file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
sequence | commit
stringclasses 4
values | url
stringclasses 4
values | start
sequence |
|---|---|---|---|---|---|---|
Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.add_mul_left_right | [
{
"state_after": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime (y + x * z) x",
"state_before": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (y + x * z)",
"tactic": "rw [isCoprime_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime (y + x * z) x",
"tactic": "exact h.symm.add_mul_left_left z"
}
] | [
295,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
293,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.pos_iff_ne_zero | [
{
"state_after": "no goals",
"state_before": "n m : ℕ\ninst✝ : NeZero n\na : Fin n\n⊢ 0 < a ↔ a ≠ 0",
"tactic": "rw [← val_fin_lt, val_zero, _root_.pos_iff_ne_zero, Ne.def, Ne.def, ext_iff, val_zero]"
}
] | [
424,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean | CategoryTheory.Presieve.FamilyOfElements.compPresheafMap_id | [] | [
355,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
] |
Mathlib/CategoryTheory/Functor/Category.lean | CategoryTheory.NatTrans.mono_of_mono_app | [
{
"state_after": "case w.h\nC : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G H I : C ⥤ D\nα : F ⟶ G\ninst✝ : ∀ (X : C), Mono (α.app X)\nZ✝ : C ⥤ D\ng h : Z✝ ⟶ F\neq : g ≫ α = h ≫ α\nX : C\n⊢ g.app X = h.app X",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G H I : C ⥤ D\nα : F ⟶ G\ninst✝ : ∀ (X : C), Mono (α.app X)\nZ✝ : C ⥤ D\ng h : Z✝ ⟶ F\neq : g ≫ α = h ≫ α\n⊢ g = h",
"tactic": "ext X"
},
{
"state_after": "no goals",
"state_before": "case w.h\nC : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G H I : C ⥤ D\nα : F ⟶ G\ninst✝ : ∀ (X : C), Mono (α.app X)\nZ✝ : C ⥤ D\ng h : Z✝ ⟶ F\neq : g ≫ α = h ≫ α\nX : C\n⊢ g.app X = h.app X",
"tactic": "rw [← cancel_mono (α.app X), ← comp_app, eq, comp_app]"
}
] | [
97,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Order/SupIndep.lean | CompleteLattice.setIndependent_iff_pairwiseDisjoint | [] | [
400,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
397,
1
] |
Mathlib/Data/Polynomial/Eval.lean | Polynomial.eval₂_at_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\n⊢ eval₂ f 0 p = ↑f (coeff p 0)",
"tactic": "simp (config := { contextual := true }) only [eval₂_eq_sum, zero_pow_eq, mul_ite, mul_zero,\n mul_one, sum, Classical.not_not, mem_support_iff, sum_ite_eq', ite_eq_left_iff,\n RingHom.map_zero, imp_true_iff, eq_self_iff_true]"
}
] | [
65,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
62,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean | CategoryTheory.Monad.algebra_equiv_of_iso_monads_comp_forget | [] | [
315,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
313,
1
] |
Mathlib/Topology/Sheaves/Forget.lean | TopCat.Presheaf.isSheaf_iff_isSheaf_comp | [
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\n⊢ IsSheafEqualizerProducts F ↔ IsSheafEqualizerProducts (F ⋙ G)",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\n⊢ IsSheaf F ↔ IsSheaf (F ⋙ G)",
"tactic": "rw [Presheaf.isSheaf_iff_isSheafEqualizerProducts,\n Presheaf.isSheaf_iff_isSheafEqualizerProducts]"
},
{
"state_after": "case mp\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\n⊢ IsSheafEqualizerProducts F → IsSheafEqualizerProducts (F ⋙ G)\n\ncase mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\n⊢ IsSheafEqualizerProducts (F ⋙ G) → IsSheafEqualizerProducts F",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\n⊢ IsSheafEqualizerProducts F ↔ IsSheafEqualizerProducts (F ⋙ G)",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"state_before": "case mp\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\n⊢ IsSheafEqualizerProducts F → IsSheafEqualizerProducts (F ⋙ G)",
"tactic": "intro S ι U"
},
{
"state_after": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"state_before": "case mp\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"tactic": "obtain ⟨t₁⟩ := S U"
},
{
"state_after": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"state_before": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"tactic": "letI := preservesSmallestLimitsOfPreservesLimits G"
},
{
"state_after": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nt₂ : IsLimit (G.mapCone (fork F U))\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"state_before": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"tactic": "have t₂ := @PreservesLimit.preserves _ _ _ _ _ _ _ G _ _ t₁"
},
{
"state_after": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nt₂ : IsLimit (G.mapCone (fork F U))\nt₃ : IsLimit ((Cones.postcompose (diagramCompPreservesLimits G F U).inv).obj (fork (F ⋙ G) U))\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"state_before": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nt₂ : IsLimit (G.mapCone (fork F U))\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"tactic": "have t₃ := IsLimit.ofIsoLimit t₂ (mapConeFork G F U)"
},
{
"state_after": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nt₂ : IsLimit (G.mapCone (fork F U))\nt₃ : IsLimit ((Cones.postcompose (diagramCompPreservesLimits G F U).inv).obj (fork (F ⋙ G) U))\nt₄ : (fun x => IsLimit (fork (F ⋙ G) U)) t₃\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"state_before": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nt₂ : IsLimit (G.mapCone (fork F U))\nt₃ : IsLimit ((Cones.postcompose (diagramCompPreservesLimits G F U).inv).obj (fork (F ⋙ G) U))\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"tactic": "have t₄ := IsLimit.postcomposeInvEquiv _ _ t₃"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts F\nι : Type v\nU : ι → Opens ↑X\nt₁ : IsLimit (fork F U)\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nt₂ : IsLimit (G.mapCone (fork F U))\nt₃ : IsLimit ((Cones.postcompose (diagramCompPreservesLimits G F U).inv).obj (fork (F ⋙ G) U))\nt₄ : (fun x => IsLimit (fork (F ⋙ G) U)) t₃\n⊢ Nonempty (IsLimit (fork (F ⋙ G) U))",
"tactic": "exact ⟨t₄⟩"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\n⊢ Nonempty (IsLimit (fork F U))",
"state_before": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\n⊢ IsSheafEqualizerProducts (F ⋙ G) → IsSheafEqualizerProducts F",
"tactic": "intro S ι U"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\n⊢ IsLimit (fork F U)",
"state_before": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\n⊢ Nonempty (IsLimit (fork F U))",
"tactic": "refine' ⟨_⟩"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\n⊢ IsLimit (fork F U)",
"state_before": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\n⊢ IsLimit (fork F U)",
"tactic": "let f := equalizer.lift _ (w F U)"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis : IsIso (G.map f)\n⊢ IsLimit (fork F U)",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis : IsIso (G.map f)\n⊢ IsLimit (fork F U)",
"tactic": "skip"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ IsLimit (fork F U)",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis : IsIso (G.map f)\n⊢ IsLimit (fork F U)",
"tactic": "haveI : IsIso f := isIso_of_reflects_iso f G"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ limit.cone (parallelPair (leftRes F U) (rightRes F U)) ≅ fork F U",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ IsLimit (fork F U)",
"tactic": "apply IsLimit.ofIsoLimit (limit.isLimit _)"
},
{
"state_after": "case I\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ fork F U ≅ limit.cone (parallelPair (leftRes F U) (rightRes F U))",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ limit.cone (parallelPair (leftRes F U) (rightRes F U)) ≅ fork F U",
"tactic": "apply Iso.symm"
},
{
"state_after": "case I.φ\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ (fork F U).pt ≅ (limit.cone (parallelPair (leftRes F U) (rightRes F U))).pt\n\ncase I.w\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ autoParam\n (∀ (j : WalkingParallelPair),\n (fork F U).π.app j = ?I.φ.hom ≫ (limit.cone (parallelPair (leftRes F U) (rightRes F U))).π.app j)\n _auto✝",
"state_before": "case I\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ fork F U ≅ limit.cone (parallelPair (leftRes F U) (rightRes F U))",
"tactic": "fapply Cones.ext"
},
{
"state_after": "case I.w\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ autoParam\n (∀ (j : WalkingParallelPair),\n (fork F U).π.app j = (asIso f).hom ≫ (limit.cone (parallelPair (leftRes F U) (rightRes F U))).π.app j)\n _auto✝",
"state_before": "case I.φ\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ (fork F U).pt ≅ (limit.cone (parallelPair (leftRes F U) (rightRes F U))).pt\n\ncase I.w\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ autoParam\n (∀ (j : WalkingParallelPair),\n (fork F U).π.app j = ?I.φ.hom ≫ (limit.cone (parallelPair (leftRes F U) (rightRes F U))).π.app j)\n _auto✝",
"tactic": "exact asIso f"
},
{
"state_after": "no goals",
"state_before": "case I.w.one\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nthis✝ : IsIso (G.map f)\nthis : IsIso f\n⊢ (fork F U).π.app WalkingParallelPair.one =\n (asIso f).hom ≫ (limit.cone (parallelPair (leftRes F U) (rightRes F U))).π.app WalkingParallelPair.one",
"tactic": "simp"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\n⊢ IsIso (G.map f)",
"state_before": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\n⊢ IsIso (G.map f)",
"tactic": "let c := fork (F ⋙ G) U"
},
{
"state_after": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\n⊢ IsIso (G.map f)",
"state_before": "case mpr\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\n⊢ IsIso (G.map f)",
"tactic": "obtain ⟨hc⟩ := S U"
},
{
"state_after": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\n⊢ IsIso (G.map f)",
"state_before": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\n⊢ IsIso (G.map f)",
"tactic": "let d := G.mapCone (equalizer.fork (leftRes.{v} F U) (rightRes F U))"
},
{
"state_after": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\n⊢ IsIso (G.map f)",
"state_before": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\n⊢ IsIso (G.map f)",
"tactic": "letI := preservesSmallestLimitsOfPreservesLimits G"
},
{
"state_after": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\n⊢ IsIso (G.map f)",
"state_before": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\n⊢ IsIso (G.map f)",
"tactic": "have hd : IsLimit d := PreservesLimit.preserves (limit.isLimit _)"
},
{
"state_after": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\n⊢ IsIso (G.map f)",
"state_before": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\n⊢ IsIso (G.map f)",
"tactic": "let d' := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d"
},
{
"state_after": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\n⊢ IsIso (G.map f)",
"state_before": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\n⊢ IsIso (G.map f)",
"tactic": "have hd' : IsLimit d' :=\n (IsLimit.postcomposeHomEquiv (diagramCompPreservesLimits G F U : _) d).symm hd"
},
{
"state_after": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nf' : c ⟶ d' := Fork.mkHom (G.map f) (_ : G.map f ≫ Fork.ι d' = Fork.ι c)\n⊢ IsIso (G.map f)",
"state_before": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\n⊢ IsIso (G.map f)",
"tactic": "let f' : c ⟶ d' := Fork.mkHom (G.map f) (by\n dsimp only [diagramCompPreservesLimits, res]\n dsimp only [Fork.ι]\n refine limit.hom_ext fun j => ?_\n dsimp\n simp only [Category.assoc, ← Functor.map_comp_assoc, equalizer.lift_ι,\n map_lift_piComparison_assoc]\n dsimp [res])"
},
{
"state_after": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis✝ : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nf' : c ⟶ d' := Fork.mkHom (G.map f) (_ : G.map f ≫ Fork.ι d' = Fork.ι c)\nthis : IsIso f'\n⊢ IsIso (G.map f)",
"state_before": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nf' : c ⟶ d' := Fork.mkHom (G.map f) (_ : G.map f ≫ Fork.ι d' = Fork.ι c)\n⊢ IsIso (G.map f)",
"tactic": "haveI : IsIso f' := IsLimit.hom_isIso hc hd' f'"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis✝ : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nf' : c ⟶ d' := Fork.mkHom (G.map f) (_ : G.map f ≫ Fork.ι d' = Fork.ι c)\nthis : IsIso f'\n⊢ IsIso (G.map f)",
"tactic": "exact IsIso.of_iso ((Cones.forget _).mapIso (asIso f'))"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\n⊢ G.map\n (equalizer.lift (Pi.lift fun i => F.map (Opens.leSupr U i).op)\n (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)) ≫\n Fork.ι\n ((Cones.postcompose\n (NatIso.ofComponents fun X_1 =>\n WalkingParallelPair.rec (PreservesProduct.iso G fun i => F.obj (U i).op)\n (PreservesProduct.iso G fun p => F.obj (U p.fst ⊓ U p.snd).op) X_1).hom).obj\n (G.mapCone (equalizer.fork (leftRes F U) (rightRes F U)))) =\n Fork.ι (fork (F ⋙ G) U)",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\n⊢ G.map f ≫ Fork.ι d' = Fork.ι c",
"tactic": "dsimp only [diagramCompPreservesLimits, res]"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\n⊢ G.map\n (equalizer.lift (Pi.lift fun i => F.map (Opens.leSupr U i).op)\n (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)) ≫\n ((Cones.postcompose\n (NatIso.ofComponents fun X_1 =>\n WalkingParallelPair.rec (PreservesProduct.iso G fun i => F.obj (U i).op)\n (PreservesProduct.iso G fun p => F.obj (U p.fst ⊓ U p.snd).op) X_1).hom).obj\n (G.mapCone (equalizer.fork (leftRes F U) (rightRes F U)))).π.app\n WalkingParallelPair.zero =\n (fork (F ⋙ G) U).π.app WalkingParallelPair.zero",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\n⊢ G.map\n (equalizer.lift (Pi.lift fun i => F.map (Opens.leSupr U i).op)\n (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)) ≫\n Fork.ι\n ((Cones.postcompose\n (NatIso.ofComponents fun X_1 =>\n WalkingParallelPair.rec (PreservesProduct.iso G fun i => F.obj (U i).op)\n (PreservesProduct.iso G fun p => F.obj (U p.fst ⊓ U p.snd).op) X_1).hom).obj\n (G.mapCone (equalizer.fork (leftRes F U) (rightRes F U)))) =\n Fork.ι (fork (F ⋙ G) U)",
"tactic": "dsimp only [Fork.ι]"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nj : Discrete ι\n⊢ (G.map\n (equalizer.lift (Pi.lift fun i => F.map (Opens.leSupr U i).op)\n (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)) ≫\n ((Cones.postcompose\n (NatIso.ofComponents fun X_1 =>\n WalkingParallelPair.rec (PreservesProduct.iso G fun i => F.obj (U i).op)\n (PreservesProduct.iso G fun p => F.obj (U p.fst ⊓ U p.snd).op) X_1).hom).obj\n (G.mapCone (equalizer.fork (leftRes F U) (rightRes F U)))).π.app\n WalkingParallelPair.zero) ≫\n limit.π (Discrete.functor fun i => (F ⋙ G).obj (U i).op) j =\n (fork (F ⋙ G) U).π.app WalkingParallelPair.zero ≫ limit.π (Discrete.functor fun i => (F ⋙ G).obj (U i).op) j",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\n⊢ G.map\n (equalizer.lift (Pi.lift fun i => F.map (Opens.leSupr U i).op)\n (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)) ≫\n ((Cones.postcompose\n (NatIso.ofComponents fun X_1 =>\n WalkingParallelPair.rec (PreservesProduct.iso G fun i => F.obj (U i).op)\n (PreservesProduct.iso G fun p => F.obj (U p.fst ⊓ U p.snd).op) X_1).hom).obj\n (G.mapCone (equalizer.fork (leftRes F U) (rightRes F U)))).π.app\n WalkingParallelPair.zero =\n (fork (F ⋙ G) U).π.app WalkingParallelPair.zero",
