ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OrthogonalFamily.sum_projection_of_mem_iSup
theorem OrthogonalFamily.sum_projection_of_mem_iSup [Fintype ι] {V : ι → Submodule 𝕜 E}
    [∀ i, CompleteSpace (V i)] (hV : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ)
    (x : E) (hx : x ∈ iSup V) : (∑ i, (orthogonalProjection (V i) x : E)) = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [Fintype ι] {V : ι → Submodule 𝕜 E} [∀ (i : ι), CompleteSpace ↥(V i)] (hV : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ) (x : E) (hx : x ∈ iSup V)
Goal: ∑ i : ι, ↑((orthogonalProjection (V i)) x) = x
-- OrthogonalFamily.projection_directSum_coeAddHom
theorem OrthogonalFamily.projection_directSum_coeAddHom [DecidableEq ι] {V : ι → Submodule 𝕜 E}
    (hV : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) (x : ⨁ i, V i) (i : ι)
    [CompleteSpace (V i)] :
    orthogonalProjection (V i) (DirectSum.coeAddMonoidHom V x) = x i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ) (x : DirectSum ι fun i => ↥(V i)) (i : ι) [CompleteSpace ↥(V i)]
Goal: (orthogonalProjection (V i)) ((DirectSum.coeAddMonoidHom V) x) = x i
-- maximal_orthonormal_iff_orthogonalComplement_eq_bot
theorem maximal_orthonormal_iff_orthogonalComplement_eq_bot (hv : Orthonormal 𝕜 ((↑) : v → E)) :
    (∀ u ⊇ v, Orthonormal 𝕜 ((↑) : u → E) → u = v) ↔ (span 𝕜 v)ᗮ = ⊥ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {v : Set E} (hv : Orthonormal 𝕜 Subtype.val)
Goal: (∀ u ⊇ v, Orthonormal 𝕜 Subtype.val → u = v) ↔ (Submodule.span 𝕜 v)ᗮ = ⊥
-- maximal_orthonormal_iff_basis_of_finiteDimensional
theorem maximal_orthonormal_iff_basis_of_finiteDimensional (hv : Orthonormal 𝕜 ((↑) : v → E)) :
    (∀ u ⊇ v, Orthonormal 𝕜 ((↑) : u → E) → u = v) ↔ ∃ b : Basis v 𝕜 E, ⇑b = ((↑) : v → E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {v : Set E} [FiniteDimensional 𝕜 E] (hv : Orthonormal 𝕜 Subtype.val)
Goal: (∀ u ⊇ v, Orthonormal 𝕜 Subtype.val → u = v) ↔ ∃ b, ⇑b = Subtype.val
-- Caratheodory.mem_convexHull_erase
theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
    {x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
    ∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 Subtype.val) {x : E} (m : x ∈ (convexHull 𝕜) ↑t)
Goal: ∃ y, x ∈ (convexHull 𝕜) ↑(t.erase ↑y)
-- Caratheodory.minCardFinsetOfMemConvexHull_subseteq
theorem minCardFinsetOfMemConvexHull_subseteq : ↑(minCardFinsetOfMemConvexHull hx) ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hx : x ∈ (convexHull 𝕜) s)
Goal: ↑(Caratheodory.minCardFinsetOfMemConvexHull hx) ⊆ s
-- Caratheodory.mem_minCardFinsetOfMemConvexHull
theorem mem_minCardFinsetOfMemConvexHull :
    x ∈ convexHull 𝕜 (minCardFinsetOfMemConvexHull hx : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hx : x ∈ (convexHull 𝕜) s)
Goal: x ∈ (convexHull 𝕜) ↑(Caratheodory.minCardFinsetOfMemConvexHull hx)
-- Caratheodory.minCardFinsetOfMemConvexHull_nonempty
theorem minCardFinsetOfMemConvexHull_nonempty : (minCardFinsetOfMemConvexHull hx).Nonempty 
Arguments: {𝕜 : Type u_1} {E : Type u} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hx : x ∈ (convexHull 𝕜) s)
Goal: (Caratheodory.minCardFinsetOfMemConvexHull hx).Nonempty
-- Caratheodory.minCardFinsetOfMemConvexHull_card_le_card
theorem minCardFinsetOfMemConvexHull_card_le_card {t : Finset E} (ht₁ : ↑t ⊆ s)
