\ninst✝ : DecidableEq α\np : Prop\nx : α\nb : Bool\nL1 L2 L3 : List (α × Bool)\nx' : α\nb' : Bool\ntail : List (α × Bool)\ny : α\nc : Bool\nr : reduce L1 = (y, c) :: (L2 ++ (x', b') :: (x', !b') :: L3)\nh : x = y ∧ b = !c\nH : tail = L2 ++ (x', b') :: (x', !b') :: L3\n⊢ p"
] | rw [H] at r | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.GCDMonoid.FinsetLemmas | {
"line": 36,
"column": 26
} | {
"line": 38,
"column": 83
} | {
"line": 40,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℕ\nh : (↑s).Pairwise (Function.onFun Nat.Coprime f)\n⊢ s.lcm f = s.prod f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Nat.Coprime",
"Function.onFun",
"congrArg",
"Nat.unique_u... | [] | by
rw [show Nat.Coprime = IsRelPrime by ext; exact Nat.coprime_iff_isRelPrime] at h
exact associated_lcm_prod h |>.eq_of_normalized (normalize_eq _) (normalize_eq _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.IntegralClosure.IntegrallyClosed | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 29
} | {
"line": 160,
"column": 30
} | [
{
"pp": "A : Type u_2\ninst✝³ : CommRing A\nC : Type u_5\ninst✝² : SetLike C A\ninst✝¹ : SubringClass C A\nS : C\nT : Subring A\ninst✝ : IsIntegrallyClosedIn (↥T) A\n⊢ (integralClosure (↥S) A).toSubring ≤ T ↔ ofClass S ≤ T",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"A : Type u_2\ninst✝³ : CommRing A\nC : Type u_5\ninst✝² : SetLike C A\ninst✝¹ : SubringClass C A\nS : C\nT : Subring A\ninst✝ : IsIntegrallyClosedIn (↥T) A\n⊢ (∀ (r : ↥S), (algebraMap (↥S) A) r ∈ T) ↔ ofClass S ≤ T"
] | integralClosure_le_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.IntegrallyClosed | {
"line": 47,
"column": 4
} | {
"line": 47,
"column": 27
} | {
"line": 48,
"column": 4
} | [
{
"pp": "case intro\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nX : FractionRing R\nx✝ : IsIntegral R X\np : R[X]\nhp₁ : p.Monic\nhp₂ : Polynomial.eval₂ (algebraMap R (FractionRing R)) X p = 0\na✝ : GCDMonoid R\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * (algebraMap... | [
"case h\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nX : FractionRing R\nx✝ : IsIntegral R X\np : R[X]\nhp₁ : p.Monic\nhp₂ : Polynomial.eval₂ (algebraMap R (FractionRing R)) X p = 0\na✝ : GCDMonoid R\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * (algebraMap R (FractionRing... | use x * (this.unit⁻¹ :) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Quaternion | {
"line": 548,
"column": 18
} | {
"line": 548,
"column": 44
} | {
"line": 549,
"column": 2
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\nc₁ c₂ c₃ r x y : R\na b : ℍ[R,c₁,c₂,c₃]\ninst✝ : CommRing R\nx✝¹ x✝ : ℍ[R,c₁,c₃]\n⊢ { re := (x✝¹ + x✝).re, imI := (x✝¹ + x✝).imJ, imJ := (x✝¹ + x✝).imI, imK := -(x✝¹ + x✝).imK } =\n { re := x✝¹.re, imI := x✝¹.imJ, imJ := x✝¹.imI, imK := -x✝¹.imK } +\n {... | [] | by ext <;> simp [add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Action.Sigma | {
"line": 85,
"column": 16
} | {
"line": 87,
"column": 26
} | {
"line": 89,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nN : Type u_3\nα : ι → Type u_4\nm : Monoid M\ninst✝ : (i : ι) → MulAction M (α i)\nx : (i : ι) × α i\n⊢ 1 • x = x",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
... | [] | by
cases x
rw [smul_mk, one_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.NatPowAssoc | {
"line": 126,
"column": 41
} | {
"line": 126,
"column": 54
} | {
"line": 126,
"column": 55
} | [
{