"tactic": "refine limit.hom_ext fun j => ?_"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nj : Discrete ι\n⊢ (G.map\n (equalizer.lift (Pi.lift fun i => F.map (Opens.leSupr U i).op)\n (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)) ≫\n G.map (equalizer.ι (leftRes F U) (rightRes F U)) ≫ piComparison G fun i => F.obj (U i).op) ≫\n limit.π (Discrete.functor fun i => G.obj (F.obj (U i).op)) j =\n res (F ⋙ G) U ≫ limit.π (Discrete.functor fun i => G.obj (F.obj (U i).op)) j",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nj : Discrete ι\n⊢ (G.map\n (equalizer.lift (Pi.lift fun i => F.map (Opens.leSupr U i).op)\n (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)) ≫\n ((Cones.postcompose\n (NatIso.ofComponents fun X_1 =>\n WalkingParallelPair.rec (PreservesProduct.iso G fun i => F.obj (U i).op)\n (PreservesProduct.iso G fun p => F.obj (U p.fst ⊓ U p.snd).op) X_1).hom).obj\n (G.mapCone (equalizer.fork (leftRes F U) (rightRes F U)))).π.app\n WalkingParallelPair.zero) ≫\n limit.π (Discrete.functor fun i => (F ⋙ G).obj (U i).op) j =\n (fork (F ⋙ G) U).π.app WalkingParallelPair.zero ≫ limit.π (Discrete.functor fun i => (F ⋙ G).obj (U i).op) j",
"tactic": "dsimp"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nj : Discrete ι\n⊢ (Pi.lift fun j => G.map (F.map (Opens.leSupr U j).op)) ≫\n limit.π (Discrete.functor fun i => G.obj (F.obj (U i).op)) j =\n res (F ⋙ G) U ≫ limit.π (Discrete.functor fun i => G.obj (F.obj (U i).op)) j",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nj : Discrete ι\n⊢ (G.map\n (equalizer.lift (Pi.lift fun i => F.map (Opens.leSupr U i).op)\n (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)) ≫\n G.map (equalizer.ι (leftRes F U) (rightRes F U)) ≫ piComparison G fun i => F.obj (U i).op) ≫\n limit.π (Discrete.functor fun i => G.obj (F.obj (U i).op)) j =\n res (F ⋙ G) U ≫ limit.π (Discrete.functor fun i => G.obj (F.obj (U i).op)) j",
"tactic": "simp only [Category.assoc, ← Functor.map_comp_assoc, equalizer.lift_ι,\n map_lift_piComparison_assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\ninst✝³ : ReflectsIsomorphisms G\ninst✝² : HasLimits C\ninst✝¹ : HasLimits D\ninst✝ : PreservesLimits G\nX : TopCat\nF : Presheaf C X\nS : IsSheafEqualizerProducts (F ⋙ G)\nι : Type v\nU : ι → Opens ↑X\nf : F.obj (iSup U).op ⟶ equalizer (leftRes F U) (rightRes F U) :=\n equalizer.lift (res F U) (_ : res F U ≫ leftRes F U = res F U ≫ rightRes F U)\nc : Fork (leftRes (F ⋙ G) U) (rightRes (F ⋙ G) U) := fork (F ⋙ G) U\nhc : IsLimit (fork (F ⋙ G) U)\nd : Cone (parallelPair (leftRes F U) (rightRes F U) ⋙ G) := G.mapCone (equalizer.fork (leftRes F U) (rightRes F U))\nthis : PreservesLimitsOfSize G := preservesSmallestLimitsOfPreservesLimits G\nhd : IsLimit d\nd' : Cone (diagram (F ⋙ G) U) := (Cones.postcompose (diagramCompPreservesLimits G F U).hom).obj d\nhd' : IsLimit d'\nj : Discrete ι\n⊢ (Pi.lift fun j => G.map (F.map (Opens.leSupr U j).op)) ≫\n limit.π (Discrete.functor fun i => G.obj (F.obj (U i).op)) j =\n res (F ⋙ G) U ≫ limit.π (Discrete.functor fun i => G.obj (F.obj (U i).op)) j",
"tactic": "dsimp [res]"
}
] | [
224,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Topology/Order/Basic.lean | exists_Ioc_subset_of_mem_nhds' | [] | [
1232,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1228,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.two_zsmul_coe_div_two | [
{
"state_after": "no goals",
"state_before": "θ : ℝ\n⊢ 2 • ↑(θ / 2) = ↑θ",
"tactic": "rw [← coe_zsmul, two_zsmul, add_halves]"
}
] | [
147,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Measure.lean | BoxIntegral.Box.coe_ae_eq_Icc | [
{
"state_after": "ι : Type u_1\nI : Box ι\ninst✝ : Fintype ι\n⊢ (pi univ fun i => Ioc (lower I i) (upper I i)) =ᵐ[volume] ↑Box.Icc I",
"state_before": "ι : Type u_1\nI : Box ι\ninst✝ : Fintype ι\n⊢ ↑I =ᵐ[volume] ↑Box.Icc I",
"tactic": "rw [coe_eq_pi]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI : Box ι\ninst✝ : Fintype ι\n⊢ (pi univ fun i => Ioc (lower I i) (upper I i)) =ᵐ[volume] ↑Box.Icc I",
"tactic": "exact Measure.univ_pi_Ioc_ae_eq_Icc"
}
] | [
79,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Order/RelIso/Group.lean | RelIso.inv_apply_self | [] | [
48,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
] |
Mathlib/Algebra/Module/LinearMap.lean | LinearMap.map_smulₛₗ | [] | [
338,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
337,
11
] |
Std/Data/Nat/Lemmas.lean | Nat.min_le_left | [] | [
179,
97
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
179,
11
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.sSup_unit_ball_eq_norm | [
{
"state_after": "no goals",
"state_before": "𝕜✝ : Type ?u.819707\n𝕜₂✝ : Type ?u.819710\n𝕜₃ : Type ?u.819713\nE✝ : Type ?u.819716\nEₗ : Type ?u.819719\nF✝ : Type ?u.819722\nFₗ : Type ?u.819725\nG : Type ?u.819728\nGₗ : Type ?u.819731\n𝓕 : Type ?u.819734\ninst✝²³ : SeminormedAddCommGroup E✝\ninst✝²² : SeminormedAddCommGroup Eₗ\ninst✝²¹ : SeminormedAddCommGroup F✝\ninst✝²⁰ : SeminormedAddCommGroup Fₗ\ninst✝¹⁹ : SeminormedAddCommGroup G\ninst✝¹⁸ : SeminormedAddCommGroup Gₗ\ninst✝¹⁷ : NontriviallyNormedField 𝕜✝\ninst✝¹⁶ : NontriviallyNormedField 𝕜₂✝\ninst✝¹⁵ : NontriviallyNormedField 𝕜₃\ninst✝¹⁴ : NormedSpace 𝕜✝ E✝\ninst✝¹³ : NormedSpace 𝕜✝ Eₗ\ninst✝¹² : NormedSpace 𝕜₂✝ F✝\ninst✝¹¹ : NormedSpace 𝕜✝ Fₗ\ninst✝¹⁰ : NormedSpace 𝕜₃ G\ninst✝⁹ : NormedSpace 𝕜✝ Gₗ\nσ₁₂✝ : 𝕜✝ →+* 𝕜₂✝\nσ₂₃ : 𝕜₂✝ →+* 𝕜₃\nσ₁₃ : 𝕜✝ →+* 𝕜₃\ninst✝⁸ : RingHomCompTriple σ₁₂✝ σ₂₃ σ₁₃\ninst✝⁷ : RingHomIsometric σ₁₂✝\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : DenselyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\ninst✝ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\n⊢ sSup ((fun x => ‖↑f x‖) '' ball 0 1) = ‖f‖",
"tactic": "simpa only [NNReal.coe_sSup, Set.image_image] using NNReal.coe_eq.2 f.sSup_unit_ball_eq_nnnorm"
}
] | [
562,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
558,
1
] |
Mathlib/RingTheory/Polynomial/Vieta.lean | Multiset.prod_X_add_C_coeff' | [
{
"state_after": "case h\nR : Type u_2\ninst✝ : CommSemiring R\nσ : Type u_1\ns : Multiset σ\nr : σ → R\nk : ℕ\nh : k ≤ ↑card s\n⊢ k ≤ ↑card s",
"state_before": "R : Type u_2\ninst✝ : CommSemiring R\nσ : Type u_1\ns : Multiset σ\nr : σ → R\nk : ℕ\nh : k ≤ ↑card s\n⊢ coeff (prod (map (fun i => X + ↑C (r i)) s)) k = esymm (map r s) (↑card s - k)",
"tactic": "erw [← map_map (fun r => X + C r) r, prod_X_add_C_coeff] <;> rw [s.card_map r]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_2\ninst✝ : CommSemiring R\nσ : Type u_1\ns : Multiset σ\nr : σ → R\nk : ℕ\nh : k ≤ ↑card s\n⊢ k ≤ ↑card s",
"tactic": "assumption"
}
] | [
80,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean | LinearMap.map_sub₂ | [] | [
167,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
166,
1
] |
Mathlib/Data/List/Zip.lean | List.zipWith_distrib_take | [
{
"state_after": "case nil\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl'✝ : List β\nn✝ : ℕ\nl' : List β\nn : ℕ\n⊢ take n (zipWith f [] l') = zipWith f (take n []) (take n l')\n\ncase cons\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl'✝ : List β\nn✝ : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nl' : List β\nn : ℕ\n⊢ take n (zipWith f (hd :: tl) l') = zipWith f (take n (hd :: tl)) (take n l')",
"state_before": "α : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn : ℕ\n⊢ take n (zipWith f l l') = zipWith f (take n l) (take n l')",
"tactic": "induction' l with hd tl hl generalizing l' n"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl'✝ : List β\nn✝ : ℕ\nl' : List β\nn : ℕ\n⊢ take n (zipWith f [] l') = zipWith f (take n []) (take n l')",
"tactic": "simp"
},
{
"state_after": "case cons.nil\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn✝ : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nn : ℕ\n⊢ take n (zipWith f (hd :: tl) []) = zipWith f (take n (hd :: tl)) (take n [])\n\ncase cons.cons\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn✝ : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nn : ℕ\nhead✝ : β\ntail✝ : List β\n⊢ take n (zipWith f (hd :: tl) (head✝ :: tail✝)) = zipWith f (take n (hd :: tl)) (take n (head✝ :: tail✝))",
"state_before": "case cons\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl'✝ : List β\nn✝ : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nl' : List β\nn : ℕ\n⊢ take n (zipWith f (hd :: tl) l') = zipWith f (take n (hd :: tl)) (take n l')",
"tactic": "cases l'"
},
{
"state_after": "no goals",
"state_before": "case cons.nil\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn✝ : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nn : ℕ\n⊢ take n (zipWith f (hd :: tl) []) = zipWith f (take n (hd :: tl)) (take n [])",
"tactic": "simp"
},
{
"state_after": "case cons.cons.zero\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nhead✝ : β\ntail✝ : List β\n⊢ take zero (zipWith f (hd :: tl) (head✝ :: tail✝)) = zipWith f (take zero (hd :: tl)) (take zero (head✝ :: tail✝))\n\ncase cons.cons.succ\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nhead✝ : β\ntail✝ : List β\nn✝ : ℕ\n⊢ take (succ n✝) (zipWith f (hd :: tl) (head✝ :: tail✝)) =\n zipWith f (take (succ n✝) (hd :: tl)) (take (succ n✝) (head✝ :: tail✝))",
"state_before": "case cons.cons\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn✝ : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nn : ℕ\nhead✝ : β\ntail✝ : List β\n⊢ take n (zipWith f (hd :: tl) (head✝ :: tail✝)) = zipWith f (take n (hd :: tl)) (take n (head✝ :: tail✝))",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.zero\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nhead✝ : β\ntail✝ : List β\n⊢ take zero (zipWith f (hd :: tl) (head✝ :: tail✝)) = zipWith f (take zero (hd :: tl)) (take zero (head✝ :: tail✝))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.succ\nα : Type u\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.191507\nε : Type ?u.191510\nf : α → β → γ\nl : List α\nl' : List β\nn : ℕ\nhd : α\ntl : List α\nhl : ∀ (l' : List β) (n : ℕ), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nhead✝ : β\ntail✝ : List β\nn✝ : ℕ\n⊢ take (succ n✝) (zipWith f (hd :: tl) (head✝ :: tail✝)) =\n zipWith f (take (succ n✝) (hd :: tl)) (take (succ n✝) (head✝ :: tail✝))",
"tactic": "simp [hl]"
}
] | [
481,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
474,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | contDiffAt_ring_inverse | [
{
"state_after": "case h0\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\n⊢ ContDiffAt 𝕜 0 Ring.inverse ↑x\n\ncase hsuc\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ContDiffAt 𝕜 (↑(Nat.succ n)) Ring.inverse ↑x\n\ncase htop\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nItop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ContDiffAt 𝕜 ⊤ Ring.inverse ↑x",
"state_before": "𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\n⊢ ContDiffAt 𝕜 n Ring.inverse ↑x",
"tactic": "induction' n using ENat.nat_induction with n IH Itop"
},
{
"state_after": "case h0\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ∃ u, u ∈ 𝓝[insert (↑x) univ] ↑x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) Ring.inverse p u",
"state_before": "case h0\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\n⊢ ContDiffAt 𝕜 0 Ring.inverse ↑x",
"tactic": "intro m hm"
},
{
"state_after": "case h0.refine'_1\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ {y | IsUnit y} ∈ 𝓝[insert (↑x) univ] ↑x\n\ncase h0.refine'_2\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ∃ p, HasFTaylorSeriesUpToOn (↑m) Ring.inverse p {y | IsUnit y}",
"state_before": "case h0\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ∃ u, u ∈ 𝓝[insert (↑x) univ] ↑x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) Ring.inverse p u",
"tactic": "refine' ⟨{ y : R | IsUnit y }, _, _⟩"
},
{
"state_after": "case h0.refine'_1\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ {y | IsUnit y} ∈ 𝓝 ↑x",
"state_before": "case h0.refine'_1\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ {y | IsUnit y} ∈ 𝓝[insert (↑x) univ] ↑x",
"tactic": "simp [nhdsWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "case h0.refine'_1\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ {y | IsUnit y} ∈ 𝓝 ↑x",
"tactic": "exact x.nhds"
},
{
"state_after": "case h0.refine'_2\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ HasFTaylorSeriesUpToOn (↑m) Ring.inverse (ftaylorSeriesWithin 𝕜 Ring.inverse univ) {y | IsUnit y}",
"state_before": "case h0.refine'_2\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ∃ p, HasFTaylorSeriesUpToOn (↑m) Ring.inverse p {y | IsUnit y}",
"tactic": "use ftaylorSeriesWithin 𝕜 inverse univ"
},
{
"state_after": "case h0.refine'_2\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ContinuousOn Ring.inverse {y | IsUnit y} ∧\n ∀ (x : R),\n x ∈ {y | IsUnit y} →\n ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin 𝕜 Ring.inverse univ x 0) = Ring.inverse x",
"state_before": "case h0.refine'_2\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ HasFTaylorSeriesUpToOn (↑m) Ring.inverse (ftaylorSeriesWithin 𝕜 Ring.inverse univ) {y | IsUnit y}",
"tactic": "rw [le_antisymm hm bot_le, hasFTaylorSeriesUpToOn_zero_iff]"
},
{
"state_after": "case h0.refine'_2.left\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ContinuousOn Ring.inverse {y | IsUnit y}\n\ncase h0.refine'_2.right\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ∀ (x : R),\n x ∈ {y | IsUnit y} →\n ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin 𝕜 Ring.inverse univ x 0) = Ring.inverse x",
"state_before": "case h0.refine'_2\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ContinuousOn Ring.inverse {y | IsUnit y} ∧\n ∀ (x : R),\n x ∈ {y | IsUnit y} →\n ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin 𝕜 Ring.inverse univ x 0) = Ring.inverse x",
"tactic": "constructor"
},
{
"state_after": "case h0.refine'_2.left.intro\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\nx' : Rˣ\n⊢ ContinuousWithinAt Ring.inverse {y | IsUnit y} ↑x'",
"state_before": "case h0.refine'_2.left\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ContinuousOn Ring.inverse {y | IsUnit y}",
"tactic": "rintro _ ⟨x', rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h0.refine'_2.left.intro\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\nx' : Rˣ\n⊢ ContinuousWithinAt Ring.inverse {y | IsUnit y} ↑x'",
"tactic": "exact (inverse_continuousAt x').continuousWithinAt"
},
{
"state_after": "no goals",
"state_before": "case h0.refine'_2.right\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ∀ (x : R),\n x ∈ {y | IsUnit y} →\n ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin 𝕜 Ring.inverse univ x 0) = Ring.inverse x",
"tactic": "simp [ftaylorSeriesWithin]"
},
{
"state_after": "case hsuc\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ∃ f', (∃ u, u ∈ 𝓝 ↑x ∧ ∀ (x : R), x ∈ u → HasFDerivAt Ring.inverse (f' x) x) ∧ ContDiffAt 𝕜 (↑n) f' ↑x",
"state_before": "case hsuc\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ContDiffAt 𝕜 (↑(Nat.succ n)) Ring.inverse ↑x",
"tactic": "apply contDiffAt_succ_iff_hasFDerivAt.mpr"
},
{
"state_after": "case hsuc.refine'_1\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ∃ u,\n u ∈ 𝓝 ↑x ∧\n ∀ (x : R),\n x ∈ u → HasFDerivAt Ring.inverse ((fun x => -↑(↑(mulLeftRight 𝕜 R) (Ring.inverse x)) (Ring.inverse x)) x) x\n\ncase hsuc.refine'_2\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ContDiffAt 𝕜 (↑n) (fun x => -↑(↑(mulLeftRight 𝕜 R) (Ring.inverse x)) (Ring.inverse x)) ↑x",
"state_before": "case hsuc\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ∃ f', (∃ u, u ∈ 𝓝 ↑x ∧ ∀ (x : R), x ∈ u → HasFDerivAt Ring.inverse (f' x) x) ∧ ContDiffAt 𝕜 (↑n) f' ↑x",
"tactic": "refine' ⟨fun x : R => -mulLeftRight 𝕜 R (inverse x) (inverse x), _, _⟩"