    (ht₂ : x ∈ convexHull 𝕜 (t : Set E)) : (minCardFinsetOfMemConvexHull hx).card ≤ t.card 
Arguments: {𝕜 : Type u_1} {E : Type u} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hx : x ∈ (convexHull 𝕜) s) {t : Finset E} (ht₁ : ↑t ⊆ s) (ht₂ : x ∈ (convexHull 𝕜) ↑t)
Goal: (Caratheodory.minCardFinsetOfMemConvexHull hx).card ≤ t.card
-- Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull
theorem affineIndependent_minCardFinsetOfMemConvexHull :
    AffineIndependent 𝕜 ((↑) : minCardFinsetOfMemConvexHull hx → E) 
Arguments: {𝕜 : Type u_1} {E : Type u} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hx : x ∈ (convexHull 𝕜) s)
Goal: AffineIndependent 𝕜 Subtype.val
-- convexHull_eq_union
theorem convexHull_eq_union : convexHull 𝕜 s =
    ⋃ (t : Finset E) (hss : ↑t ⊆ s) (hai : AffineIndependent 𝕜 ((↑) : t → E)), convexHull 𝕜 ↑t 
Arguments: {𝕜 : Type u_1} {E : Type u} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: (convexHull 𝕜) s = ⋃ t, ⋃ (_ : ↑t ⊆ s), ⋃ (_ : AffineIndependent 𝕜 Subtype.val), (convexHull 𝕜) ↑t
-- eq_pos_convex_span_of_mem_convexHull
theorem eq_pos_convex_span_of_mem_convexHull {x : E} (hx : x ∈ convexHull 𝕜 s) :
    ∃ (ι : Sort (u + 1)) (_ : Fintype ι),
      ∃ (z : ι → E) (w : ι → 𝕜), Set.range z ⊆ s ∧ AffineIndependent 𝕜 z ∧ (∀ i, 0 < w i) ∧
        ∑ i, w i = 1 ∧ ∑ i, w i • z i = x 
Arguments: {𝕜 : Type u_1} {E : Type u} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hx : x ∈ (convexHull 𝕜) s)
Goal: ∃ ι x_1 z w, Set.range z ⊆ s ∧ AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ ∑ i : ι, w i = 1 ∧ ∑ i : ι, w i • z i = x
-- strictConvex_iff_openSegment_subset
theorem strictConvex_iff_openSegment_subset :
    StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E}
Goal: StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s
-- StrictConvex.openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
    (h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {x y : E} (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (h : x ≠ y)
Goal: openSegment 𝕜 x y ⊆ interior s
-- strictConvex_empty
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [SMul 𝕜 E]
Goal: StrictConvex 𝕜 ∅
-- strictConvex_univ
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [SMul 𝕜 E]
Goal: StrictConvex 𝕜 Set.univ
-- StrictConvex.eq
protected nonrec theorem StrictConvex.eq (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
    (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s) : x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {x y : E} {a b : 𝕜} (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s)
Goal: x = y
-- StrictConvex.inter
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
    StrictConvex 𝕜 (s ∩ t) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [SMul 𝕜 E] {s t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t)
Goal: StrictConvex 𝕜 (s ∩ t)
-- Directed.strictConvex_iUnion
theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
    (hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [SMul 𝕜 E] {ι : Sort u_6} {s : ι → Set E} (hdir : Directed (fun x x_1 => x ⊆ x_1) s) (hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i))
Goal: StrictConvex 𝕜 (⋃ i, s i)
-- DirectedOn.strictConvex_sUnion
theorem DirectedOn.strictConvex_sUnion {S : Set (Set E)} (hdir : DirectedOn (· ⊆ ·) S)
