ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgebraicGeometry.PresheafedSpace.ofRestrict_c_app
@[simps]
def ofRestrict {U : TopCat} (X : PresheafedSpace C) {f : U ⟶ (X : TopCat)}
    (h : OpenEmbedding f) : X.restrict h ⟶ X 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {U : TopCat} (X : AlgebraicGeometry.PresheafedSpace C) {f : U ⟶ ↑X} (h : OpenEmbedding ⇑f) (V : (TopologicalSpace.Opens ↑↑X)ᵒᵖ)
Goal: (X.ofRestrict h).c.app V = X.presheaf.map (⋯.adjunction.counit.app (Opposite.unop V)).op
-- AlgebraicGeometry.PresheafedSpace.ofRestrict_base
@[simps]
def ofRestrict {U : TopCat} (X : PresheafedSpace C) {f : U ⟶ (X : TopCat)}
    (h : OpenEmbedding f) : X.restrict h ⟶ X 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {U : TopCat} (X : AlgebraicGeometry.PresheafedSpace C) {f : U ⟶ ↑X} (h : OpenEmbedding ⇑f)
Goal: (X.ofRestrict h).base = f
-- AlgebraicGeometry.PresheafedSpace.restrict_top_presheaf
theorem restrict_top_presheaf (X : PresheafedSpace C) :
    (X.restrict (Opens.openEmbedding ⊤)).presheaf =
      (Opens.inclusionTopIso X.carrier).inv _* X.presheaf 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : AlgebraicGeometry.PresheafedSpace C)
Goal: (X.restrict ⋯).presheaf = (TopCat.Presheaf.pushforward C (TopologicalSpace.Opens.inclusionTopIso ↑X).inv).obj X.presheaf
-- AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c
theorem ofRestrict_top_c (X : PresheafedSpace C) :
    (X.ofRestrict (Opens.openEmbedding ⊤)).c =
      eqToHom
        (by
          rw [restrict_top_presheaf, ← Presheaf.Pushforward.comp_eq]
          erw [Iso.inv_hom_id]
          rw [Presheaf.id_pushforward]
          dsimp) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : AlgebraicGeometry.PresheafedSpace C)
Goal: (X.ofRestrict ⋯).c = CategoryTheory.eqToHom ⋯
-- AlgebraicGeometry.PresheafedSpace.toRestrictTop_c
@[simps]
def toRestrictTop (X : PresheafedSpace C) : X ⟶ X.restrict (Opens.openEmbedding ⊤) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : AlgebraicGeometry.PresheafedSpace C)
Goal: X.toRestrictTop.c = CategoryTheory.eqToHom ⋯
-- AlgebraicGeometry.PresheafedSpace.toRestrictTop_base
@[simps]
def toRestrictTop (X : PresheafedSpace C) : X ⟶ X.restrict (Opens.openEmbedding ⊤) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : AlgebraicGeometry.PresheafedSpace C)
Goal: X.toRestrictTop.base = (TopologicalSpace.Opens.inclusionTopIso ↑X).inv
-- AlgebraicGeometry.PresheafedSpace.restrictTopIso_inv
@[simps]
def restrictTopIso (X : PresheafedSpace C) : X.restrict (Opens.openEmbedding ⊤) ≅ X 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : AlgebraicGeometry.PresheafedSpace C)
Goal: X.restrictTopIso.inv = X.toRestrictTop
-- AlgebraicGeometry.PresheafedSpace.restrictTopIso_hom
@[simps]