},
{
"state_after": "case hsuc.refine'_1\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ∀ (x : R),\n x ∈ {y | IsUnit y} →\n HasFDerivAt Ring.inverse ((fun x => -↑(↑(mulLeftRight 𝕜 R) (Ring.inverse x)) (Ring.inverse x)) x) x",
"state_before": "case hsuc.refine'_1\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ∃ u,\n u ∈ 𝓝 ↑x ∧\n ∀ (x : R),\n x ∈ u → HasFDerivAt Ring.inverse ((fun x => -↑(↑(mulLeftRight 𝕜 R) (Ring.inverse x)) (Ring.inverse x)) x) x",
"tactic": "refine' ⟨{ y : R | IsUnit y }, x.nhds, _⟩"
},
{
"state_after": "case hsuc.refine'_1.intro\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\ny : Rˣ\n⊢ HasFDerivAt Ring.inverse ((fun x => -↑(↑(mulLeftRight 𝕜 R) (Ring.inverse x)) (Ring.inverse x)) ↑y) ↑y",
"state_before": "case hsuc.refine'_1\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ∀ (x : R),\n x ∈ {y | IsUnit y} →\n HasFDerivAt Ring.inverse ((fun x => -↑(↑(mulLeftRight 𝕜 R) (Ring.inverse x)) (Ring.inverse x)) x) x",
"tactic": "rintro _ ⟨y, rfl⟩"
},
{
"state_after": "case hsuc.refine'_1.intro\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\ny : Rˣ\n⊢ HasFDerivAt Ring.inverse (-↑(↑(mulLeftRight 𝕜 R) ↑y⁻¹) ↑y⁻¹) ↑y",
"state_before": "case hsuc.refine'_1.intro\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\ny : Rˣ\n⊢ HasFDerivAt Ring.inverse ((fun x => -↑(↑(mulLeftRight 𝕜 R) (Ring.inverse x)) (Ring.inverse x)) ↑y) ↑y",
"tactic": "simp_rw [inverse_unit]"
},
{
"state_after": "no goals",
"state_before": "case hsuc.refine'_1.intro\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\ny : Rˣ\n⊢ HasFDerivAt Ring.inverse (-↑(↑(mulLeftRight 𝕜 R) ↑y⁻¹) ↑y⁻¹) ↑y",
"tactic": "exact hasFDerivAt_ring_inverse y"
},
{
"state_after": "no goals",
"state_before": "case hsuc.refine'_2\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ContDiffAt 𝕜 (↑n) (fun x => -↑(↑(mulLeftRight 𝕜 R) (Ring.inverse x)) (Ring.inverse x)) ↑x",
"tactic": "convert (mulLeftRight_isBoundedBilinear 𝕜 R).contDiff.neg.comp_contDiffAt (x : R)\n (IH.prod IH)"
},
{
"state_after": "no goals",
"state_before": "case htop\n𝕜 : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹² : NormedAddCommGroup D\ninst✝¹¹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nX : Type ?u.2234009\ninst✝⁴ : NormedAddCommGroup X\ninst✝³ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nItop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) Ring.inverse ↑x\n⊢ ContDiffAt 𝕜 ⊤ Ring.inverse ↑x",
"tactic": "exact contDiffAt_top.mpr Itop"
}
] | [
1682,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1661,
1
] |
Std/Data/List/Lemmas.lean | List.find?_cons_of_neg | [
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\np : α✝ → Bool\na : α✝\nl : List α✝\nh : ¬p a = true\n⊢ find? p (a :: l) = find? p l",
"tactic": "simp [find?, h]"
}
] | [
1257,
21
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1256,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.minpoly_powerBasis_gen_of_monic | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\ng : R[X]\ninst✝ : Field K\nf : K[X]\nhf : Monic f\nhf' : optParam (f ≠ 0) (_ : f ≠ 0)\n⊢ minpoly K (powerBasis hf').gen = f",
"tactic": "rw [minpoly_powerBasis_gen hf', hf.leadingCoeff, inv_one, C.map_one, mul_one]"
}
] | [
604,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
602,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.inv | [
{
"state_after": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Group G\ninst✝ : AddGroup A\ni : ℤ\nx : G\nhx : IsOfFinOrder x\nn : ℕ\nnpos : 0 < n\nhn : x ^ n = 1\n⊢ ∃ n, 0 < n ∧ x⁻¹ ^ n = 1",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝¹ : Group G\ninst✝ : AddGroup A\ni : ℤ\nx : G\nhx : IsOfFinOrder x\n⊢ ∃ n, 0 < n ∧ x⁻¹ ^ n = 1",
"tactic": "rcases(isOfFinOrder_iff_pow_eq_one x).mp hx with ⟨n, npos, hn⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Group G\ninst✝ : AddGroup A\ni : ℤ\nx : G\nhx : IsOfFinOrder x\nn : ℕ\nnpos : 0 < n\nhn : x ^ n = 1\n⊢ ∃ n, 0 < n ∧ x⁻¹ ^ n = 1",
"tactic": "refine' ⟨n, npos, by simp_rw [inv_pow, hn, inv_one]⟩"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Group G\ninst✝ : AddGroup A\ni : ℤ\nx : G\nhx : IsOfFinOrder x\nn : ℕ\nnpos : 0 < n\nhn : x ^ n = 1\n⊢ x⁻¹ ^ n = 1",
"tactic": "simp_rw [inv_pow, hn, inv_one]"
}
] | [
544,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | DoubleCentralizer.zero_toProd | [] | [
223,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean | SimpleGraph.isEmpty_of_chromaticNumber_eq_zero | [
{
"state_after": "V : Type u\nG✝ : SimpleGraph V\nα : Type v\nC : Coloring G✝ α\nG : SimpleGraph V\ninst✝ : Finite V\nh : chromaticNumber G = 0\nh' : Colorable G (chromaticNumber G)\n⊢ IsEmpty V",
"state_before": "V : Type u\nG✝ : SimpleGraph V\nα : Type v\nC : Coloring G✝ α\nG : SimpleGraph V\ninst✝ : Finite V\nh : chromaticNumber G = 0\n⊢ IsEmpty V",
"tactic": "have h' := G.colorable_chromaticNumber_of_fintype"
},
{
"state_after": "V : Type u\nG✝ : SimpleGraph V\nα : Type v\nC : Coloring G✝ α\nG : SimpleGraph V\ninst✝ : Finite V\nh : chromaticNumber G = 0\nh' : Colorable G 0\n⊢ IsEmpty V",
"state_before": "V : Type u\nG✝ : SimpleGraph V\nα : Type v\nC : Coloring G✝ α\nG : SimpleGraph V\ninst✝ : Finite V\nh : chromaticNumber G = 0\nh' : Colorable G (chromaticNumber G)\n⊢ IsEmpty V",
"tactic": "rw [h] at h'"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nG✝ : SimpleGraph V\nα : Type v\nC : Coloring G✝ α\nG : SimpleGraph V\ninst✝ : Finite V\nh : chromaticNumber G = 0\nh' : Colorable G 0\n⊢ IsEmpty V",
"tactic": "exact G.isEmpty_of_colorable_zero h'"
}
] | [
301,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
297,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean | Bool.of_decide_false | [] | [
152,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/Analysis/Convex/Between.lean | affineSegment_const_vadd_image | [] | [
95,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/RingTheory/Algebraic.lean | exists_integral_multiple | [
{
"state_after": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"state_before": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\nhz : IsAlgebraic R z\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"tactic": "rcases hz with ⟨p, p_ne_zero, px⟩"
},
{
"state_after": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\na : R := leadingCoeff p\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"state_before": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"tactic": "set a := p.leadingCoeff"
},
{
"state_after": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\na : R := leadingCoeff p\na_ne_zero : a ≠ 0\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"state_before": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\na : R := leadingCoeff p\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"tactic": "have a_ne_zero : a ≠ 0 := mt Polynomial.leadingCoeff_eq_zero.mp p_ne_zero"
},
{
"state_after": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\na : R := leadingCoeff p\na_ne_zero : a ≠ 0\nx_integral : IsIntegral R (z * ↑(algebraMap R S) a)\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"state_before": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\na : R := leadingCoeff p\na_ne_zero : a ≠ 0\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"tactic": "have x_integral : IsIntegral R (z * algebraMap R S a) :=\n ⟨p.integralNormalization, monic_integralNormalization p_ne_zero,\n integralNormalization_aeval_eq_zero px inj⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nz : S\ninj : ∀ (x : R), ↑(algebraMap R S) x = 0 → x = 0\np : R[X]\np_ne_zero : p ≠ 0\npx : ↑(aeval z) p = 0\na : R := leadingCoeff p\na_ne_zero : a ≠ 0\nx_integral : IsIntegral R (z * ↑(algebraMap R S) a)\n⊢ ∃ x y x_1, z * ↑(algebraMap R S) y = ↑x",
"tactic": "exact ⟨⟨_, x_integral⟩, a, a_ne_zero, rfl⟩"
}
] | [
318,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
] |
Mathlib/Data/Polynomial/Derivative.lean | Polynomial.derivative_X_add_C_pow | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommSemiring R\nc : R\nm : ℕ\n⊢ ↑derivative ((X + ↑C c) ^ m) = ↑C ↑m * (X + ↑C c) ^ (m - 1)",
"tactic": "rw [derivative_pow, derivative_X_add_C, mul_one]"
}
] | [
526,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
524,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.two_zsmul_toReal_eq_two_mul | [
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ toReal (2 • θ) = 2 * toReal θ ↔ toReal θ ∈ Set.Ioc (-π / 2) (π / 2)",
"tactic": "rw [two_zsmul, ← two_nsmul, two_nsmul_toReal_eq_two_mul]"
}
] | [
678,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
676,
1
] |
Mathlib/Data/Set/Function.lean | Monotone.rangeFactorization | [] | [
1337,
4
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1336,
11
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.mem_comap | [] | [
1369,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1368,
1
] |
Std/Data/String/Lemmas.lean | Substring.ValidFor.nextn_stop | [
{
"state_after": "no goals",
"state_before": "l m r : List Char\nx✝ : Substring\nh : ValidFor l m r x✝\nn : Nat\n⊢ nextn x✝ (n + 1) { byteIdx := utf8Len m } = { byteIdx := utf8Len m }",
"tactic": "simp [Substring.nextn, h.next_stop, h.nextn_stop n]"
}
] | [
854,
72
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
852,
1
] |
Mathlib/GroupTheory/DoubleCoset.lean | Doset.bot_rel_eq_leftRel | [
{
"state_after": "case h.h.a\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ Setoid.Rel (setoid ↑⊥ ↑H) a b ↔ Setoid.Rel (QuotientGroup.leftRel H) a b",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\n⊢ Setoid.Rel (setoid ↑⊥ ↑H) = Setoid.Rel (QuotientGroup.leftRel H)",
"tactic": "ext (a b)"
},
{
"state_after": "case h.h.a\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ (∃ a_1, a_1 ∈ ⊥ ∧ ∃ b_1, b_1 ∈ H ∧ b = a_1 * a * b_1) ↔ a⁻¹ * b ∈ H",
"state_before": "case h.h.a\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ Setoid.Rel (setoid ↑⊥ ↑H) a b ↔ Setoid.Rel (QuotientGroup.leftRel H) a b",
"tactic": "rw [rel_iff, Setoid.Rel, QuotientGroup.leftRel_apply]"
},
{
"state_after": "case h.h.a.mp\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ (∃ a_1, a_1 ∈ ⊥ ∧ ∃ b_1, b_1 ∈ H ∧ b = a_1 * a * b_1) → a⁻¹ * b ∈ H\n\ncase h.h.a.mpr\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ a⁻¹ * b ∈ H → ∃ a_2, a_2 ∈ ⊥ ∧ ∃ b_1, b_1 ∈ H ∧ b = a_2 * a * b_1",
"state_before": "case h.h.a\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ (∃ a_1, a_1 ∈ ⊥ ∧ ∃ b_1, b_1 ∈ H ∧ b = a_1 * a * b_1) ↔ a⁻¹ * b ∈ H",
"tactic": "constructor"
},
{
"state_after": "case h.h.a.mp.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\nhb : b ∈ H\n⊢ a⁻¹ * (1 * a * b) ∈ H",
"state_before": "case h.h.a.mp\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ (∃ a_1, a_1 ∈ ⊥ ∧ ∃ b_1, b_1 ∈ H ∧ b = a_1 * a * b_1) → a⁻¹ * b ∈ H",
"tactic": "rintro ⟨a, rfl : a = 1, b, hb, rfl⟩"
},
{
"state_after": "case h.h.a.mp.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\nhb : b ∈ H\n⊢ a⁻¹ * (1 * a * b) ∈ H",
"state_before": "case h.h.a.mp.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\nhb : b ∈ H\n⊢ a⁻¹ * (1 * a * b) ∈ H",
"tactic": "change a⁻¹ * (1 * a * b) ∈ H"
},
{
"state_after": "no goals",
"state_before": "case h.h.a.mp.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\nhb : b ∈ H\n⊢ a⁻¹ * (1 * a * b) ∈ H",
"tactic": "rwa [one_mul, inv_mul_cancel_left]"
},
{
"state_after": "case h.h.a.mpr\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\nh : a⁻¹ * b ∈ H\n⊢ ∃ a_1, a_1 ∈ ⊥ ∧ ∃ b_1, b_1 ∈ H ∧ b = a_1 * a * b_1",
"state_before": "case h.h.a.mpr\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\n⊢ a⁻¹ * b ∈ H → ∃ a_2, a_2 ∈ ⊥ ∧ ∃ b_1, b_1 ∈ H ∧ b = a_2 * a * b_1",
"tactic": "rintro (h : a⁻¹ * b ∈ H)"
},
{
"state_after": "no goals",
"state_before": "case h.h.a.mpr\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\nh : a⁻¹ * b ∈ H\n⊢ ∃ a_1, a_1 ∈ ⊥ ∧ ∃ b_1, b_1 ∈ H ∧ b = a_1 * a * b_1",
"tactic": "exact ⟨1, rfl, a⁻¹ * b, h, by rw [one_mul, mul_inv_cancel_left]⟩"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.15522\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na b : G\nh : a⁻¹ * b ∈ H\n⊢ b = 1 * a * (a⁻¹ * b)",
"tactic": "rw [one_mul, mul_inv_cancel_left]"
}
] | [
103,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Analysis/Convex/Hull.lean | Set.Nonempty.convexHull | [] | [
122,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
11
] |
Mathlib/Combinatorics/Additive/Energy.lean | Finset.multiplicativeEnergy_univ_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : CommGroup α\ninst✝ : Fintype α\ns t : Finset α\n⊢ multiplicativeEnergy s univ = Fintype.card α * card s ^ 2",
"tactic": "rw [multiplicativeEnergy_comm, multiplicativeEnergy_univ_left]"
}
] | [
169,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean | CategoryTheory.CosimplicialObject.σ_naturality | [] | [
589,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
587,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearEquiv.coe_toHomeomorph | [] | [
1834,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1833,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.div_toGerm | [] | [
768,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
767,
1
] |
Mathlib/Order/UpperLower/Basic.lean | LowerSet.disjoint_coe | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.78350\nγ : Type ?u.78353\nι : Sort ?u.78356\nκ : ι → Sort ?u.78361\ninst✝ : LE α\nS : Set (LowerSet α)\ns t : LowerSet α\na : α\n⊢ Disjoint ↑s ↑t ↔ Disjoint s t",
"tactic": "simp [disjoint_iff, SetLike.ext'_iff]"
}
] | [
775,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
774,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | WithTop.isGLB_sInf' | [
{
"state_after": "case left\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ sInf s ∈ lowerBounds s\n\ncase right\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ sInf s ∈ upperBounds (lowerBounds s)",
"state_before": "α : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ IsGLB s (sInf s)",
"tactic": "constructor"
},
{
"state_after": "case left\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ (if s ⊆ {⊤} then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ lowerBounds s",
"state_before": "case left\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ sInf s ∈ lowerBounds s",
"tactic": "show ite _ _ _ ∈ _"
},
{
"state_after": "case left.inl\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : s ⊆ {⊤}\n⊢ ⊤ ∈ lowerBounds s\n\ncase left.inr\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\n⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ∈ lowerBounds s",
"state_before": "case left\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ (if s ⊆ {⊤} then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ lowerBounds s",
"tactic": "split_ifs with h"
},
{
"state_after": "case left.inl\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : s ⊆ {⊤}\na : WithTop β\nha : a ∈ s\n⊢ ⊤ ≤ a",
"state_before": "case left.inl\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : s ⊆ {⊤}\n⊢ ⊤ ∈ lowerBounds s",
"tactic": "intro a ha"
},
{
"state_after": "no goals",
"state_before": "case left.inl\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : s ⊆ {⊤}\na : WithTop β\nha : a ∈ s\n⊢ ⊤ ≤ a",
"tactic": "exact top_le_iff.2 (Set.mem_singleton_iff.1 (h ha))"
},
{
"state_after": "case left.inr.none\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ s\n⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ≤ none\n\ncase left.inr.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\n⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ≤ Option.some a",
