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Mathlib/CategoryTheory/CofilteredSystem.lean
|
CategoryTheory.Functor.IsMittagLeffler.toPreimages
|
[
{
"state_after": "case intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\n⊢ ∃ i_1 f,\n ∀ ⦃k : J⦄ (g : k ⟶ i_1), range ((Functor.toPreimages F s).map f) ⊆ range ((Functor.toPreimages F s).map (g ≫ f))",
"state_before": "J : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj : J\n⊢ ∃ i_1 f,\n ∀ ⦃k : J⦄ (g : k ⟶ i_1), range ((Functor.toPreimages F s).map f) ⊆ range ((Functor.toPreimages F s).map (g ≫ f))",
"tactic": "obtain ⟨j₁, g₁, f₁, -⟩ := IsCofilteredOrEmpty.cone_objs i j"
},
{
"state_after": "case intro.intro.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\n⊢ ∃ i_1 f,\n ∀ ⦃k : J⦄ (g : k ⟶ i_1), range ((Functor.toPreimages F s).map f) ⊆ range ((Functor.toPreimages F s).map (g ≫ f))",
"state_before": "case intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\n⊢ ∃ i_1 f,\n ∀ ⦃k : J⦄ (g : k ⟶ i_1), range ((Functor.toPreimages F s).map f) ⊆ range ((Functor.toPreimages F s).map (g ≫ f))",
"tactic": "obtain ⟨j₂, f₂, h₂⟩ := F.isMittagLeffler_iff_eventualRange.1 h j₁"
},
{
"state_after": "case intro.intro.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\n⊢ range ((Functor.toPreimages F s).map (f₂ ≫ f₁)) ⊆ range ((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁))",
"state_before": "case intro.intro.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\n⊢ ∃ i_1 f,\n ∀ ⦃k : J⦄ (g : k ⟶ i_1), range ((Functor.toPreimages F s).map f) ⊆ range ((Functor.toPreimages F s).map (g ≫ f))",
"tactic": "refine' ⟨j₂, f₂ ≫ f₁, fun j₃ f₃ => _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\n⊢ (Functor.toPreimages F s).map (f₂ ≫ f₁) { val := x, property := hx } ∈\n range ((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁))",
"state_before": "case intro.intro.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\n⊢ range ((Functor.toPreimages F s).map (f₂ ≫ f₁)) ⊆ range ((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁))",
"tactic": "rintro _ ⟨⟨x, hx⟩, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\n⊢ (Functor.toPreimages F s).map (f₂ ≫ f₁) { val := x, property := hx } ∈\n range ((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁))",
"state_before": "case intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\n⊢ (Functor.toPreimages F s).map (f₂ ≫ f₁) { val := x, property := hx } ∈\n range ((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁))",
"tactic": "have : F.map f₂ x ∈ F.eventualRange j₁ := by\n rw [h₂]\n exact ⟨_, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\n⊢ (Functor.toPreimages F s).map (f₂ ≫ f₁) { val := x, property := hx } ∈\n range ((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁))",
"state_before": "case intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\n⊢ (Functor.toPreimages F s).map (f₂ ≫ f₁) { val := x, property := hx } ∈\n range ((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁))",
"tactic": "obtain ⟨y, hy, h₃⟩ := h.subset_image_eventualRange F (f₃ ≫ f₂) this"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\ng₂ : j₃ ⟶ i\n⊢ y ∈ F.map g₂ ⁻¹' s\n\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_2\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\n⊢ ↑((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁)\n { val := y, property := (_ : y ∈ ⋂ (i_1 : j₃ ⟶ i), F.map i_1 ⁻¹' s) }) =\n ↑((Functor.toPreimages F s).map (f₂ ≫ f₁) { val := x, property := hx })",
"state_before": "case intro.intro.intro.intro.intro.intro.mk.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\n⊢ (Functor.toPreimages F s).map (f₂ ≫ f₁) { val := x, property := hx } ∈\n range ((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁))",
"tactic": "refine' ⟨⟨y, mem_iInter.2 fun g₂ => _⟩, Subtype.ext _⟩"
},
{
"state_after": "J : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\n⊢ F.map f₂ x ∈ range (F.map f₂)",
"state_before": "J : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\n⊢ F.map f₂ x ∈ eventualRange F j₁",
"tactic": "rw [h₂]"
},
{
"state_after": "no goals",
"state_before": "J : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\n⊢ F.map f₂ x ∈ range (F.map f₂)",
"tactic": "exact ⟨_, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\ng₂ : j₃ ⟶ i\nj₄ : J\nf₄ : j₄ ⟶ j₃\nh₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁\n⊢ y ∈ F.map g₂ ⁻¹' s",
"state_before": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\ng₂ : j₃ ⟶ i\n⊢ y ∈ F.map g₂ ⁻¹' s",
"tactic": "obtain ⟨j₄, f₄, h₄⟩ := IsCofilteredOrEmpty.cone_maps g₂ ((f₃ ≫ f₂) ≫ g₁)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ng₂ : j₃ ⟶ i\nj₄ : J\nf₄ : j₄ ⟶ j₃\nh₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁\ny : F.obj j₄\nhy : F.map f₄ y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) (F.map f₄ y) = F.map f₂ x\n⊢ F.map f₄ y ∈ F.map g₂ ⁻¹' s",
"state_before": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\ng₂ : j₃ ⟶ i\nj₄ : J\nf₄ : j₄ ⟶ j₃\nh₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁\n⊢ y ∈ F.map g₂ ⁻¹' s",
"tactic": "obtain ⟨y, rfl⟩ := F.mem_eventualRange_iff.1 hy f₄"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ng₂ : j₃ ⟶ i\nj₄ : J\nf₄ : j₄ ⟶ j₃\nh₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁\ny : F.obj j₄\nhy : F.map f₄ y ∈ eventualRange F j₃\nh₃ : F.map (f₄ ≫ f₃ ≫ f₂) y = F.map f₂ x\n⊢ F.map f₄ y ∈ F.map g₂ ⁻¹' s",
"state_before": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ng₂ : j₃ ⟶ i\nj₄ : J\nf₄ : j₄ ⟶ j₃\nh₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁\ny : F.obj j₄\nhy : F.map f₄ y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) (F.map f₄ y) = F.map f₂ x\n⊢ F.map f₄ y ∈ F.map g₂ ⁻¹' s",
"tactic": "rw [← map_comp_apply] at h₃"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ng₂ : j₃ ⟶ i\nj₄ : J\nf₄ : j₄ ⟶ j₃\nh₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁\ny : F.obj j₄\nhy : F.map f₄ y ∈ eventualRange F j₃\nh₃ : F.map (f₄ ≫ f₃ ≫ f₂) y = F.map f₂ x\n⊢ F.map (f₂ ≫ g₁) x ∈ s",
"state_before": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ng₂ : j₃ ⟶ i\nj₄ : J\nf₄ : j₄ ⟶ j₃\nh₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁\ny : F.obj j₄\nhy : F.map f₄ y ∈ eventualRange F j₃\nh₃ : F.map (f₄ ≫ f₃ ≫ f₂) y = F.map f₂ x\n⊢ F.map f₄ y ∈ F.map g₂ ⁻¹' s",
"tactic": "rw [mem_preimage, ← map_comp_apply, h₄, ← Category.assoc, map_comp_apply, h₃,\n ← map_comp_apply]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ng₂ : j₃ ⟶ i\nj₄ : J\nf₄ : j₄ ⟶ j₃\nh₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁\ny : F.obj j₄\nhy : F.map f₄ y ∈ eventualRange F j₃\nh₃ : F.map (f₄ ≫ f₃ ≫ f₂) y = F.map f₂ x\n⊢ F.map (f₂ ≫ g₁) x ∈ s",
"tactic": "apply mem_iInter.1 hx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_2\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\n⊢ F.map (f₃ ≫ f₂ ≫ f₁) y = F.map (f₂ ≫ f₁) x",
"state_before": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_2\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\n⊢ ↑((Functor.toPreimages F s).map (f₃ ≫ f₂ ≫ f₁)\n { val := y, property := (_ : y ∈ ⋂ (i_1 : j₃ ⟶ i), F.map i_1 ⁻¹' s) }) =\n ↑((Functor.toPreimages F s).map (f₂ ≫ f₁) { val := x, property := hx })",
"tactic": "simp_rw [toPreimages_map, MapsTo.val_restrict_apply, Subtype.coe_mk]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_2\nJ : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : eventualRange F j₁ = range (F.map f₂)\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\nhx : x ∈ ⋂ (f : j₂ ⟶ i), F.map f ⁻¹' s\nthis : F.map f₂ x ∈ eventualRange F j₁\ny : F.obj j₃\nhy : y ∈ eventualRange F j₃\nh₃ : F.map (f₃ ≫ f₂) y = F.map f₂ x\n⊢ F.map (f₃ ≫ f₂ ≫ f₁) y = F.map (f₂ ≫ f₁) x",
"tactic": "rw [← Category.assoc, map_comp_apply, h₃, map_comp_apply]"
}
] |
[
253,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
AntitoneOn.Iio
|
[] |
[
101,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
11
] |
Mathlib/Data/Set/Finite.lean
|
Finset.bddAbove
|
[] |
[
1641,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1640,
11
] |
Mathlib/GroupTheory/DoubleCoset.lean
|
Doset.mem_doset_of_not_disjoint
|
[
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b : G\nh : ∃ x, x ∈ doset a ↑H ↑K ∧ x ∈ doset b ↑H ↑K\n⊢ b ∈ doset a ↑H ↑K",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b : G\nh : ¬Disjoint (doset a ↑H ↑K) (doset b ↑H ↑K)\n⊢ b ∈ doset a ↑H ↑K",
"tactic": "rw [Set.not_disjoint_iff] at h"
},
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b : G\nh : ∃ x, (∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y) ∧ ∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * b * y\n⊢ ∃ x, x ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ b = x * a * y",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b : G\nh : ∃ x, x ∈ doset a ↑H ↑K ∧ x ∈ doset b ↑H ↑K\n⊢ b ∈ doset a ↑H ↑K",
"tactic": "simp only [mem_doset] at *"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b l : G\nhl : l ∈ ↑H\nr : G\nhr : r ∈ ↑K\ny : G\nhy : y ∈ ↑H\nr' : G\nhr' : r' ∈ ↑K\nhrx : y * b * r' = l * a * r\n⊢ ∃ x, x ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ b = x * a * y",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b : G\nh : ∃ x, (∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y) ∧ ∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * b * y\n⊢ ∃ x, x ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ b = x * a * y",
"tactic": "obtain ⟨x, ⟨l, hl, r, hr, hrx⟩, y, hy, ⟨r', hr', rfl⟩⟩ := h"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b l : G\nhl : l ∈ ↑H\nr : G\nhr : r ∈ ↑K\ny : G\nhy : y ∈ ↑H\nr' : G\nhr' : r' ∈ ↑K\nhrx : y * b * r' = l * a * r\n⊢ b = y⁻¹ * l * a * (r * r'⁻¹)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b l : G\nhl : l ∈ ↑H\nr : G\nhr : r ∈ ↑K\ny : G\nhy : y ∈ ↑H\nr' : G\nhr' : r' ∈ ↑K\nhrx : y * b * r' = l * a * r\n⊢ ∃ x, x ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ b = x * a * y",
"tactic": "refine' ⟨y⁻¹ * l, H.mul_mem (H.inv_mem hy) hl, r * r'⁻¹, K.mul_mem hr (K.inv_mem hr'), _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.6410\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na b l : G\nhl : l ∈ ↑H\nr : G\nhr : r ∈ ↑K\ny : G\nhy : y ∈ ↑H\nr' : G\nhr' : r' ∈ ↑K\nhrx : y * b * r' = l * a * r\n⊢ b = y⁻¹ * l * a * (r * r'⁻¹)",
"tactic": "rwa [mul_assoc, mul_assoc, eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc, eq_mul_inv_iff_mul_eq]"
}
] |
[
67,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
Prod.fst_inv
|
[] |
[
138,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.nat_cast_lt
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ ↑m < ↑n ↔ m < n",
"tactic": "simp only [lt_iff_le_not_le, nat_cast_le]"
}
] |
[
2311,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2310,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.map_restriction
|
[
{
"state_after": "no goals",
"state_before": "R✝ : Type u\nS : Type ?u.71352\ninst✝¹ : Ring R✝\nR : Type u\ninst✝ : CommRing R\np : R[X]\nn : ℕ\n⊢ coeff (map (algebraMap { x // x ∈ Subring.closure ↑(frange p) } R) (restriction p)) n = coeff p n",
"tactic": "rw [coeff_map, Algebra.algebraMap_ofSubring_apply, coeff_restriction]"
}
] |
[
302,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.add_apply
|
[] |
[
214,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
div_add_div
|
[] |
[
210,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
|
ENNReal.eventually_pow_one_div_le
|
[
{
"state_after": "case intro\ny : ℝ≥0∞\nhy : 1 < y\nx : ℝ≥0\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ y",
"state_before": "x : ℝ≥0∞\nhx : x ≠ ⊤\ny : ℝ≥0∞\nhy : 1 < y\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y",
"tactic": "lift x to ℝ≥0 using hx"
},
{
"state_after": "case pos\ny : ℝ≥0∞\nhy : 1 < y\nx : ℝ≥0\nh : y = ⊤\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ y\n\ncase neg\ny : ℝ≥0∞\nhy : 1 < y\nx : ℝ≥0\nh : ¬y = ⊤\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ y",
"state_before": "case intro\ny : ℝ≥0∞\nhy : 1 < y\nx : ℝ≥0\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ y",
"tactic": "by_cases y = ∞"
},
{
"state_after": "no goals",
"state_before": "case pos\ny : ℝ≥0∞\nhy : 1 < y\nx : ℝ≥0\nh : y = ⊤\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ y",
"tactic": "exact eventually_of_forall fun n => h.symm ▸ le_top"
},
{
"state_after": "case neg.intro\nx y : ℝ≥0\nhy : 1 < ↑y\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ ↑y",