    (hS : ∀ s ∈ S, StrictConvex 𝕜 s) : StrictConvex 𝕜 (⋃₀ S) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [SMul 𝕜 E] {S : Set (Set E)} (hdir : DirectedOn (fun x x_1 => x ⊆ x_1) S) (hS : ∀ s ∈ S, StrictConvex 𝕜 s)
Goal: StrictConvex 𝕜 (⋃₀ S)
-- StrictConvex.convex
protected theorem StrictConvex.convex (hs : StrictConvex 𝕜 s) : Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : StrictConvex 𝕜 s)
Goal: Convex 𝕜 s
-- Convex.strictConvex_of_isOpen
protected theorem Convex.strictConvex_of_isOpen (h : IsOpen s) (hs : Convex 𝕜 s) :
    StrictConvex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (h : IsOpen s) (hs : Convex 𝕜 s)
Goal: StrictConvex 𝕜 s
-- IsOpen.strictConvex_iff
theorem IsOpen.strictConvex_iff (h : IsOpen s) : StrictConvex 𝕜 s ↔ Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (h : IsOpen s)
Goal: StrictConvex 𝕜 s ↔ Convex 𝕜 s
-- strictConvex_singleton
theorem strictConvex_singleton (c : E) : StrictConvex 𝕜 ({c} : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [Module 𝕜 E] (c : E)
Goal: StrictConvex 𝕜 {c}
-- Set.Subsingleton.strictConvex
theorem Set.Subsingleton.strictConvex (hs : s.Subsingleton) : StrictConvex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : s.Subsingleton)
Goal: StrictConvex 𝕜 s
-- StrictConvex.linear_image
theorem StrictConvex.linear_image [Semiring 𝕝] [Module 𝕝 E] [Module 𝕝 F]
    [LinearMap.CompatibleSMul E F 𝕜 𝕝] (hs : StrictConvex 𝕜 s) (f : E →ₗ[𝕝] F) (hf : IsOpenMap f) :
    StrictConvex 𝕜 (f '' s) 
Arguments: {𝕜 : Type u_1} {𝕝 : Type u_2} {E : Type u_3} {F : Type u_4} [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {s : Set E} [Semiring 𝕝] [Module 𝕝 E] [Module 𝕝 F] [LinearMap.CompatibleSMul E F 𝕜 𝕝] (hs : StrictConvex 𝕜 s) (f : E →ₗ[𝕝] F) (hf : IsOpenMap ⇑f)
Goal: StrictConvex 𝕜 (⇑f '' s)
-- StrictConvex.is_linear_image
theorem StrictConvex.is_linear_image (hs : StrictConvex 𝕜 s) {f : E → F} (h : IsLinearMap 𝕜 f)
    (hf : IsOpenMap f) : StrictConvex 𝕜 (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {s : Set E} (hs : StrictConvex 𝕜 s) {f : E → F} (h : IsLinearMap 𝕜 f) (hf : IsOpenMap f)
Goal: StrictConvex 𝕜 (f '' s)
-- StrictConvex.linear_preimage
theorem StrictConvex.linear_preimage {s : Set F} (hs : StrictConvex 𝕜 s) (f : E →ₗ[𝕜] F)
    (hf : Continuous f) (hfinj : Injective f) : StrictConvex 𝕜 (s.preimage f) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {s : Set F} (hs : StrictConvex 𝕜 s) (f : E →ₗ[𝕜] F) (hf : Continuous ⇑f) (hfinj : Function.Injective ⇑f)
Goal: StrictConvex 𝕜 (⇑f ⁻¹' s)
-- StrictConvex.is_linear_preimage
theorem StrictConvex.is_linear_preimage {s : Set F} (hs : StrictConvex 𝕜 s) {f : E → F}
    (h : IsLinearMap 𝕜 f) (hf : Continuous f) (hfinj : Injective f) :
    StrictConvex 𝕜 (s.preimage f) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {s : Set F} (hs : StrictConvex 𝕜 s) {f : E → F} (h : IsLinearMap 𝕜 f) (hf : Continuous f) (hfinj : Function.Injective f)
Goal: StrictConvex 𝕜 (f ⁻¹' s)
-- Set.OrdConnected.strictConvex
protected theorem Set.OrdConnected.strictConvex {s : Set β} (hs : OrdConnected s) :
    StrictConvex 𝕜 s 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set β} (hs : s.OrdConnected)
Goal: StrictConvex 𝕜 s
-- strictConvex_Iic
theorem strictConvex_Iic (r : β) : StrictConvex 𝕜 (Iic r) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r : β)
Goal: StrictConvex 𝕜 (Set.Iic r)
-- strictConvex_Ici
theorem strictConvex_Ici (r : β) : StrictConvex 𝕜 (Ici r) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r : β)
