ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mem_extremePoints
theorem mem_extremePoints : x ∈ A.extremePoints 𝕜 ↔
    x ∈ A ∧ ∀ᵉ (x₁ ∈ A) (x₂ ∈ A), x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A : Set E} {x : E}
Goal: x ∈ Set.extremePoints 𝕜 A ↔ x ∈ A ∧ ∀ x₁ ∈ A, ∀ x₂ ∈ A, x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x
-- isExtreme_singleton
@[simp] lemma isExtreme_singleton : IsExtreme 𝕜 A {x} ↔ x ∈ A.extremePoints 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A : Set E} {x : E}
Goal: IsExtreme 𝕜 A {x} ↔ x ∈ Set.extremePoints 𝕜 A
-- extremePoints_subset
theorem extremePoints_subset : A.extremePoints 𝕜 ⊆ A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A : Set E}
Goal: Set.extremePoints 𝕜 A ⊆ A
-- extremePoints_empty
@[simp]
theorem extremePoints_empty : (∅ : Set E).extremePoints 𝕜 = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E]
Goal: Set.extremePoints 𝕜 ∅ = ∅
-- extremePoints_singleton
@[simp]
theorem extremePoints_singleton : ({x} : Set E).extremePoints 𝕜 = {x} 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E}
Goal: Set.extremePoints 𝕜 {x} = {x}
-- inter_extremePoints_subset_extremePoints_of_subset
theorem inter_extremePoints_subset_extremePoints_of_subset (hBA : B ⊆ A) :
    B ∩ A.extremePoints 𝕜 ⊆ B.extremePoints 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A B : Set E} (hBA : B ⊆ A)
Goal: B ∩ Set.extremePoints 𝕜 A ⊆ Set.extremePoints 𝕜 B
-- IsExtreme.extremePoints_subset_extremePoints
theorem IsExtreme.extremePoints_subset_extremePoints (hAB : IsExtreme 𝕜 A B) :
    B.extremePoints 𝕜 ⊆ A.extremePoints 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A B : Set E} (hAB : IsExtreme 𝕜 A B)
Goal: Set.extremePoints 𝕜 B ⊆ Set.extremePoints 𝕜 A
-- IsExtreme.extremePoints_eq
theorem IsExtreme.extremePoints_eq (hAB : IsExtreme 𝕜 A B) :
    B.extremePoints 𝕜 = B ∩ A.extremePoints 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A B : Set E} (hAB : IsExtreme 𝕜 A B)
Goal: Set.extremePoints 𝕜 B = B ∩ Set.extremePoints 𝕜 A
-- IsExtreme.convex_diff
theorem IsExtreme.convex_diff (hA : Convex 𝕜 A) (hAB : IsExtreme 𝕜 A B) : Convex 𝕜 (A \ B) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {A B : Set E} (hA : Convex 𝕜 A) (hAB : IsExtreme 𝕜 A B)
Goal: Convex 𝕜 (A \ B)
-- extremePoints_prod
@[simp]
theorem extremePoints_prod (s : Set E) (t : Set F) :
    (s ×ˢ t).extremePoints 𝕜 = s.extremePoints 𝕜 ×ˢ t.extremePoints 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] (s : Set E) (t : Set F)
Goal: Set.extremePoints 𝕜 (s ×ˢ t) = Set.extremePoints 𝕜 s ×ˢ Set.extremePoints 𝕜 t
-- extremePoints_pi
@[simp]
theorem extremePoints_pi (s : ∀ i, Set (π i)) :
    (univ.pi s).extremePoints 𝕜 = univ.pi fun i ↦ (s i).extremePoints 𝕜 
Arguments: {𝕜 : Type u_1} {ι : Type u_4} {π : ι → Type u_5} [OrderedSemiring 𝕜] [(i : ι) → AddCommGroup (π i)] [(i : ι) → Module 𝕜 (π i)] (s : (i : ι) → Set (π i))
Goal: Set.extremePoints 𝕜 (Set.univ.pi s) = Set.univ.pi fun i => Set.extremePoints 𝕜 (s i)
-- image_extremePoints
lemma image_extremePoints (f : L) (s : Set E) :
    f '' extremePoints 𝕜 s = extremePoints 𝕜 (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {L : Type u_6} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [EquivLike L E F] [LinearEquivClass L 𝕜 E F] (f : L) (s : Set E)
Goal: ⇑f '' Set.extremePoints 𝕜 s = Set.extremePoints 𝕜 (⇑f '' s)
-- mem_extremePoints_iff_forall_segment
theorem mem_extremePoints_iff_forall_segment : x ∈ A.extremePoints 𝕜 ↔