"state_before": "case neg\ny : ℝ≥0∞\nhy : 1 < y\nx : ℝ≥0\nh : ¬y = ⊤\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ y",
"tactic": "lift y to ℝ≥0 using h"
},
{
"state_after": "case neg.intro\nx y : ℝ≥0\nhy : 1 < ↑y\nthis : ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ ↑y",
"state_before": "case neg.intro\nx y : ℝ≥0\nhy : 1 < ↑y\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ ↑y",
"tactic": "have := NNReal.eventually_pow_one_div_le x (by exact_mod_cast hy : 1 < y)"
},
{
"state_after": "case neg.intro\nx y : ℝ≥0\nhy : 1 < ↑y\nthis : ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y\nn : ℕ\n⊢ x ^ (1 / ↑n) ≤ y ↔ ↑x ^ (1 / ↑n) ≤ ↑y",
"state_before": "case neg.intro\nx y : ℝ≥0\nhy : 1 < ↑y\nthis : ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y\n⊢ ∀ᶠ (n : ℕ) in atTop, ↑x ^ (1 / ↑n) ≤ ↑y",
"tactic": "refine' this.congr (eventually_of_forall fun n => _)"
},
{
"state_after": "no goals",
"state_before": "case neg.intro\nx y : ℝ≥0\nhy : 1 < ↑y\nthis : ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y\nn : ℕ\n⊢ x ^ (1 / ↑n) ≤ y ↔ ↑x ^ (1 / ↑n) ≤ ↑y",
"tactic": "rw [coe_rpow_of_nonneg x (by positivity : 0 ≤ (1 / n : ℝ)), coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ≥0\nhy : 1 < ↑y\n⊢ 1 < y",
"tactic": "exact_mod_cast hy"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ≥0\nhy : 1 < ↑y\nthis : ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y\nn : ℕ\n⊢ 0 ≤ 1 / ↑n",
"tactic": "positivity"
}
] |
[
500,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arcsin_nonneg
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ sin 0 ≤ x ↔ 0 ≤ x",
"tactic": "rw [sin_zero]"
}
] |
[
204,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiCountable.dist_le_dist_pi_of_dist_lt
|
[
{
"state_after": "no goals",
"state_before": "E : ℕ → Type ?u.306614\nι : Type u_1\ninst✝¹ : Encodable ι\nF : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (F i)\nx y : (i : ι) → F i\ni : ι\nh : dist x y < (1 / 2) ^ encode i\n⊢ dist (x i) (y i) ≤ dist x y",
"tactic": "simpa only [not_le.2 h, false_or_iff] using min_le_iff.1 (min_dist_le_dist_pi x y i)"
}
] |
[
850,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
848,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_sum_le
|
[] |
[
1478,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1476,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.map_mul_right_eq_self
|
[] |
[
93,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
|
[
{
"state_after": "𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.1266934\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx✝ y : E\nr R : ℝ≥0\nk l : ℕ\nx : E\n⊢ ‖↑(changeOriginSeries p k l) fun x_1 => x‖₊ ≤ (∑' (x : { s // Finset.card s = l }), ‖p (k + l)‖₊) * ∏ _i : Fin l, ‖x‖₊",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.1266934\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx✝ y : E\nr R : ℝ≥0\nk l : ℕ\nx : E\n⊢ ‖↑(changeOriginSeries p k l) fun x_1 => x‖₊ ≤ ∑' (x_1 : { s // Finset.card s = l }), ‖p (k + l)‖₊ * ‖x‖₊ ^ l",
"tactic": "rw [NNReal.tsum_mul_right, ← Fin.prod_const]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.1266934\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx✝ y : E\nr R : ℝ≥0\nk l : ℕ\nx : E\n⊢ ‖↑(changeOriginSeries p k l) fun x_1 => x‖₊ ≤ (∑' (x : { s // Finset.card s = l }), ‖p (k + l)‖₊) * ∏ _i : Fin l, ‖x‖₊",
"tactic": "exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)"
}
] |
[
1155,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1151,
1
] |
Mathlib/LinearAlgebra/CrossProduct.lean
|
triple_product_eq_det
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nu v w : Fin 3 → R\n⊢ u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 +\n u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 +\n u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 =\n vecCons u ![v, w] 0 0 * vecCons u ![v, w] 1 1 * vecCons u ![v, w] 2 2 -\n vecCons u ![v, w] 0 0 * vecCons u ![v, w] 1 2 * vecCons u ![v, w] 2 1 -\n vecCons u ![v, w] 0 1 * vecCons u ![v, w] 1 0 * vecCons u ![v, w] 2 2 +\n vecCons u ![v, w] 0 1 * vecCons u ![v, w] 1 2 * vecCons u ![v, w] 2 0 +\n vecCons u ![v, w] 0 2 * vecCons u ![v, w] 1 0 * vecCons u ![v, w] 2 1 -\n vecCons u ![v, w] 0 2 * vecCons u ![v, w] 1 1 * vecCons u ![v, w] 2 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nu v w : Fin 3 → R\n⊢ u ⬝ᵥ ↑(↑crossProduct v) w = det ![u, v, w]",
"tactic": "rw [vec3_dotProduct, cross_apply, det_fin_three]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nu v w : Fin 3 → R\n⊢ u 0 * (v 1 * w 2 - v 2 * w 1) + u 1 * (v 2 * w 0 - v 0 * w 2) + u 2 * (v 0 * w 1 - v 1 * w 0) =\n u 0 * v 1 * w 2 - u 0 * v 2 * w 1 - u 1 * v 0 * w 2 + u 1 * v 2 * w 0 + u 2 * v 0 * w 1 - u 2 * v 1 * w 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nu v w : Fin 3 → R\n⊢ u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 +\n u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 +\n u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 =\n vecCons u ![v, w] 0 0 * vecCons u ![v, w] 1 1 * vecCons u ![v, w] 2 2 -\n vecCons u ![v, w] 0 0 * vecCons u ![v, w] 1 2 * vecCons u ![v, w] 2 1 -\n vecCons u ![v, w] 0 1 * vecCons u ![v, w] 1 0 * vecCons u ![v, w] 2 2 +\n vecCons u ![v, w] 0 1 * vecCons u ![v, w] 1 2 * vecCons u ![v, w] 2 0 +\n vecCons u ![v, w] 0 2 * vecCons u ![v, w] 1 0 * vecCons u ![v, w] 2 1 -\n vecCons u ![v, w] 0 2 * vecCons u ![v, w] 1 1 * vecCons u ![v, w] 2 0",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nu v w : Fin 3 → R\n⊢ u 0 * (v 1 * w 2 - v 2 * w 1) + u 1 * (v 2 * w 0 - v 0 * w 2) + u 2 * (v 0 * w 1 - v 1 * w 0) =\n u 0 * v 1 * w 2 - u 0 * v 2 * w 1 - u 1 * v 0 * w 2 + u 1 * v 2 * w 0 + u 2 * v 0 * w 1 - u 2 * v 1 * w 0",
"tactic": "ring"
}
] |
[
124,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Algebra/Regular/Basic.lean
|
isRegular_mul_iff
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommSemigroup R\na b : R\n⊢ IsRegular (a * b) ↔ IsRegular (a * b) ∧ IsRegular (b * a)",
"state_before": "R : Type u_1\ninst✝ : CommSemigroup R\na b : R\n⊢ IsRegular (a * b) ↔ IsRegular a ∧ IsRegular b",
"tactic": "refine' Iff.trans _ isRegular_mul_and_mul_iff"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommSemigroup R\na b : R\n⊢ IsRegular (a * b) ↔ IsRegular (a * b) ∧ IsRegular (b * a)",
"tactic": "refine' ⟨fun ab => ⟨ab, by rwa [mul_comm]⟩, fun rab => rab.1⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommSemigroup R\na b : R\nab : IsRegular (a * b)\n⊢ IsRegular (b * a)",
"tactic": "rwa [mul_comm]"
}
] |
[
296,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Logic/Denumerable.lean
|
nonempty_denumerable
|
[] |
[
382,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infinitesimal_inv_of_infinite
|
[] |
[
756,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.ascFactorial_of_sub
|
[
{
"state_after": "n k : ℕ\nh : k < n\nt : ℕ := n - succ k\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)",
"state_before": "n k : ℕ\nh : k < n\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)",
"tactic": "let t := n - k.succ"
},
{
"state_after": "n k : ℕ\nh : k < n\nt : ℕ := n - succ k\nht : t = n - succ k := rfl\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)",
"state_before": "n k : ℕ\nh : k < n\nt : ℕ := n - succ k\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)",
"tactic": "let ht : t = n - k.succ := rfl"
},
{
"state_after": "n k : ℕ\nh : k < n\nt : ℕ := n - succ k\nht : t = n - succ k := rfl\nh' : n - k = succ t\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)\n\ncase h'\nn k : ℕ\nh : k < n\nt : ℕ := n - succ k\nht : t = n - succ k := rfl\n⊢ n - k = succ t",
"state_before": "n k : ℕ\nh : k < n\nt : ℕ := n - succ k\nht : t = n - succ k := rfl\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)",
"tactic": "suffices h' : n - k = t.succ"
},
{
"state_after": "no goals",
"state_before": "case h'\nn k : ℕ\nh : k < n\nt : ℕ := n - succ k\nht : t = n - succ k := rfl\n⊢ n - k = succ t",
"tactic": "rw [ht, succ_eq_add_one, ← tsub_tsub_assoc (succ_le_of_lt h) (succ_pos _), succ_sub_one]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\nh : k < n\nt : ℕ := n - succ k\nht : t = n - succ k := rfl\nh' : n - k = succ t\n⊢ (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)",
"tactic": "rw [← ht, h', succ_ascFactorial, ascFactorial_succ]"
}
] |
[
282,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.spanFinset_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.1280273\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1280480\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nι : Type u_1\ns : Finset ι\nf : ι → K\n⊢ spanFinset R₁ s f ≠ 0 ↔ ∃ j, j ∈ s ∧ f j ≠ 0",
"tactic": "simp"
}
] |
[
1281,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1280,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.coeff_inv_aux
|
[
{
"state_after": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\n⊢ (if single () n = 0 then a\n else\n -a *\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0) =\n if n = 0 then a\n else\n -a *\n ∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0",
"state_before": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\n⊢ ↑(coeff R n) (inv.aux a φ) =\n if n = 0 then a\n else\n -a *\n ∑ x in Finset.Nat.antidiagonal n, if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (inv.aux a φ) else 0",
"tactic": "rw [coeff, inv.aux, MvPowerSeries.coeff_inv_aux]"
},
{
"state_after": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\n⊢ (if n = 0 then a\n else\n -a *\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0) =\n if n = 0 then a\n else\n -a *\n ∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0",
"state_before": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\n⊢ (if single () n = 0 then a\n else\n -a *\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0) =\n if n = 0 then a\n else\n -a *\n ∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0",
"tactic": "simp only [Finsupp.single_eq_zero]"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : n = 0\n⊢ a = a\n\ncase inr\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ (-a *\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0) =\n -a *\n ∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0",
"state_before": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\n⊢ (if n = 0 then a\n else\n -a *\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0) =\n if n = 0 then a\n else\n -a *\n ∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0",
"tactic": "split_ifs"
},
{
"state_after": "case inr.e_a\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ (∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0) =\n ∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0",
"state_before": "case inr\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ (-a *\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0) =\n -a *\n ∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0",
"tactic": "congr 1"
},
{
"state_after": "case inr.e_a\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ (∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0) =\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0",
"state_before": "case inr.e_a\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ (∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0) =\n ∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0",
"tactic": "symm"
},
{
"state_after": "case inr.e_a.hi\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ ∀ (a : ℕ × ℕ), a ∈ Finset.Nat.antidiagonal n → (single () a.fst, single () a.snd) ∈ Finsupp.antidiagonal (single () n)\n\ncase inr.e_a.h\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ ∀ (a_1 : ℕ × ℕ),\n a_1 ∈ Finset.Nat.antidiagonal n →\n (if a_1.snd < n then ↑(coeff R a_1.fst) φ * ↑(coeff R a_1.snd) (MvPowerSeries.inv.aux a φ) else 0) =\n if (single () a_1.fst, single () a_1.snd).snd < single () n then\n ↑(MvPowerSeries.coeff R (single () a_1.fst, single () a_1.snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () a_1.fst, single () a_1.snd).snd) (MvPowerSeries.inv.aux a φ)\n else 0\n\ncase inr.e_a.i_inj\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ ∀ (a₁ a₂ : ℕ × ℕ),\n a₁ ∈ Finset.Nat.antidiagonal n →\n a₂ ∈ Finset.Nat.antidiagonal n →\n (single () a₁.fst, single () a₁.snd) = (single () a₂.fst, single () a₂.snd) → a₁ = a₂\n\ncase inr.e_a.i_surj\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ ∀ (b : (Unit →₀ ℕ) × (Unit →₀ ℕ)),\n b ∈ Finsupp.antidiagonal (single () n) → ∃ a ha, b = (single () a.fst, single () a.snd)",