"state_before": "case left.inr\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\n⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ∈ lowerBounds s",
"tactic": "rintro (⟨⟩ | a) ha"
},
{
"state_after": "case left.inr.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\n⊢ BddBelow ((fun a => ↑a) ⁻¹' s)",
"state_before": "case left.inr.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\n⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ≤ Option.some a",
"tactic": "refine' some_le_some.2 (csInf_le _ ha)"
},
{
"state_after": "case left.inr.some.intro.none\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\n⊢ BddBelow ((fun a => ↑a) ⁻¹' s)\n\ncase left.inr.some.intro.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nb : β\nhb : Option.some b ∈ lowerBounds s\n⊢ BddBelow ((fun a => ↑a) ⁻¹' s)",
"state_before": "case left.inr.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\n⊢ BddBelow ((fun a => ↑a) ⁻¹' s)",
"tactic": "rcases hs with ⟨⟨⟩ | b, hb⟩"
},
{
"state_after": "case left.inr.some.intro.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nb : β\nhb : Option.some b ∈ lowerBounds s\n⊢ b ∈ lowerBounds ((fun a => ↑a) ⁻¹' s)",
"state_before": "case left.inr.some.intro.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nb : β\nhb : Option.some b ∈ lowerBounds s\n⊢ BddBelow ((fun a => ↑a) ⁻¹' s)",
"tactic": "use b"
},
{
"state_after": "case left.inr.some.intro.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nb : β\nhb : Option.some b ∈ lowerBounds s\nc : β\nhc : c ∈ (fun a => ↑a) ⁻¹' s\n⊢ b ≤ c",
"state_before": "case left.inr.some.intro.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nb : β\nhb : Option.some b ∈ lowerBounds s\n⊢ b ∈ lowerBounds ((fun a => ↑a) ⁻¹' s)",
"tactic": "intro c hc"
},
{
"state_after": "no goals",
"state_before": "case left.inr.some.intro.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nb : β\nhb : Option.some b ∈ lowerBounds s\nc : β\nhc : c ∈ (fun a => ↑a) ⁻¹' s\n⊢ b ≤ c",
"tactic": "exact some_le_some.1 (hb hc)"
},
{
"state_after": "no goals",
"state_before": "case left.inr.none\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ s\n⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ≤ none",
"tactic": "exact le_top"
},
{
"state_after": "case left.inr.some.intro.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\n⊢ False",
"state_before": "case left.inr.some.intro.none\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\n⊢ BddBelow ((fun a => ↑a) ⁻¹' s)",
"tactic": "exfalso"
},
{
"state_after": "case left.inr.some.intro.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\n⊢ s ⊆ {⊤}",
"state_before": "case left.inr.some.intro.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\n⊢ False",
"tactic": "apply h"
},
{
"state_after": "case left.inr.some.intro.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\nc : WithTop β\nhc : c ∈ s\n⊢ c ∈ {⊤}",
"state_before": "case left.inr.some.intro.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\n⊢ s ⊆ {⊤}",
"tactic": "intro c hc"
},
{
"state_after": "case left.inr.some.intro.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\nc : WithTop β\nhc : c ∈ s\n⊢ ⊤ ≤ c",
"state_before": "case left.inr.some.intro.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\nc : WithTop β\nhc : c ∈ s\n⊢ c ∈ {⊤}",
"tactic": "rw [mem_singleton_iff, ← top_le_iff]"
},
{
"state_after": "no goals",
"state_before": "case left.inr.some.intro.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ s\nhb : none ∈ lowerBounds s\nc : WithTop β\nhc : c ∈ s\n⊢ ⊤ ≤ c",
"tactic": "exact hb hc"
},
{
"state_after": "case right\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ (if s ⊆ {⊤} then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ upperBounds (lowerBounds s)",
"state_before": "case right\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ sInf s ∈ upperBounds (lowerBounds s)",
"tactic": "show ite _ _ _ ∈ _"
},
{
"state_after": "case right.inl\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : s ⊆ {⊤}\n⊢ ⊤ ∈ upperBounds (lowerBounds s)\n\ncase right.inr\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\n⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ∈ upperBounds (lowerBounds s)",
"state_before": "case right\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\n⊢ (if s ⊆ {⊤} then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ upperBounds (lowerBounds s)",
"tactic": "split_ifs with h"
},
{
"state_after": "case right.inl\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : s ⊆ {⊤}\na✝¹ : WithTop β\na✝ : a✝¹ ∈ lowerBounds s\n⊢ a✝¹ ≤ ⊤",
"state_before": "case right.inl\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : s ⊆ {⊤}\n⊢ ⊤ ∈ upperBounds (lowerBounds s)",
"tactic": "intro _ _"
},
{
"state_after": "no goals",
"state_before": "case right.inl\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : s ⊆ {⊤}\na✝¹ : WithTop β\na✝ : a✝¹ ∈ lowerBounds s\n⊢ a✝¹ ≤ ⊤",
"tactic": "exact le_top"
},
{
"state_after": "case right.inr.none\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ lowerBounds s\n⊢ none ≤ ↑(sInf ((fun a => ↑a) ⁻¹' s))\n\ncase right.inr.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\n⊢ Option.some a ≤ ↑(sInf ((fun a => ↑a) ⁻¹' s))",
"state_before": "case right.inr\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\n⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ∈ upperBounds (lowerBounds s)",
"tactic": "rintro (⟨⟩ | a) ha"
},
{
"state_after": "case right.inr.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ lowerBounds s\n⊢ False",
"state_before": "case right.inr.none\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ lowerBounds s\n⊢ none ≤ ↑(sInf ((fun a => ↑a) ⁻¹' s))",
"tactic": "exfalso"
},
{
"state_after": "case right.inr.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ lowerBounds s\n⊢ s ⊆ {⊤}",
"state_before": "case right.inr.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ lowerBounds s\n⊢ False",
"tactic": "apply h"
},
{
"state_after": "case right.inr.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ lowerBounds s\nb : WithTop β\nhb : b ∈ s\n⊢ b ∈ {⊤}",
"state_before": "case right.inr.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ lowerBounds s\n⊢ s ⊆ {⊤}",
"tactic": "intro b hb"
},
{
"state_after": "no goals",
"state_before": "case right.inr.none.h\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\nha : none ∈ lowerBounds s\nb : WithTop β\nhb : b ∈ s\n⊢ b ∈ {⊤}",
"tactic": "exact Set.mem_singleton_iff.2 (top_le_iff.1 (ha hb))"
},
{
"state_after": "case right.inr.some.refine'_1\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\n⊢ Set.Nonempty ((fun a => ↑a) ⁻¹' s)\n\ncase right.inr.some.refine'_2\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\n⊢ ∀ (b : β), b ∈ (fun a => ↑a) ⁻¹' s → a ≤ b",
"state_before": "case right.inr.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\n⊢ Option.some a ≤ ↑(sInf ((fun a => ↑a) ⁻¹' s))",
"tactic": "refine' some_le_some.2 (le_csInf _ _)"
},
{
"state_after": "case right.inr.some.refine'_1\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\na : β\nha : Option.some a ∈ lowerBounds s\nh : ¬Set.Nonempty ((fun a => ↑a) ⁻¹' s)\n⊢ s ⊆ {⊤}",
"state_before": "case right.inr.some.refine'_1\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\n⊢ Set.Nonempty ((fun a => ↑a) ⁻¹' s)",
"tactic": "contrapose! h"
},
{
"state_after": "case right.inr.some.refine'_1.none\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\na : β\nha✝ : Option.some a ∈ lowerBounds s\nh : ¬Set.Nonempty ((fun a => ↑a) ⁻¹' s)\nha : none ∈ s\n⊢ none ∈ {⊤}\n\ncase right.inr.some.refine'_1.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\na✝ : β\nha✝ : Option.some a✝ ∈ lowerBounds s\nh : ¬Set.Nonempty ((fun a => ↑a) ⁻¹' s)\na : β\nha : Option.some a ∈ s\n⊢ Option.some a ∈ {⊤}",
"state_before": "case right.inr.some.refine'_1\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\na : β\nha : Option.some a ∈ lowerBounds s\nh : ¬Set.Nonempty ((fun a => ↑a) ⁻¹' s)\n⊢ s ⊆ {⊤}",
"tactic": "rintro (⟨⟩ | a) ha"
},
{
"state_after": "no goals",
"state_before": "case right.inr.some.refine'_1.none\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\na : β\nha✝ : Option.some a ∈ lowerBounds s\nh : ¬Set.Nonempty ((fun a => ↑a) ⁻¹' s)\nha : none ∈ s\n⊢ none ∈ {⊤}",
"tactic": "exact mem_singleton ⊤"
},
{
"state_after": "no goals",
"state_before": "case right.inr.some.refine'_1.some\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\na✝ : β\nha✝ : Option.some a✝ ∈ lowerBounds s\nh : ¬Set.Nonempty ((fun a => ↑a) ⁻¹' s)\na : β\nha : Option.some a ∈ s\n⊢ Option.some a ∈ {⊤}",
"tactic": "exact (h ⟨a, ha⟩).elim"
},
{
"state_after": "case right.inr.some.refine'_2\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\nb : β\nhb : b ∈ (fun a => ↑a) ⁻¹' s\n⊢ a ≤ b",
"state_before": "case right.inr.some.refine'_2\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\n⊢ ∀ (b : β), b ∈ (fun a => ↑a) ⁻¹' s → a ≤ b",
"tactic": "intro b hb"
},
{
"state_after": "case right.inr.some.refine'_2\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\nb : β\nhb : b ∈ (fun a => ↑a) ⁻¹' s\n⊢ Option.some a ≤ Option.some b",
"state_before": "case right.inr.some.refine'_2\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\nb : β\nhb : b ∈ (fun a => ↑a) ⁻¹' s\n⊢ a ≤ b",
"tactic": "rw [← some_le_some]"
},
{
"state_after": "no goals",
"state_before": "case right.inr.some.refine'_2\nα : Type ?u.90006\nβ✝ : Type ?u.90009\nγ : Type ?u.90012\nι : Sort ?u.90015\ninst✝¹ : ConditionallyCompleteLinearOrderBot α\nβ : Type u_1\ninst✝ : ConditionallyCompleteLattice β\ns : Set (WithTop β)\nhs : BddBelow s\nh : ¬s ⊆ {⊤}\na : β\nha : Option.some a ∈ lowerBounds s\nb : β\nhb : b ∈ (fun a => ↑a) ⁻¹' s\n⊢ Option.some a ≤ Option.some b",
"tactic": "exact ha hb"
}
] | [
1201,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1164,
1
] |
Mathlib/LinearAlgebra/Matrix/Basis.lean | Basis.toMatrix_eq_toMatrix_constr | [
{
"state_after": "case a.h\nι : Type u_1\nι' : Type ?u.55138\nκ : Type ?u.55141\nκ' : Type ?u.55144\nR : Type u_2\nM : Type u_3\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nR₂ : Type ?u.55337\nM₂ : Type ?u.55340\ninst✝⁴ : CommRing R₂\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R₂ M₂\ne : Basis ι R M\nv✝ : ι' → M\ni : ι\nj : ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nv : ι → M\ni✝ x✝ : ι\n⊢ toMatrix e v i✝ x✝ = ↑(LinearMap.toMatrix e e) (↑(constr e ℕ) v) i✝ x✝",
"state_before": "ι : Type u_1\nι' : Type ?u.55138\nκ : Type ?u.55141\nκ' : Type ?u.55144\nR : Type u_2\nM : Type u_3\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nR₂ : Type ?u.55337\nM₂ : Type ?u.55340\ninst✝⁴ : CommRing R₂\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R₂ M₂\ne : Basis ι R M\nv✝ : ι' → M\ni : ι\nj : ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nv : ι → M\n⊢ toMatrix e v = ↑(LinearMap.toMatrix e e) (↑(constr e ℕ) v)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nι : Type u_1\nι' : Type ?u.55138\nκ : Type ?u.55141\nκ' : Type ?u.55144\nR : Type u_2\nM : Type u_3\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nR₂ : Type ?u.55337\nM₂ : Type ?u.55340\ninst✝⁴ : CommRing R₂\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R₂ M₂\ne : Basis ι R M\nv✝ : ι' → M\ni : ι\nj : ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nv : ι → M\ni✝ x✝ : ι\n⊢ toMatrix e v i✝ x✝ = ↑(LinearMap.toMatrix e e) (↑(constr e ℕ) v) i✝ x✝",
"tactic": "rw [Basis.toMatrix_apply, LinearMap.toMatrix_apply, Basis.constr_basis]"
}
] | [
76,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Order/Filter/Pointwise.lean | Filter.inv_le_iff_le_inv | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.32470\nα : Type u_1\nβ : Type ?u.32476\nγ : Type ?u.32479\nδ : Type ?u.32482\nε : Type ?u.32485\ninst✝ : InvolutiveInv α\nf g : Filter α\ns : Set α\n⊢ f⁻¹ ≤ g ↔ f ≤ g⁻¹",
"tactic": "rw [← Filter.inv_le_inv_iff, inv_inv]"
}
] | [
262,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
262,
1
] |
Mathlib/Order/Antichain.lean | IsGreatest.antichain_iff | [] | [
249,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Data/Nat/PartENat.lean | PartENat.coe_le_iff | [
{
"state_after": "n : ℕ\nx : PartENat\n⊢ ↑n ≤ x ↔ ∀ (h : x.Dom), n ≤ Part.get x h",
"state_before": "n : ℕ\nx : PartENat\n⊢ ↑n ≤ x ↔ ∀ (h : x.Dom), n ≤ Part.get x h",
"tactic": "rw [← some_eq_natCast]"
},
{
"state_after": "n : ℕ\nx : PartENat\n⊢ (∀ (hy : x.Dom), Part.get ↑n (_ : (↑n).Dom) ≤ Part.get x hy) ↔ ∀ (h : x.Dom), n ≤ Part.get x h",
"state_before": "n : ℕ\nx : PartENat\n⊢ ↑n ≤ x ↔ ∀ (h : x.Dom), n ≤ Part.get x h",
"tactic": "simp only [le_def, exists_prop_of_true, dom_some, forall_true_iff]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nx : PartENat\n⊢ (∀ (hy : x.Dom), Part.get ↑n (_ : (↑n).Dom) ≤ Part.get x hy) ↔ ∀ (h : x.Dom), n ≤ Part.get x h",
"tactic": "rfl"
}
] | [
303,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.mul_self_le_self | [
{
"state_after": "case h.e'_4\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nhI : I ≤ 1\n⊢ I = (fun x x_1 => x * x_1) I 1",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nhI : I ≤ 1\n⊢ I * I ≤ I",
"tactic": "convert mul_left_mono I hI"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nhI : I ≤ 1\n⊢ I = (fun x x_1 => x * x_1) I 1",
"tactic": "exact (mul_one I).symm"
}
] | [
650,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
648,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean | Set.einfsep_pair | [
{
"state_after": "α : Type u_1\nβ : Type ?u.39983\ninst✝ : PseudoEMetricSpace α\nx y z : α\ns t : Set α\nhxy : x ≠ y\n⊢ einfsep {x, y} = min (edist x y) (edist x y)",
"state_before": "α : Type u_1\nβ : Type ?u.39983\ninst✝ : PseudoEMetricSpace α\nx y z : α\ns t : Set α\nhxy : x ≠ y\n⊢ einfsep {x, y} = edist x y",
"tactic": "nth_rw 1 [← min_self (edist x y)]"
},
{
"state_after": "case h.e'_3.h.e'_2\nα : Type u_1\nβ : Type ?u.39983\ninst✝ : PseudoEMetricSpace α\nx y z : α\ns t : Set α\nhxy : x ≠ y\n⊢ edist x y = edist y x",
"state_before": "α : Type u_1\nβ : Type ?u.39983\ninst✝ : PseudoEMetricSpace α\nx y z : α\ns t : Set α\nhxy : x ≠ y\n⊢ einfsep {x, y} = min (edist x y) (edist x y)",
"tactic": "convert einfsep_pair_eq_inf hxy using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_2\nα : Type u_1\nβ : Type ?u.39983\ninst✝ : PseudoEMetricSpace α\nx y z : α\ns t : Set α\nhxy : x ≠ y\n⊢ edist x y = edist y x",
"tactic": "rw [edist_comm]"
}
] | [
220,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | Polynomial.cyclotomic_dvd_geom_sum_of_dvd | [
{
"state_after": "R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\n⊢ cyclotomic d ℤ ∣ ∑ i in range n, X ^ i",
"state_before": "R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\n⊢ cyclotomic d R ∣ ∑ i in range n, X ^ i",
"tactic": "suffices cyclotomic d ℤ ∣ ∑ i in Finset.range n, X ^ i by\n simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_X] using\n map_dvd (Int.castRingHom R) this"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝ : Ring R\nd : ℕ\nhd : d ≠ 1\nhdn : d ∣ 0\n⊢ cyclotomic d ℤ ∣ ∑ i in range 0, X ^ i\n\ncase inr\nR : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nhn : n > 0\n⊢ cyclotomic d ℤ ∣ ∑ i in range n, X ^ i",
"state_before": "R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\n⊢ cyclotomic d ℤ ∣ ∑ i in range n, X ^ i",
"tactic": "rcases n.eq_zero_or_pos with (rfl | hn)"
},
{
"state_after": "case inr\nR : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nhn : n > 0\n⊢ cyclotomic d ℤ ∣ ∏ i in Finset.erase (Nat.divisors n) 1, cyclotomic i ℤ",