Goal: StrictConvex 𝕜 (Set.Ici r)
-- strictConvex_Iio
theorem strictConvex_Iio (r : β) : StrictConvex 𝕜 (Iio r) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r : β)
Goal: StrictConvex 𝕜 (Set.Iio r)
-- strictConvex_Ioi
theorem strictConvex_Ioi (r : β) : StrictConvex 𝕜 (Ioi r) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r : β)
Goal: StrictConvex 𝕜 (Set.Ioi r)
-- strictConvex_Icc
theorem strictConvex_Icc (r s : β) : StrictConvex 𝕜 (Icc r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: StrictConvex 𝕜 (Set.Icc r s)
-- strictConvex_Ioo
theorem strictConvex_Ioo (r s : β) : StrictConvex 𝕜 (Ioo r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: StrictConvex 𝕜 (Set.Ioo r s)
-- strictConvex_Ico
theorem strictConvex_Ico (r s : β) : StrictConvex 𝕜 (Ico r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: StrictConvex 𝕜 (Set.Ico r s)
-- strictConvex_Ioc
theorem strictConvex_Ioc (r s : β) : StrictConvex 𝕜 (Ioc r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: StrictConvex 𝕜 (Set.Ioc r s)
-- strictConvex_uIcc
theorem strictConvex_uIcc (r s : β) : StrictConvex 𝕜 (uIcc r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: StrictConvex 𝕜 (Set.uIcc r s)
-- strictConvex_uIoc
theorem strictConvex_uIoc (r s : β) : StrictConvex 𝕜 (uIoc r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [TopologicalSpace β] [LinearOrderedCancelAddCommMonoid β] [OrderTopology β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: StrictConvex 𝕜 (Ι r s)
-- StrictConvex.preimage_add_right
theorem StrictConvex.preimage_add_right (hs : StrictConvex 𝕜 s) (z : E) :
    StrictConvex 𝕜 ((fun x => z + x) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCancelCommMonoid E] [ContinuousAdd E] [Module 𝕜 E] {s : Set E} (hs : StrictConvex 𝕜 s) (z : E)
Goal: StrictConvex 𝕜 ((fun x => z + x) ⁻¹' s)
-- StrictConvex.preimage_add_left
theorem StrictConvex.preimage_add_left (hs : StrictConvex 𝕜 s) (z : E) :
    StrictConvex 𝕜 ((fun x => x + z) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCancelCommMonoid E] [ContinuousAdd E] [Module 𝕜 E] {s : Set E} (hs : StrictConvex 𝕜 s) (z : E)
Goal: StrictConvex 𝕜 ((fun x => x + z) ⁻¹' s)
-- StrictConvex.add
theorem StrictConvex.add (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
    StrictConvex 𝕜 (s + t) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] [ContinuousAdd E] {s t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t)
Goal: StrictConvex 𝕜 (s + t)
-- StrictConvex.add_left
theorem StrictConvex.add_left (hs : StrictConvex 𝕜 s) (z : E) :
    StrictConvex 𝕜 ((fun x => z + x) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] [ContinuousAdd E] {s : Set E} (hs : StrictConvex 𝕜 s) (z : E)
Goal: StrictConvex 𝕜 ((fun x => z + x) '' s)
-- StrictConvex.add_right
theorem StrictConvex.add_right (hs : StrictConvex 𝕜 s) (z : E) :
    StrictConvex 𝕜 ((fun x => x + z) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] [ContinuousAdd E] {s : Set E} (hs : StrictConvex 𝕜 s) (z : E)
Goal: StrictConvex 𝕜 ((fun x => x + z) '' s)
-- StrictConvex.vadd
theorem StrictConvex.vadd (hs : StrictConvex 𝕜 s) (x : E) : StrictConvex 𝕜 (x +ᵥ s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] [ContinuousAdd E] {s : Set E} (hs : StrictConvex 𝕜 s) (x : E)
Goal: StrictConvex 𝕜 (x +ᵥ s)
-- StrictConvex.smul