"state_before": "case inr.e_a\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ (∑ x in Finset.Nat.antidiagonal n,\n if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (MvPowerSeries.inv.aux a φ) else 0) =\n ∑ x in Finsupp.antidiagonal (single () n),\n if x.snd < single () n then\n ↑(MvPowerSeries.coeff R x.fst) φ * ↑(MvPowerSeries.coeff R x.snd) (MvPowerSeries.inv.aux a φ)\n else 0",
"tactic": "apply Finset.sum_bij fun (p : ℕ × ℕ) _h => (single () p.1, single () p.2)"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : n = 0\n⊢ a = a",
"tactic": "rfl"
},
{
"state_after": "case inr.e_a.hi.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal n\n⊢ (single () (i, j).fst, single () (i, j).snd) ∈ Finsupp.antidiagonal (single () n)",
"state_before": "case inr.e_a.hi\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ ∀ (a : ℕ × ℕ), a ∈ Finset.Nat.antidiagonal n → (single () a.fst, single () a.snd) ∈ Finsupp.antidiagonal (single () n)",
"tactic": "rintro ⟨i, j⟩ hij"
},
{
"state_after": "case inr.e_a.hi.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = n\n⊢ (single () (i, j).fst, single () (i, j).snd) ∈ Finsupp.antidiagonal (single () n)",
"state_before": "case inr.e_a.hi.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal n\n⊢ (single () (i, j).fst, single () (i, j).snd) ∈ Finsupp.antidiagonal (single () n)",
"tactic": "rw [Finset.Nat.mem_antidiagonal] at hij"
},
{
"state_after": "no goals",
"state_before": "case inr.e_a.hi.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = n\n⊢ (single () (i, j).fst, single () (i, j).snd) ∈ Finsupp.antidiagonal (single () n)",
"tactic": "rw [Finsupp.mem_antidiagonal, ← Finsupp.single_add, hij]"
},
{
"state_after": "case inr.e_a.h.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\n⊢ (if (i, j).snd < n then ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) (MvPowerSeries.inv.aux a φ) else 0) =\n if (single () (i, j).fst, single () (i, j).snd).snd < single () n then\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).snd) (MvPowerSeries.inv.aux a φ)\n else 0",
"state_before": "case inr.e_a.h\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ ∀ (a_1 : ℕ × ℕ),\n a_1 ∈ Finset.Nat.antidiagonal n →\n (if a_1.snd < n then ↑(coeff R a_1.fst) φ * ↑(coeff R a_1.snd) (MvPowerSeries.inv.aux a φ) else 0) =\n if (single () a_1.fst, single () a_1.snd).snd < single () n then\n ↑(MvPowerSeries.coeff R (single () a_1.fst, single () a_1.snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () a_1.fst, single () a_1.snd).snd) (MvPowerSeries.inv.aux a φ)\n else 0",
"tactic": "rintro ⟨i, j⟩ _hij"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ (if (i, j).snd < n then ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) (MvPowerSeries.inv.aux a φ) else 0) =\n if (single () (i, j).fst, single () (i, j).snd).snd < single () n then\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).snd) (MvPowerSeries.inv.aux a φ)\n else 0\n\ncase neg\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n⊢ (if (i, j).snd < n then ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) (MvPowerSeries.inv.aux a φ) else 0) =\n if (single () (i, j).fst, single () (i, j).snd).snd < single () n then\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).snd) (MvPowerSeries.inv.aux a φ)\n else 0",
"state_before": "case inr.e_a.h.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\n⊢ (if (i, j).snd < n then ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) (MvPowerSeries.inv.aux a φ) else 0) =\n if (single () (i, j).fst, single () (i, j).snd).snd < single () n then\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).snd) (MvPowerSeries.inv.aux a φ)\n else 0",
"tactic": "by_cases H : j < n"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) (MvPowerSeries.inv.aux a φ) =\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).snd) (MvPowerSeries.inv.aux a φ)\n\ncase pos.hc\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ (single () (i, j).fst, single () (i, j).snd).snd < single () n",
"state_before": "case pos\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ (if (i, j).snd < n then ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) (MvPowerSeries.inv.aux a φ) else 0) =\n if (single () (i, j).fst, single () (i, j).snd).snd < single () n then\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).snd) (MvPowerSeries.inv.aux a φ)\n else 0",
"tactic": "rw [if_pos H, if_pos]"
},
{
"state_after": "case pos.hc.left\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd i_1) (↑(single () n) i_1)\n\ncase pos.hc.right\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ ¬∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)",
"state_before": "case pos.hc\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ (single () (i, j).fst, single () (i, j).snd).snd < single () n",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) (MvPowerSeries.inv.aux a φ) =\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).snd) (MvPowerSeries.inv.aux a φ)",
"tactic": "rfl"
},
{
"state_after": "case pos.hc.left.unit\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd PUnit.unit) (↑(single () n) PUnit.unit)",
"state_before": "case pos.hc.left\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd i_1) (↑(single () n) i_1)",
"tactic": "rintro ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.hc.left.unit\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd PUnit.unit) (↑(single () n) PUnit.unit)",
"tactic": "simpa [Finsupp.single_eq_same] using le_of_lt H"
},
{
"state_after": "case pos.hc.right\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\nhh :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ False",
"state_before": "case pos.hc.right\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\n⊢ ¬∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)",
"tactic": "intro hh"
},
{
"state_after": "case pos.hc.right\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j ≥ n\nhh :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ False",
"state_before": "case pos.hc.right\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : j < n\nhh :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ False",
"tactic": "rw [lt_iff_not_ge] at H"
},
{
"state_after": "case pos.hc.right\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j ≥ n\nhh :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ j ≥ n",
"state_before": "case pos.hc.right\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j ≥ n\nhh :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ False",
"tactic": "apply H"
},
{
"state_after": "no goals",
"state_before": "case pos.hc.right\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j ≥ n\nhh :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ j ≥ n",
"tactic": "simpa [Finsupp.single_eq_same] using hh ()"
},
{
"state_after": "case neg.hnc\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n⊢ ¬(single () (i, j).fst, single () (i, j).snd).snd < single () n",
"state_before": "case neg\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n⊢ (if (i, j).snd < n then ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) (MvPowerSeries.inv.aux a φ) else 0) =\n if (single () (i, j).fst, single () (i, j).snd).snd < single () n then\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).fst) φ *\n ↑(MvPowerSeries.coeff R (single () (i, j).fst, single () (i, j).snd).snd) (MvPowerSeries.inv.aux a φ)\n else 0",
"tactic": "rw [if_neg H, if_neg]"
},
{
"state_after": "case neg.hnc.intro\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n_h₁ :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd i_1) (↑(single () n) i_1)\nh₂ :\n ¬∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ False",
"state_before": "case neg.hnc\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n⊢ ¬(single () (i, j).fst, single () (i, j).snd).snd < single () n",
"tactic": "rintro ⟨_h₁, h₂⟩"
},
{
"state_after": "case neg.hnc.intro\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n_h₁ :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd i_1) (↑(single () n) i_1)\nh₂ :\n ¬∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)",
"state_before": "case neg.hnc.intro\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n_h₁ :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd i_1) (↑(single () n) i_1)\nh₂ :\n ¬∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ False",
"tactic": "apply h₂"
},
{
"state_after": "case neg.hnc.intro.unit\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n_h₁ :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd i_1) (↑(single () n) i_1)\nh₂ :\n ¬∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ PartialOrder.toPreorder.1.1 (↑(single () n) PUnit.unit) (↑(single () (i, j).fst, single () (i, j).snd).snd PUnit.unit)",
"state_before": "case neg.hnc.intro\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n_h₁ :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd i_1) (↑(single () n) i_1)\nh₂ :\n ¬∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)",
"tactic": "rintro ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.hnc.intro.unit\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\nH : ¬j < n\n_h₁ :\n ∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () (i, j).fst, single () (i, j).snd).snd i_1) (↑(single () n) i_1)\nh₂ :\n ¬∀ (i_1 : Unit),\n PartialOrder.toPreorder.1.1 (↑(single () n) i_1) (↑(single () (i, j).fst, single () (i, j).snd).snd i_1)\n⊢ PartialOrder.toPreorder.1.1 (↑(single () n) PUnit.unit) (↑(single () (i, j).fst, single () (i, j).snd).snd PUnit.unit)",
"tactic": "simpa [Finsupp.single_eq_same] using not_lt.1 H"
},
{
"state_after": "case inr.e_a.i_inj.mk.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j k l : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\n_hkl : (k, l) ∈ Finset.Nat.antidiagonal n\n⊢ (single () (i, j).fst, single () (i, j).snd) = (single () (k, l).fst, single () (k, l).snd) → (i, j) = (k, l)",
"state_before": "case inr.e_a.i_inj\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ ∀ (a₁ a₂ : ℕ × ℕ),\n a₁ ∈ Finset.Nat.antidiagonal n →\n a₂ ∈ Finset.Nat.antidiagonal n →\n (single () a₁.fst, single () a₁.snd) = (single () a₂.fst, single () a₂.snd) → a₁ = a₂",
"tactic": "rintro ⟨i, j⟩ ⟨k, l⟩ _hij _hkl"
},
{
"state_after": "no goals",
"state_before": "case inr.e_a.i_inj.mk.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\ni j k l : ℕ\n_hij : (i, j) ∈ Finset.Nat.antidiagonal n\n_hkl : (k, l) ∈ Finset.Nat.antidiagonal n\n⊢ (single () (i, j).fst, single () (i, j).snd) = (single () (k, l).fst, single () (k, l).snd) → (i, j) = (k, l)",
"tactic": "simpa only [Prod.mk.inj_iff, Finsupp.unique_single_eq_iff] using id"
},
{
"state_after": "case inr.e_a.i_surj.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ ∃ a ha, (f, g) = (single () a.fst, single () a.snd)",
"state_before": "case inr.e_a.i_surj\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\n⊢ ∀ (b : (Unit →₀ ℕ) × (Unit →₀ ℕ)),\n b ∈ Finsupp.antidiagonal (single () n) → ∃ a ha, b = (single () a.fst, single () a.snd)",
"tactic": "rintro ⟨f, g⟩ hfg"
},
{
"state_after": "case inr.e_a.i_surj.mk.refine'_1\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ (↑f (), ↑g ()) ∈ Finset.Nat.antidiagonal n\n\ncase inr.e_a.i_surj.mk.refine'_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ (f, g) = (single () (↑f (), ↑g ()).fst, single () (↑f (), ↑g ()).snd)",
"state_before": "case inr.e_a.i_surj.mk\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ ∃ a ha, (f, g) = (single () a.fst, single () a.snd)",
"tactic": "refine' ⟨(f (), g ()), _, _⟩"
},
{
"state_after": "case inr.e_a.i_surj.mk.refine'_1\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g).fst + (f, g).snd = single () n\n⊢ (↑f (), ↑g ()) ∈ Finset.Nat.antidiagonal n",
"state_before": "case inr.e_a.i_surj.mk.refine'_1\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ (↑f (), ↑g ()) ∈ Finset.Nat.antidiagonal n",
"tactic": "rw [Finsupp.mem_antidiagonal] at hfg"
},
{
"state_after": "no goals",
"state_before": "case inr.e_a.i_surj.mk.refine'_1\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g).fst + (f, g).snd = single () n\n⊢ (↑f (), ↑g ()) ∈ Finset.Nat.antidiagonal n",
"tactic": "rw [Finset.Nat.mem_antidiagonal, ← Finsupp.add_apply, hfg, Finsupp.single_eq_same]"
},
{
"state_after": "case inr.e_a.i_surj.mk.refine'_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ f = single () (↑f (), ↑g ()).fst ∧ g = single () (↑f (), ↑g ()).snd",