"pp": "case succ\nR : Type u_2\ninst✝² : NonAssocSemiring R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nn m : ℕ\nih : ↑(n ^ m) = ↑n ^ m\n⊢ ↑(n ^ m * n ^ 1) = ↑n ^ m * ↑n ^ 1",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"case succ\nR : Type u_2\ninst✝² : NonAssocSemiring R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nn m : ℕ\nih : ↑(n ^ m) = ↑n ^ m\n⊢ ↑(n ^ m) * ↑(n ^ 1) = ↑n ^ m * ↑n ^ 1"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Finsupp | {
"line": 23,
"column": 2
} | {
"line": 32,
"column": 20
} | {
"line": 34,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nι : Type u_2\nf : ι → M\nx : M\nhx : x ∈ closure (Set.range f)\n⊢ ∃ a, x = a.prod fun x1 x2 ↦ f x1 ^ x2",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClas... | [] | classical
induction hx using closure_induction with
| mem x h => obtain ⟨i, rfl⟩ := h; exact ⟨Finsupp.single i 1, by simp⟩
| one => use 0; simp
| mul x y hx hy hx' hy' =>
obtain ⟨⟨v, rfl⟩, w, rfl⟩ := And.intro hx' hy'
use v + w
rw [Finsupp.prod_add_index]
· simp
· simp [pow_add] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.Group.Submonoid.Finsupp | {
"line": 23,
"column": 2
} | {
"line": 32,
"column": 20
} | {
"line": 34,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nι : Type u_2\nf : ι → M\nx : M\nhx : x ∈ closure (Set.range f)\n⊢ ∃ a, x = a.prod fun x1 x2 ↦ f x1 ^ x2",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClas... | [] | classical
induction hx using closure_induction with
| mem x h => obtain ⟨i, rfl⟩ := h; exact ⟨Finsupp.single i 1, by simp⟩
| one => use 0; simp
| mul x y hx hy hx' hy' =>
obtain ⟨⟨v, rfl⟩, w, rfl⟩ := And.intro hx' hy'
use v + w
rw [Finsupp.prod_add_index]
· simp
· simp [pow_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Finsupp | {
"line": 23,
"column": 2
} | {
"line": 32,
"column": 20
} | {
"line": 34,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nι : Type u_2\nf : ι → M\nx : M\nhx : x ∈ closure (Set.range f)\n⊢ ∃ a, x = a.prod fun x1 x2 ↦ f x1 ^ x2",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClas... | [] | classical
induction hx using closure_induction with
| mem x h => obtain ⟨i, rfl⟩ := h; exact ⟨Finsupp.single i 1, by simp⟩
| one => use 0; simp
| mul x y hx hy hx' hy' =>
obtain ⟨⟨v, rfl⟩, w, rfl⟩ := And.intro hx' hy'
use v + w
rw [Finsupp.prod_add_index]
· simp
· simp [pow_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset | {
"line": 145,
"column": 4
} | {
"line": 147,
"column": 27
} | {
"line": 149,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq β\ninst✝² : GroupWithZero α\ninst✝¹ : MulAction α β\ninst✝ : Fintype β\ns : Finset α\nhs : ¬s ⊆ 0\n⊢ ↑(s • univ) = ↑univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"... | [] | rw [← coe_subset] at hs
push_cast at hs ⊢
exact Set.smul_univ₀ hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset | {
"line": 145,
"column": 4
} | {
"line": 147,
"column": 27
} | {
"line": 149,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq β\ninst✝² : GroupWithZero α\ninst✝¹ : MulAction α β\ninst✝ : Fintype β\ns : Finset α\nhs : ¬s ⊆ 0\n⊢ ↑(s • univ) = ↑univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"... | [] | rw [← coe_subset] at hs
push_cast at hs ⊢
exact Set.smul_univ₀ hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Torsion | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 32
} | {
"line": 29,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : NoZeroDivisors M\nih : ∀ (x : M), x ≠ 0 → ∀ (y : M), y ≠ 0 → ∀ (n : ℕ), n ≠ 0 → x ^ n = y ^ n → x = y\n⊢ IsMulTorsionFree M",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"IsMulTorsionFree.mk",