"state_before": "case inr\nR : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nhn : n > 0\n⊢ cyclotomic d ℤ ∣ ∑ i in range n, X ^ i",
"tactic": "rw [← prod_cyclotomic_eq_geom_sum hn]"
},
{
"state_after": "case inr.ha\nR : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nhn : n > 0\n⊢ d ∈ Finset.erase (Nat.divisors n) 1",
"state_before": "case inr\nR : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nhn : n > 0\n⊢ cyclotomic d ℤ ∣ ∏ i in Finset.erase (Nat.divisors n) 1, cyclotomic i ℤ",
"tactic": "apply Finset.dvd_prod_of_mem"
},
{
"state_after": "no goals",
"state_before": "case inr.ha\nR : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nhn : n > 0\n⊢ d ∈ Finset.erase (Nat.divisors n) 1",
"tactic": "simp [hd, hdn, hn.ne']"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nthis : cyclotomic d ℤ ∣ ∑ i in range n, X ^ i\n⊢ cyclotomic d R ∣ ∑ i in range n, X ^ i",
"tactic": "simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_X] using\n map_dvd (Int.castRingHom R) this"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝ : Ring R\nd : ℕ\nhd : d ≠ 1\nhdn : d ∣ 0\n⊢ cyclotomic d ℤ ∣ ∑ i in range 0, X ^ i",
"tactic": "simp"
}
] | [
431,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
422,
1
] |
Mathlib/Topology/CompactOpen.lean | Homeomorph.continuousMapOfUnique_symm_apply | [] | [
468,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
466,
1
] |
Mathlib/Algebra/Star/SelfAdjoint.lean | selfAdjoint.val_smul | [] | [
437,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
436,
1
] |
Mathlib/Topology/Constructions.lean | IsOpenMap.restrict | [] | [
1033,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1031,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.mem_one | [] | [
86,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
] |
Mathlib/RingTheory/JacobsonIdeal.lean | Ideal.isUnit_of_sub_one_mem_jacobson_bot | [
{
"state_after": "case intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nr : R\nh : r - 1 ∈ jacobson ⊥\ns : R\nhs : s * r - 1 ∈ ⊥\n⊢ IsUnit r",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nr : R\nh : r - 1 ∈ jacobson ⊥\n⊢ IsUnit r",
"tactic": "cases' exists_mul_sub_mem_of_sub_one_mem_jacobson r h with s hs"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nr : R\nh : r - 1 ∈ jacobson ⊥\ns : R\nhs : r * s = 1\n⊢ IsUnit r",
"state_before": "case intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nr : R\nh : r - 1 ∈ jacobson ⊥\ns : R\nhs : s * r - 1 ∈ ⊥\n⊢ IsUnit r",
"tactic": "rw [mem_bot, sub_eq_zero, mul_comm] at hs"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nr : R\nh : r - 1 ∈ jacobson ⊥\ns : R\nhs : r * s = 1\n⊢ IsUnit r",
"tactic": "exact isUnit_of_mul_eq_one _ _ hs"
}
] | [
258,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
] |
Mathlib/FieldTheory/IntermediateField.lean | AlgHom.mem_fieldRange | [] | [
485,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
484,
1
] |
Mathlib/ModelTheory/FinitelyGenerated.lean | FirstOrder.Language.Substructure.FG.cg | [
{
"state_after": "case intro.intro\nL : Language\nM : Type u_3\ninst✝ : Structure L M\ns : Set M\nhf : Set.Finite s\nh : FG (LowerAdjoint.toFun (closure L) s)\n⊢ CG (LowerAdjoint.toFun (closure L) s)",
"state_before": "L : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\nh : FG N\n⊢ CG N",
"tactic": "obtain ⟨s, hf, rfl⟩ := fg_def.1 h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nL : Language\nM : Type u_3\ninst✝ : Structure L M\ns : Set M\nhf : Set.Finite s\nh : FG (LowerAdjoint.toFun (closure L) s)\n⊢ CG (LowerAdjoint.toFun (closure L) s)",
"tactic": "refine' ⟨s, hf.countable, rfl⟩"
}
] | [
116,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
114,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.blimsup_eq_iInf_biSup | [
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\n⊢ (range fun s => ⨅ (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), u b) ⊆ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\n\ncase refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\n⊢ a ≤ a'",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\n⊢ blimsup u f p = ⨅ (s : Set β) (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), u b",
"tactic": "refine' le_antisymm (sInf_le_sInf _) (iInf_le_iff.mpr fun a ha => le_sInf_iff.mpr fun a' ha' => _)"
},
{
"state_after": "case refine'_1.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ns : Set β\n⊢ (fun s => ⨅ (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), u b) s ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\n⊢ (range fun s => ⨅ (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), u b) ⊆ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}",
"tactic": "rintro - ⟨s, rfl⟩"
},
{
"state_after": "case refine'_1.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ns : Set β\n⊢ ∀ᶠ (x : β) in f, p x → s ∈ f → u x ≤ ⨆ (b : β) (_ : p b ∧ b ∈ s), u b",
"state_before": "case refine'_1.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ns : Set β\n⊢ (fun s => ⨅ (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), u b) s ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}",
"tactic": "simp only [mem_setOf_eq, le_iInf_iff]"
},
{
"state_after": "case refine'_1.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ns : Set β\n⊢ ∀ᶠ (x : β) in f, s ∈ f → p x → u x ≤ ⨆ (b : β) (_ : p b ∧ b ∈ s), u b",
"state_before": "case refine'_1.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ns : Set β\n⊢ ∀ᶠ (x : β) in f, p x → s ∈ f → u x ≤ ⨆ (b : β) (_ : p b ∧ b ∈ s), u b",
"tactic": "conv =>\n congr\n ext\n rw [Imp.swap]"
},
{
"state_after": "case refine'_1.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ns : Set β\nh : s ∈ f\nx : β\nh₁ : x ∈ s\nh₂ : p x\n⊢ u x ≤ ⨆ (b : β) (_ : p b ∧ b ∈ s), u b",
"state_before": "case refine'_1.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ns : Set β\n⊢ ∀ᶠ (x : β) in f, s ∈ f → p x → u x ≤ ⨆ (b : β) (_ : p b ∧ b ∈ s), u b",
"tactic": "refine'\n eventually_imp_distrib_left.mpr fun h => eventually_iff_exists_mem.2 ⟨s, h, fun x h₁ h₂ => _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ns : Set β\nh : s ∈ f\nx : β\nh₁ : x ∈ s\nh₂ : p x\n⊢ u x ≤ ⨆ (b : β) (_ : p b ∧ b ∈ s), u b",
"tactic": "exact @le_iSup₂ α β (fun b => p b ∧ b ∈ s) _ (fun b _ => u b) x ⟨h₂, h₁⟩"
},
{
"state_after": "case refine'_2.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\ns : Set β\nhs : s ∈ f\nhs' : ∀ (y : β), y ∈ s → p y → u y ≤ a'\n⊢ a ≤ a'",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\n⊢ a ≤ a'",
"tactic": "obtain ⟨s, hs, hs'⟩ := eventually_iff_exists_mem.mp ha'"
},
{
"state_after": "case refine'_2.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\ns : Set β\nhs : s ∈ f\nhs' : ∀ (y : β), y ∈ s → p y → u y ≤ a'\nthis : ∀ (y : β), p y → y ∈ s → u y ≤ a'\n⊢ a ≤ a'",
"state_before": "case refine'_2.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\ns : Set β\nhs : s ∈ f\nhs' : ∀ (y : β), y ∈ s → p y → u y ≤ a'\n⊢ a ≤ a'",
"tactic": "have : ∀ (y : β), p y → y ∈ s → u y ≤ a' := fun y ↦ by rw [Imp.swap]; exact hs' y"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\ns : Set β\nhs : s ∈ f\nhs' : ∀ (y : β), y ∈ s → p y → u y ≤ a'\nthis : ∀ (y : β), p y → y ∈ s → u y ≤ a'\n⊢ a ≤ a'",
"tactic": "exact (le_iInf_iff.mp (ha s) hs).trans (by simpa only [iSup₂_le_iff, and_imp] )"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\ns : Set β\nhs : s ∈ f\nhs' : ∀ (y : β), y ∈ s → p y → u y ≤ a'\ny : β\n⊢ y ∈ s → p y → u y ≤ a'",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\ns : Set β\nhs : s ∈ f\nhs' : ∀ (y : β), y ∈ s → p y → u y ≤ a'\ny : β\n⊢ p y → y ∈ s → u y ≤ a'",
"tactic": "rw [Imp.swap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\ns : Set β\nhs : s ∈ f\nhs' : ∀ (y : β), y ∈ s → p y → u y ≤ a'\ny : β\n⊢ y ∈ s → p y → u y ≤ a'",
"tactic": "exact hs' y"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.111946\nι : Type ?u.111949\ninst✝ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\na : α\nha : ∀ (i : Set β), a ≤ ⨅ (_ : i ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ i), u b\na' : α\nha' : a' ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a}\ns : Set β\nhs : s ∈ f\nhs' : ∀ (y : β), y ∈ s → p y → u y ≤ a'\nthis : ∀ (y : β), p y → y ∈ s → u y ≤ a'\n⊢ (⨆ (b : β) (_ : p b ∧ b ∈ s), u b) ≤ a'",
"tactic": "simpa only [iSup₂_le_iff, and_imp]"
}
] | [
741,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
727,
1
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_eq_zero' | [
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\n⊢ ↑(weightedHomogeneousComponent w n) φ = 0",
"tactic": "classical\nrw [weightedHomogeneousComponent_apply, sum_eq_zero]\nintro d hd; rw [mem_filter] at hd\nexfalso; exact h _ hd.1 hd.2"
},
{
"state_after": "R : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\n⊢ ∀ (x : σ →₀ ℕ), x ∈ Finset.filter (fun d => ↑(weightedDegree' w) d = n) (support φ) → ↑(monomial x) (coeff x φ) = 0",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\n⊢ ↑(weightedHomogeneousComponent w n) φ = 0",
"tactic": "rw [weightedHomogeneousComponent_apply, sum_eq_zero]"
},
{
"state_after": "R : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\nd : σ →₀ ℕ\nhd : d ∈ Finset.filter (fun d => ↑(weightedDegree' w) d = n) (support φ)\n⊢ ↑(monomial d) (coeff d φ) = 0",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\n⊢ ∀ (x : σ →₀ ℕ), x ∈ Finset.filter (fun d => ↑(weightedDegree' w) d = n) (support φ) → ↑(monomial x) (coeff x φ) = 0",
"tactic": "intro d hd"
},
{
"state_after": "R : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\nd : σ →₀ ℕ\nhd : d ∈ support φ ∧ ↑(weightedDegree' w) d = n\n⊢ ↑(monomial d) (coeff d φ) = 0",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\nd : σ →₀ ℕ\nhd : d ∈ Finset.filter (fun d => ↑(weightedDegree' w) d = n) (support φ)\n⊢ ↑(monomial d) (coeff d φ) = 0",
"tactic": "rw [mem_filter] at hd"
},
{
"state_after": "case h\nR : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\nd : σ →₀ ℕ\nhd : d ∈ support φ ∧ ↑(weightedDegree' w) d = n\n⊢ False",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\nd : σ →₀ ℕ\nhd : d ∈ support φ ∧ ↑(weightedDegree' w) d = n\n⊢ ↑(monomial d) (coeff d φ) = 0",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_1\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ↑(weightedDegree' w) d ≠ n\nd : σ →₀ ℕ\nhd : d ∈ support φ ∧ ↑(weightedDegree' w) d = n\n⊢ False",
"tactic": "exact h _ hd.1 hd.2"
}
] | [
373,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
367,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | sq_le_one_iff_abs_le_one | [
{
"state_after": "no goals",
"state_before": "β : Type ?u.268169\nA : Type ?u.268172\nG : Type ?u.268175\nM : Type ?u.268178\nR : Type u_1\ninst✝ : LinearOrderedRing R\nx✝ y x : R\n⊢ x ^ 2 ≤ 1 ↔ abs x ≤ 1",
"tactic": "simpa only [one_pow, abs_one] using @sq_le_sq _ _ x 1"
}
] | [
731,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
730,
1
] |
Mathlib/Topology/Connected.lean | isTotallyDisconnected_of_clopen_set | [
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\n⊢ Set.Subsingleton S",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\n⊢ IsTotallyDisconnected univ",
"tactic": "rintro S - hS"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\n⊢ ∀ ⦃x : X⦄, x ∈ S → ∀ ⦃y : X⦄, y ∈ S → x = y",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\n⊢ Set.Subsingleton S",
"tactic": "unfold Set.Subsingleton"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\nh_contra : Exists fun ⦃x⦄ => x ∈ S ∧ Exists fun ⦃y⦄ => y ∈ S ∧ x ≠ y\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\n⊢ ∀ ⦃x : X⦄, x ∈ S → ∀ ⦃y : X⦄, y ∈ S → x = y",
"tactic": "by_contra' h_contra"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\nhxy : x ≠ y\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\nh_contra : Exists fun ⦃x⦄ => x ∈ S ∧ Exists fun ⦃y⦄ => y ∈ S ∧ x ≠ y\n⊢ False",
"tactic": "rcases h_contra with ⟨x, hx, y, hy, hxy⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\nhxy : x ≠ y\nU : Set X\nh_clopen : IsClopen U\nhxU : x ∈ U\nhyU : ¬y ∈ U\n⊢ False",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\nhxy : x ≠ y\n⊢ False",
"tactic": "obtain ⟨U, h_clopen, hxU, hyU⟩ := hX hxy"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\nhxy : x ≠ y\nU : Set X\nh_clopen : IsClopen U\nhxU : x ∈ U\nhyU : ¬y ∈ U\nhS : Set.Nonempty (S ∩ (U ∩ Uᶜ))\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nhS : IsPreconnected S\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\nhxy : x ≠ y\nU : Set X\nh_clopen : IsClopen U\nhxU : x ∈ U\nhyU : ¬y ∈ U\n⊢ False",
"tactic": "specialize\n hS U (Uᶜ) h_clopen.1 h_clopen.compl.1 (fun a _ => em (a ∈ U)) ⟨x, hx, hxU⟩ ⟨y, hy, hyU⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\nhxy : x ≠ y\nU : Set X\nh_clopen : IsClopen U\nhxU : x ∈ U\nhyU : ¬y ∈ U\nhS : Set.Nonempty ∅\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\nhxy : x ≠ y\nU : Set X\nh_clopen : IsClopen U\nhxU : x ∈ U\nhyU : ¬y ∈ U\nhS : Set.Nonempty (S ∩ (U ∩ Uᶜ))\n⊢ False",
"tactic": "rw [inter_compl_self, Set.inter_empty] at hS"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.120754\nπ : ι → Type ?u.120759\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\nX : Type u_1\ninst✝ : TopologicalSpace X\nhX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ ¬y ∈ U\nS : Set X\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\nhxy : x ≠ y\nU : Set X\nh_clopen : IsClopen U\nhxU : x ∈ U\nhyU : ¬y ∈ U\nhS : Set.Nonempty ∅\n⊢ False",
"tactic": "exact Set.not_nonempty_empty hS"
}
] | [
1291,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1280,
1
] |
Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints | [
{
"state_after": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∃ᶠ (x : C(X, ℝ)) in 𝓝 f, x ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\n⊢ ∃ g, ‖↑g - f‖ < ε",
"tactic": "have w :=\n mem_closure_iff_frequently.mp (continuousMap_mem_subalgebra_closure_of_separatesPoints A w f)"
},
{
"state_after": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∃ᶠ (x : C(X, ℝ)) in 𝓝 f, x ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"tactic": "rw [Metric.nhds_basis_ball.frequently_iff] at w"
},
{
"state_after": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A\ng : C(X, ℝ)\nH : g ∈ Metric.ball f ε\nm : g ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"tactic": "obtain ⟨g, H, m⟩ := w ε pos"
},
{
"state_after": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A\ng : C(X, ℝ)\nH : ‖g - f‖ < ε\nm : g ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"state_before": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A\ng : C(X, ℝ)\nH : g ∈ Metric.ball f ε\nm : g ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"tactic": "rw [Metric.mem_ball, dist_eq_norm] at H"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : Subalgebra.SeparatesPoints A\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A\ng : C(X, ℝ)\nH : ‖g - f‖ < ε\nm : g ∈ ↑A\n⊢ ∃ g, ‖↑g - f‖ < ε",
"tactic": "exact ⟨⟨g, m⟩, H⟩"
}
] | [
315,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
] |
Mathlib/Analysis/Complex/Basic.lean | Complex.nnnorm_int | [] | [
196,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
195,
1
] |
Mathlib/RingTheory/Localization/NumDen.lean | IsFractionRing.num_mul_den_eq_num_iff_eq | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.59265\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.59471\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.59725\ninst✝⁶ : CommRing P\nA : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : x * ↑(algebraMap A K) ↑(den A y) = ↑(algebraMap A K) (num A y)\n⊢ x = y",
