ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isIntegralCurveAt_comp_add
lemma isIntegralCurveAt_comp_add {dt : ℝ} :
    IsIntegralCurveAt γ v t₀ ↔ IsIntegralCurveAt (γ ∘ (· + dt)) v (t₀ - dt) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ dt : ℝ}
Goal: IsIntegralCurveAt γ v t₀ ↔ IsIntegralCurveAt (γ ∘ fun x => x + dt) v (t₀ - dt)
-- IsIntegralCurve.comp_add
lemma IsIntegralCurve.comp_add (hγ : IsIntegralCurve γ v) (dt : ℝ) :
    IsIntegralCurve (γ ∘ (· + dt)) v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} (hγ : IsIntegralCurve γ v) (dt : ℝ)
Goal: IsIntegralCurve (γ ∘ fun x => x + dt) v
-- isIntegralCurve_comp_add
lemma isIntegralCurve_comp_add {dt : ℝ} :
    IsIntegralCurve γ v ↔ IsIntegralCurve (γ ∘ (· + dt)) v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {dt : ℝ}
Goal: IsIntegralCurve γ v ↔ IsIntegralCurve (γ ∘ fun x => x + dt) v
-- IsIntegralCurveOn.comp_mul
lemma IsIntegralCurveOn.comp_mul (hγ : IsIntegralCurveOn γ v s) (a : ℝ) :
    IsIntegralCurveOn (γ ∘ (· * a)) (a • v) { t | t * a ∈ s } 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} (hγ : IsIntegralCurveOn γ v s) (a : ℝ)
Goal: IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}
-- isIntegralCurveOn_comp_mul_ne_zero
lemma isIntegralCurveOn_comp_mul_ne_zero {a : ℝ} (ha : a ≠ 0) :
    IsIntegralCurveOn γ v s ↔ IsIntegralCurveOn (γ ∘ (· * a)) (a • v) { t | t * a ∈ s } 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} {a : ℝ} (ha : a ≠ 0)
Goal: IsIntegralCurveOn γ v s ↔ IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}
-- IsIntegralCurveAt.comp_mul_ne_zero
lemma IsIntegralCurveAt.comp_mul_ne_zero (hγ : IsIntegralCurveAt γ v t₀) {a : ℝ} (ha : a ≠ 0) :
    IsIntegralCurveAt (γ ∘ (· * a)) (a • v) (t₀ / a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} (hγ : IsIntegralCurveAt γ v t₀) {a : ℝ} (ha : a ≠ 0)
Goal: IsIntegralCurveAt (γ ∘ fun x => x * a) (a • v) (t₀ / a)
-- isIntegralCurveAt_comp_mul_ne_zero
lemma isIntegralCurveAt_comp_mul_ne_zero {a : ℝ} (ha : a ≠ 0) :
    IsIntegralCurveAt γ v t₀ ↔ IsIntegralCurveAt (γ ∘ (· * a)) (a • v) (t₀ / a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ a : ℝ} (ha : a ≠ 0)
Goal: IsIntegralCurveAt γ v t₀ ↔ IsIntegralCurveAt (γ ∘ fun x => x * a) (a • v) (t₀ / a)
-- IsIntegralCurve.comp_mul
lemma IsIntegralCurve.comp_mul (hγ : IsIntegralCurve γ v) (a : ℝ) :
    IsIntegralCurve (γ ∘ (· * a)) (a • v) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} (hγ : IsIntegralCurve γ v) (a : ℝ)
Goal: IsIntegralCurve (γ ∘ fun x => x * a) (a • v)
-- isIntegralCurve_comp_mul_ne_zero
lemma isIntegralCurve_comp_mul_ne_zero {a : ℝ} (ha : a ≠ 0) :
    IsIntegralCurve γ v ↔ IsIntegralCurve (γ ∘ (· * a)) (a • v) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {a : ℝ} (ha : a ≠ 0)
Goal: IsIntegralCurve γ v ↔ IsIntegralCurve (γ ∘ fun x => x * a) (a • v)
-- isIntegralCurve_const
lemma isIntegralCurve_const {x : M} (h : v x = 0) : IsIntegralCurve (fun _ ↦ x) v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {v : (x : M) → TangentSpace I x} {x : M} (h : v x = 0)
Goal: IsIntegralCurve (fun x_1 => x) v
-- exists_isIntegralCurveAt_of_contMDiffAt