def restrictTopIso (X : PresheafedSpace C) : X.restrict (Opens.openEmbedding ⊤) ≅ X 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : AlgebraicGeometry.PresheafedSpace C)
Goal: X.restrictTopIso.hom = X.ofRestrict ⋯
-- AlgebraicGeometry.PresheafedSpace.Γ_map
@[simps]
def Γ : (PresheafedSpace C)ᵒᵖ ⥤ C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C]
Goal: ∀ {X Y : (AlgebraicGeometry.PresheafedSpace C)ᵒᵖ} (f : X ⟶ Y),
  AlgebraicGeometry.PresheafedSpace.Γ.map f = f.unop.c.app (Opposite.op ⊤)
-- AlgebraicGeometry.PresheafedSpace.Γ_obj
@[simps]
def Γ : (PresheafedSpace C)ᵒᵖ ⥤ C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : (AlgebraicGeometry.PresheafedSpace C)ᵒᵖ)
Goal: AlgebraicGeometry.PresheafedSpace.Γ.obj X = (Opposite.unop X).presheaf.obj (Opposite.op ⊤)
-- AlgebraicGeometry.PresheafedSpace.Γ_obj_op
theorem Γ_obj_op (X : PresheafedSpace C) : Γ.obj (op X) = X.presheaf.obj (op ⊤) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : AlgebraicGeometry.PresheafedSpace C)
Goal: AlgebraicGeometry.PresheafedSpace.Γ.obj (Opposite.op X) = X.presheaf.obj (Opposite.op ⊤)
-- AlgebraicGeometry.PresheafedSpace.Γ_map_op
theorem Γ_map_op {X Y : PresheafedSpace C} (f : X ⟶ Y) : Γ.map f.op = f.c.app (op ⊤) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X Y : AlgebraicGeometry.PresheafedSpace C} (f : X ⟶ Y)
Goal: AlgebraicGeometry.PresheafedSpace.Γ.map f.op = f.c.app (Opposite.op ⊤)
-- CategoryTheory.Functor.mapPresheaf_obj_X
@[simp]
theorem mapPresheaf_obj_X (F : C ⥤ D) (X : PresheafedSpace C) :
    (F.mapPresheaf.obj X : TopCat) = (X : TopCat) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_4, u_2} D] (F : C ⥤ D) (X : AlgebraicGeometry.PresheafedSpace C)
Goal: ↑(F.mapPresheaf.obj X) = ↑X
-- CategoryTheory.Functor.mapPresheaf_obj_presheaf
@[simp]
theorem mapPresheaf_obj_presheaf (F : C ⥤ D) (X : PresheafedSpace C) :
    (F.mapPresheaf.obj X).presheaf = X.presheaf ⋙ F 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_4, u_2} D] (F : C ⥤ D) (X : AlgebraicGeometry.PresheafedSpace C)
Goal: (F.mapPresheaf.obj X).presheaf = X.presheaf ⋙ F
-- CategoryTheory.Functor.mapPresheaf_map_f
@[simp]
theorem mapPresheaf_map_f (F : C ⥤ D) {X Y : PresheafedSpace C} (f : X ⟶ Y) :
    (F.mapPresheaf.map f).base = f.base 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_4, u_2} D] (F : C ⥤ D) {X Y : AlgebraicGeometry.PresheafedSpace C} (f : X ⟶ Y)
Goal: (F.mapPresheaf.map f).base = f.base
-- CategoryTheory.Functor.mapPresheaf_map_c
@[simp]
theorem mapPresheaf_map_c (F : C ⥤ D) {X Y : PresheafedSpace C} (f : X ⟶ Y) :
    (F.mapPresheaf.map f).c = whiskerRight f.c F 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_4, u_2} D] (F : C ⥤ D) {X Y : AlgebraicGeometry.PresheafedSpace C} (f : X ⟶ Y)
Goal: (F.mapPresheaf.map f).c = CategoryTheory.whiskerRight f.c F