"state_before": "case inr.e_a.i_surj.mk.refine'_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ (f, g) = (single () (↑f (), ↑g ()).fst, single () (↑f (), ↑g ()).snd)",
"tactic": "rw [Prod.mk.inj_iff]"
},
{
"state_after": "case inr.e_a.i_surj.mk.refine'_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ f = single () (↑f ()) ∧ g = single () (↑g ())",
"state_before": "case inr.e_a.i_surj.mk.refine'_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ f = single () (↑f (), ↑g ()).fst ∧ g = single () (↑f (), ↑g ()).snd",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case inr.e_a.i_surj.mk.refine'_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : PowerSeries R\nh✝ : ¬n = 0\nf g : Unit →₀ ℕ\nhfg : (f, g) ∈ Finsupp.antidiagonal (single () n)\n⊢ f = single () (↑f ()) ∧ g = single () (↑g ())",
"tactic": "exact ⟨Finsupp.unique_single f, Finsupp.unique_single g⟩"
}
] |
[
1907,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1867,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_diff
|
[] |
[
444,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
441,
1
] |
Mathlib/Topology/Order/Basic.lean
|
pi_Ici_mem_nhds'
|
[] |
[
1473,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1472,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.coe_sub
|
[] |
[
137,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
DifferentiableAt.rpow_const
|
[] |
[
475,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_mul_noncomm
|
[
{
"state_after": "case ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nq r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nhf : ∀ (k : ℕ), Commute (↑f (coeff q k)) x\np : AddMonoidAlgebra R ℕ\n⊢ eval₂ f x ({ toFinsupp := p } * q) = eval₂ f x { toFinsupp := p } * eval₂ f x q",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nhf : ∀ (k : ℕ), Commute (↑f (coeff q k)) x\n⊢ eval₂ f x (p * q) = eval₂ f x p * eval₂ f x q",
"tactic": "rcases p with ⟨p⟩"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nr : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\np q : AddMonoidAlgebra R ℕ\nhf : ∀ (k : ℕ), Commute (↑f (coeff { toFinsupp := q } k)) x\n⊢ eval₂ f x ({ toFinsupp := p } * { toFinsupp := q }) = eval₂ f x { toFinsupp := p } * eval₂ f x { toFinsupp := q }",
"state_before": "case ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nq r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nhf : ∀ (k : ℕ), Commute (↑f (coeff q k)) x\np : AddMonoidAlgebra R ℕ\n⊢ eval₂ f x ({ toFinsupp := p } * q) = eval₂ f x { toFinsupp := p } * eval₂ f x q",
"tactic": "rcases q with ⟨q⟩"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nr : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\np q : AddMonoidAlgebra R ℕ\nhf : ∀ (k : ℕ), Commute (↑f (↑q k)) x\n⊢ eval₂ f x ({ toFinsupp := p } * { toFinsupp := q }) = eval₂ f x { toFinsupp := p } * eval₂ f x { toFinsupp := q }",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nr : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\np q : AddMonoidAlgebra R ℕ\nhf : ∀ (k : ℕ), Commute (↑f (coeff { toFinsupp := q } k)) x\n⊢ eval₂ f x ({ toFinsupp := p } * { toFinsupp := q }) = eval₂ f x { toFinsupp := p } * eval₂ f x { toFinsupp := q }",
"tactic": "simp only [coeff] at hf"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nr : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\np q : AddMonoidAlgebra R ℕ\nhf : ∀ (k : ℕ), Commute (↑f (↑q k)) x\n⊢ ↑(liftNC ↑f ↑(↑(powersHom S) x)) (p * q) = ↑(liftNC ↑f ↑(↑(powersHom S) x)) p * ↑(liftNC ↑f ↑(↑(powersHom S) x)) q",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nr : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\np q : AddMonoidAlgebra R ℕ\nhf : ∀ (k : ℕ), Commute (↑f (↑q k)) x\n⊢ eval₂ f x ({ toFinsupp := p } * { toFinsupp := q }) = eval₂ f x { toFinsupp := p } * eval₂ f x { toFinsupp := q }",
"tactic": "simp only [← ofFinsupp_mul, eval₂_ofFinsupp]"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\nr : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\np q : AddMonoidAlgebra R ℕ\nhf : ∀ (k : ℕ), Commute (↑f (↑q k)) x\n⊢ ↑(liftNC ↑f ↑(↑(powersHom S) x)) (p * q) = ↑(liftNC ↑f ↑(↑(powersHom S) x)) p * ↑(liftNC ↑f ↑(↑(powersHom S) x)) q",
"tactic": "exact liftNC_mul _ _ p q fun {k n} _hn => (hf k).pow_right n"
}
] |
[
184,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Algebra/Group/Conj.lean
|
ConjClasses.exists_rep
|
[] |
[
204,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Data/List/Rdrop.lean
|
List.rdrop_eq_reverse_drop_reverse
|
[
{
"state_after": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ take (length l - n) l = reverse (drop n (reverse l))",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ rdrop l n = reverse (drop n (reverse l))",
"tactic": "rw [rdrop]"
},
{
"state_after": "case H0\nα : Type u_1\np : α → Bool\nl : List α\nn✝ n : ℕ\n⊢ take (length [] - n) [] = reverse (drop n (reverse []))\n\ncase H1\nα : Type u_1\np : α → Bool\nl : List α\nn✝ : ℕ\nxs : List α\nx : α\nIH : ∀ (n : ℕ), take (length xs - n) xs = reverse (drop n (reverse xs))\nn : ℕ\n⊢ take (length (xs ++ [x]) - n) (xs ++ [x]) = reverse (drop n (reverse (xs ++ [x])))",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ take (length l - n) l = reverse (drop n (reverse l))",
"tactic": "induction' l using List.reverseRecOn with xs x IH generalizing n"
},
{
"state_after": "no goals",
"state_before": "case H0\nα : Type u_1\np : α → Bool\nl : List α\nn✝ n : ℕ\n⊢ take (length [] - n) [] = reverse (drop n (reverse []))",
"tactic": "simp"
},
{
"state_after": "case H1.zero\nα : Type u_1\np : α → Bool\nl : List α\nn : ℕ\nxs : List α\nx : α\nIH : ∀ (n : ℕ), take (length xs - n) xs = reverse (drop n (reverse xs))\n⊢ take (length (xs ++ [x]) - Nat.zero) (xs ++ [x]) = reverse (drop Nat.zero (reverse (xs ++ [x])))\n\ncase H1.succ\nα : Type u_1\np : α → Bool\nl : List α\nn : ℕ\nxs : List α\nx : α\nIH : ∀ (n : ℕ), take (length xs - n) xs = reverse (drop n (reverse xs))\nn✝ : ℕ\n⊢ take (length (xs ++ [x]) - Nat.succ n✝) (xs ++ [x]) = reverse (drop (Nat.succ n✝) (reverse (xs ++ [x])))",
"state_before": "case H1\nα : Type u_1\np : α → Bool\nl : List α\nn✝ : ℕ\nxs : List α\nx : α\nIH : ∀ (n : ℕ), take (length xs - n) xs = reverse (drop n (reverse xs))\nn : ℕ\n⊢ take (length (xs ++ [x]) - n) (xs ++ [x]) = reverse (drop n (reverse (xs ++ [x])))",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case H1.zero\nα : Type u_1\np : α → Bool\nl : List α\nn : ℕ\nxs : List α\nx : α\nIH : ∀ (n : ℕ), take (length xs - n) xs = reverse (drop n (reverse xs))\n⊢ take (length (xs ++ [x]) - Nat.zero) (xs ++ [x]) = reverse (drop Nat.zero (reverse (xs ++ [x])))",
"tactic": "simp [take_append]"
},
{
"state_after": "no goals",
"state_before": "case H1.succ\nα : Type u_1\np : α → Bool\nl : List α\nn : ℕ\nxs : List α\nx : α\nIH : ∀ (n : ℕ), take (length xs - n) xs = reverse (drop n (reverse xs))\nn✝ : ℕ\n⊢ take (length (xs ++ [x]) - Nat.succ n✝) (xs ++ [x]) = reverse (drop (Nat.succ n✝) (reverse (xs ++ [x])))",
"tactic": "simp [take_append_eq_append_take, IH]"
}
] |
[
62,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.mul_inv_cancel
|
[
{
"state_after": "no goals",
"state_before": "z : ℂ\nh : z ≠ 0\n⊢ z * z⁻¹ = 1",
"tactic": "rw [inv_def, ← mul_assoc, mul_conj, ← ofReal_mul, mul_inv_cancel (mt normSq_eq_zero.1 h),\n ofReal_one]"
}
] |
[
757,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.pi_pos
|
[
{
"state_after": "no goals",
"state_before": "⊢ 0 < 2",
"tactic": "norm_num"
}
] |
[
166,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.gcd_to_nat
|
[
{
"state_after": "a b : ZNum\n⊢ Nat.gcd (natAbs ↑a) (natAbs ↑b) = Int.gcd ↑a ↑b",
"state_before": "a b : ZNum\n⊢ Nat.gcd ↑(abs a) ↑(abs b) = Int.gcd ↑a ↑b",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "a b : ZNum\n⊢ Nat.gcd (natAbs ↑a) (natAbs ↑b) = Int.gcd ↑a ↑b",
"tactic": "rfl"
}
] |
[
1774,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1773,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_val
|
[] |
[
316,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousWithinAt_insert_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.325537\nδ : Type ?u.325540\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\nx : α\ns : Set α\n⊢ ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x",
"tactic": "simp only [← singleton_union, continuousWithinAt_union, continuousWithinAt_singleton,\n true_and_iff]"
}
] |
[
766,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
763,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.toSubtype_of_subtype
|
[
{
"state_after": "case a.h\nn : ℕ\nα : TypeVec n\np : α ⟹ repeat n Prop\ni : Fin2 n\nx : Subtype_ p i\n⊢ (toSubtype p ⊚ ofSubtype p) i x = id i x",
"state_before": "n : ℕ\nα : TypeVec n\np : α ⟹ repeat n Prop\n⊢ toSubtype p ⊚ ofSubtype p = id",
"tactic": "ext (i x)"
},
{
"state_after": "case a.h.fs\nn n✝ : ℕ\na✝ : Fin2 n✝\na_ih✝ : ∀ {α : TypeVec n✝} (p : α ⟹ repeat n✝ Prop) (x : Subtype_ p a✝), toSubtype p a✝ (ofSubtype p a✝ x) = x\nα : TypeVec (Nat.succ n✝)\np : α ⟹ repeat (Nat.succ n✝) Prop\nx : Subtype_ p (Fin2.fs a✝)\n⊢ toSubtype (dropFun p) a✝ (ofSubtype (dropFun p) a✝ x) = x",
"state_before": "case a.h\nn : ℕ\nα : TypeVec n\np : α ⟹ repeat n Prop\ni : Fin2 n\nx : Subtype_ p i\n⊢ (toSubtype p ⊚ ofSubtype p) i x = id i x",
"tactic": "induction i <;> dsimp only [id, toSubtype, comp, ofSubtype] at *"
},
{
"state_after": "no goals",
"state_before": "case a.h.fs\nn n✝ : ℕ\na✝ : Fin2 n✝\na_ih✝ : ∀ {α : TypeVec n✝} (p : α ⟹ repeat n✝ Prop) (x : Subtype_ p a✝), toSubtype p a✝ (ofSubtype p a✝ x) = x\nα : TypeVec (Nat.succ n✝)\np : α ⟹ repeat (Nat.succ n✝) Prop\nx : Subtype_ p (Fin2.fs a✝)\n⊢ toSubtype (dropFun p) a✝ (ofSubtype (dropFun p) a✝ x) = x",
"tactic": "simp [*]"
}
] |
[
798,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
794,
1
] |
Mathlib/Init/CcLemmas.lean
|
if_eq_of_eq_true
|
[] |
[
86,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.ringHom_rightInverse
|
[] |
[
1172,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1170,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
derivWithin_sub
|
[] |
[
317,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
pow_lt_pow_of_lt_one
|
[] |
[
523,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
522,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_mul
|
[] |
[
135,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.coeFn_mul
|
[] |
[
646,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
645,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousWithinAt_pi
|
[] |
[
573,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.oreDiv_mul_oreDiv
|
[] |
[
257,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Dynamics/FixedPoints/Basic.lean
|
Function.fixedPoints_subset_range
|
[] |
[
154,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Algebra/GCDMonoid/Multiset.lean
|
Multiset.gcd_singleton
|
[] |
[
148,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.lift_prod
|
[
{
"state_after": "case intro\nα β ι : Type u\nc : ι → Type v\n⊢ lift (prod fun i => #c i) = prod fun i => lift ((fun i => #c i) i)",
"state_before": "α β ι : Type u\nc : ι → Cardinal\n⊢ lift (prod c) = prod fun i => lift (c i)",
"tactic": "lift c to ι → Type v using fun _ => trivial"
},
{
"state_after": "case intro\nα β ι : Type u\nc : ι → Type v\n⊢ (#ULift ((i : ι) → c i)) = (#(i : ι) → ULift (c i))",
"state_before": "case intro\nα β ι : Type u\nc : ι → Type v\n⊢ lift (prod fun i => #c i) = prod fun i => lift ((fun i => #c i) i)",
"tactic": "simp only [← mk_pi, ← mk_uLift]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα β ι : Type u\nc : ι → Type v\n⊢ (#ULift ((i : ι) → c i)) = (#(i : ι) → ULift (c i))",
"tactic": "exact mk_congr (Equiv.ulift.trans <| Equiv.piCongrRight fun i => Equiv.ulift.symm)"
}
] |
[
1076,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1072,
1
] |
Mathlib/RingTheory/Localization/FractionRing.lean
|
IsFractionRing.mk'_eq_div
|
[] |
[
165,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Std/Data/List/Lemmas.lean
|
List.isInfix.trans
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl l₁ r₁ l₂ r₂ : List α\n⊢ l₂ ++ l₁ ++ l ++ (r₁ ++ r₂) = l₂ ++ (l₁ ++ l ++ r₁) ++ r₂",
"tactic": "simp only [append_assoc]"
}
] |
[
1598,
95
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1597,
1
] |
Mathlib/CategoryTheory/Sites/CoverPreserving.lean
|
CategoryTheory.compatiblePreservingOfDownwardsClosed
|
[
{