theorem exists_isIntegralCurveAt_of_contMDiffAt
    (hv : ContMDiffAt I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)) x₀)
    (hx : I.IsInteriorPoint x₀) :
    ∃ γ : ℝ → M, γ t₀ = x₀ ∧ IsIntegralCurveAt γ v t₀ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {v : (x : M) → TangentSpace I x} (t₀ : ℝ) {x₀ : M} (hv : ContMDiffAt I I.tangent 1 (fun x => { proj := x, snd := v x }) x₀) (hx : I.IsInteriorPoint x₀)
Goal: ∃ γ, γ t₀ = x₀ ∧ IsIntegralCurveAt γ v t₀
-- exists_isIntegralCurveAt_of_contMDiffAt_boundaryless
lemma exists_isIntegralCurveAt_of_contMDiffAt_boundaryless [BoundarylessManifold I M]
    (hv : ContMDiffAt I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)) x₀) :
    ∃ γ : ℝ → M, γ t₀ = x₀ ∧ IsIntegralCurveAt γ v t₀ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {v : (x : M) → TangentSpace I x} (t₀ : ℝ) {x₀ : M} [BoundarylessManifold I M] (hv : ContMDiffAt I I.tangent 1 (fun x => { proj := x, snd := v x }) x₀)
Goal: ∃ γ, γ t₀ = x₀ ∧ IsIntegralCurveAt γ v t₀
-- isIntegralCurveAt_eventuallyEq_of_contMDiffAt
theorem isIntegralCurveAt_eventuallyEq_of_contMDiffAt (hγt₀ : I.IsInteriorPoint (γ t₀))
    (hv : ContMDiffAt I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)) (γ t₀))
    (hγ : IsIntegralCurveAt γ v t₀) (hγ' : IsIntegralCurveAt γ' v t₀) (h : γ t₀ = γ' t₀) :
    γ =ᶠ[𝓝 t₀] γ' 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ γ' : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} (hγt₀ : I.IsInteriorPoint (γ t₀)) (hv : ContMDiffAt I I.tangent 1 (fun x => { proj := x, snd := v x }) (γ t₀)) (hγ : IsIntegralCurveAt γ v t₀) (hγ' : IsIntegralCurveAt γ' v t₀) (h : γ t₀ = γ' t₀)
Goal: γ =ᶠ[𝓝 t₀] γ'
-- isIntegralCurveAt_eventuallyEq_of_contMDiffAt_boundaryless
theorem isIntegralCurveAt_eventuallyEq_of_contMDiffAt_boundaryless [BoundarylessManifold I M]
    (hv : ContMDiffAt I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)) (γ t₀))
    (hγ : IsIntegralCurveAt γ v t₀) (hγ' : IsIntegralCurveAt γ' v t₀) (h : γ t₀ = γ' t₀) :
    γ =ᶠ[𝓝 t₀] γ' 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ γ' : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} [BoundarylessManifold I M] (hv : ContMDiffAt I I.tangent 1 (fun x => { proj := x, snd := v x }) (γ t₀)) (hγ : IsIntegralCurveAt γ v t₀) (hγ' : IsIntegralCurveAt γ' v t₀) (h : γ t₀ = γ' t₀)
Goal: γ =ᶠ[𝓝 t₀] γ'
-- isIntegralCurveOn_Ioo_eqOn_of_contMDiff
theorem isIntegralCurveOn_Ioo_eqOn_of_contMDiff (ht₀ : t₀ ∈ Ioo a b)
    (hγt : ∀ t ∈ Ioo a b, I.IsInteriorPoint (γ t))
    (hv : ContMDiff I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)))
    (hγ : IsIntegralCurveOn γ v (Ioo a b)) (hγ' : IsIntegralCurveOn γ' v (Ioo a b))
    (h : γ t₀ = γ' t₀) : EqOn γ γ' (Ioo a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ γ' : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} [T2Space M] {a b : ℝ} (ht₀ : t₀ ∈ Set.Ioo a b) (hγt : ∀ t ∈ Set.Ioo a b, I.IsInteriorPoint (γ t)) (hv : ContMDiff I I.tangent 1 fun x => { proj := x, snd := v x }) (hγ : IsIntegralCurveOn γ v (Set.Ioo a b)) (hγ' : IsIntegralCurveOn γ' v (Set.Ioo a b)) (h : γ t₀ = γ' t₀)
Goal: Set.EqOn γ γ' (Set.Ioo a b)
-- isIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless
theorem isIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless [BoundarylessManifold I M]
    (ht₀ : t₀ ∈ Ioo a b)
    (hv : ContMDiff I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)))
    (hγ : IsIntegralCurveOn γ v (Ioo a b)) (hγ' : IsIntegralCurveOn γ' v (Ioo a b))
    (h : γ t₀ = γ' t₀) : EqOn γ γ' (Ioo a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ γ' : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} [T2Space M] {a b : ℝ} [BoundarylessManifold I M] (ht₀ : t₀ ∈ Set.Ioo a b) (hv : ContMDiff I I.tangent 1 fun x => { proj := x, snd := v x }) (hγ : IsIntegralCurveOn γ v (Set.Ioo a b)) (hγ' : IsIntegralCurveOn γ' v (Set.Ioo a b)) (h : γ t₀ = γ' t₀)
Goal: Set.EqOn γ γ' (Set.Ioo a b)
-- isIntegralCurve_eq_of_contMDiff
theorem isIntegralCurve_eq_of_contMDiff (hγt : ∀ t, I.IsInteriorPoint (γ t))
    (hv : ContMDiff I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)))
    (hγ : IsIntegralCurve γ v) (hγ' : IsIntegralCurve γ' v) (h : γ t₀ = γ' t₀) : γ = γ' 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ γ' : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} [T2Space M] (hγt : ∀ (t : ℝ), I.IsInteriorPoint (γ t)) (hv : ContMDiff I I.tangent 1 fun x => { proj := x, snd := v x }) (hγ : IsIntegralCurve γ v) (hγ' : IsIntegralCurve γ' v) (h : γ t₀ = γ' t₀)
Goal: γ = γ'
-- isIntegralCurve_Ioo_eq_of_contMDiff_boundaryless
theorem isIntegralCurve_Ioo_eq_of_contMDiff_boundaryless [BoundarylessManifold I M]
    (hv : ContMDiff I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)))
    (hγ : IsIntegralCurve γ v) (hγ' : IsIntegralCurve γ' v) (h : γ t₀ = γ' t₀) : γ = γ' 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ γ' : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} [T2Space M] [BoundarylessManifold I M] (hv : ContMDiff I I.tangent 1 fun x => { proj := x, snd := v x }) (hγ : IsIntegralCurve γ v) (hγ' : IsIntegralCurve γ' v) (h : γ t₀ = γ' t₀)
Goal: γ = γ'
-- IsIntegralCurve.periodic_of_eq
lemma IsIntegralCurve.periodic_of_eq [BoundarylessManifold I M]
    (hγ : IsIntegralCurve γ v)
    (hv : ContMDiff I I.tangent 1 (fun x => (⟨x, v x⟩ : TangentBundle I M)))
    (heq : γ a = γ b) : Periodic γ (a - b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} [T2Space M] {a b : ℝ} [BoundarylessManifold I M] (hγ : IsIntegralCurve γ v) (hv : ContMDiff I I.tangent 1 fun x => { proj := x, snd := v x }) (heq : γ a = γ b)
Goal: Function.Periodic γ (a - b)
-- IsIntegralCurve.periodic_xor_injective
lemma IsIntegralCurve.periodic_xor_injective [BoundarylessManifold I M]
    (hγ : IsIntegralCurve γ v)
    (hv : ContMDiff I I.tangent 1 (fun x => (⟨x, v x⟩ : TangentBundle I M))) :
    Xor' (∃ T > 0, Periodic γ T) (Injective γ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} [T2Space M] [BoundarylessManifold I M] (hγ : IsIntegralCurve γ v) (hv : ContMDiff I I.tangent 1 fun x => { proj := x, snd := v x })
Goal: Xor' (∃ T > 0, Function.Periodic γ T) (Function.Injective γ)
-- AlgebraicGeometry.SheafedSpace.isColimit_exists_rep
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
    ∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimits C] {J : Type v} [CategoryTheory.Category.{v, v} J] (F : CategoryTheory.Functor J (AlgebraicGeometry.SheafedSpace C)) {c : CategoryTheory.Limits.Cocone F} (hc : CategoryTheory.Limits.IsColimit c) (x : ↑↑c.pt.toPresheafedSpace)
Goal: ∃ i y, (c.ι.app i).base y = x
-- AlgebraicGeometry.SheafedSpace.colimit_exists_rep
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
    ∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimits C] {J : Type v} [CategoryTheory.Category.{v, v} J] (F : CategoryTheory.Functor J (AlgebraicGeometry.SheafedSpace C)) (x : ↑↑(CategoryTheory.Limits.colimit F).toPresheafedSpace)
Goal: ∃ i y, (CategoryTheory.Limits.colimit.ι F i).base y = x
-- AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_preimage :
    (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
      (imageBasicOpen f g U s).1 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f g : X ⟶ Y) (U : TopologicalSpace.Opens ↑↑(CategoryTheory.Limits.coequalizer f.val g.val).toPresheafedSpace) (s : ↑((CategoryTheory.Limits.coequalizer f.val g.val).presheaf.obj (Opposite.op U)))
Goal: ⇑(CategoryTheory.Limits.coequalizer.π f.val g.val).base ⁻¹'
    (⇑(CategoryTheory.Limits.coequalizer.π f.val g.val).base ''
      (AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen f g U s).carrier) =
  (AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen f g U s).carrier
-- AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
theorem imageBasicOpen_image_open :
    IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f g : X ⟶ Y) (U : TopologicalSpace.Opens ↑↑(CategoryTheory.Limits.coequalizer f.val g.val).toPresheafedSpace) (s : ↑((CategoryTheory.Limits.coequalizer f.val g.val).presheaf.obj (Opposite.op U)))
Goal: IsOpen
  (⇑(CategoryTheory.Limits.coequalizer.π f.val g.val).base ''
    (AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen f g U s).carrier)
-- AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
    (h : IsLocalRingHom (f.stalkMap x)) :
    IsLocalRingHom (g.stalkMap x) 
Arguments: {X Y : AlgebraicGeometry.RingedSpace} (f g : X ⟶ Y) (H : f = g) (x : ↑↑X.toPresheafedSpace) (h : IsLocalRingHom (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap f x))
Goal: IsLocalRingHom (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap g x)
-- EuclideanGeometry.inversion_mem_perpBisector_inversion_iff
theorem inversion_mem_perpBisector_inversion_iff (hR : R ≠ 0) (hx : x ≠ c) (hy : y ≠ c) :
    inversion c R x ∈ perpBisector c (inversion c R y) ↔ dist x y = dist y c 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c x y : P} {R : ℝ} (hR : R ≠ 0) (hx : x ≠ c) (hy : y ≠ c)
Goal: EuclideanGeometry.inversion c R x ∈ AffineSubspace.perpBisector c (EuclideanGeometry.inversion c R y) ↔
  dist x y = dist y c
-- EuclideanGeometry.inversion_mem_perpBisector_inversion_iff'
theorem inversion_mem_perpBisector_inversion_iff' (hR : R ≠ 0) (hy : y ≠ c) :
    inversion c R x ∈ perpBisector c (inversion c R y) ↔ dist x y = dist y c ∧ x ≠ c 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c x y : P} {R : ℝ} (hR : R ≠ 0) (hy : y ≠ c)
Goal: EuclideanGeometry.inversion c R x ∈ AffineSubspace.perpBisector c (EuclideanGeometry.inversion c R y) ↔