-- ContDiffWithinAt.inversion
protected theorem ContDiffWithinAt.inversion (hc : ContDiffWithinAt ℝ n c s a)
    (hR : ContDiffWithinAt ℝ n R s a) (hx : ContDiffWithinAt ℝ n x s a) (hne : x a ≠ c a) :
    ContDiffWithinAt ℝ n (fun a ↦ inversion (c a) (R a) (x a)) s a 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : E → F} {R : E → ℝ} {s : Set E} {a : E} {n : ℕ∞} (hc : ContDiffWithinAt ℝ n c s a) (hR : ContDiffWithinAt ℝ n R s a) (hx : ContDiffWithinAt ℝ n x s a) (hne : x a ≠ c a)
Goal: ContDiffWithinAt ℝ n (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) s a
-- ContDiffOn.inversion
protected theorem ContDiffOn.inversion (hc : ContDiffOn ℝ n c s) (hR : ContDiffOn ℝ n R s)
    (hx : ContDiffOn ℝ n x s) (hne : ∀ a ∈ s, x a ≠ c a) :
    ContDiffOn ℝ n (fun a ↦ inversion (c a) (R a) (x a)) s 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : E → F} {R : E → ℝ} {s : Set E} {n : ℕ∞} (hc : ContDiffOn ℝ n c s) (hR : ContDiffOn ℝ n R s) (hx : ContDiffOn ℝ n x s) (hne : ∀ a ∈ s, x a ≠ c a)
Goal: ContDiffOn ℝ n (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) s
-- ContDiffAt.inversion
protected nonrec theorem ContDiffAt.inversion (hc : ContDiffAt ℝ n c a) (hR : ContDiffAt ℝ n R a)
    (hx : ContDiffAt ℝ n x a) (hne : x a ≠ c a) :
    ContDiffAt ℝ n (fun a ↦ inversion (c a) (R a) (x a)) a 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : E → F} {R : E → ℝ} {a : E} {n : ℕ∞} (hc : ContDiffAt ℝ n c a) (hR : ContDiffAt ℝ n R a) (hx : ContDiffAt ℝ n x a) (hne : x a ≠ c a)
Goal: ContDiffAt ℝ n (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) a
-- ContDiff.inversion
protected nonrec theorem ContDiff.inversion (hc : ContDiff ℝ n c) (hR : ContDiff ℝ n R)
    (hx : ContDiff ℝ n x) (hne : ∀ a, x a ≠ c a) :
    ContDiff ℝ n (fun a ↦ inversion (c a) (R a) (x a)) 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : E → F} {R : E → ℝ} {n : ℕ∞} (hc : ContDiff ℝ n c) (hR : ContDiff ℝ n R) (hx : ContDiff ℝ n x) (hne : ∀ (a : E), x a ≠ c a)
Goal: ContDiff ℝ n fun a => EuclideanGeometry.inversion (c a) (R a) (x a)
-- DifferentiableWithinAt.inversion
protected theorem DifferentiableWithinAt.inversion (hc : DifferentiableWithinAt ℝ c s a)
    (hR : DifferentiableWithinAt ℝ R s a) (hx : DifferentiableWithinAt ℝ x s a) (hne : x a ≠ c a) :
    DifferentiableWithinAt ℝ (fun a ↦ inversion (c a) (R a) (x a)) s a 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : E → F} {R : E → ℝ} {s : Set E} {a : E} (hc : DifferentiableWithinAt ℝ c s a) (hR : DifferentiableWithinAt ℝ R s a) (hx : DifferentiableWithinAt ℝ x s a) (hne : x a ≠ c a)
Goal: DifferentiableWithinAt ℝ (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) s a
-- DifferentiableOn.inversion
protected theorem DifferentiableOn.inversion (hc : DifferentiableOn ℝ c s)