"state_after": "case Compatible\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op G ⋙ ℱ.val) T\nh : Compatible x\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\n⊢ ∀ (ℱ : SheafOfTypes K) {Z : C} {T : Presieve Z} {x : FamilyOfElements (Functor.op F ⋙ ℱ.val) T},\n Compatible x →\n ∀ {Y₁ Y₂ : C} {X : D} (f₁ : X ⟶ F.obj Y₁) (f₂ : X ⟶ F.obj Y₂) {g₁ : Y₁ ⟶ Z} {g₂ : Y₂ ⟶ Z} (hg₁ : T g₁)\n (hg₂ : T g₂), f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂ → ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op G ⋙ ℱ.val) T\nh : Compatible x\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\n⊢ CompatiblePreserving K F",
"tactic": "constructor"
},
{
"state_after": "case Compatible\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)",
"state_before": "case Compatible\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op G ⋙ ℱ.val) T\nh : Compatible x\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\n⊢ ∀ (ℱ : SheafOfTypes K) {Z : C} {T : Presieve Z} {x : FamilyOfElements (Functor.op F ⋙ ℱ.val) T},\n Compatible x →\n ∀ {Y₁ Y₂ : C} {X : D} (f₁ : X ⟶ F.obj Y₁) (f₂ : X ⟶ F.obj Y₂) {g₁ : Y₁ ⟶ Z} {g₂ : Y₂ ⟶ Z} (hg₁ : T g₁)\n (hg₂ : T g₂), f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂ → ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)",
"tactic": "introv hx he"
},
{
"state_after": "case Compatible.mk\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)",
"state_before": "case Compatible\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)",
"tactic": "obtain ⟨X', e⟩ := hF f₁"
},
{
"state_after": "case Compatible.mk.a\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ↑(ℱ.val.mapIso (Iso.op e)).toEquiv (ℱ.val.map f₁.op (x g₁ hg₁)) =\n ↑(ℱ.val.mapIso (Iso.op e)).toEquiv (ℱ.val.map f₂.op (x g₂ hg₂))",
"state_before": "case Compatible.mk\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)",
"tactic": "apply (ℱ.1.mapIso e.op).toEquiv.injective"
},
{
"state_after": "case Compatible.mk.a\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ℱ.val.map (f₁.op ≫ e.hom.op) (x g₁ hg₁) = ℱ.val.map (f₂.op ≫ e.hom.op) (x g₂ hg₂)",
"state_before": "case Compatible.mk.a\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ↑(ℱ.val.mapIso (Iso.op e)).toEquiv (ℱ.val.map f₁.op (x g₁ hg₁)) =\n ↑(ℱ.val.mapIso (Iso.op e)).toEquiv (ℱ.val.map f₂.op (x g₂ hg₂))",
"tactic": "simp only [Iso.op_hom, Iso.toEquiv_fun, ℱ.1.mapIso_hom, ← FunctorToTypes.map_comp_apply]"
},
{
"state_after": "no goals",
"state_before": "case Compatible.mk.a\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ℱ.val.map (f₁.op ≫ e.hom.op) (x g₁ hg₁) = ℱ.val.map (f₂.op ≫ e.hom.op) (x g₂ hg₂)",
"tactic": "simpa using\n hx (F.preimage <| e.hom ≫ f₁) (F.preimage <| e.hom ≫ f₂) hg₁ hg₂\n (F.map_injective <| by simpa using he)"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nA : Type u₃\ninst✝² : Category A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (Functor.op G ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (Functor.op F ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ F.map (F.preimage (e.hom ≫ f₁) ≫ g₁) = F.map (F.preimage (e.hom ≫ f₂) ≫ g₂)",
"tactic": "simpa using he"
}
] |
[
171,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Real.tendsto_comp_exp_atBot
|
[
{
"state_after": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf : ℝ → α\n⊢ Tendsto (fun x => f (exp x)) atBot l ↔ Tendsto (f ∘ exp) atBot l",
"state_before": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf : ℝ → α\n⊢ Tendsto (fun x => f (exp x)) atBot l ↔ Tendsto f (𝓝[Set.Ioi 0] 0) l",
"tactic": "rw [← map_exp_atBot, tendsto_map'_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf : ℝ → α\n⊢ Tendsto (fun x => f (exp x)) atBot l ↔ Tendsto (f ∘ exp) atBot l",
"tactic": "rfl"
}
] |
[
334,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.continuousOn_primitive_interval_left
|
[
{
"state_after": "ι : Type ?u.18752176\n𝕜 : Type ?u.18752179\nE : Type u_1\nF : Type ?u.18752185\nA : Type ?u.18752188\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b b₀ b₁ b₂ : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f [[b, a]]\n⊢ ContinuousOn (fun x => ∫ (t : ℝ) in x..b, f t ∂μ) [[b, a]]",
"state_before": "ι : Type ?u.18752176\n𝕜 : Type ?u.18752179\nE : Type u_1\nF : Type ?u.18752185\nA : Type ?u.18752188\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b b₀ b₁ b₂ : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f [[a, b]]\n⊢ ContinuousOn (fun x => ∫ (t : ℝ) in x..b, f t ∂μ) [[a, b]]",
"tactic": "rw [uIcc_comm a b] at h_int ⊢"
},
{
"state_after": "ι : Type ?u.18752176\n𝕜 : Type ?u.18752179\nE : Type u_1\nF : Type ?u.18752185\nA : Type ?u.18752188\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b b₀ b₁ b₂ : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f [[b, a]]\n⊢ ContinuousOn (fun x => -∫ (x : ℝ) in b..x, f x ∂μ) [[b, a]]",
"state_before": "ι : Type ?u.18752176\n𝕜 : Type ?u.18752179\nE : Type u_1\nF : Type ?u.18752185\nA : Type ?u.18752188\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b b₀ b₁ b₂ : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f [[b, a]]\n⊢ ContinuousOn (fun x => ∫ (t : ℝ) in x..b, f t ∂μ) [[b, a]]",
"tactic": "simp only [integral_symm b]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.18752176\n𝕜 : Type ?u.18752179\nE : Type u_1\nF : Type ?u.18752185\nA : Type ?u.18752188\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b b₀ b₁ b₂ : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f [[b, a]]\n⊢ ContinuousOn (fun x => -∫ (x : ℝ) in b..x, f x ∂μ) [[b, a]]",
"tactic": "exact (continuousOn_primitive_interval h_int).neg"
}
] |
[
1236,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1232,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.blsub_le
|
[] |
[
1812,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1811,
1
] |
Mathlib/Data/List/Sort.lean
|
List.length_split_lt
|
[
{
"state_after": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁ l₂ : List α\nh : split (a :: b :: l) = (l₁, l₂)\nl₁' l₂' : List α\ne : split l = (l₁', l₂')\n⊢ length l₁ < length (a :: b :: l) ∧ length l₂ < length (a :: b :: l)",
"state_before": "α : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁ l₂ : List α\nh : split (a :: b :: l) = (l₁, l₂)\n⊢ length l₁ < length (a :: b :: l) ∧ length l₂ < length (a :: b :: l)",
"tactic": "cases' e : split l with l₁' l₂'"
},
{
"state_after": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁ l₂ : List α\nh : split (a :: b :: l) = (l₁, l₂)\nl₁' l₂' : List α\ne : split l = (l₁', l₂')\nfst_eq✝ : a :: l₁' = l₁\nsnd_eq✝ : b :: l₂' = l₂\n⊢ length l₁ < length (a :: b :: l) ∧ length l₂ < length (a :: b :: l)",
"state_before": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁ l₂ : List α\nh : split (a :: b :: l) = (l₁, l₂)\nl₁' l₂' : List α\ne : split l = (l₁', l₂')\n⊢ length l₁ < length (a :: b :: l) ∧ length l₂ < length (a :: b :: l)",
"tactic": "injection (split_cons_of_eq _ (split_cons_of_eq _ e)).symm.trans h"
},
{
"state_after": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁' l₂' : List α\ne : split l = (l₁', l₂')\nh : split (a :: b :: l) = (a :: l₁', b :: l₂')\n⊢ length (a :: l₁') < length (a :: b :: l) ∧ length (b :: l₂') < length (a :: b :: l)",
"state_before": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁ l₂ : List α\nh : split (a :: b :: l) = (l₁, l₂)\nl₁' l₂' : List α\ne : split l = (l₁', l₂')\nfst_eq✝ : a :: l₁' = l₁\nsnd_eq✝ : b :: l₂' = l₂\n⊢ length l₁ < length (a :: b :: l) ∧ length l₂ < length (a :: b :: l)",
"tactic": "substs l₁ l₂"
},
{
"state_after": "case mk.intro\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁' l₂' : List α\ne : split l = (l₁', l₂')\nh : split (a :: b :: l) = (a :: l₁', b :: l₂')\nh₁ : length l₁' ≤ length l\nh₂ : length l₂' ≤ length l\n⊢ length (a :: l₁') < length (a :: b :: l) ∧ length (b :: l₂') < length (a :: b :: l)",
"state_before": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁' l₂' : List α\ne : split l = (l₁', l₂')\nh : split (a :: b :: l) = (a :: l₁', b :: l₂')\n⊢ length (a :: l₁') < length (a :: b :: l) ∧ length (b :: l₂') < length (a :: b :: l)",
"tactic": "cases' length_split_le e with h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nα : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na b : α\nl l₁' l₂' : List α\ne : split l = (l₁', l₂')\nh : split (a :: b :: l) = (a :: l₁', b :: l₂')\nh₁ : length l₁' ≤ length l\nh₂ : length l₂' ≤ length l\n⊢ length (a :: l₁') < length (a :: b :: l) ∧ length (b :: l₂') < length (a :: b :: l)",
"tactic": "exact ⟨Nat.succ_le_succ (Nat.succ_le_succ h₁), Nat.succ_le_succ (Nat.succ_le_succ h₂)⟩"
}
] |
[
333,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
measurable_const_smul_iff
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.3449887\nβ : Type u_2\nα : Type u_1\ninst✝⁹ : MeasurableSpace M\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : Monoid M\ninst✝⁶ : MulAction M β\ninst✝⁵ : MeasurableSMul M β\ninst✝⁴ : MeasurableSpace α\nf : α → β\nμ : MeasureTheory.Measure α\nG : Type u_3\ninst✝³ : Group G\ninst✝² : MeasurableSpace G\ninst✝¹ : MulAction G β\ninst✝ : MeasurableSMul G β\nc : G\nh : Measurable fun x => c • f x\n⊢ Measurable f",
"tactic": "simpa only [inv_smul_smul] using h.const_smul' c⁻¹"
}
] |
[
716,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.symm_toNonUnitalRingHom_apply_toNonUnitalRingHom_apply
|
[] |
[
642,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
640,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.cyclotomic'.monic
|
[] |
[
102,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
DoubleQuot.quotQuotEquivComm_algebraMap
|
[] |
[
668,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/Data/List/Prime.lean
|
Prime.not_dvd_prod
|
[] |
[
46,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Topology/Algebra/Constructions.lean
|
Units.continuous_val
|
[] |
[
155,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Order/BoundedOrder.lean
|
top_inf_eq
|
[] |
[
492,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
491,
1
] |
Std/Data/List/Basic.lean
|
List.sections_eq_sectionsTR
|
[
{
"state_after": "case h.h\nα : Type u_1\nL : List (List α)\n⊢ sections L = sectionsTR L",
"state_before": "⊢ @sections = @sectionsTR",
"tactic": "funext α L"
},
{
"state_after": "case h.h\nα : Type u_1\nL : List (List α)\n⊢ sections L = bif any L isEmpty then [] else (foldr sectionsTR.go #[[]] L).data",
"state_before": "case h.h\nα : Type u_1\nL : List (List α)\n⊢ sections L = sectionsTR L",
"tactic": "simp [sectionsTR]"
},
{
"state_after": "case h.h.false\nα : Type u_1\nL : List (List α)\ne : any L isEmpty = false\n⊢ sections L = (foldr sectionsTR.go #[[]] L).data",
"state_before": "case h.h\nα : Type u_1\nL : List (List α)\n⊢ sections L = bif any L isEmpty then [] else (foldr sectionsTR.go #[[]] L).data",
"tactic": "cases e : L.any isEmpty <;> simp [sections_eq_nil_of_isEmpty, *]"
},
{
"state_after": "case h.h.false\nα : Type u_1\nL : List (List α)\n⊢ sections L = (foldr sectionsTR.go #[[]] L).data",
"state_before": "case h.h.false\nα : Type u_1\nL : List (List α)\ne : any L isEmpty = false\n⊢ sections L = (foldr sectionsTR.go #[[]] L).data",
"tactic": "clear e"
},
{
"state_after": "case h.h.false.cons\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ sections (l :: L) = (foldr sectionsTR.go #[[]] (l :: L)).data",
"state_before": "case h.h.false\nα : Type u_1\nL : List (List α)\n⊢ sections L = (foldr sectionsTR.go #[[]] L).data",
"tactic": "induction L with | nil => rfl | cons l L IH => ?_"
},
{
"state_after": "case h.h.false.cons\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ (List.bind\n (foldr (fun l acc => foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc.data)\n #[[]] L).data\n fun s => map (fun a => a :: s) l) =\n (foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[]\n (foldr (fun l acc => foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc.data)\n #[[]] L).data).data",
"state_before": "case h.h.false.cons\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ sections (l :: L) = (foldr sectionsTR.go #[[]] (l :: L)).data",
"tactic": "simp [IH, sectionsTR.go, Array.foldl_eq_foldl_data]"
},
{