"tactic": "simpa only [mk'_num_den] using eq_mk'_iff_mul_eq.mpr h"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.59265\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.59471\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.59725\ninst✝⁶ : CommRing P\nA : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : x = y\n⊢ x = mk' K (num A y) (den A y)",
"tactic": "rw [h, mk'_num_den]"
}
] | [
84,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.sup_mul_le_mul_sup_of_nonneg | [] | [
1893,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1889,
1
] |
Mathlib/Order/CompleteLattice.lean | le_iInf_const | [] | [
1040,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1039,
1
] |
Mathlib/Analysis/NormedSpace/CompactOperator.lean | IsCompactOperator.clm_comp | [] | [
276,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.sup_closed_of_sup_closed | [] | [
998,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
996,
1
] |
Mathlib/Data/Seq/Seq.lean | Stream'.Seq.eq_or_mem_of_mem_cons | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na b : α\nf : Stream' (Option α)\nal : IsSeq f\nh✝ : a ∈ cons b { val := f, property := al }\nh : some a = some b\n⊢ a = b",
"tactic": "injection h"
}
] | [
199,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.coe_filterMap | [] | [
2110,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2109,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean | Polynomial.aeval_eq_sum_range | [
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1271761\nB' : Type ?u.1271764\na b : R\nn : ℕ\ninst✝⁸ : CommSemiring A'\ninst✝⁷ : Semiring B'\ninst✝⁶ : CommSemiring R\np✝ q : R[X]\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB : Type ?u.1271977\ninst✝³ : Semiring B\ninst✝² : Algebra R B\nx✝ : A\ninst✝¹ : Semiring S\nf : R →+* S\ninst✝ : Algebra R S\np : R[X]\nx : S\n⊢ ↑(aeval x) p = ∑ x_1 in range (natDegree p + 1), ↑(algebraMap R S) (coeff p x_1) * x ^ x_1",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1271761\nB' : Type ?u.1271764\na b : R\nn : ℕ\ninst✝⁸ : CommSemiring A'\ninst✝⁷ : Semiring B'\ninst✝⁶ : CommSemiring R\np✝ q : R[X]\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB : Type ?u.1271977\ninst✝³ : Semiring B\ninst✝² : Algebra R B\nx✝ : A\ninst✝¹ : Semiring S\nf : R →+* S\ninst✝ : Algebra R S\np : R[X]\nx : S\n⊢ ↑(aeval x) p = ∑ i in range (natDegree p + 1), coeff p i • x ^ i",
"tactic": "simp_rw [Algebra.smul_def]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1271761\nB' : Type ?u.1271764\na b : R\nn : ℕ\ninst✝⁸ : CommSemiring A'\ninst✝⁷ : Semiring B'\ninst✝⁶ : CommSemiring R\np✝ q : R[X]\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB : Type ?u.1271977\ninst✝³ : Semiring B\ninst✝² : Algebra R B\nx✝ : A\ninst✝¹ : Semiring S\nf : R →+* S\ninst✝ : Algebra R S\np : R[X]\nx : S\n⊢ ↑(aeval x) p = ∑ x_1 in range (natDegree p + 1), ↑(algebraMap R S) (coeff p x_1) * x ^ x_1",
"tactic": "exact eval₂_eq_sum_range (algebraMap R S) x"
}
] | [
358,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
355,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_insert | [] | [
1451,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1449,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean | CategoryTheory.MonoOver.bot_arrow | [] | [
103,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean | NormedSpace.isBounded_iff_subset_smul_closedBall | [
{
"state_after": "case mp\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ Bornology.IsBounded s → ∃ a, s ⊆ a • Metric.closedBall 0 1\n\ncase mpr\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ (∃ a, s ⊆ a • Metric.closedBall 0 1) → Bornology.IsBounded s",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.closedBall 0 1",
"tactic": "constructor"
},
{
"state_after": "case mp\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ (∃ a, s ⊆ a • Metric.ball 0 1) → ∃ a, s ⊆ a • Metric.closedBall 0 1",
"state_before": "case mp\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ Bornology.IsBounded s → ∃ a, s ⊆ a • Metric.closedBall 0 1",
"tactic": "rw [isBounded_iff_subset_smul_ball 𝕜]"
},
{
"state_after": "no goals",
"state_before": "case mp\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ (∃ a, s ⊆ a • Metric.ball 0 1) → ∃ a, s ⊆ a • Metric.closedBall 0 1",
"tactic": "exact Exists.imp fun a ha => ha.trans <| Set.smul_set_mono <| Metric.ball_subset_closedBall"
},
{
"state_after": "case mpr\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ (∃ a, s ⊆ a • Metric.closedBall 0 1) → Bornology.IsVonNBounded 𝕜 s",
"state_before": "case mpr\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ (∃ a, s ⊆ a • Metric.closedBall 0 1) → Bornology.IsBounded s",
"tactic": "rw [← isVonNBounded_iff 𝕜]"
},
{
"state_after": "case mpr.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\na : 𝕜\nha : s ⊆ a • Metric.closedBall 0 1\n⊢ Bornology.IsVonNBounded 𝕜 s",
"state_before": "case mpr\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ (∃ a, s ⊆ a • Metric.closedBall 0 1) → Bornology.IsVonNBounded 𝕜 s",
"tactic": "rintro ⟨a, ha⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.270507\nE : Type u_1\nE' : Type ?u.270513\nF : Type ?u.270516\nι : Type ?u.270519\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\na : 𝕜\nha : s ⊆ a • Metric.closedBall 0 1\n⊢ Bornology.IsVonNBounded 𝕜 s",
"tactic": "exact ((isVonNBounded_closedBall 𝕜 E 1).image (a • (1 : E →L[𝕜] E))).subset ha"
}
] | [
338,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
] |
Mathlib/MeasureTheory/Measure/Haar/Basic.lean | MeasureTheory.Measure.haar.index_union_eq | [
{
"state_after": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index (K₁.carrier ∪ K₂.carrier) V = index K₁.carrier V + index K₂.carrier V",
"tactic": "apply le_antisymm (index_union_le K₁ K₂ hV)"
},
{
"state_after": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V",
"tactic": "rcases index_elim (K₁.2.union K₂.2) hV with ⟨s, h1s, h2s⟩"
},
{
"state_after": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\n⊢ index K₁.carrier V + index K₂.carrier V ≤ Finset.card s",
"state_before": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V",
"tactic": "rw [← h2s]"
},
{
"state_after": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s) +\n Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s) ≤\n Finset.card s",
"state_before": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ index K₁.carrier V + index K₂.carrier V ≤ Finset.card s",
"tactic": "refine'\n le_trans\n (add_le_add (this K₁.1 <| Subset.trans (subset_union_left _ _) h1s)\n (this K₂.1 <| Subset.trans (subset_union_right _ _) h1s)) _"
},
{
"state_after": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ Finset.card\n (Finset.filter\n (fun x =>\n Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₁.carrier) ∨ Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₂.carrier))\n s) ≤\n Finset.card s\n\ncase intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ Disjoint (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s)\n (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s)",
"state_before": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s) +\n Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s) ≤\n Finset.card s",
"tactic": "rw [← Finset.card_union_eq, Finset.filter_union_right]"
},
{
"state_after": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ Disjoint (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s)\n (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s)",
"state_before": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ Finset.card\n (Finset.filter\n (fun x =>\n Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₁.carrier) ∨ Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₂.carrier))\n s) ≤\n Finset.card s\n\ncase intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ Disjoint (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s)\n (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s)",
"tactic": "exact s.card_filter_le _"
},
{
"state_after": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ ∀ (x : G),\n x ∈ s → Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₁.carrier) → ¬Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₂.carrier)",
"state_before": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ Disjoint (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s)\n (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s)",
"tactic": "apply Finset.disjoint_filter.mpr"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh1g₂ : g₂ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh1g₃ : g₃ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₃ : g₃ ∈ K₂.carrier\n⊢ False",
"state_before": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\n⊢ ∀ (x : G),\n x ∈ s → Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₁.carrier) → ¬Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₂.carrier)",
"tactic": "rintro g₁ _ ⟨g₂, h1g₂, h2g₂⟩ ⟨g₃, h1g₃, h2g₃⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh1g₂ : g₂ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh1g₃ : g₃ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₃ : g₃ ∈ K₂.carrier\n⊢ False",
"tactic": "simp only [mem_preimage] at h1g₃ h1g₂"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹ ⊓ K₂.carrier * V⁻¹",
"state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ False",
"tactic": "refine' h.le_bot (_ : g₁⁻¹ ∈ _)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.left\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ ∃ x, x ∈ K₁.carrier ∧ ∃ x_1, x_1⁻¹ ∈ V ∧ x * x_1 = g₁⁻¹\n\ncase intro.intro.intro.intro.intro.intro.right\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ ∃ x, x ∈ K₂.carrier ∧ ∃ x_1, x_1⁻¹ ∈ V ∧ x * x_1 = g₁⁻¹",
"state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹ ⊓ K₂.carrier * V⁻¹",
"tactic": "constructor <;> simp only [Set.mem_inv, Set.mem_mul, exists_exists_and_eq_and, exists_and_left]"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\n⊢ index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\n⊢ ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)",
"tactic": "intro K hK"
},
{
"state_after": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\n⊢ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s) ∈\n Finset.card '' {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\n⊢ index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)",
"tactic": "apply Nat.sInf_le"
},
{
"state_after": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\n⊢ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s ∈\n {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}",
"state_before": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\n⊢ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s) ∈\n Finset.card '' {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}",
"tactic": "refine' ⟨_, _, rfl⟩"
},
{
"state_after": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\n⊢ K ⊆ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V",
"state_before": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\n⊢ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s ∈\n {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}",
"tactic": "rw [mem_setOf_eq]"
},
{
"state_after": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\n⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V",
"state_before": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\n⊢ K ⊆ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V",
"tactic": "intro g hg"
},
{
"state_after": "case hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g ∈ (fun h => (fun h => g₀ * h) ⁻¹' V) h1g₀\n⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V",
"state_before": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\n⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V",
"tactic": "rcases hK hg with ⟨_, ⟨g₀, rfl⟩, _, ⟨h1g₀, rfl⟩, h2g₀⟩"
},
{
"state_after": "case hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V",
"state_before": "case hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g ∈ (fun h => (fun h => g₀ * h) ⁻¹' V) h1g₀\n⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V",
"tactic": "simp only [mem_preimage] at h2g₀"
},
{
"state_after": "case hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ ∃ i i_1, g ∈ (fun h => i * h) ⁻¹' V",
"state_before": "case hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V",
"tactic": "simp only [mem_iUnion]"
},
{
"state_after": "case hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ ∃ i, g ∈ (fun h => g₀ * h) ⁻¹' V",
"state_before": "case hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ ∃ i i_1, g ∈ (fun h => i * h) ⁻¹' V",
"tactic": "use g₀"
},
{
"state_after": "case hm.intro.intro.intro.intro.intro.intro.h\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g ∈ (fun h => g₀ * h) ⁻¹' V\n\ncase hm.intro.intro.intro.intro.intro.intro.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g₀ ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s",
"state_before": "case hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ ∃ i, g ∈ (fun h => g₀ * h) ⁻¹' V",
"tactic": "constructor"
},
{
"state_after": "case hm.intro.intro.intro.intro.intro.intro.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g₀ ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s\n\ncase hm.intro.intro.intro.intro.intro.intro.h\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g ∈ (fun h => g₀ * h) ⁻¹' V",
"state_before": "case hm.intro.intro.intro.intro.intro.intro.h\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g ∈ (fun h => g₀ * h) ⁻¹' V\n\ncase hm.intro.intro.intro.intro.intro.intro.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g₀ ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s",
"tactic": "swap"
},
{
"state_after": "no goals",
"state_before": "case hm.intro.intro.intro.intro.intro.intro.h\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g ∈ (fun h => g₀ * h) ⁻¹' V",
"tactic": "exact h2g₀"
},
{
"state_after": "case hm.intro.intro.intro.intro.intro.intro.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ Set.Nonempty ((fun h => g₀ * h) ⁻¹' V ∩ K)",
"state_before": "case hm.intro.intro.intro.intro.intro.intro.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g₀ ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s",
"tactic": "simp only [Finset.mem_filter, h1g₀, true_and_iff]"
},
{
"state_after": "case hm.intro.intro.intro.intro.intro.intro.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g ∈ (fun h => g₀ * h) ⁻¹' V ∩ K",
"state_before": "case hm.intro.intro.intro.intro.intro.intro.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ Set.Nonempty ((fun h => g₀ * h) ⁻¹' V ∩ K)",
"tactic": "use g"
},
{
"state_after": "no goals",
"state_before": "case hm.intro.intro.intro.intro.intro.intro.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nK : Set G\nhK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\ng : G\nhg : g ∈ K\ng₀ : G\nh1g₀ : g₀ ∈ s\nh2g₀ : g₀ * g ∈ V\n⊢ g ∈ (fun h => g₀ * h) ⁻¹' V ∩ K",
"tactic": "simp only [hg, h2g₀, mem_inter_iff, mem_preimage, and_self_iff]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.left.refine'_1\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ (g₁ * g₂)⁻¹⁻¹ ∈ V\n\ncase intro.intro.intro.intro.intro.intro.left.refine'_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹",