    (hR : DifferentiableOn ℝ R s) (hx : DifferentiableOn ℝ x s) (hne : ∀ a ∈ s, x a ≠ c a) :
    DifferentiableOn ℝ (fun a ↦ inversion (c a) (R a) (x a)) s 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : E → F} {R : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ c s) (hR : DifferentiableOn ℝ R s) (hx : DifferentiableOn ℝ x s) (hne : ∀ a ∈ s, x a ≠ c a)
Goal: DifferentiableOn ℝ (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) s
-- DifferentiableAt.inversion
protected theorem DifferentiableAt.inversion (hc : DifferentiableAt ℝ c a)
    (hR : DifferentiableAt ℝ R a) (hx : DifferentiableAt ℝ x a) (hne : x a ≠ c a) :
    DifferentiableAt ℝ (fun a ↦ inversion (c a) (R a) (x a)) a 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : E → F} {R : E → ℝ} {a : E} (hc : DifferentiableAt ℝ c a) (hR : DifferentiableAt ℝ R a) (hx : DifferentiableAt ℝ x a) (hne : x a ≠ c a)
Goal: DifferentiableAt ℝ (fun a => EuclideanGeometry.inversion (c a) (R a) (x a)) a
-- Differentiable.inversion
protected theorem Differentiable.inversion (hc : Differentiable ℝ c)
    (hR : Differentiable ℝ R) (hx : Differentiable ℝ x) (hne : ∀ a, x a ≠ c a) :
    Differentiable ℝ (fun a ↦ inversion (c a) (R a) (x a)) 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : E → F} {R : E → ℝ} (hc : Differentiable ℝ c) (hR : Differentiable ℝ R) (hx : Differentiable ℝ x) (hne : ∀ (a : E), x a ≠ c a)
Goal: Differentiable ℝ fun a => EuclideanGeometry.inversion (c a) (R a) (x a)
-- EuclideanGeometry.hasFDerivAt_inversion
theorem hasFDerivAt_inversion (hx : x ≠ c) :
    HasFDerivAt (inversion c R)
      ((R / dist x c) ^ 2 • (reflection (ℝ ∙ (x - c))ᗮ : F →L[ℝ] F)) x 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {c x : F} {R : ℝ} (hx : x ≠ c)
Goal: HasFDerivAt (EuclideanGeometry.inversion c R)
  ((R / dist x c) ^ 2 •
    ↑{ toLinearEquiv := (reflection (Submodule.span ℝ {x - c})ᗮ).toLinearEquiv, continuous_toFun := ⋯,
        continuous_invFun := ⋯ })
  x
-- EuclideanGeometry.Sphere.mk_center
theorem Sphere.mk_center (c : P) (r : ℝ) : (⟨c, r⟩ : Sphere P).center = c 
Arguments: {P : Type u_2} [MetricSpace P] (c : P) (r : ℝ)
Goal: { center := c, radius := r }.center = c
-- EuclideanGeometry.Sphere.mk_radius
theorem Sphere.mk_radius (c : P) (r : ℝ) : (⟨c, r⟩ : Sphere P).radius = r 
Arguments: {P : Type u_2} [MetricSpace P] (c : P) (r : ℝ)
Goal: { center := c, radius := r }.radius = r
-- EuclideanGeometry.Sphere.mk_center_radius
@[simp]
theorem Sphere.mk_center_radius (s : Sphere P) : (⟨s.center, s.radius⟩ : Sphere P) = s 
Arguments: {P : Type u_2} [MetricSpace P] (s : EuclideanGeometry.Sphere P)
Goal: { center := s.center, radius := s.radius } = s
-- EuclideanGeometry.Sphere.coe_mk
@[simp]
theorem Sphere.coe_mk (c : P) (r : ℝ) : ↑(⟨c, r⟩ : Sphere P) = Metric.sphere c r 