theorem StrictConvex.smul (hs : StrictConvex 𝕜 s) (c : 𝕝) : StrictConvex 𝕜 (c • s) 
Arguments: {𝕜 : Type u_1} {𝕝 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] [LinearOrderedField 𝕝] [Module 𝕝 E] [ContinuousConstSMul 𝕝 E] [LinearMap.CompatibleSMul E E 𝕜 𝕝] {s : Set E} (hs : StrictConvex 𝕜 s) (c : 𝕝)
Goal: StrictConvex 𝕜 (c • s)
-- StrictConvex.affinity
theorem StrictConvex.affinity [ContinuousAdd E] (hs : StrictConvex 𝕜 s) (z : E) (c : 𝕝) :
    StrictConvex 𝕜 (z +ᵥ c • s) 
Arguments: {𝕜 : Type u_1} {𝕝 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] [LinearOrderedField 𝕝] [Module 𝕝 E] [ContinuousConstSMul 𝕝 E] [LinearMap.CompatibleSMul E E 𝕜 𝕝] {s : Set E} [ContinuousAdd E] (hs : StrictConvex 𝕜 s) (z : E) (c : 𝕝)
Goal: StrictConvex 𝕜 (z +ᵥ c • s)
-- StrictConvex.preimage_smul
theorem StrictConvex.preimage_smul (hs : StrictConvex 𝕜 s) (c : 𝕜) :
    StrictConvex 𝕜 ((fun z => c • z) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedCommSemiring 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] [NoZeroSMulDivisors 𝕜 E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : StrictConvex 𝕜 s) (c : 𝕜)
Goal: StrictConvex 𝕜 ((fun z => c • z) ⁻¹' s)
-- StrictConvex.eq_of_openSegment_subset_frontier
theorem StrictConvex.eq_of_openSegment_subset_frontier [Nontrivial 𝕜] [DenselyOrdered 𝕜]
    (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (h : openSegment 𝕜 x y ⊆ frontier s) :
    x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedRing 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x y : E} [Nontrivial 𝕜] [DenselyOrdered 𝕜] (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (h : openSegment 𝕜 x y ⊆ frontier s)
Goal: x = y
-- StrictConvex.add_smul_mem
theorem StrictConvex.add_smul_mem (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hxy : x + y ∈ s)
    (hy : y ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) : x + t • y ∈ interior s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedRing 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x y : E} (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hxy : x + y ∈ s) (hy : y ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1)
Goal: x + t • y ∈ interior s
-- StrictConvex.smul_mem_of_zero_mem
theorem StrictConvex.smul_mem_of_zero_mem (hs : StrictConvex 𝕜 s) (zero_mem : (0 : E) ∈ s)
    (hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) : t • x ∈ interior s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedRing 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hs : StrictConvex 𝕜 s) (zero_mem : 0 ∈ s) (hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1)
Goal: t • x ∈ interior s
-- StrictConvex.add_smul_sub_mem
theorem StrictConvex.add_smul_sub_mem (h : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y)
    {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) : x + t • (y - x) ∈ interior s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedRing 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x y : E} (h : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1)
Goal: x + t • (y - x) ∈ interior s
-- StrictConvex.affine_preimage
theorem StrictConvex.affine_preimage {s : Set F} (hs : StrictConvex 𝕜 s) {f : E →ᵃ[𝕜] F}
    (hf : Continuous f) (hfinj : Injective f) : StrictConvex 𝕜 (f ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedRing 𝕜] [TopologicalSpace E] [TopologicalSpace F] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] {s : Set F} (hs : StrictConvex 𝕜 s) {f : E →ᵃ[𝕜] F} (hf : Continuous ⇑f) (hfinj : Function.Injective ⇑f)