    x ∈ A ∧ ∀ᵉ (x₁ ∈ A) (x₂ ∈ A), x ∈ segment 𝕜 x₁ x₂ → x₁ = x ∨ x₂ = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E] {A : Set E} {x : E}
Goal: x ∈ Set.extremePoints 𝕜 A ↔ x ∈ A ∧ ∀ x₁ ∈ A, ∀ x₂ ∈ A, x ∈ segment 𝕜 x₁ x₂ → x₁ = x ∨ x₂ = x
-- Convex.mem_extremePoints_iff_convex_diff
theorem Convex.mem_extremePoints_iff_convex_diff (hA : Convex 𝕜 A) :
    x ∈ A.extremePoints 𝕜 ↔ x ∈ A ∧ Convex 𝕜 (A \ {x}) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E] {A : Set E} {x : E} (hA : Convex 𝕜 A)
Goal: x ∈ Set.extremePoints 𝕜 A ↔ x ∈ A ∧ Convex 𝕜 (A \ {x})
-- Convex.mem_extremePoints_iff_mem_diff_convexHull_diff
theorem Convex.mem_extremePoints_iff_mem_diff_convexHull_diff (hA : Convex 𝕜 A) :
    x ∈ A.extremePoints 𝕜 ↔ x ∈ A \ convexHull 𝕜 (A \ {x}) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E] {A : Set E} {x : E} (hA : Convex 𝕜 A)
Goal: x ∈ Set.extremePoints 𝕜 A ↔ x ∈ A \ (convexHull 𝕜) (A \ {x})
-- extremePoints_convexHull_subset
theorem extremePoints_convexHull_subset : (convexHull 𝕜 A).extremePoints 𝕜 ⊆ A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E] {A : Set E}
Goal: Set.extremePoints 𝕜 ((convexHull 𝕜) A) ⊆ A
-- starConvex_iff_segment_subset
theorem starConvex_iff_segment_subset : StarConvex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → [x -[𝕜] y] ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {s : Set E}
Goal: StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → [x-[𝕜]y] ⊆ s
-- StarConvex.segment_subset
theorem StarConvex.segment_subset (h : StarConvex 𝕜 x s) {y : E} (hy : y ∈ s) : [x -[𝕜] y] ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {s : Set E} (h : StarConvex 𝕜 x s) {y : E} (hy : y ∈ s)
Goal: [x-[𝕜]y] ⊆ s
-- StarConvex.openSegment_subset
theorem StarConvex.openSegment_subset (h : StarConvex 𝕜 x s) {y : E} (hy : y ∈ s) :
    openSegment 𝕜 x y ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {s : Set E} (h : StarConvex 𝕜 x s) {y : E} (hy : y ∈ s)
Goal: openSegment 𝕜 x y ⊆ s
-- starConvex_iff_pointwise_add_subset
theorem starConvex_iff_pointwise_add_subset :
    StarConvex 𝕜 x s ↔ ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • {x} + b • s ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {s : Set E}
Goal: StarConvex 𝕜 x s ↔ ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • {x} + b • s ⊆ s
-- starConvex_empty
theorem starConvex_empty (x : E) : StarConvex 𝕜 x ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x : E)
Goal: StarConvex 𝕜 x ∅
-- starConvex_univ
theorem starConvex_univ (x : E) : StarConvex 𝕜 x univ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x : E)
Goal: StarConvex 𝕜 x Set.univ
-- StarConvex.inter
theorem StarConvex.inter (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 x t) : StarConvex 𝕜 x (s ∩ t) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {s t : Set E} (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 x t)
Goal: StarConvex 𝕜 x (s ∩ t)
-- starConvex_sInter
theorem starConvex_sInter {S : Set (Set E)} (h : ∀ s ∈ S, StarConvex 𝕜 x s) :
    StarConvex 𝕜 x (⋂₀ S) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {S : Set (Set E)} (h : ∀ s ∈ S, StarConvex 𝕜 x s)
Goal: StarConvex 𝕜 x (⋂₀ S)
-- starConvex_iInter
theorem starConvex_iInter {ι : Sort*} {s : ι → Set E} (h : ∀ i, StarConvex 𝕜 x (s i)) :
    StarConvex 𝕜 x (⋂ i, s i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {ι : Sort u_4} {s : ι → Set E} (h : ∀ (i : ι), StarConvex 𝕜 x (s i))
Goal: StarConvex 𝕜 x (⋂ i, s i)
-- StarConvex.union
theorem StarConvex.union (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 x t) :