  dist x y = dist y c ∧ x ≠ c
-- EuclideanGeometry.preimage_inversion_perpBisector_inversion
theorem preimage_inversion_perpBisector_inversion (hR : R ≠ 0) (hy : y ≠ c) :
    inversion c R ⁻¹' perpBisector c (inversion c R y) = sphere y (dist y c) \ {c} 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c y : P} {R : ℝ} (hR : R ≠ 0) (hy : y ≠ c)
Goal: EuclideanGeometry.inversion c R ⁻¹' ↑(AffineSubspace.perpBisector c (EuclideanGeometry.inversion c R y)) =
  Metric.sphere y (dist y c) \ {c}
-- EuclideanGeometry.preimage_inversion_perpBisector
theorem preimage_inversion_perpBisector (hR : R ≠ 0) (hy : y ≠ c) :
    inversion c R ⁻¹' perpBisector c y = sphere (inversion c R y) (R ^ 2 / dist y c) \ {c} 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c y : P} {R : ℝ} (hR : R ≠ 0) (hy : y ≠ c)
Goal: EuclideanGeometry.inversion c R ⁻¹' ↑(AffineSubspace.perpBisector c y) =
  Metric.sphere (EuclideanGeometry.inversion c R y) (R ^ 2 / dist y c) \ {c}
-- EuclideanGeometry.image_inversion_perpBisector
theorem image_inversion_perpBisector (hR : R ≠ 0) (hy : y ≠ c) :
    inversion c R '' perpBisector c y = sphere (inversion c R y) (R ^ 2 / dist y c) \ {c} 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c y : P} {R : ℝ} (hR : R ≠ 0) (hy : y ≠ c)
Goal: EuclideanGeometry.inversion c R '' ↑(AffineSubspace.perpBisector c y) =
  Metric.sphere (EuclideanGeometry.inversion c R y) (R ^ 2 / dist y c) \ {c}
-- EuclideanGeometry.preimage_inversion_sphere_dist_center
theorem preimage_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) :
    inversion c R ⁻¹' sphere y (dist y c) =
      insert c (perpBisector c (inversion c R y) : Set P) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c y : P} {R : ℝ} (hR : R ≠ 0) (hy : y ≠ c)
Goal: EuclideanGeometry.inversion c R ⁻¹' Metric.sphere y (dist y c) =
  insert c ↑(AffineSubspace.perpBisector c (EuclideanGeometry.inversion c R y))
-- EuclideanGeometry.image_inversion_sphere_dist_center
theorem image_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) :
    inversion c R '' sphere y (dist y c) = insert c (perpBisector c (inversion c R y) : Set P) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c y : P} {R : ℝ} (hR : R ≠ 0) (hy : y ≠ c)
Goal: EuclideanGeometry.inversion c R '' Metric.sphere y (dist y c) =
  insert c ↑(AffineSubspace.perpBisector c (EuclideanGeometry.inversion c R y))
-- EuclideanGeometry.mapsTo_inversion_affineSubspace_of_mem
theorem mapsTo_inversion_affineSubspace_of_mem {p : AffineSubspace ℝ P} (hp : c ∈ p) :
    MapsTo (inversion c R) p p 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c : P} {R : ℝ} {p : AffineSubspace ℝ P} (hp : c ∈ p)
Goal: Set.MapsTo (EuclideanGeometry.inversion c R) ↑p ↑p
-- EuclideanGeometry.image_inversion_affineSubspace_of_mem
theorem image_inversion_affineSubspace_of_mem {p : AffineSubspace ℝ P} (hR : R ≠ 0) (hp : c ∈ p) :
    inversion c R '' p = p 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c : P} {R : ℝ} {p : AffineSubspace ℝ P} (hR : R ≠ 0) (hp : c ∈ p)
Goal: EuclideanGeometry.inversion c R '' ↑p = ↑p
-- InnerProductGeometry.continuousAt_angle
theorem continuousAt_angle {x : V × V} (hx1 : x.1 ≠ 0) (hx2 : x.2 ≠ 0) :