Arguments: {P : Type u_2} [MetricSpace P] (c : P) (r : ℝ)
Goal: Metric.sphere { center := c, radius := r }.center { center := c, radius := r }.radius = Metric.sphere c r
-- EuclideanGeometry.Sphere.mem_coe
theorem Sphere.mem_coe {p : P} {s : Sphere P} : p ∈ (s : Set P) ↔ p ∈ s 
Arguments: {P : Type u_2} [MetricSpace P] {p : P} {s : EuclideanGeometry.Sphere P}
Goal: p ∈ Metric.sphere s.center s.radius ↔ p ∈ s
-- EuclideanGeometry.Sphere.mem_coe'
@[simp]
theorem Sphere.mem_coe' {p : P} {s : Sphere P} : dist p s.center = s.radius ↔ p ∈ s 
Arguments: {P : Type u_2} [MetricSpace P] {p : P} {s : EuclideanGeometry.Sphere P}
Goal: dist p s.center = s.radius ↔ p ∈ s
-- EuclideanGeometry.mem_sphere
theorem mem_sphere {p : P} {s : Sphere P} : p ∈ s ↔ dist p s.center = s.radius 
Arguments: {P : Type u_2} [MetricSpace P] {p : P} {s : EuclideanGeometry.Sphere P}
Goal: p ∈ s ↔ dist p s.center = s.radius
-- EuclideanGeometry.mem_sphere'
theorem mem_sphere' {p : P} {s : Sphere P} : p ∈ s ↔ dist s.center p = s.radius 
Arguments: {P : Type u_2} [MetricSpace P] {p : P} {s : EuclideanGeometry.Sphere P}
Goal: p ∈ s ↔ dist s.center p = s.radius
-- EuclideanGeometry.subset_sphere
theorem subset_sphere {ps : Set P} {s : Sphere P} : ps ⊆ s ↔ ∀ p ∈ ps, p ∈ s 
Arguments: {P : Type u_2} [MetricSpace P] {ps : Set P} {s : EuclideanGeometry.Sphere P}
Goal: ps ⊆ Metric.sphere s.center s.radius ↔ ∀ p ∈ ps, p ∈ s
-- EuclideanGeometry.dist_of_mem_subset_sphere
theorem dist_of_mem_subset_sphere {p : P} {ps : Set P} {s : Sphere P} (hp : p ∈ ps)
    (hps : ps ⊆ (s : Set P)) : dist p s.center = s.radius 
Arguments: {P : Type u_2} [MetricSpace P] {p : P} {ps : Set P} {s : EuclideanGeometry.Sphere P} (hp : p ∈ ps) (hps : ps ⊆ Metric.sphere s.center s.radius)
Goal: dist p s.center = s.radius
-- EuclideanGeometry.dist_of_mem_subset_mk_sphere
theorem dist_of_mem_subset_mk_sphere {p c : P} {ps : Set P} {r : ℝ} (hp : p ∈ ps)
    (hps : ps ⊆ ↑(⟨c, r⟩ : Sphere P)) : dist p c = r 
Arguments: {P : Type u_2} [MetricSpace P] {p c : P} {ps : Set P} {r : ℝ} (hp : p ∈ ps) (hps : ps ⊆ Metric.sphere { center := c, radius := r }.center { center := c, radius := r }.radius)
Goal: dist p c = r
-- EuclideanGeometry.Sphere.ne_iff
theorem Sphere.ne_iff {s₁ s₂ : Sphere P} :
    s₁ ≠ s₂ ↔ s₁.center ≠ s₂.center ∨ s₁.radius ≠ s₂.radius 
Arguments: {P : Type u_2} [MetricSpace P] {s₁ s₂ : EuclideanGeometry.Sphere P}
Goal: s₁ ≠ s₂ ↔ s₁.center ≠ s₂.center ∨ s₁.radius ≠ s₂.radius
-- EuclideanGeometry.Sphere.center_eq_iff_eq_of_mem
theorem Sphere.center_eq_iff_eq_of_mem {s₁ s₂ : Sphere P} {p : P} (hs₁ : p ∈ s₁) (hs₂ : p ∈ s₂) :
    s₁.center = s₂.center ↔ s₁ = s₂ 
Arguments: {P : Type u_2} [MetricSpace P] {s₁ s₂ : EuclideanGeometry.Sphere P} {p : P} (hs₁ : p ∈ s₁) (hs₂ : p ∈ s₂)