    StarConvex 𝕜 x (s ∪ t) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {s t : Set E} (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 x t)
Goal: StarConvex 𝕜 x (s ∪ t)
-- starConvex_iUnion
theorem starConvex_iUnion {ι : Sort*} {s : ι → Set E} (hs : ∀ i, StarConvex 𝕜 x (s i)) :
    StarConvex 𝕜 x (⋃ i, s i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {ι : Sort u_4} {s : ι → Set E} (hs : ∀ (i : ι), StarConvex 𝕜 x (s i))
Goal: StarConvex 𝕜 x (⋃ i, s i)
-- starConvex_sUnion
theorem starConvex_sUnion {S : Set (Set E)} (hS : ∀ s ∈ S, StarConvex 𝕜 x s) :
    StarConvex 𝕜 x (⋃₀ S) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {S : Set (Set E)} (hS : ∀ s ∈ S, StarConvex 𝕜 x s)
Goal: StarConvex 𝕜 x (⋃₀ S)
-- StarConvex.prod
theorem StarConvex.prod {y : F} {s : Set E} {t : Set F} (hs : StarConvex 𝕜 x s)
    (ht : StarConvex 𝕜 y t) : StarConvex 𝕜 (x, y) (s ×ˢ t) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [SMul 𝕜 E] [SMul 𝕜 F] {x : E} {y : F} {s : Set E} {t : Set F} (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 y t)
Goal: StarConvex 𝕜 (x, y) (s ×ˢ t)
-- starConvex_pi
theorem starConvex_pi {ι : Type*} {E : ι → Type*} [∀ i, AddCommMonoid (E i)] [∀ i, SMul 𝕜 (E i)]
    {x : ∀ i, E i} {s : Set ι} {t : ∀ i, Set (E i)} (ht : ∀ ⦃i⦄, i ∈ s → StarConvex 𝕜 (x i) (t i)) :
    StarConvex 𝕜 x (s.pi t) 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {ι : Type u_4} {E : ι → Type u_5} [(i : ι) → AddCommMonoid (E i)] [(i : ι) → SMul 𝕜 (E i)] {x : (i : ι) → E i} {s : Set ι} {t : (i : ι) → Set (E i)} (ht : ∀ ⦃i : ι⦄, i ∈ s → StarConvex 𝕜 (x i) (t i))
Goal: StarConvex 𝕜 x (s.pi t)
-- StarConvex.mem
theorem StarConvex.mem (hs : StarConvex 𝕜 x s) (h : s.Nonempty) : x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} (hs : StarConvex 𝕜 x s) (h : s.Nonempty)
Goal: x ∈ s
-- starConvex_iff_forall_pos
theorem starConvex_iff_forall_pos (hx : x ∈ s) : StarConvex 𝕜 x s ↔
    ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} (hx : x ∈ s)
Goal: StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s
-- starConvex_iff_forall_ne_pos
theorem starConvex_iff_forall_ne_pos (hx : x ∈ s) :
    StarConvex 𝕜 x s ↔
      ∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} (hx : x ∈ s)
Goal: StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s
-- starConvex_iff_openSegment_subset
theorem starConvex_iff_openSegment_subset (hx : x ∈ s) :
    StarConvex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → openSegment 𝕜 x y ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} (hx : x ∈ s)
Goal: StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → openSegment 𝕜 x y ⊆ s
-- starConvex_singleton
theorem starConvex_singleton (x : E) : StarConvex 𝕜 x {x} 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (x : E)
Goal: StarConvex 𝕜 x {x}
-- StarConvex.linear_image
theorem StarConvex.linear_image (hs : StarConvex 𝕜 x s) (f : E →ₗ[𝕜] F) :
    StarConvex 𝕜 (f x) (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {x : E} {s : Set E} (hs : StarConvex 𝕜 x s) (f : E →ₗ[𝕜] F)
Goal: StarConvex 𝕜 (f x) (⇑f '' s)
-- StarConvex.is_linear_image
theorem StarConvex.is_linear_image (hs : StarConvex 𝕜 x s) {f : E → F} (hf : IsLinearMap 𝕜 f) :
    StarConvex 𝕜 (f x) (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {x : E} {s : Set E} (hs : StarConvex 𝕜 x s) {f : E → F} (hf : IsLinearMap 𝕜 f)
Goal: StarConvex 𝕜 (f x) (f '' s)
-- StarConvex.linear_preimage
theorem StarConvex.linear_preimage {s : Set F} (f : E →ₗ[𝕜] F) (hs : StarConvex 𝕜 (f x) s) :