Goal: StrictConvex 𝕜 (⇑f ⁻¹' s)
-- StrictConvex.affine_image
theorem StrictConvex.affine_image (hs : StrictConvex 𝕜 s) {f : E →ᵃ[𝕜] F} (hf : IsOpenMap f) :
    StrictConvex 𝕜 (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedRing 𝕜] [TopologicalSpace E] [TopologicalSpace F] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] {s : Set E} (hs : StrictConvex 𝕜 s) {f : E →ᵃ[𝕜] F} (hf : IsOpenMap ⇑f)
Goal: StrictConvex 𝕜 (⇑f '' s)
-- StrictConvex.neg
theorem StrictConvex.neg (hs : StrictConvex 𝕜 s) : StrictConvex 𝕜 (-s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedRing 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] {s : Set E} [TopologicalAddGroup E] (hs : StrictConvex 𝕜 s)
Goal: StrictConvex 𝕜 (-s)
-- StrictConvex.sub
theorem StrictConvex.sub (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) : StrictConvex 𝕜 (s - t) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [OrderedRing 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} [TopologicalAddGroup E] (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t)
Goal: StrictConvex 𝕜 (s - t)
-- strictConvex_iff_div
theorem strictConvex_iff_div :
    StrictConvex 𝕜 s ↔
      s.Pairwise fun x y =>
        ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → (a / (a + b)) • x + (b / (a + b)) • y ∈ interior s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [LinearOrderedField 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: StrictConvex 𝕜 s ↔ s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → (a / (a + b)) • x + (b / (a + b)) • y ∈ interior s
-- StrictConvex.mem_smul_of_zero_mem
theorem StrictConvex.mem_smul_of_zero_mem (hs : StrictConvex 𝕜 s) (zero_mem : (0 : E) ∈ s)
    (hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht : 1 < t) : x ∈ t • interior s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [LinearOrderedField 𝕜] [TopologicalSpace E] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hs : StrictConvex 𝕜 s) (zero_mem : 0 ∈ s) (hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht : 1 < t)
Goal: x ∈ t • interior s
-- strictConvex_iff_convex
@[simp]
theorem strictConvex_iff_convex : StrictConvex 𝕜 s ↔ Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {s : Set 𝕜}
Goal: StrictConvex 𝕜 s ↔ Convex 𝕜 s
-- strictConvex_iff_ordConnected
theorem strictConvex_iff_ordConnected : StrictConvex 𝕜 s ↔ s.OrdConnected 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {s : Set 𝕜}
Goal: StrictConvex 𝕜 s ↔ s.OrdConnected
-- Complex.hasStrictDerivAt_tan
theorem hasStrictDerivAt_tan {x : ℂ} (h : cos x ≠ 0) : HasStrictDerivAt tan (1 / cos x ^ 2) x 
Arguments: {x : ℂ} (h : Complex.cos x ≠ 0)
Goal: HasStrictDerivAt Complex.tan (1 / Complex.cos x ^ 2) x
-- Complex.hasDerivAt_tan
theorem hasDerivAt_tan {x : ℂ} (h : cos x ≠ 0) : HasDerivAt tan (1 / cos x ^ 2) x 
Arguments: {x : ℂ} (h : Complex.cos x ≠ 0)
Goal: HasDerivAt Complex.tan (1 / Complex.cos x ^ 2) x
-- Complex.tendsto_abs_tan_of_cos_eq_zero
theorem tendsto_abs_tan_of_cos_eq_zero {x : ℂ} (hx : cos x = 0) :
    Tendsto (fun x => abs (tan x)) (𝓝[≠] x) atTop 
Arguments: {x : ℂ} (hx : Complex.cos x = 0)
Goal: Filter.Tendsto (fun x => Complex.abs (Complex.tan x)) (nhdsWithin x {x}ᶜ) Filter.atTop