"state_after": "case h.h.false.cons\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ (List.bind\n (foldr (fun l acc => foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc.data)\n #[[]] L).data\n fun s => map (fun a => a :: s) l) =\n #[].data ++\n List.bind\n (foldr (fun l acc => foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc.data)\n #[[]] L).data\n ?h.h.false.cons.G\n\ncase h.h.false.cons.G\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ List α → List (List α)\n\ncase h.h.false.cons.H\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ ∀ (acc : Array (List α)) (a : List α),\n (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ ?h.h.false.cons.G a",
"state_before": "case h.h.false.cons\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ (List.bind\n (foldr (fun l acc => foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc.data)\n #[[]] L).data\n fun s => map (fun a => a :: s) l) =\n (foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[]\n (foldr (fun l acc => foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc.data)\n #[[]] L).data).data",
"tactic": "rw [Array.foldl_data_eq_bind]"
},
{
"state_after": "case h.h.false.cons.H\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ ∀ (acc : Array (List α)) (a : List α),\n (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ map (fun a_1 => a_1 :: a) l",
"state_before": "case h.h.false.cons\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ (List.bind\n (foldr (fun l acc => foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc.data)\n #[[]] L).data\n fun s => map (fun a => a :: s) l) =\n #[].data ++\n List.bind\n (foldr (fun l acc => foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc.data)\n #[[]] L).data\n ?h.h.false.cons.G\n\ncase h.h.false.cons.G\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ List α → List (List α)\n\ncase h.h.false.cons.H\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ ∀ (acc : Array (List α)) (a : List α),\n (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ ?h.h.false.cons.G a",
"tactic": "rfl"
},
{
"state_after": "case h.h.false.cons.H\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\nacc✝ : Array (List α)\na✝ : List α\n⊢ (foldl (fun acc' a => Array.push acc' (a :: a✝)) acc✝ l).data = acc✝.data ++ map (fun a => a :: a✝) l",
"state_before": "case h.h.false.cons.H\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\n⊢ ∀ (acc : Array (List α)) (a : List α),\n (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ map (fun a_1 => a_1 :: a) l",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case h.h.false.cons.H\nα : Type u_1\nl : List α\nL : List (List α)\nIH : sections L = (foldr sectionsTR.go #[[]] L).data\nacc✝ : Array (List α)\na✝ : List α\n⊢ (foldl (fun acc' a => Array.push acc' (a :: a✝)) acc✝ l).data = acc✝.data ++ map (fun a => a :: a✝) l",
"tactic": "apply Array.foldl_data_eq_map"
},
{
"state_after": "no goals",
"state_before": "case h.h.false.nil\nα : Type u_1\n⊢ sections [] = (foldr sectionsTR.go #[[]] []).data",
"tactic": "rfl"
}
] |
[
928,
40
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
922,
10
] |
Mathlib/Algebra/Group/Prod.lean
|
Units.embedProduct_injective
|
[] |
[
780,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
779,
1
] |
Mathlib/Topology/Algebra/Constructions.lean
|
Units.continuous_coe_inv
|
[] |
[
170,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Algebra/Invertible.lean
|
mul_invOf_eq_iff_eq_mul_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nc a b : α\ninst✝¹ : Monoid α\ninst✝ : Invertible c\n⊢ a * ⅟c = b ↔ a = b * c",
"tactic": "rw [← mul_right_inj_of_invertible (c := c), mul_invOf_mul_self_cancel]"
}
] |
[
317,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
|
Subalgebra.mul_self
|
[
{
"state_after": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ ↑toSubmodule S * ↑toSubmodule S ≤ ↑toSubmodule S\n\ncase a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ ↑toSubmodule S ≤ ↑toSubmodule S * ↑toSubmodule S",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ ↑toSubmodule S * ↑toSubmodule S = ↑toSubmodule S",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ ↑toSubmodule (S ⊔ S) = ↑toSubmodule S",
"state_before": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ ↑toSubmodule S * ↑toSubmodule S ≤ ↑toSubmodule S",
"tactic": "refine' (mul_toSubmodule_le _ _).trans_eq _"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ ↑toSubmodule (S ⊔ S) = ↑toSubmodule S",
"tactic": "rw [sup_idem]"
},
{
"state_after": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ ↑toSubmodule S\n⊢ x ∈ ↑toSubmodule S * ↑toSubmodule S",
"state_before": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ ↑toSubmodule S ≤ ↑toSubmodule S * ↑toSubmodule S",
"tactic": "intro x hx1"
},
{
"state_after": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ ↑toSubmodule S\n⊢ x * 1 ∈ ↑toSubmodule S * ↑toSubmodule S",
"state_before": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ ↑toSubmodule S\n⊢ x ∈ ↑toSubmodule S * ↑toSubmodule S",
"tactic": "rw [← mul_one x]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ ↑toSubmodule S\n⊢ x * 1 ∈ ↑toSubmodule S * ↑toSubmodule S",
"tactic": "exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)"
}
] |
[
47,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
ContinuousOrderHom.cancel_right
|
[] |
[
192,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.comp_zero
|
[] |
[
334,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.tan_ofReal_im
|
[
{
"state_after": "no goals",
"state_before": "x✝ y : ℂ\nx : ℝ\n⊢ (tan ↑x).im = 0",
"tactic": "rw [← ofReal_tan_ofReal_re, ofReal_im]"
}
] |
[
1002,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.Nodup.mem_erase_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8207\nγ : Type ?u.8210\nr : α → α → Prop\ns t : Multiset α\na✝ : α\ninst✝ : DecidableEq α\na b : α\nl : Multiset α\nd : Nodup l\n⊢ a ∈ Multiset.erase l b ↔ a ≠ b ∧ a ∈ l",
"tactic": "rw [d.erase_eq_filter b, mem_filter, and_comm]"
}
] |
[
183,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
HomogeneousLocalization.NumDenSameDeg.den_smul
|
[] |
[
276,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.image_subset_sups_right
|
[] |
[
105,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.fp_unbounded
|
[
{
"state_after": "case h.e'_3\nf : Ordinal → Ordinal\nH : IsNormal f\n⊢ fixedPoints f = ⋂ (i : Unit), fixedPoints f",
"state_before": "f : Ordinal → Ordinal\nH : IsNormal f\n⊢ Set.Unbounded (fun x x_1 => x < x_1) (fixedPoints f)",
"tactic": "convert fp_family_unbounded fun _ : Unit => H"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nf : Ordinal → Ordinal\nH : IsNormal f\n⊢ fixedPoints f = ⋂ (i : Unit), fixedPoints f",
"tactic": "exact (Set.iInter_const _).symm"
}
] |
[
502,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.coe_map
|
[] |
[
1167,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1166,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Subperm.perm_of_length_le
|
[] |
[
443,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Data/Num/Bitwise.lean
|
PosNum.shiftl_succ_eq_bit0_shiftl
|
[] |
[
111,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
fderivWithin_zero_of_not_differentiableWithinAt
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.37783\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.37878\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : ¬DifferentiableWithinAt 𝕜 f s x\nthis : ¬∃ f', HasFDerivWithinAt f f' s x\n⊢ fderivWithin 𝕜 f s x = 0",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.37783\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.37878\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : ¬DifferentiableWithinAt 𝕜 f s x\n⊢ fderivWithin 𝕜 f s x = 0",
"tactic": "have : ¬∃ f', HasFDerivWithinAt f f' s x := h"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.37783\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.37878\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : ¬DifferentiableWithinAt 𝕜 f s x\nthis : ¬∃ f', HasFDerivWithinAt f f' s x\n⊢ fderivWithin 𝕜 f s x = 0",
"tactic": "simp [fderivWithin, this]"
}
] |
[
222,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
strictMonoOn_Iic_of_lt_succ
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ny : α\nhy : y ∈ Iic n\nhxy : x < y\n⊢ ψ x < ψ y",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\n⊢ StrictMonoOn ψ (Iic n)",
"tactic": "intro x hx y hy hxy"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy : (succ^[i]) x ∈ Iic n\nhxy : x < (succ^[i]) x\n⊢ ψ x < ψ ((succ^[i]) x)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ny : α\nhy : y ∈ Iic n\nhxy : x < y\n⊢ ψ x < ψ y",
"tactic": "obtain ⟨i, rfl⟩ := hxy.le.exists_succ_iterate"
},
{
"state_after": "case intro.zero\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy✝ : (succ^[i]) x ∈ Iic n\nhxy✝ : x < (succ^[i]) x\nhy : (succ^[Nat.zero]) x ∈ Iic n\nhxy : x < (succ^[Nat.zero]) x\n⊢ ψ x < ψ ((succ^[Nat.zero]) x)\n\ncase intro.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy✝ : (succ^[i]) x ∈ Iic n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : (succ^[k]) x ∈ Iic n → x < (succ^[k]) x → ψ x < ψ ((succ^[k]) x)\nhy : (succ^[Nat.succ k]) x ∈ Iic n\nhxy : x < (succ^[Nat.succ k]) x\n⊢ ψ x < ψ ((succ^[Nat.succ k]) x)",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy : (succ^[i]) x ∈ Iic n\nhxy : x < (succ^[i]) x\n⊢ ψ x < ψ ((succ^[i]) x)",
"tactic": "induction' i with k ih"
},
{
"state_after": "case intro.succ.zero\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy✝ : (succ^[i]) x ∈ Iic n\nhxy✝ : x < (succ^[i]) x\nih : (succ^[Nat.zero]) x ∈ Iic n → x < (succ^[Nat.zero]) x → ψ x < ψ ((succ^[Nat.zero]) x)\nhy : (succ^[Nat.succ Nat.zero]) x ∈ Iic n\nhxy : x < (succ^[Nat.succ Nat.zero]) x\n⊢ ψ x < ψ ((succ^[Nat.succ Nat.zero]) x)\n\ncase intro.succ.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy✝ : (succ^[i]) x ∈ Iic n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : (succ^[Nat.succ k]) x ∈ Iic n → x < (succ^[Nat.succ k]) x → ψ x < ψ ((succ^[Nat.succ k]) x)\nhy : (succ^[Nat.succ (Nat.succ k)]) x ∈ Iic n\nhxy : x < (succ^[Nat.succ (Nat.succ k)]) x\n⊢ ψ x < ψ ((succ^[Nat.succ (Nat.succ k)]) x)",
"state_before": "case intro.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy✝ : (succ^[i]) x ∈ Iic n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : (succ^[k]) x ∈ Iic n → x < (succ^[k]) x → ψ x < ψ ((succ^[k]) x)\nhy : (succ^[Nat.succ k]) x ∈ Iic n\nhxy : x < (succ^[Nat.succ k]) x\n⊢ ψ x < ψ ((succ^[Nat.succ k]) x)",
"tactic": "cases' k with k"
},
{
"state_after": "case intro.succ.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : (succ^[Nat.succ k]) x ≤ n → x < (succ^[Nat.succ k]) x → ψ x < ψ ((succ^[Nat.succ k]) x)\nhy : (succ^[Nat.succ (Nat.succ k)]) x ≤ n\nhxy : x < (succ^[Nat.succ (Nat.succ k)]) x\n⊢ ψ x < ψ ((succ^[Nat.succ (Nat.succ k)]) x)",
"state_before": "case intro.succ.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy✝ : (succ^[i]) x ∈ Iic n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : (succ^[Nat.succ k]) x ∈ Iic n → x < (succ^[Nat.succ k]) x → ψ x < ψ ((succ^[Nat.succ k]) x)\nhy : (succ^[Nat.succ (Nat.succ k)]) x ∈ Iic n\nhxy : x < (succ^[Nat.succ (Nat.succ k)]) x\n⊢ ψ x < ψ ((succ^[Nat.succ (Nat.succ k)]) x)",
"tactic": "rw [Set.mem_Iic] at *"
},
{
"state_after": "case intro.succ.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))",
"state_before": "case intro.succ.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : (succ^[Nat.succ k]) x ≤ n → x < (succ^[Nat.succ k]) x → ψ x < ψ ((succ^[Nat.succ k]) x)\nhy : (succ^[Nat.succ (Nat.succ k)]) x ≤ n\nhxy : x < (succ^[Nat.succ (Nat.succ k)]) x\n⊢ ψ x < ψ ((succ^[Nat.succ (Nat.succ k)]) x)",
"tactic": "simp only [Function.iterate_succ', Function.comp_apply] at ih hxy hy⊢"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : IsMax ((succ^[k]) x)\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))\n\ncase neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))",
"state_before": "case intro.succ.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))",
"tactic": "by_cases hmax : IsMax ((succ^[k]) x)"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : IsMax (succ ((succ^[k]) x))\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))\n\ncase neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : ¬IsMax (succ ((succ^[k]) x))\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))",
"tactic": "by_cases hmax' : IsMax (succ ((succ^[k]) x))"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : ¬IsMax (succ ((succ^[k]) x))\n⊢ ψ (succ ((succ^[k]) x)) < ψ (succ (succ ((succ^[k]) x)))",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : ¬IsMax (succ ((succ^[k]) x))\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))",