Goal: s₁.center = s₂.center ↔ s₁ = s₂
-- EuclideanGeometry.Sphere.center_ne_iff_ne_of_mem
theorem Sphere.center_ne_iff_ne_of_mem {s₁ s₂ : Sphere P} {p : P} (hs₁ : p ∈ s₁) (hs₂ : p ∈ s₂) :
    s₁.center ≠ s₂.center ↔ s₁ ≠ s₂ 
Arguments: {P : Type u_2} [MetricSpace P] {s₁ s₂ : EuclideanGeometry.Sphere P} {p : P} (hs₁ : p ∈ s₁) (hs₂ : p ∈ s₂)
Goal: s₁.center ≠ s₂.center ↔ s₁ ≠ s₂
-- EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere
theorem dist_center_eq_dist_center_of_mem_sphere {p₁ p₂ : P} {s : Sphere P} (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) : dist p₁ s.center = dist p₂ s.center 
Arguments: {P : Type u_2} [MetricSpace P] {p₁ p₂ : P} {s : EuclideanGeometry.Sphere P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s)
Goal: dist p₁ s.center = dist p₂ s.center
-- EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere'
theorem dist_center_eq_dist_center_of_mem_sphere' {p₁ p₂ : P} {s : Sphere P} (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) : dist s.center p₁ = dist s.center p₂ 
Arguments: {P : Type u_2} [MetricSpace P] {p₁ p₂ : P} {s : EuclideanGeometry.Sphere P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s)
Goal: dist s.center p₁ = dist s.center p₂
-- EuclideanGeometry.cospherical_def
theorem cospherical_def (ps : Set P) :
    Cospherical ps ↔ ∃ (center : P) (radius : ℝ), ∀ p ∈ ps, dist p center = radius 
Arguments: {P : Type u_2} [MetricSpace P] (ps : Set P)
Goal: EuclideanGeometry.Cospherical ps ↔ ∃ center radius, ∀ p ∈ ps, dist p center = radius
-- EuclideanGeometry.cospherical_iff_exists_sphere
theorem cospherical_iff_exists_sphere {ps : Set P} :
    Cospherical ps ↔ ∃ s : Sphere P, ps ⊆ (s : Set P) 
Arguments: {P : Type u_2} [MetricSpace P] {ps : Set P}
Goal: EuclideanGeometry.Cospherical ps ↔ ∃ s, ps ⊆ Metric.sphere s.center s.radius
-- EuclideanGeometry.Sphere.cospherical
theorem Sphere.cospherical (s : Sphere P) : Cospherical (s : Set P) 
Arguments: {P : Type u_2} [MetricSpace P] (s : EuclideanGeometry.Sphere P)
Goal: EuclideanGeometry.Cospherical (Metric.sphere s.center s.radius)
-- EuclideanGeometry.Cospherical.subset
theorem Cospherical.subset {ps₁ ps₂ : Set P} (hs : ps₁ ⊆ ps₂) (hc : Cospherical ps₂) :
    Cospherical ps₁ 
Arguments: {P : Type u_2} [MetricSpace P] {ps₁ ps₂ : Set P} (hs : ps₁ ⊆ ps₂) (hc : EuclideanGeometry.Cospherical ps₂)
Goal: EuclideanGeometry.Cospherical ps₁
-- EuclideanGeometry.cospherical_empty
theorem cospherical_empty [Nonempty P] : Cospherical (∅ : Set P) 
Arguments: {P : Type u_2} [MetricSpace P] [Nonempty P]
Goal: EuclideanGeometry.Cospherical ∅
-- EuclideanGeometry.cospherical_singleton
theorem cospherical_singleton (p : P) : Cospherical ({p} : Set P) 
Arguments: {P : Type u_2} [MetricSpace P] (p : P)
Goal: EuclideanGeometry.Cospherical {p}