    StarConvex 𝕜 x (f ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {x : E} {s : Set F} (f : E →ₗ[𝕜] F) (hs : StarConvex 𝕜 (f x) s)
Goal: StarConvex 𝕜 x (⇑f ⁻¹' s)
-- StarConvex.is_linear_preimage
theorem StarConvex.is_linear_preimage {s : Set F} {f : E → F} (hs : StarConvex 𝕜 (f x) s)
    (hf : IsLinearMap 𝕜 f) : StarConvex 𝕜 x (preimage f s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {x : E} {s : Set F} {f : E → F} (hs : StarConvex 𝕜 (f x) s) (hf : IsLinearMap 𝕜 f)
Goal: StarConvex 𝕜 x (f ⁻¹' s)
-- StarConvex.add
theorem StarConvex.add {t : Set E} (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 y t) :
    StarConvex 𝕜 (x + y) (s + t) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x y : E} {s t : Set E} (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 y t)
Goal: StarConvex 𝕜 (x + y) (s + t)
-- StarConvex.add_left
theorem StarConvex.add_left (hs : StarConvex 𝕜 x s) (z : E) :
    StarConvex 𝕜 (z + x) ((fun x => z + x) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} (hs : StarConvex 𝕜 x s) (z : E)
Goal: StarConvex 𝕜 (z + x) ((fun x => z + x) '' s)
-- StarConvex.add_right
theorem StarConvex.add_right (hs : StarConvex 𝕜 x s) (z : E) :
    StarConvex 𝕜 (x + z) ((fun x => x + z) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} (hs : StarConvex 𝕜 x s) (z : E)
Goal: StarConvex 𝕜 (x + z) ((fun x => x + z) '' s)
-- StarConvex.preimage_add_right
theorem StarConvex.preimage_add_right (hs : StarConvex 𝕜 (z + x) s) :
    StarConvex 𝕜 x ((fun x => z + x) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x z : E} {s : Set E} (hs : StarConvex 𝕜 (z + x) s)
Goal: StarConvex 𝕜 x ((fun x => z + x) ⁻¹' s)
-- StarConvex.preimage_add_left
theorem StarConvex.preimage_add_left (hs : StarConvex 𝕜 (x + z) s) :
    StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x z : E} {s : Set E} (hs : StarConvex 𝕜 (x + z) s)
Goal: StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s)
-- StarConvex.sub'
theorem StarConvex.sub' {s : Set (E × E)} (hs : StarConvex 𝕜 (x, y) s) :
    StarConvex 𝕜 (x - y) ((fun x : E × E => x.1 - x.2) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y : E} {s : Set (E × E)} (hs : StarConvex 𝕜 (x, y) s)
Goal: StarConvex 𝕜 (x - y) ((fun x => x.1 - x.2) '' s)
-- StarConvex.smul
theorem StarConvex.smul (hs : StarConvex 𝕜 x s) (c : 𝕜) : StarConvex 𝕜 (c • x) (c • s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedCommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} (hs : StarConvex 𝕜 x s) (c : 𝕜)
Goal: StarConvex 𝕜 (c • x) (c • s)
-- StarConvex.preimage_smul
theorem StarConvex.preimage_smul {c : 𝕜} (hs : StarConvex 𝕜 (c • x) s) :
    StarConvex 𝕜 x ((fun z => c • z) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedCommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} {c : 𝕜} (hs : StarConvex 𝕜 (c • x) s)
Goal: StarConvex 𝕜 x ((fun z => c • z) ⁻¹' s)
-- StarConvex.affinity
theorem StarConvex.affinity (hs : StarConvex 𝕜 x s) (z : E) (c : 𝕜) :
    StarConvex 𝕜 (z + c • x) ((fun x => z + c • x) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedCommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {s : Set E} (hs : StarConvex 𝕜 x s) (z : E) (c : 𝕜)
Goal: StarConvex 𝕜 (z + c • x) ((fun x => z + c • x) '' s)
-- starConvex_zero_iff
theorem starConvex_zero_iff :
    StarConvex 𝕜 0 s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommMonoid E] [SMulWithZero 𝕜 E] {s : Set E}
Goal: StarConvex 𝕜 0 s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s