"tactic": "refine'\n lt_trans\n (ih (le_trans (le_succ _) hy)\n (lt_of_le_of_lt (le_succ_iterate k _) (lt_succ_iff_not_isMax.2 hmax)))\n _"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : ¬IsMax (succ ((succ^[k]) x))\n⊢ ψ ((succ^[Nat.succ k]) x) < ψ (succ ((succ^[Nat.succ k]) x))",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : ¬IsMax (succ ((succ^[k]) x))\n⊢ ψ (succ ((succ^[k]) x)) < ψ (succ (succ ((succ^[k]) x)))",
"tactic": "rw [← Function.comp_apply (f := succ), ← Function.iterate_succ']"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : ¬IsMax (succ ((succ^[k]) x))\n⊢ (succ^[Nat.succ k]) x < succ (succ ((succ^[k]) x))",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : ¬IsMax (succ ((succ^[k]) x))\n⊢ ψ ((succ^[Nat.succ k]) x) < ψ (succ ((succ^[Nat.succ k]) x))",
"tactic": "refine' hψ _ (lt_of_lt_of_le _ hy)"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : ¬IsMax (succ ((succ^[k]) x))\n⊢ (succ^[Nat.succ k]) x < succ (succ ((succ^[k]) x))",
"tactic": "rwa [Function.iterate_succ', Function.comp_apply, lt_succ_iff_not_isMax]"
},
{
"state_after": "no goals",
"state_before": "case intro.zero\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy✝ : (succ^[i]) x ∈ Iic n\nhxy✝ : x < (succ^[i]) x\nhy : (succ^[Nat.zero]) x ∈ Iic n\nhxy : x < (succ^[Nat.zero]) x\n⊢ ψ x < ψ ((succ^[Nat.zero]) x)",
"tactic": "simp at hxy"
},
{
"state_after": "no goals",
"state_before": "case intro.succ.zero\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ∈ Iic n\ni : ℕ\nhy✝ : (succ^[i]) x ∈ Iic n\nhxy✝ : x < (succ^[i]) x\nih : (succ^[Nat.zero]) x ∈ Iic n → x < (succ^[Nat.zero]) x → ψ x < ψ ((succ^[Nat.zero]) x)\nhy : (succ^[Nat.succ Nat.zero]) x ∈ Iic n\nhxy : x < (succ^[Nat.succ Nat.zero]) x\n⊢ ψ x < ψ ((succ^[Nat.succ Nat.zero]) x)",
"tactic": "exact hψ _ (lt_of_lt_of_le hxy hy)"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ ((succ^[k]) x)\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : IsMax ((succ^[k]) x)\n⊢ ψ x < ψ (succ ((succ^[k]) x))",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : IsMax ((succ^[k]) x)\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))",
"tactic": "rw [succ_eq_iff_isMax.2 hmax] at hxy ⊢"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ ((succ^[k]) x)\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : IsMax ((succ^[k]) x)\n⊢ ψ x < ψ (succ ((succ^[k]) x))",
"tactic": "exact ih (le_trans (le_succ _) hy) hxy"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ ((succ^[k]) x)\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : IsMax (succ ((succ^[k]) x))\n⊢ ψ x < ψ (succ ((succ^[k]) x))",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ (succ ((succ^[k]) x))\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : IsMax (succ ((succ^[k]) x))\n⊢ ψ x < ψ (succ (succ ((succ^[k]) x)))",
"tactic": "rw [succ_eq_iff_isMax.2 hmax'] at hxy ⊢"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : Preorder β\nψ : α → β\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nn : α\nhψ : ∀ (m : α), m < n → ψ m < ψ (succ m)\nx : α\nhx : x ≤ n\ni : ℕ\nhy✝ : (succ^[i]) x ≤ n\nhxy✝ : x < (succ^[i]) x\nk : ℕ\nih : succ ((succ^[k]) x) ≤ n → x < succ ((succ^[k]) x) → ψ x < ψ (succ ((succ^[k]) x))\nhxy : x < succ ((succ^[k]) x)\nhy : succ (succ ((succ^[k]) x)) ≤ n\nhmax : ¬IsMax ((succ^[k]) x)\nhmax' : IsMax (succ ((succ^[k]) x))\n⊢ ψ x < ψ (succ ((succ^[k]) x))",
"tactic": "exact ih (le_trans (le_succ _) hy) hxy"
}
] |
[
256,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Logic/Basic.lean
|
funext₂
|
[] |
[
602,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
differentiableOn_sub_const_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.509886\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.509981\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\n⊢ DifferentiableOn 𝕜 (fun y => f y - c) s ↔ DifferentiableOn 𝕜 f s",
"tactic": "simp only [sub_eq_add_neg, differentiableOn_add_const_iff]"
}
] |
[
576,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Topology/Sequences.lean
|
IsCompact.tendsto_subseq
|
[] |
[
302,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.Bounded.prod
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.492392\nι : Type ?u.492395\ninst✝¹ : PseudoMetricSpace α\nx : α\ns✝ t✝ : Set α\nr : ℝ\ninst✝ : PseudoMetricSpace β\ns : Set α\nt : Set β\nhs : IsBounded s\nht : IsBounded t\n⊢ IsBounded (s ×ˢ t)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.492392\nι : Type ?u.492395\ninst✝¹ : PseudoMetricSpace α\nx : α\ns✝ t✝ : Set α\nr : ℝ\ninst✝ : PseudoMetricSpace β\ns : Set α\nt : Set β\nhs : Bounded s\nht : Bounded t\n⊢ Bounded (s ×ˢ t)",
"tactic": "rw [bounded_iff_isBounded] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.492392\nι : Type ?u.492395\ninst✝¹ : PseudoMetricSpace α\nx : α\ns✝ t✝ : Set α\nr : ℝ\ninst✝ : PseudoMetricSpace β\ns : Set α\nt : Set β\nhs : IsBounded s\nht : IsBounded t\n⊢ IsBounded (s ×ˢ t)",
"tactic": "exact hs.prod ht"
}
] |
[
2389,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2386,
11
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.isSome_find_iff
|
[
{
"state_after": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝ : DecidablePred p\nh : ∃ a, find p = some a\n⊢ ∃ i, p i",
"state_before": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝ : DecidablePred p\nh : Option.isSome (find p) = true\n⊢ ∃ i, p i",
"tactic": "rw [Option.isSome_iff_exists] at h"
},
{
"state_after": "case intro\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝ : DecidablePred p\ni : Fin (n + 1)\nhi : find p = some i\n⊢ ∃ i, p i",
"state_before": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝ : DecidablePred p\nh : ∃ a, find p = some a\n⊢ ∃ i, p i",
"tactic": "cases' h with i hi"
},
{
"state_after": "no goals",
"state_before": "case intro\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝ : DecidablePred p\ni : Fin (n + 1)\nhi : find p = some i\n⊢ ∃ i, p i",
"tactic": "exact ⟨i, find_spec _ hi⟩"
},
{
"state_after": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\n⊢ Option.isSome\n (Option.rec (if p (last (n + 0)) then some (last (n + 0)) else none)\n (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0))))\n (find fun i => p (castLT i (_ : ↑i < Nat.succ (n + 0))))) =\n true",
"state_before": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\n⊢ Option.isSome (find p) = true",
"tactic": "dsimp [find]"
},
{
"state_after": "case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\n⊢ Option.isSome\n (Option.rec (if p (last (n + 0)) then some (last (n + 0)) else none)\n (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) =\n true\n\ncase some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nj : Fin n\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ Option.isSome\n (Option.rec (if p (last (n + 0)) then some (last (n + 0)) else none)\n (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) (some j)) =\n true",
"state_before": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\n⊢ Option.isSome\n (Option.rec (if p (last (n + 0)) then some (last (n + 0)) else none)\n (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0))))\n (find fun i => p (castLT i (_ : ↑i < Nat.succ (n + 0))))) =\n true",
"tactic": "cases' h : find fun i : Fin n ↦ p (i.castLT (Nat.lt_succ_of_lt i.2)) with j"
},
{
"state_after": "case none.inl\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : p (last (n + 0))\n⊢ Option.isSome (Option.rec (some (last (n + 0))) (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) =\n true\n\ncase none.inr\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\n⊢ Option.isSome (Option.rec none (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) = true",
"state_before": "case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\n⊢ Option.isSome\n (Option.rec (if p (last (n + 0)) then some (last (n + 0)) else none)\n (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) =\n true",
"tactic": "split_ifs with hl"
},
{
"state_after": "no goals",
"state_before": "case none.inl\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : p (last (n + 0))\n⊢ Option.isSome (Option.rec (some (last (n + 0))) (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) =\n true",
"tactic": "exact Option.isSome_some"
},
{
"state_after": "case none.inr\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\nthis : Option.isSome (find fun x => p (castLT x (_ : ↑x < Nat.succ n))) = true\n⊢ Option.isSome (Option.rec none (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) = true",
"state_before": "case none.inr\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\n⊢ Option.isSome (Option.rec none (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) = true",
"tactic": "have := (@isSome_find_iff n (fun x ↦ p (x.castLT (Nat.lt_succ_of_lt x.2))) _).2\n ⟨⟨i, lt_of_le_of_ne (Nat.le_of_lt_succ hin) fun h ↦ by cases h; exact hl hi⟩, hi⟩"
},
{
"state_after": "case none.inr\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\nthis : Option.isSome none = true\n⊢ Option.isSome (Option.rec none (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) = true",
"state_before": "case none.inr\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\nthis : Option.isSome (find fun x => p (castLT x (_ : ↑x < Nat.succ n))) = true\n⊢ Option.isSome (Option.rec none (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) = true",
"tactic": "rw [h] at this"
},
{
"state_after": "no goals",
"state_before": "case none.inr\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\nthis : Option.isSome none = true\n⊢ Option.isSome (Option.rec none (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) none) = true",
"tactic": "exact this"
},
{
"state_after": "case refl\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\nhin : n < n + 1\nhi : p { val := n, isLt := hin }\n⊢ False",
"state_before": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nh✝ : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\nh : i = n\n⊢ False",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last (n + 0))\nhin : n < n + 1\nhi : p { val := n, isLt := hin }\n⊢ False",
"tactic": "exact hl hi"
},
{
"state_after": "no goals",
"state_before": "case some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nx✝¹ : DecidablePred p\nx✝ : ∃ i, p i\ni : ℕ\nhin : i < n + 1\nhi : p { val := i, isLt := hin }\nj : Fin n\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ Option.isSome\n (Option.rec (if p (last (n + 0)) then some (last (n + 0)) else none)\n (fun val => some (castLT val (_ : ↑val < Nat.succ (n + 0)))) (some j)) =\n true",
"tactic": "simp"
}
] |
[
884,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
868,
1
] |
Mathlib/LinearAlgebra/Matrix/Circulant.lean
|
Matrix.circulant_mul_comm
|
[
{
"state_after": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j : n\n⊢ (circulant v ⬝ circulant w) i j = (circulant w ⬝ circulant v) i j",
"state_before": "α : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\n⊢ circulant v ⬝ circulant w = circulant w ⬝ circulant v",
"tactic": "ext (i j)"
},
{
"state_after": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j : n\n⊢ ∑ x : n, v (i - x) * w (x - j) = ∑ x : n, w (i - x) * v (x - j)",
"state_before": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j : n\n⊢ (circulant v ⬝ circulant w) i j = (circulant w ⬝ circulant v) i j",
"tactic": "simp only [mul_apply, circulant_apply, mul_comm]"
},
{
"state_after": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j : n\n⊢ ∀ (x : n),\n v (i - x) * w (x - j) =\n w (i - ↑((Equiv.subLeft i).trans (Equiv.addRight j)) x) * v (↑((Equiv.subLeft i).trans (Equiv.addRight j)) x - j)",
"state_before": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j : n\n⊢ ∑ x : n, v (i - x) * w (x - j) = ∑ x : n, w (i - x) * v (x - j)",
"tactic": "refine' Fintype.sum_equiv ((Equiv.subLeft i).trans (Equiv.addRight j)) _ _ _"
},
{
"state_after": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j x : n\n⊢ v (i - x) * w (x - j) =\n w (i - ↑((Equiv.subLeft i).trans (Equiv.addRight j)) x) * v (↑((Equiv.subLeft i).trans (Equiv.addRight j)) x - j)",
"state_before": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j : n\n⊢ ∀ (x : n),\n v (i - x) * w (x - j) =\n w (i - ↑((Equiv.subLeft i).trans (Equiv.addRight j)) x) * v (↑((Equiv.subLeft i).trans (Equiv.addRight j)) x - j)",
"tactic": "intro x"
},
{
"state_after": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j x : n\n⊢ v (i - x) * w (x - j) = v (i - x) * w (i - (i - x + j))",