-- EuclideanGeometry.cospherical_pair
theorem cospherical_pair (p₁ p₂ : P) : Cospherical ({p₁, p₂} : Set P) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ p₂ : P)
Goal: EuclideanGeometry.Cospherical {p₁, p₂}
-- EuclideanGeometry.Concyclic.Coplanar
structure Concyclic (ps : Set P) : Prop where
  Cospherical : Cospherical ps
  Coplanar : Coplanar ℝ ps
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {ps : Set P} (self : EuclideanGeometry.Concyclic ps)
Goal: Coplanar ℝ ps
-- EuclideanGeometry.Concyclic.Cospherical
structure Concyclic (ps : Set P) : Prop where
  Cospherical : Cospherical ps
  Coplanar : Coplanar ℝ ps
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {ps : Set P} (self : EuclideanGeometry.Concyclic ps)
Goal: EuclideanGeometry.Cospherical ps
-- EuclideanGeometry.Concyclic.subset
theorem Concyclic.subset {ps₁ ps₂ : Set P} (hs : ps₁ ⊆ ps₂) (h : Concyclic ps₂) : Concyclic ps₁ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {ps₁ ps₂ : Set P} (hs : ps₁ ⊆ ps₂) (h : EuclideanGeometry.Concyclic ps₂)
Goal: EuclideanGeometry.Concyclic ps₁
-- EuclideanGeometry.concyclic_empty
theorem concyclic_empty : Concyclic (∅ : Set P) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P]
Goal: EuclideanGeometry.Concyclic ∅
-- EuclideanGeometry.concyclic_singleton
theorem concyclic_singleton (p : P) : Concyclic ({p} : Set P) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p : P)
Goal: EuclideanGeometry.Concyclic {p}
-- EuclideanGeometry.concyclic_pair
theorem concyclic_pair (p₁ p₂ : P) : Concyclic ({p₁, p₂} : Set P) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ p₂ : P)
Goal: EuclideanGeometry.Concyclic {p₁, p₂}
-- EuclideanGeometry.Cospherical.affineIndependent
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
    (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : Set P} (hs : EuclideanGeometry.Cospherical s) {p : Fin 3 → P} (hps : Set.range p ⊆ s) (hpi : Function.Injective p)
Goal: AffineIndependent ℝ p
-- EuclideanGeometry.Cospherical.affineIndependent_of_mem_of_ne
theorem Cospherical.affineIndependent_of_mem_of_ne {s : Set P} (hs : Cospherical s) {p₁ p₂ p₃ : P}
    (h₁ : p₁ ∈ s) (h₂ : p₂ ∈ s) (h₃ : p₃ ∈ s) (h₁₂ : p₁ ≠ p₂) (h₁₃ : p₁ ≠ p₃) (h₂₃ : p₂ ≠ p₃) :
    AffineIndependent ℝ ![p₁, p₂, p₃] 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : Set P} (hs : EuclideanGeometry.Cospherical s) {p₁ p₂ p₃ : P} (h₁ : p₁ ∈ s) (h₂ : p₂ ∈ s) (h₃ : p₃ ∈ s) (h₁₂ : p₁ ≠ p₂) (h₁₃ : p₁ ≠ p₃) (h₂₃ : p₂ ≠ p₃)
Goal: AffineIndependent ℝ ![p₁, p₂, p₃]
-- EuclideanGeometry.Cospherical.affineIndependent_of_ne
theorem Cospherical.affineIndependent_of_ne {p₁ p₂ p₃ : P} (hs : Cospherical ({p₁, p₂, p₃} : Set P))
    (h₁₂ : p₁ ≠ p₂) (h₁₃ : p₁ ≠ p₃) (h₂₃ : p₂ ≠ p₃) : AffineIndependent ℝ ![p₁, p₂, p₃] 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (hs : EuclideanGeometry.Cospherical {p₁, p₂, p₃}) (h₁₂ : p₁ ≠ p₂) (h₁₃ : p₁ ≠ p₃) (h₂₃ : p₂ ≠ p₃)