"Ne",
"instOfNa... | [
"M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : NoZeroDivisors M\nih : ∀ (x : M), x ≠ 0 → ∀ (y : M), y ≠ 0 → ∀ (n : ℕ), n ≠ 0 → x ^ n = y ^ n → x = y\nn : ℕ\nhn : n ≠ 0\nx y : M\nhxy : (fun a ↦ a ^ n) x = (fun a ↦ a ^ n) y\n⊢ x = y"
] | refine ⟨fun n hn x y hxy ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 83,
"column": 21
} | {
"line": 83,
"column": 25
} | {
"line": 83,
"column": 26
} | [
{
"pp": "case h\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : MonoidWithZero B\nf : A →*₀ B\na✝ b✝ : Bˣ\ny : A\nhy : f y = ↑a✝\ny' : A\nhy' : f y' = ↑b✝\n⊢ ↑a✝ * f y' = ↑(a✝ * b✝)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.... | [
"case h\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : MonoidWithZero B\nf : A →*₀ B\na✝ b✝ : Bˣ\ny : A\nhy : f y = ↑a✝\ny' : A\nhy' : f y' = ↑b✝\n⊢ ↑a✝ * ↑b✝ = ↑(a✝ * b✝)"
] | hy', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 246,
"column": 4
} | {
"line": 258,
"column": 11
} | {
"line": 260,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : CommGroupWithZero B\nf : A →*₀ B\ny : Bˣ\nx✝ : ∃ a, f a ≠ 0 ∧ ∃ x, f a * ↑y = f x\na : A\nha : f a ≠ 0\nx : A\nhy : f a * ↑y = f x\n⊢ y ∈ f.valueGroup",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | have hv : f x ≠ 0 := by
simp only [← hy, ne_eq, mul_eq_zero, ha, Units.ne_zero, or_self, not_false_eq_true]
let v := (Ne.isUnit hv).unit
have hv₀ : f x = ↑v := IsUnit.unit_spec (Ne.isUnit hv)
let u := (Ne.isUnit ha).unit
have ha₀ : f a = ↑u := IsUnit.unit_spec (Ne.isUnit ha)
rw_mod_cast [hv₀, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 246,
"column": 4
} | {
"line": 258,
"column": 11
} | {
"line": 260,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : CommGroupWithZero B\nf : A →*₀ B\ny : Bˣ\nx✝ : ∃ a, f a ≠ 0 ∧ ∃ x, f a * ↑y = f x\na : A\nha : f a ≠ 0\nx : A\nhy : f a * ↑y = f x\n⊢ y ∈ f.valueGroup",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | have hv : f x ≠ 0 := by
simp only [← hy, ne_eq, mul_eq_zero, ha, Units.ne_zero, or_self, not_false_eq_true]
let v := (Ne.isUnit hv).unit
have hv₀ : f x = ↑v := IsUnit.unit_spec (Ne.isUnit hv)
let u := (Ne.isUnit ha).unit
have ha₀ : f a = ↑u := IsUnit.unit_spec (Ne.isUnit ha)
rw_mod_cast [hv₀, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 16
} | {
"line": 113,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : SimplicialObject C\nn : ℕ\nh : n = n\n⊢ X.eqToIso h = Iso.refl (X _⦋n⦌)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Opposite",
"congrArg",
"CategoryTheory.Functor.mapIso_refl",
"CategoryTheory.Iso.refl",
... | [] | simp [eqToIso] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 16
} | {
"line": 113,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : SimplicialObject C\nn : ℕ\nh : n = n\n⊢ X.eqToIso h = Iso.refl (X _⦋n⦌)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Opposite",
"congrArg",
"CategoryTheory.Functor.mapIso_refl",
"CategoryTheory.Iso.refl",
... | [] | simp [eqToIso] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 16
} | {
"line": 113,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : SimplicialObject C\nn : ℕ\nh : n = n\n⊢ X.eqToIso h = Iso.refl (X _⦋n⦌)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Opposite",
"congrArg",
"CategoryTheory.Functor.mapIso_refl",
"CategoryTheory.Iso.refl",
... | [] | simp [eqToIso] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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