"state_before": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j x : n\n⊢ v (i - x) * w (x - j) =\n w (i - ↑((Equiv.subLeft i).trans (Equiv.addRight j)) x) * v (↑((Equiv.subLeft i).trans (Equiv.addRight j)) x - j)",
"tactic": "simp [mul_comm]"
},
{
"state_after": "case a.h.e_a.e_a\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j x : n\n⊢ x - j = i - (i - x + j)",
"state_before": "case a.h\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j x : n\n⊢ v (i - x) * w (x - j) = v (i - x) * w (i - (i - x + j))",
"tactic": "congr 2"
},
{
"state_after": "no goals",
"state_before": "case a.h.e_a.e_a\nα : Type u_1\nβ : Type ?u.24908\nm : Type ?u.24911\nn : Type u_2\nR : Type ?u.24917\ninst✝³ : CommSemigroup α\ninst✝² : AddCommMonoid α\ninst✝¹ : Fintype n\ninst✝ : AddCommGroup n\nv w : n → α\ni j x : n\n⊢ x - j = i - (i - x + j)",
"tactic": "abel"
}
] |
[
152,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/RingTheory/Localization/Away/Basic.lean
|
selfZpow_sub_cast_nat
|
[
{
"state_after": "case pos\nR : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℕ\nh : m ≤ n\n⊢ selfZpow x B (↑n - ↑m) = mk' B (x ^ n) (Submonoid.pow x m)\n\ncase neg\nR : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℕ\nh : ¬m ≤ n\n⊢ selfZpow x B (↑n - ↑m) = mk' B (x ^ n) (Submonoid.pow x m)",
"state_before": "R : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℕ\n⊢ selfZpow x B (↑n - ↑m) = mk' B (x ^ n) (Submonoid.pow x m)",
"tactic": "by_cases h : m ≤ n"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℕ\nh : m ≤ n\n⊢ selfZpow x B (↑n - ↑m) = mk' B (x ^ n) (Submonoid.pow x m)",
"tactic": "rw [IsLocalization.eq_mk'_iff_mul_eq, Submonoid.pow_apply, Subtype.coe_mk, ← Int.ofNat_sub h,\n selfZpow_coe_nat, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]"
},
{
"state_after": "case neg\nR : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℕ\nh : ¬m ≤ n\n⊢ ↑(algebraMap R B) (↑(Submonoid.pow x m) * 1) = ↑(algebraMap R B) (↑(Submonoid.pow x (m - n)) * x ^ n)",
"state_before": "case neg\nR : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℕ\nh : ¬m ≤ n\n⊢ selfZpow x B (↑n - ↑m) = mk' B (x ^ n) (Submonoid.pow x m)",
"tactic": "rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZpow_neg_coe_nat,\n IsLocalization.mk'_eq_iff_eq]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℕ\nh : ¬m ≤ n\n⊢ ↑(algebraMap R B) (↑(Submonoid.pow x m) * 1) = ↑(algebraMap R B) (↑(Submonoid.pow x (m - n)) * x ^ n)",
"tactic": "simp [Submonoid.pow_apply, ← pow_add, Nat.sub_add_cancel (le_of_not_le h)]"
}
] |
[
254,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Std/Logic.lean
|
or_congr
|
[] |
[
255,
99
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
255,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.coe_lt_coe
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.87442\nγ : Type ?u.87445\nδ : Type ?u.87448\ninst✝ : LT α\na b : α\n⊢ ↑a < ↑b ↔ a < b",
"tactic": "simp only [← toDual_lt_toDual_iff, toDual_apply_coe, WithBot.coe_lt_coe, toDual_lt_toDual]"
}
] |
[
1056,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
PerfectClosure.mk_inv
|
[] |
[
504,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.cast_neg
|
[] |
[
1057,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1056,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.filterMap_le_filterMap
|
[] |
[
2185,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2183,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.inclusion.injective
|
[
{
"state_after": "ι : Sort ?u.172133\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nx y : Subgraph G\nh✝ : x ≤ y\nv w : ↑x.verts\nh : ↑(inclusion h✝) v = ↑(inclusion h✝) w\n⊢ v = w",
"state_before": "ι : Sort ?u.172133\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nx y : Subgraph G\nh : x ≤ y\n⊢ Function.Injective ↑(inclusion h)",
"tactic": "intro v w h"
},
{
"state_after": "ι : Sort ?u.172133\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nx y : Subgraph G\nh✝ : x ≤ y\nv w : ↑x.verts\nh :\n ↑(RelHomClass.toFunLike.1\n { toFun := fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) },\n map_rel' :=\n (_ :\n ∀ {a b : ↑x.verts},\n SimpleGraph.Adj (Subgraph.coe x) a b →\n Adj y ↑((fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) }) a)\n ↑((fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) }) b)) }\n v) =\n ↑(RelHomClass.toFunLike.1\n { toFun := fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) },\n map_rel' :=\n (_ :\n ∀ {a b : ↑x.verts},\n SimpleGraph.Adj (Subgraph.coe x) a b →\n Adj y ↑((fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) }) a)\n ↑((fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) }) b)) }\n w)\n⊢ v = w",
"state_before": "ι : Sort ?u.172133\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nx y : Subgraph G\nh✝ : x ≤ y\nv w : ↑x.verts\nh : ↑(inclusion h✝) v = ↑(inclusion h✝) w\n⊢ v = w",
"tactic": "rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.172133\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nx y : Subgraph G\nh✝ : x ≤ y\nv w : ↑x.verts\nh :\n ↑(RelHomClass.toFunLike.1\n { toFun := fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) },\n map_rel' :=\n (_ :\n ∀ {a b : ↑x.verts},\n SimpleGraph.Adj (Subgraph.coe x) a b →\n Adj y ↑((fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) }) a)\n ↑((fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) }) b)) }\n v) =\n ↑(RelHomClass.toFunLike.1\n { toFun := fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) },\n map_rel' :=\n (_ :\n ∀ {a b : ↑x.verts},\n SimpleGraph.Adj (Subgraph.coe x) a b →\n Adj y ↑((fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) }) a)\n ↑((fun v => { val := ↑v, property := (_ : ↑v ∈ y.verts) }) b)) }\n w)\n⊢ v = w",
"tactic": "exact Subtype.ext h"
}
] |
[
732,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
729,
1
] |
Mathlib/Data/Finset/Slice.lean
|
Set.Sized.subsingleton'
|
[] |
[
86,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.isCauSeq_iff_lift
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nf : ℕ → ℚ\nH : IsCauSeq abs f\nε : ℝ\nε0 : ε > 0\nδ : ℚ\nδ0 : 0 < δ\nδε : ↑δ < ε\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abs (f j - f i) < δ\nj : ℕ\nij : j ≥ i\n⊢ abs ((fun i => ↑(f i)) j - (fun i => ↑(f i)) i) < ↑δ",
"tactic": "simpa using (@Rat.cast_lt ℝ _ _ _).2 (hi _ ij)"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nf : ℕ → ℚ\nH : IsCauSeq abs fun i => ↑(f i)\nε : ℚ\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abs ((fun i => ↑(f i)) j - (fun i => ↑(f i)) i) < ↑ε\nj : ℕ\nij : j ≥ i\n⊢ ↑(abs (f j - f i)) < ↑ε",
"tactic": "simpa using hi _ ij"
}
] |
[
662,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/FieldTheory/Separable.lean
|
Polynomial.separable_prod'
|
[
{
"state_after": "R : Type u\ninst✝ : CommRing R\nι : Type u_1\nf : ι → R[X]\ns✝ : Finset ι\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih :\n (∀ (x : ι), x ∈ s → ∀ (y : ι), y ∈ s → x ≠ y → IsCoprime (f x) (f y)) →\n (∀ (x : ι), x ∈ s → Separable (f x)) → Separable (∏ x in s, f x)\nh2 : Separable (f a) ∧ ∀ (x : ι), x ∈ s → Separable (f x)\nh1 :\n ((a ≠ a → IsCoprime (f a) (f a)) ∧ ∀ (x : ι), x ∈ s → a ≠ x → IsCoprime (f a) (f x)) ∧\n (∀ (x : ι), x ∈ s → x ≠ a → IsCoprime (f x) (f a)) ∧\n ∀ (x : ι), x ∈ s → ∀ (x_2 : ι), x_2 ∈ s → x ≠ x_2 → IsCoprime (f x) (f x_2)\n⊢ Separable (∏ x in insert a s, f x)",
"state_before": "R : Type u\ninst✝ : CommRing R\nι : Type u_1\nf : ι → R[X]\ns✝ : Finset ι\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih :\n (∀ (x : ι), x ∈ s → ∀ (y : ι), y ∈ s → x ≠ y → IsCoprime (f x) (f y)) →\n (∀ (x : ι), x ∈ s → Separable (f x)) → Separable (∏ x in s, f x)\nh1 : ∀ (x : ι), x ∈ insert a s → ∀ (y : ι), y ∈ insert a s → x ≠ y → IsCoprime (f x) (f y)\nh2 : ∀ (x : ι), x ∈ insert a s → Separable (f x)\n⊢ Separable (∏ x in insert a s, f x)",
"tactic": "simp_rw [Finset.forall_mem_insert, forall_and] at h1 h2"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nι : Type u_1\nf : ι → R[X]\ns✝ : Finset ι\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih :\n (∀ (x : ι), x ∈ s → ∀ (y : ι), y ∈ s → x ≠ y → IsCoprime (f x) (f y)) →\n (∀ (x : ι), x ∈ s → Separable (f x)) → Separable (∏ x in s, f x)\nh2 : Separable (f a) ∧ ∀ (x : ι), x ∈ s → Separable (f x)\nh1 :\n ((a ≠ a → IsCoprime (f a) (f a)) ∧ ∀ (x : ι), x ∈ s → a ≠ x → IsCoprime (f a) (f x)) ∧\n (∀ (x : ι), x ∈ s → x ≠ a → IsCoprime (f x) (f a)) ∧\n ∀ (x : ι), x ∈ s → ∀ (x_2 : ι), x_2 ∈ s → x ≠ x_2 → IsCoprime (f x) (f x_2)\n⊢ Separable (f a * ∏ x in s, f x)",
"state_before": "R : Type u\ninst✝ : CommRing R\nι : Type u_1\nf : ι → R[X]\ns✝ : Finset ι\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih :\n (∀ (x : ι), x ∈ s → ∀ (y : ι), y ∈ s → x ≠ y → IsCoprime (f x) (f y)) →\n (∀ (x : ι), x ∈ s → Separable (f x)) → Separable (∏ x in s, f x)\nh2 : Separable (f a) ∧ ∀ (x : ι), x ∈ s → Separable (f x)\nh1 :\n ((a ≠ a → IsCoprime (f a) (f a)) ∧ ∀ (x : ι), x ∈ s → a ≠ x → IsCoprime (f a) (f x)) ∧\n (∀ (x : ι), x ∈ s → x ≠ a → IsCoprime (f x) (f a)) ∧\n ∀ (x : ι), x ∈ s → ∀ (x_2 : ι), x_2 ∈ s → x ≠ x_2 → IsCoprime (f x) (f x_2)\n⊢ Separable (∏ x in insert a s, f x)",
"tactic": "rw [prod_insert has]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nι : Type u_1\nf : ι → R[X]\ns✝ : Finset ι\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih :\n (∀ (x : ι), x ∈ s → ∀ (y : ι), y ∈ s → x ≠ y → IsCoprime (f x) (f y)) →\n (∀ (x : ι), x ∈ s → Separable (f x)) → Separable (∏ x in s, f x)\nh2 : Separable (f a) ∧ ∀ (x : ι), x ∈ s → Separable (f x)\nh1 :\n ((a ≠ a → IsCoprime (f a) (f a)) ∧ ∀ (x : ι), x ∈ s → a ≠ x → IsCoprime (f a) (f x)) ∧\n (∀ (x : ι), x ∈ s → x ≠ a → IsCoprime (f x) (f a)) ∧\n ∀ (x : ι), x ∈ s → ∀ (x_2 : ι), x_2 ∈ s → x ≠ x_2 → IsCoprime (f x) (f x_2)\n⊢ Separable (f a * ∏ x in s, f x)",
"tactic": "exact\n h2.1.mul (ih h1.2.2 h2.2)\n (IsCoprime.prod_right fun i his => h1.1.2 i his <| Ne.symm <| ne_of_mem_of_not_mem his has)"
}
] |
[
193,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Data/FunLike/Equiv.lean
|
EquivLike.comp_surjective
|
[] |
[
224,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.isSuccLimit_iff_succ_lt
|
[] |
[
161,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/GroupWithZero/Power.lean
|
zpow_add_one₀
|
[
{
"state_after": "no goals",
"state_before": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (↑n + 1) = a ^ ↑n * a",
"tactic": "simp only [← Int.ofNat_succ, zpow_ofNat, pow_succ']"
},
{
"state_after": "no goals",
"state_before": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\n⊢ a ^ (-[0+1] + 1) = a ^ -[0+1] * a",
"tactic": "erw [zpow_zero, zpow_negSucc, pow_one, inv_mul_cancel ha]"
},
{
"state_after": "no goals",
"state_before": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (-[n + 1+1] + 1) = a ^ -[n + 1+1] * a",
"tactic": "rw [Int.negSucc_eq, zpow_neg, neg_add, neg_add_cancel_right, zpow_neg, ← Int.ofNat_succ,\n zpow_ofNat, zpow_ofNat, pow_succ _ (n + 1), mul_inv_rev, mul_assoc, inv_mul_cancel ha,\n mul_one]"
}
] |
[
83,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.uniformContinuousOn_iff
|
[] |
[
827,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
824,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
Monotone.Icc
|
[] |
[
106,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
11
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
Filter.Tendsto.inner
|
[] |
[
2262,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2260,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
|
strictConvexOn_pow
|
[
{
"state_after": "n : ℕ\nhn : 2 ≤ n\n⊢ StrictMonoOn (deriv fun x => x ^ n) (interior (Ici 0))",
"state_before": "n : ℕ\nhn : 2 ≤ n\n⊢ StrictConvexOn ℝ (Ici 0) fun x => x ^ n",
"tactic": "apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)"
},
{
"state_after": "n : ℕ\nhn : 2 ≤ n\n⊢ StrictMonoOn (fun x => ↑n * x ^ (n - 1)) (Ioi 0)",
"state_before": "n : ℕ\nhn : 2 ≤ n\n⊢ StrictMonoOn (deriv fun x => x ^ n) (interior (Ici 0))",
"tactic": "rw [deriv_pow', interior_Ici]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : 2 ≤ n\n⊢ StrictMonoOn (fun x => ↑n * x ^ (n - 1)) (Ioi 0)",
"tactic": "exact fun x (hx : 0 < x) y hy hxy =>\n mul_lt_mul_of_pos_left (pow_lt_pow_of_lt_left hxy hx.le <| Nat.sub_pos_of_lt hn)\n (Nat.cast_pos.2 <| zero_lt_two.trans_le hn)"
}
] |
[
48,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
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