-- StarConvex.add_smul_mem
theorem StarConvex.add_smul_mem (hs : StarConvex 𝕜 x s) (hy : x + y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t)
    (ht₁ : t ≤ 1) : x + t • y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y : E} {s : Set E} (hs : StarConvex 𝕜 x s) (hy : x + y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t) (ht₁ : t ≤ 1)
Goal: x + t • y ∈ s
-- StarConvex.smul_mem
theorem StarConvex.smul_mem (hs : StarConvex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t)
    (ht₁ : t ≤ 1) : t • x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x : E} {s : Set E} (hs : StarConvex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t) (ht₁ : t ≤ 1)
Goal: t • x ∈ s
-- StarConvex.add_smul_sub_mem
theorem StarConvex.add_smul_sub_mem (hs : StarConvex 𝕜 x s) (hy : y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t)
    (ht₁ : t ≤ 1) : x + t • (y - x) ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y : E} {s : Set E} (hs : StarConvex 𝕜 x s) (hy : y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t) (ht₁ : t ≤ 1)
Goal: x + t • (y - x) ∈ s
-- StarConvex.affine_preimage
theorem StarConvex.affine_preimage (f : E →ᵃ[𝕜] F) {s : Set F} (hs : StarConvex 𝕜 (f x) s) :
    StarConvex 𝕜 x (f ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] {x : E} (f : E →ᵃ[𝕜] F) {s : Set F} (hs : StarConvex 𝕜 (f x) s)
Goal: StarConvex 𝕜 x (⇑f ⁻¹' s)
-- StarConvex.affine_image
theorem StarConvex.affine_image (f : E →ᵃ[𝕜] F) {s : Set E} (hs : StarConvex 𝕜 x s) :
    StarConvex 𝕜 (f x) (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] {x : E} (f : E →ᵃ[𝕜] F) {s : Set E} (hs : StarConvex 𝕜 x s)
Goal: StarConvex 𝕜 (f x) (⇑f '' s)
-- StarConvex.neg
theorem StarConvex.neg (hs : StarConvex 𝕜 x s) : StarConvex 𝕜 (-x) (-s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x : E} {s : Set E} (hs : StarConvex 𝕜 x s)
Goal: StarConvex 𝕜 (-x) (-s)
-- StarConvex.sub
theorem StarConvex.sub (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 y t) :
    StarConvex 𝕜 (x - y) (s - t) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y : E} {s t : Set E} (hs : StarConvex 𝕜 x s) (ht : StarConvex 𝕜 y t)
Goal: StarConvex 𝕜 (x - y) (s - t)
-- starConvex_compl_Iic
lemma starConvex_compl_Iic (h : x < y) : StarConvex 𝕜 y (Iic x)ᶜ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {x y : E} (h : x < y)
Goal: StarConvex 𝕜 y (Set.Iic x)ᶜ
-- starConvex_compl_Ici
lemma starConvex_compl_Ici (h : x < y) : StarConvex 𝕜 x (Ici y)ᶜ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {x y : E} (h : x < y)
Goal: StarConvex 𝕜 x (Set.Ici y)ᶜ
-- starConvex_iff_div
theorem starConvex_iff_div : StarConvex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s →
    ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {x : E} {s : Set E}
Goal: StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s
-- StarConvex.mem_smul
theorem StarConvex.mem_smul (hs : StarConvex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht : 1 ≤ t) :
    x ∈ t • s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {x : E} {s : Set E} (hs : StarConvex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht : 1 ≤ t)
Goal: x ∈ t • s
-- Set.OrdConnected.starConvex
theorem Set.OrdConnected.starConvex [OrderedSemiring 𝕜] [OrderedAddCommMonoid E] [Module 𝕜 E]
    [OrderedSMul 𝕜 E] {x : E} {s : Set E} (hs : s.OrdConnected) (hx : x ∈ s)
    (h : ∀ y ∈ s, x ≤ y ∨ y ≤ x) : StarConvex 𝕜 x s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [OrderedAddCommMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {x : E} {s : Set E} (hs : s.OrdConnected) (hx : x ∈ s) (h : ∀ y ∈ s, x ≤ y ∨ y ≤ x)