"state_before": "case intro.intro.intro.intro.intro.intro.left\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ ∃ x, x ∈ K₁.carrier ∧ ∃ x_1, x_1⁻¹ ∈ V ∧ x * x_1 = g₁⁻¹",
"tactic": "refine' ⟨_, h2g₂, (g₁ * g₂)⁻¹, _, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.left.refine'_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹",
"state_before": "case intro.intro.intro.intro.intro.intro.left.refine'_1\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ (g₁ * g₂)⁻¹⁻¹ ∈ V\n\ncase intro.intro.intro.intro.intro.intro.left.refine'_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹",
"tactic": "simp only [inv_inv, h1g₂]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.left.refine'_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹",
"tactic": "simp only [mul_inv_rev, mul_inv_cancel_left]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.right.refine'_1\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ (g₁ * g₃)⁻¹⁻¹ ∈ V\n\ncase intro.intro.intro.intro.intro.intro.right.refine'_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₃ * (g₁ * g₃)⁻¹ = g₁⁻¹",
"state_before": "case intro.intro.intro.intro.intro.intro.right\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ ∃ x, x ∈ K₂.carrier ∧ ∃ x_1, x_1⁻¹ ∈ V ∧ x * x_1 = g₁⁻¹",
"tactic": "refine' ⟨_, h2g₃, (g₁ * g₃)⁻¹, _, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.right.refine'_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₃ * (g₁ * g₃)⁻¹ = g₁⁻¹",
"state_before": "case intro.intro.intro.intro.intro.intro.right.refine'_1\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ (g₁ * g₃)⁻¹⁻¹ ∈ V\n\ncase intro.intro.intro.intro.intro.intro.right.refine'_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₃ * (g₁ * g₃)⁻¹ = g₁⁻¹",
"tactic": "simp only [inv_inv, h1g₃]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.right.refine'_2\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V\nh2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n ∀ (K : Set G),\n (K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V) →\n index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₃ * (g₁ * g₃)⁻¹ = g₁⁻¹",
"tactic": "simp only [mul_inv_rev, mul_inv_cancel_left]"
}
] | [
269,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
239,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean | UniqueFactorizationMonoid.normalizedFactors_pos | [
{
"state_after": "case mp\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\n⊢ 0 < normalizedFactors x → ¬IsUnit x\n\ncase mpr\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\n⊢ ¬IsUnit x → 0 < normalizedFactors x",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\n⊢ 0 < normalizedFactors x ↔ ¬IsUnit x",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx✝ : x ≠ 0\nh : 0 < normalizedFactors x\nhx : IsUnit x\n⊢ False",
"state_before": "case mp\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\n⊢ 0 < normalizedFactors x → ¬IsUnit x",
"tactic": "intro h hx"
},
{
"state_after": "case mp.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx✝ : x ≠ 0\nh : 0 < normalizedFactors x\nhx : IsUnit x\np : α\nhp : p ∈ normalizedFactors x\n⊢ False",
"state_before": "case mp\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx✝ : x ≠ 0\nh : 0 < normalizedFactors x\nhx : IsUnit x\n⊢ False",
"tactic": "obtain ⟨p, hp⟩ := Multiset.exists_mem_of_ne_zero h.ne'"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx✝ : x ≠ 0\nh : 0 < normalizedFactors x\nhx : IsUnit x\np : α\nhp : p ∈ normalizedFactors x\n⊢ False",
"tactic": "exact\n (prime_of_normalized_factor _ hp).not_unit\n (isUnit_of_dvd_unit (dvd_of_mem_normalizedFactors hp) hx)"
},
{
"state_after": "case mpr\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\n⊢ 0 < normalizedFactors x",
"state_before": "case mpr\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\n⊢ ¬IsUnit x → 0 < normalizedFactors x",
"tactic": "intro h"
},
{
"state_after": "case mpr.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\np : α\nhp : p ∈ normalizedFactors x\n⊢ 0 < normalizedFactors x",
"state_before": "case mpr\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\n⊢ 0 < normalizedFactors x",
"tactic": "obtain ⟨p, hp⟩ := exists_mem_normalizedFactors hx h"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\np : α\nhp : p ∈ normalizedFactors x\n⊢ 0 < normalizedFactors x",
"tactic": "exact\n bot_lt_iff_ne_bot.mpr\n (mt Multiset.eq_zero_iff_forall_not_mem.mp (not_forall.mpr ⟨p, not_not.mpr hp⟩))"
}
] | [
789,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
778,
1
] |
Mathlib/Algebra/Algebra/Spectrum.lean | spectrum.preimage_units_mul_eq_swap_mul | [] | [
250,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.coe_map | [] | [
318,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean | Dfinsupp.zipWith_neLocus_eq_left | [
{
"state_after": "case a\nα : Type u_4\nN : α → Type u_1\ninst✝⁵ : DecidableEq α\nM : α → Type u_3\nP : α → Type u_2\ninst✝⁴ : (a : α) → Zero (N a)\ninst✝³ : (a : α) → Zero (M a)\ninst✝² : (a : α) → Zero (P a)\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → DecidableEq (P a)\nF : (a : α) → M a → N a → P a\nF0 : ∀ (a : α), F a 0 0 = 0\nf : Π₀ (a : α), M a\ng₁ g₂ : Π₀ (a : α), N a\nhF : ∀ (a : α) (f : M a), Function.Injective fun g => F a f g\na : α\n⊢ a ∈ neLocus (zipWith F F0 f g₁) (zipWith F F0 f g₂) ↔ a ∈ neLocus g₁ g₂",
"state_before": "α : Type u_4\nN : α → Type u_1\ninst✝⁵ : DecidableEq α\nM : α → Type u_3\nP : α → Type u_2\ninst✝⁴ : (a : α) → Zero (N a)\ninst✝³ : (a : α) → Zero (M a)\ninst✝² : (a : α) → Zero (P a)\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → DecidableEq (P a)\nF : (a : α) → M a → N a → P a\nF0 : ∀ (a : α), F a 0 0 = 0\nf : Π₀ (a : α), M a\ng₁ g₂ : Π₀ (a : α), N a\nhF : ∀ (a : α) (f : M a), Function.Injective fun g => F a f g\n⊢ neLocus (zipWith F F0 f g₁) (zipWith F F0 f g₂) = neLocus g₁ g₂",
"tactic": "ext a"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_4\nN : α → Type u_1\ninst✝⁵ : DecidableEq α\nM : α → Type u_3\nP : α → Type u_2\ninst✝⁴ : (a : α) → Zero (N a)\ninst✝³ : (a : α) → Zero (M a)\ninst✝² : (a : α) → Zero (P a)\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → DecidableEq (P a)\nF : (a : α) → M a → N a → P a\nF0 : ∀ (a : α), F a 0 0 = 0\nf : Π₀ (a : α), M a\ng₁ g₂ : Π₀ (a : α), N a\nhF : ∀ (a : α) (f : M a), Function.Injective fun g => F a f g\na : α\n⊢ a ∈ neLocus (zipWith F F0 f g₁) (zipWith F F0 f g₂) ↔ a ∈ neLocus g₁ g₂",
"tactic": "simpa only [mem_neLocus] using (hF a _).ne_iff"
}
] | [
102,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
] |
Mathlib/RingTheory/TensorProduct.lean | TensorProduct.AlgebraTensorModule.lift_apply | [] | [
151,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
149,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.card_set_walk_length_eq | [
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu v : V\nn : ℕ\np : Walk G u v\n⊢ p ∈ finsetWalkLength G n u v ↔ p ∈ {p | Walk.length p = n}",
"tactic": "rw [← Finset.mem_coe, coe_finsetWalkLength_eq]"
}
] | [
2391,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2388,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.mem_top | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.57666\nι : Sort x\nf g : Filter α\ns✝ t s : Set α\n⊢ s ∈ ⊤ ↔ s = univ",
"tactic": "rw [mem_top_iff_forall, eq_univ_iff_forall]"
}
] | [
461,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
460,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.Ici_iSup | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.295226\nγ : Type ?u.295229\nι : Sort u_2\nι' : Sort ?u.295235\nι₂ : Sort ?u.295238\nκ : ι → Sort ?u.295243\nκ₁ : ι → Sort ?u.295248\nκ₂ : ι → Sort ?u.295253\nκ' : ι' → Sort ?u.295258\ninst✝ : CompleteLattice α\nf : ι → α\nx✝ : α\n⊢ x✝ ∈ Ici (⨆ (i : ι), f i) ↔ x✝ ∈ ⋂ (i : ι), Ici (f i)",
"tactic": "simp only [mem_Ici, iSup_le_iff, mem_iInter]"
}
] | [
2123,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2122,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.eventuallyEq_of_toReal_eventuallyEq | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.27073\nγ : Type ?u.27076\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nl : Filter α\nf g : α → ℝ≥0∞\nhfi : ∀ᶠ (x : α) in l, f x ≠ ⊤\nhgi : ∀ᶠ (x : α) in l, g x ≠ ⊤\nhfg : (fun x => ENNReal.toReal (f x)) =ᶠ[l] fun x => ENNReal.toReal (g x)\na✝¹ : α\nhfx : f a✝¹ ≠ ⊤\nhgx : g a✝¹ ≠ ⊤\na✝ : ENNReal.toReal (f a✝¹) = ENNReal.toReal (g a✝¹)\n⊢ f a✝¹ = g a✝¹",
"state_before": "α : Type u_1\nβ : Type ?u.27073\nγ : Type ?u.27076\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nl : Filter α\nf g : α → ℝ≥0∞\nhfi : ∀ᶠ (x : α) in l, f x ≠ ⊤\nhgi : ∀ᶠ (x : α) in l, g x ≠ ⊤\nhfg : (fun x => ENNReal.toReal (f x)) =ᶠ[l] fun x => ENNReal.toReal (g x)\n⊢ f =ᶠ[l] g",
"tactic": "filter_upwards [hfi, hgi, hfg]with _ hfx hgx _"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.27073\nγ : Type ?u.27076\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nl : Filter α\nf g : α → ℝ≥0∞\nhfi : ∀ᶠ (x : α) in l, f x ≠ ⊤\nhgi : ∀ᶠ (x : α) in l, g x ≠ ⊤\nhfg : (fun x => ENNReal.toReal (f x)) =ᶠ[l] fun x => ENNReal.toReal (g x)\na✝¹ : α\nhfx : f a✝¹ ≠ ⊤\nhgx : g a✝¹ ≠ ⊤\na✝ : ENNReal.toReal (f a✝¹) = ENNReal.toReal (g a✝¹)\n⊢ f a✝¹ = g a✝¹",
"tactic": "rwa [← ENNReal.toReal_eq_toReal hfx hgx]"
}
] | [
127,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | zpow_strictMono_right | [
{
"state_after": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : OrderedCommGroup α\nm✝ n✝ : ℤ\na b : α\nha : 1 < a\nm n : ℤ\nh : m < n\n⊢ a ^ (m + (n - m)) = a ^ n",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : OrderedCommGroup α\nm✝ n✝ : ℤ\na b : α\nha : 1 < a\nm n : ℤ\nh : m < n\n⊢ a ^ m * a ^ (n - m) = a ^ n",
"tactic": "rw [← zpow_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : OrderedCommGroup α\nm✝ n✝ : ℤ\na b : α\nha : 1 < a\nm n : ℤ\nh : m < n\n⊢ a ^ (m + (n - m)) = a ^ n",
"tactic": "simp"
}
] | [
334,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
330,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean | OrderRingHom.id_apply | [] | [
283,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Data/Bool/AllAny.lean | List.any_iff_exists | [
{
"state_after": "case nil\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl : List α\na : α\np : α → Bool\n⊢ any [] p = true ↔ ∃ a, a ∈ [] ∧ p a = true\n\ncase cons\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl✝ : List α\na✝ : α\np : α → Bool\na : α\nl : List α\nih : any l p = true ↔ ∃ a, a ∈ l ∧ p a = true\n⊢ any (a :: l) p = true ↔ ∃ a_1, a_1 ∈ a :: l ∧ p a_1 = true",
"state_before": "α : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl : List α\na : α\np : α → Bool\n⊢ any l p = true ↔ ∃ a, a ∈ l ∧ p a = true",
"tactic": "induction' l with a l ih"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl✝ : List α\na✝ : α\np : α → Bool\na : α\nl : List α\nih : any l p = true ↔ ∃ a, a ∈ l ∧ p a = true\n⊢ any (a :: l) p = true ↔ ∃ a_1, a_1 ∈ a :: l ∧ p a_1 = true",
"tactic": "simp only [any_cons, Bool.or_coe_iff, ih, exists_mem_cons_iff]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl : List α\na : α\np : α → Bool\n⊢ any [] p = true ↔ ∃ a, a ∈ [] ∧ p a = true",
"tactic": "exact iff_of_false Bool.not_false' (not_exists_mem_nil _)"
}
] | [
48,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
] |
Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearEquiv.coe_toContinuousLinearEquiv | [] | [
370,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
369,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean | PiLp.nnnorm_eq_sum | [
{
"state_after": "case a\np✝ : ℝ≥0∞\n𝕜 : Type ?u.229074\n𝕜' : Type ?u.229077\nι : Type u_2\nα : ι → Type ?u.229085\nβ✝ : ι → Type ?u.229090\ninst✝³ : Fintype ι\ninst✝² : Fact (1 ≤ p✝)\np : ℝ≥0∞\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\nhp : p ≠ ⊤\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nf : PiLp p β\n⊢ ↑‖f‖₊ = ↑((∑ i : ι, ‖f i‖₊ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p))",
"state_before": "p✝ : ℝ≥0∞\n𝕜 : Type ?u.229074\n𝕜' : Type ?u.229077\nι : Type u_2\nα : ι → Type ?u.229085\nβ✝ : ι → Type ?u.229090\ninst✝³ : Fintype ι\ninst✝² : Fact (1 ≤ p✝)\np : ℝ≥0∞\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\nhp : p ≠ ⊤\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nf : PiLp p β\n⊢ ‖f‖₊ = (∑ i : ι, ‖f i‖₊ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\np✝ : ℝ≥0∞\n𝕜 : Type ?u.229074\n𝕜' : Type ?u.229077\nι : Type u_2\nα : ι → Type ?u.229085\nβ✝ : ι → Type ?u.229090\ninst✝³ : Fintype ι\ninst✝² : Fact (1 ≤ p✝)\np : ℝ≥0∞\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\nhp : p ≠ ⊤\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nf : PiLp p β\n⊢ ↑‖f‖₊ = ↑((∑ i : ι, ‖f i‖₊ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p))",
"tactic": "simp [NNReal.coe_sum, norm_eq_sum (p.toReal_pos_iff_ne_top.mpr hp)]"
}
] | [
564,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
560,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.replicate_add | [] | [
898,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
897,
1
] |
Mathlib/Data/Sym/Sym2.lean | Sym2.mem_iff | [] | [
339,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | pow_four_le_pow_two_of_pow_two_le | [] | [
750,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
749,
1
] |
Mathlib/Order/Antisymmetrization.lean | ofAntisymmetrization_lt_ofAntisymmetrization_iff | [] | [
203,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/RingTheory/EuclideanDomain.lean | EuclideanDomain.isCoprime_of_dvd | [] | [
96,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Topology/UrysohnsLemma.lean | Urysohns.CU.continuous_lim | [
{
"state_after": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\n⊢ Continuous (CU.lim c)",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\n⊢ Continuous (CU.lim c)",
"tactic": "obtain ⟨h0, h1234, h1⟩ : 0 < (2⁻¹ : ℝ) ∧ (2⁻¹ : ℝ) < 3 / 4 ∧ (3 / 4 : ℝ) < 1 := by norm_num"
},
{
"state_after": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, CU.lim c x_1 ∈ Metric.closedBall (CU.lim c x) ((3 / 4) ^ n)",
"state_before": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\n⊢ Continuous (CU.lim c)",
"tactic": "refine'\n continuous_iff_continuousAt.2 fun x =>\n (Metric.nhds_basis_closedBall_pow (h0.trans h1234) h1).tendsto_right_iff.2 fun n _ => _"
},
{
"state_after": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n",
"state_before": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, CU.lim c x_1 ∈ Metric.closedBall (CU.lim c x) ((3 / 4) ^ n)",
"tactic": "simp only [Metric.mem_closedBall]"
},
{
"state_after": "case intro.intro.zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nc : CU X\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.zero\n\ncase intro.intro.succ\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case intro.intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n",
"tactic": "induction' n with n ihn generalizing c"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\n⊢ 0 < 2⁻¹ ∧ 2⁻¹ < 3 / 4 ∧ 3 / 4 < 1",
"tactic": "norm_num"
},
{
"state_after": "case intro.intro.zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nc : CU X\ny : X\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.zero",
"state_before": "case intro.intro.zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nc : CU X\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.zero",
"tactic": "refine' eventually_of_forall fun y => _"
},
{
"state_after": "case intro.intro.zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nc : CU X\ny : X\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ 1",