Goal: AffineIndependent ℝ ![p₁, p₂, p₃]
-- EuclideanGeometry.inner_vsub_vsub_of_mem_sphere_of_mem_sphere
theorem inner_vsub_vsub_of_mem_sphere_of_mem_sphere {p₁ p₂ : P} {s₁ s₂ : Sphere P} (hp₁s₁ : p₁ ∈ s₁)
    (hp₂s₁ : p₂ ∈ s₁) (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂) :
    ⟪s₂.center -ᵥ s₁.center, p₂ -ᵥ p₁⟫ = 0 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ : P} {s₁ s₂ : EuclideanGeometry.Sphere P} (hp₁s₁ : p₁ ∈ s₁) (hp₂s₁ : p₂ ∈ s₁) (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂)
Goal: ⟪s₂.center -ᵥ s₁.center, p₂ -ᵥ p₁⟫_ℝ = 0
-- EuclideanGeometry.eq_of_mem_sphere_of_mem_sphere_of_mem_of_finrank_eq_two
theorem eq_of_mem_sphere_of_mem_sphere_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {s₁ s₂ : Sphere P}
    {p₁ p₂ p : P} (hs₁ : s₁.center ∈ s) (hs₂ : s₂.center ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s)
    (hps : p ∈ s) (hs : s₁ ≠ s₂) (hp : p₁ ≠ p₂) (hp₁s₁ : p₁ ∈ s₁) (hp₂s₁ : p₂ ∈ s₁) (hps₁ : p ∈ s₁)
    (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂) (hps₂ : p ∈ s₂) : p = p₁ ∨ p = p₂ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} [FiniteDimensional ℝ ↥s.direction] (hd : FiniteDimensional.finrank ℝ ↥s.direction = 2) {s₁ s₂ : EuclideanGeometry.Sphere P} {p₁ p₂ p : P} (hs₁ : s₁.center ∈ s) (hs₂ : s₂.center ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) (hs : s₁ ≠ s₂) (hp : p₁ ≠ p₂) (hp₁s₁ : p₁ ∈ s₁) (hp₂s₁ : p₂ ∈ s₁) (hps₁ : p ∈ s₁) (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂) (hps₂ : p ∈ s₂)
Goal: p = p₁ ∨ p = p₂
-- EuclideanGeometry.eq_of_mem_sphere_of_mem_sphere_of_finrank_eq_two
theorem eq_of_mem_sphere_of_mem_sphere_of_finrank_eq_two [FiniteDimensional ℝ V]
    (hd : finrank ℝ V = 2) {s₁ s₂ : Sphere P} {p₁ p₂ p : P} (hs : s₁ ≠ s₂) (hp : p₁ ≠ p₂)
    (hp₁s₁ : p₁ ∈ s₁) (hp₂s₁ : p₂ ∈ s₁) (hps₁ : p ∈ s₁) (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂)
    (hps₂ : p ∈ s₂) : p = p₁ ∨ p = p₂ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [FiniteDimensional ℝ V] (hd : FiniteDimensional.finrank ℝ V = 2) {s₁ s₂ : EuclideanGeometry.Sphere P} {p₁ p₂ p : P} (hs : s₁ ≠ s₂) (hp : p₁ ≠ p₂) (hp₁s₁ : p₁ ∈ s₁) (hp₂s₁ : p₂ ∈ s₁) (hps₁ : p ∈ s₁) (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂) (hps₂ : p ∈ s₂)
Goal: p = p₁ ∨ p = p₂
-- EuclideanGeometry.inner_pos_or_eq_of_dist_le_radius
theorem inner_pos_or_eq_of_dist_le_radius {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
    (hp₂ : dist p₂ s.center ≤ s.radius) : 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫ ∨ p₁ = p₂ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center ≤ s.radius)
Goal: 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫_ℝ ∨ p₁ = p₂