Goal: StarConvex 𝕜 x s
-- starConvex_iff_ordConnected
theorem starConvex_iff_ordConnected [LinearOrderedField 𝕜] {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) :
    StarConvex 𝕜 x s ↔ s.OrdConnected 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s)
Goal: StarConvex 𝕜 x s ↔ s.OrdConnected
-- Seminorm.smul'
structure Seminorm (𝕜 : Type*) (E : Type*) [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] extends
  AddGroupSeminorm E where
    smul' : ∀ (a : 𝕜) (x : E), toFun (a • x) = ‖a‖ * toFun x
Arguments: {𝕜 : Type u_13} {E : Type u_14} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] (self : Seminorm 𝕜 E) (a : 𝕜) (x : E)
Goal: self.toFun (a • x) = ‖a‖ * self.toFun x
-- SeminormClass.map_smul_eq_mul
class SeminormClass (F : Type*) (𝕜 E : outParam Type*) [SeminormedRing 𝕜] [AddGroup E]
  [SMul 𝕜 E] [FunLike F E ℝ] extends AddGroupSeminormClass F E ℝ : Prop where
    map_smul_eq_mul (f : F) (a : 𝕜) (x : E) : f (a • x) = ‖a‖ * f x
Arguments: {F : Type u_13} {𝕜 : outParam (Type u_14)} {E : outParam (Type u_15)} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] [FunLike F E ℝ] [self : SeminormClass F 𝕜 E] (f : F) (a : 𝕜) (x : E)
Goal: f (a • x) = ‖a‖ * f x
-- Seminorm.ext
@[ext]
theorem ext {p q : Seminorm 𝕜 E} (h : ∀ x, (p : E → ℝ) x = q x) : p = q 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] {p q : Seminorm 𝕜 E} (h : ∀ (x : E), p x = q x)
Goal: p = q
-- Seminorm.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : Seminorm 𝕜 E) = 0 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E]
Goal: ⇑0 = 0
-- Seminorm.zero_apply
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm 𝕜 E) x = 0 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] (x : E)
Goal: 0 x = 0
-- Seminorm.coe_smul
theorem coe_smul [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : Seminorm 𝕜 E) :
    ⇑(r • p) = r • ⇑p 
Arguments: {R : Type u_1} {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : Seminorm 𝕜 E)
Goal: ⇑(r • p) = r • ⇑p
-- Seminorm.smul_apply
@[simp]
theorem smul_apply [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : Seminorm 𝕜 E)
    (x : E) : (r • p) x = r • p x 
Arguments: {R : Type u_1} {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : Seminorm 𝕜 E) (x : E)
Goal: (r • p) x = r • p x
-- Seminorm.coe_add
theorem coe_add (p q : Seminorm 𝕜 E) : ⇑(p + q) = p + q 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] (p q : Seminorm 𝕜 E)
Goal: ⇑(p + q) = ⇑p + ⇑q
-- Seminorm.add_apply
@[simp]
theorem add_apply (p q : Seminorm 𝕜 E) (x : E) : (p + q) x = p x + q x 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] (p q : Seminorm 𝕜 E) (x : E)
Goal: (p + q) x = p x + q x
-- Seminorm.coeFnAddMonoidHom_apply
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm 𝕜 E) (E → ℝ) 
Arguments: (𝕜 : Type u_3) (E : Type u_7) [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E]
Goal: ∀ (a : Seminorm 𝕜 E) (a_1 : E), (Seminorm.coeFnAddMonoidHom 𝕜 E) a a_1 = a a_1
-- Seminorm.coeFnAddMonoidHom_injective
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom 𝕜 E) 
Arguments: (𝕜 : Type u_3) (E : Type u_7) [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E]
Goal: Function.Injective ⇑(Seminorm.coeFnAddMonoidHom 𝕜 E)
-- Seminorm.coe_sup
@[simp]
theorem coe_sup (p q : Seminorm 𝕜 E) : ⇑(p ⊔ q) = (p : E → ℝ) ⊔ (q : E → ℝ) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] (p q : Seminorm 𝕜 E)
Goal: ⇑(p ⊔ q) = ⇑p ⊔ ⇑q