"state_before": "case intro.intro.zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nc : CU X\ny : X\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.zero",
"tactic": "rw [pow_zero]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nc : CU X\ny : X\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ 1",
"tactic": "exact Real.dist_le_of_mem_Icc_01 (c.lim_mem_Icc _) (c.lim_mem_Icc _)"
},
{
"state_after": "case pos\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n\n\ncase neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : ¬x ∈ (left c).U\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case intro.intro.succ\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "by_cases hxl : x ∈ c.left.U"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\na✝ : X\nhyl : a✝ ∈ (left c).U\nhyd : dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ (3 / 4) ^ n\n⊢ dist (CU.lim c a✝) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case pos\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "filter_upwards [IsOpen.mem_nhds c.left.open_U hxl, ihn c.left]with _ hyl hyd"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\na✝ : X\nhyl : a✝ ∈ (left c).U\nhyd : dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ (3 / 4) ^ n\n⊢ dist (midpoint ℝ (CU.lim (left c) a✝) 0) (midpoint ℝ (CU.lim (left c) x) 0) ≤ 3 / 4 * (3 / 4) ^ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\na✝ : X\nhyl : a✝ ∈ (left c).U\nhyd : dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ (3 / 4) ^ n\n⊢ dist (CU.lim c a✝) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "rw [pow_succ, c.lim_eq_midpoint, c.lim_eq_midpoint,\n c.right.lim_of_mem_C _ (c.left_U_subset_right_C hyl),\n c.right.lim_of_mem_C _ (c.left_U_subset_right_C hxl)]"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\na✝ : X\nhyl : a✝ ∈ (left c).U\nhyd : dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ (3 / 4) ^ n\n⊢ (dist (CU.lim (left c) a✝) (CU.lim (left c) x) + dist 0 0) / 2 ≤ 3 / 4 * (3 / 4) ^ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\na✝ : X\nhyl : a✝ ∈ (left c).U\nhyd : dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ (3 / 4) ^ n\n⊢ dist (midpoint ℝ (CU.lim (left c) a✝) 0) (midpoint ℝ (CU.lim (left c) x) 0) ≤ 3 / 4 * (3 / 4) ^ n",
"tactic": "refine' (dist_midpoint_midpoint_le _ _ _ _).trans _"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\na✝ : X\nhyl : a✝ ∈ (left c).U\nhyd : dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ (3 / 4) ^ n\n⊢ 2⁻¹ * dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ 3 / 4 * (3 / 4) ^ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\na✝ : X\nhyl : a✝ ∈ (left c).U\nhyd : dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ (3 / 4) ^ n\n⊢ (dist (CU.lim (left c) a✝) (CU.lim (left c) x) + dist 0 0) / 2 ≤ 3 / 4 * (3 / 4) ^ n",
"tactic": "rw [dist_self, add_zero, div_eq_inv_mul]"
},
{
"state_after": "no goals",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (left c).U\na✝ : X\nhyl : a✝ ∈ (left c).U\nhyd : dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ (3 / 4) ^ n\n⊢ 2⁻¹ * dist (CU.lim (left c) a✝) (CU.lim (left c) x) ≤ 3 / 4 * (3 / 4) ^ n",
"tactic": "gcongr"
},
{
"state_after": "case hxl\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : ¬x ∈ (left c).U\n⊢ x ∈ (right (left c)).Cᶜ\n\ncase neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (right (left c)).Cᶜ\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : ¬x ∈ (left c).U\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "replace hxl : x ∈ c.left.right.Cᶜ"
},
{
"state_after": "case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (right (left c)).Cᶜ\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case hxl\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : ¬x ∈ (left c).U\n⊢ x ∈ (right (left c)).Cᶜ\n\ncase neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (right (left c)).Cᶜ\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "exact compl_subset_compl.2 c.left.right.subset hxl"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (right (left c)).Cᶜ\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (right (left c)).Cᶜ\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "filter_upwards [IsOpen.mem_nhds (isOpen_compl_iff.2 c.left.right.closed_C) hxl,\n ihn c.left.right, ihn c.right]with y hyl hydl hydr"
},
{
"state_after": "case hxl\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (right (left c)).Cᶜ\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\n⊢ ¬x ∈ (left (left c)).U\n\ncase h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (right (left c)).Cᶜ\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "replace hxl : x ∉ c.left.left.U"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case hxl\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\nhxl : x ∈ (right (left c)).Cᶜ\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\n⊢ ¬x ∈ (left (left c)).U\n\ncase h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "exact compl_subset_compl.2 c.left.left_U_subset_right_C hxl"
},
{
"state_after": "case hyl\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\n⊢ ¬y ∈ (left (left c)).U\n\ncase h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "replace hyl : y ∉ c.left.left.U"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"state_before": "case hyl\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y ∈ (right (left c)).Cᶜ\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\n⊢ ¬y ∈ (left (left c)).U\n\ncase h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "exact compl_subset_compl.2 c.left.left_U_subset_right_C hyl"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ dist (midpoint ℝ (midpoint ℝ 1 (CU.lim (right (left c)) y)) (CU.lim (right c) y))\n (midpoint ℝ (midpoint ℝ 1 (CU.lim (right (left c)) x)) (CU.lim (right c) x)) ≤\n 3 / 4 * (3 / 4) ^ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ dist (CU.lim c y) (CU.lim c x) ≤ (3 / 4) ^ Nat.succ n",
"tactic": "simp only [pow_succ, c.lim_eq_midpoint, c.left.lim_eq_midpoint,\n c.left.left.lim_of_nmem_U _ hxl, c.left.left.lim_of_nmem_U _ hyl]"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ (dist (midpoint ℝ 1 (CU.lim (right (left c)) y)) (midpoint ℝ 1 (CU.lim (right (left c)) x)) +\n dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n 3 / 4 * (3 / 4) ^ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ dist (midpoint ℝ (midpoint ℝ 1 (CU.lim (right (left c)) y)) (CU.lim (right c) y))\n (midpoint ℝ (midpoint ℝ 1 (CU.lim (right (left c)) x)) (CU.lim (right c) x)) ≤\n 3 / 4 * (3 / 4) ^ n",
"tactic": "refine' (dist_midpoint_midpoint_le _ _ _ _).trans _"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ ((dist 1 1 + dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x)) / 2 +\n dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n 3 / 4 * (3 / 4) ^ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ (dist (midpoint ℝ 1 (CU.lim (right (left c)) y)) (midpoint ℝ 1 (CU.lim (right (left c)) x)) +\n dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n 3 / 4 * (3 / 4) ^ n",
"tactic": "refine' (div_le_div_of_le_of_nonneg (add_le_add_right (dist_midpoint_midpoint_le _ _ _ _) _)\n zero_le_two).trans _"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ (dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) / 2 + dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n 3 / 4 * (3 / 4) ^ n",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ ((dist 1 1 + dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x)) / 2 +\n dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n 3 / 4 * (3 / 4) ^ n",
"tactic": "rw [dist_self, zero_add]"
},
{
"state_after": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nc : CU X\ny : X\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\nr : ℝ := (3 / 4) ^ n\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ r\n⊢ (dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) / 2 + dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n 3 / 4 * r",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ (3 / 4) ^ n\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\n⊢ (dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) / 2 + dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n 3 / 4 * (3 / 4) ^ n",
"tactic": "set r := (3 / 4 : ℝ) ^ n"
},
{
"state_after": "no goals",
"state_before": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nc : CU X\ny : X\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\nr : ℝ := (3 / 4) ^ n\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ r\n⊢ (dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) / 2 + dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n 3 / 4 * r",
"tactic": "calc _ ≤ (r / 2 + r) / 2 := by gcongr\n _ = _ := by field_simp; ring"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nc : CU X\ny : X\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\nr : ℝ := (3 / 4) ^ n\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ r\n⊢ (dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) / 2 + dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 ≤\n (r / 2 + r) / 2",
"tactic": "gcongr"
},
{
"state_after": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nc : CU X\ny : X\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\nr : ℝ := (3 / 4) ^ n\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ r\n⊢ (3 ^ n * 4 ^ n + 3 ^ n * (4 ^ n * 2)) * (4 * 4 ^ n) = 3 * 3 ^ n * (4 ^ n * 2 * 4 ^ n * 2)",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nc : CU X\ny : X\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\nr : ℝ := (3 / 4) ^ n\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ r\n⊢ (r / 2 + r) / 2 = 3 / 4 * r",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nc : CU X\ny : X\nhxl : ¬x ∈ (left (left c)).U\nhyl : ¬y ∈ (left (left c)).U\nr : ℝ := (3 / 4) ^ n\nihn : ∀ (c : CU X), ∀ᶠ (x_1 : X) in 𝓝 x, dist (CU.lim c x_1) (CU.lim c x) ≤ r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) ≤ r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) ≤ r\n⊢ (3 ^ n * 4 ^ n + 3 ^ n * (4 ^ n * 2)) * (4 * 4 ^ n) = 3 * 3 ^ n * (4 ^ n * 2 * 4 ^ n * 2)",
"tactic": "ring"
}
] | [
292,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | Monotone.measurable | [] | [
1171,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1168,
11
] |
Mathlib/Data/Int/Bitwise.lean | Int.lnot_bit | [
{
"state_after": "no goals",
"state_before": "b : Bool\nn : ℕ\n⊢ lnot (bit b ↑n) = bit (!b) (lnot ↑n)",
"tactic": "simp [lnot]"
},
{
"state_after": "no goals",
"state_before": "b : Bool\nn : ℕ\n⊢ lnot (bit b -[n+1]) = bit (!b) (lnot -[n+1])",
"tactic": "simp [lnot]"
}
] | [
331,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
329,
1
] |
Mathlib/Topology/Order/Basic.lean | isOpen_iff_generate_intervals | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\ns : Set α\n⊢ IsOpen s ↔ GenerateOpen {s | ∃ a, s = Ioi a ∨ s = Iio a} s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\ns : Set α\n⊢ IsOpen s ↔ GenerateOpen {s | ∃ a, s = Ioi a ∨ s = Iio a} s",
"tactic": "rw [t.topology_eq_generate_intervals]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\ns : Set α\n⊢ IsOpen s ↔ GenerateOpen {s | ∃ a, s = Ioi a ∨ s = Iio a} s",
"tactic": "rfl"
}
] | [
900,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
898,
1
] |
Mathlib/Order/CompleteLattice.lean | iInf_lt_iff | [] | [
614,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
613,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.mapRange_single | [
{
"state_after": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\ni' : ι\nh : i = i'\n⊢ ↑(mapRange f hf (single i b)) i' = ↑(single i (f i b)) i'\n\ncase neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\ni' : ι\nh : ¬i = i'\n⊢ ↑(mapRange f hf (single i b)) i' = ↑(single i (f i b)) i'",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\ni' : ι\n⊢ ↑(mapRange f hf (single i b)) i' = ↑(single i (f i b)) i'",
"tactic": "by_cases h : i = i'"
},
{
"state_after": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\n⊢ ↑(mapRange f hf (single i b)) i = ↑(single i (f i b)) i",
"state_before": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\ni' : ι\nh : i = i'\n⊢ ↑(mapRange f hf (single i b)) i' = ↑(single i (f i b)) i'",
"tactic": "subst i'"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\n⊢ ↑(mapRange f hf (single i b)) i = ↑(single i (f i b)) i",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\ni' : ι\nh : ¬i = i'\n⊢ ↑(mapRange f hf (single i b)) i' = ↑(single i (f i b)) i'",
"tactic": "simp [h, hf]"
}
] | [
1190,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1184,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean | PadicInt.coe_mul | [] | [
129,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean | exists_norm_eq | [
{
"state_after": "case intro\nα : Type ?u.296964\nβ : Type ?u.296967\nγ : Type ?u.296970\nι : Type ?u.296973\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.297066\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nc : ℝ\nhc : 0 ≤ c\nx : E\nhx : x ≠ 0\n⊢ ∃ x, ‖x‖ = c",
"state_before": "α : Type ?u.296964\nβ : Type ?u.296967\nγ : Type ?u.296970\nι : Type ?u.296973\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.297066\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nc : ℝ\nhc : 0 ≤ c\n⊢ ∃ x, ‖x‖ = c",
"tactic": "rcases exists_ne (0 : E) with ⟨x, hx⟩"
},
{
"state_after": "case intro\nα : Type ?u.296964\nβ : Type ?u.296967\nγ : Type ?u.296970\nι : Type ?u.296973\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.297066\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nc : ℝ\nhc : 0 ≤ c\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ∃ x, ‖x‖ = c",
"state_before": "case intro\nα : Type ?u.296964\nβ : Type ?u.296967\nγ : Type ?u.296970\nι : Type ?u.296973\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.297066\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nc : ℝ\nhc : 0 ≤ c\nx : E\nhx : x ≠ 0\n⊢ ∃ x, ‖x‖ = c",
"tactic": "rw [← norm_ne_zero_iff] at hx"
},
{
"state_after": "case intro\nα : Type ?u.296964\nβ : Type ?u.296967\nγ : Type ?u.296970\nι : Type ?u.296973\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.297066\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nc : ℝ\nhc : 0 ≤ c\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ‖c • ‖x‖⁻¹ • x‖ = c",
"state_before": "case intro\nα : Type ?u.296964\nβ : Type ?u.296967\nγ : Type ?u.296970\nι : Type ?u.296973\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.297066\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nc : ℝ\nhc : 0 ≤ c\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ∃ x, ‖x‖ = c",
"tactic": "use c • ‖x‖⁻¹ • x"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.296964\nβ : Type ?u.296967\nγ : Type ?u.296970\nι : Type ?u.296973\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.297066\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nc : ℝ\nhc : 0 ≤ c\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ‖c • ‖x‖⁻¹ • x‖ = c",
"tactic": "simp [norm_smul, Real.norm_of_nonneg hc, abs_of_nonneg hc, inv_mul_cancel hx]"
}
] | [
357,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
] |
Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | CategoryTheory.MonoidalCategory.pentagon_hom_inv | [
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nW X Y Z : C\n⊢ (α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) = (α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (X ⊗ Y) Z).hom",
"tactic": "coherence"
}
] | [
83,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.