    ContinuousAt (fun y : V × V => angle y.1 y.2) x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x : V × V} (hx1 : x.1 ≠ 0) (hx2 : x.2 ≠ 0)
Goal: ContinuousAt (fun y => InnerProductGeometry.angle y.1 y.2) x
-- InnerProductGeometry.angle_smul_smul
theorem angle_smul_smul {c : ℝ} (hc : c ≠ 0) (x y : V) : angle (c • x) (c • y) = angle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {c : ℝ} (hc : c ≠ 0) (x y : V)
Goal: InnerProductGeometry.angle (c • x) (c • y) = InnerProductGeometry.angle x y
-- LinearIsometry.angle_map
@[simp]
theorem _root_.LinearIsometry.angle_map {E F : Type*} [NormedAddCommGroup E] [NormedAddCommGroup F]
    [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] (f : E →ₗᵢ[ℝ] F) (u v : E) :
    angle (f u) (f v) = angle u v 
Arguments: {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] (f : E →ₗᵢ[ℝ] F) (u v : E)
Goal: InnerProductGeometry.angle (f u) (f v) = InnerProductGeometry.angle u v
-- Submodule.angle_coe
@[simp, norm_cast]
theorem _root_.Submodule.angle_coe {s : Submodule ℝ V} (x y : s) :
    angle (x : V) (y : V) = angle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {s : Submodule ℝ V} (x y : ↥s)
Goal: InnerProductGeometry.angle ↑x ↑y = InnerProductGeometry.angle x y
-- InnerProductGeometry.cos_angle
theorem cos_angle (x y : V) : Real.cos (angle x y) = ⟪x, y⟫ / (‖x‖ * ‖y‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: Real.cos (InnerProductGeometry.angle x y) = ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)
-- InnerProductGeometry.angle_comm
theorem angle_comm (x y : V) : angle x y = angle y x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: InnerProductGeometry.angle x y = InnerProductGeometry.angle y x
-- InnerProductGeometry.angle_neg_neg
@[simp]
theorem angle_neg_neg (x y : V) : angle (-x) (-y) = angle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: InnerProductGeometry.angle (-x) (-y) = InnerProductGeometry.angle x y
-- InnerProductGeometry.angle_nonneg
theorem angle_nonneg (x y : V) : 0 ≤ angle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: 0 ≤ InnerProductGeometry.angle x y
-- InnerProductGeometry.angle_le_pi
theorem angle_le_pi (x y : V) : angle x y ≤ π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: InnerProductGeometry.angle x y ≤ π
-- InnerProductGeometry.angle_neg_right
theorem angle_neg_right (x y : V) : angle x (-y) = π - angle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: InnerProductGeometry.angle x (-y) = π - InnerProductGeometry.angle x y
-- InnerProductGeometry.angle_neg_left
theorem angle_neg_left (x y : V) : angle (-x) y = π - angle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: InnerProductGeometry.angle (-x) y = π - InnerProductGeometry.angle x y
-- InnerProductGeometry.angle_zero_left
@[simp]
theorem angle_zero_left (x : V) : angle 0 x = π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x : V)
Goal: InnerProductGeometry.angle 0 x = π / 2
-- InnerProductGeometry.angle_zero_right
@[simp]
theorem angle_zero_right (x : V) : angle x 0 = π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x : V)
Goal: InnerProductGeometry.angle x 0 = π / 2