ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- segment_translate_image
theorem segment_translate_image (a b c : E) : (fun x => a + x) '' [b -[𝕜] c] = [a + b -[𝕜] a + c] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (a b c : E)
Goal: (fun x => a + x) '' [b-[𝕜]c] = [a + b-[𝕜]a + c]
-- openSegment_translate_image
theorem openSegment_translate_image (a b c : E) :
    (fun x => a + x) '' openSegment 𝕜 b c = openSegment 𝕜 (a + b) (a + c) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (a b c : E)
Goal: (fun x => a + x) '' openSegment 𝕜 b c = openSegment 𝕜 (a + b) (a + c)
-- segment_inter_eq_endpoint_of_linearIndependent_sub
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c])
Goal: [c-[𝕜]x] ∩ [c-[𝕜]y] = {c}
-- sameRay_of_mem_segment
theorem sameRay_of_mem_segment [StrictOrderedCommRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y z : E}
    (h : x ∈ [y -[𝕜] z]) : SameRay 𝕜 (x - y) (z - x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [StrictOrderedCommRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y z : E} (h : x ∈ [y-[𝕜]z])
Goal: SameRay 𝕜 (x - y) (z - x)
-- segment_inter_eq_endpoint_of_linearIndependent_of_ne
lemma segment_inter_eq_endpoint_of_linearIndependent_of_ne [OrderedCommRing 𝕜] [NoZeroDivisors 𝕜]
    [AddCommGroup E] [Module 𝕜 E]
    {x y : E} (h : LinearIndependent 𝕜 ![x, y]) {s t : 𝕜} (hs : s ≠ t) (c : E) :
    [c + x -[𝕜] c + t • y] ∩ [c + x -[𝕜] c + s • y] = {c + x} 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedCommRing 𝕜] [NoZeroDivisors 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y : E} (h : LinearIndependent 𝕜 ![x, y]) {s t : 𝕜} (hs : s ≠ t) (c : E)
Goal: [c + x-[𝕜]c + t • y] ∩ [c + x-[𝕜]c + s • y] = {c + x}
-- midpoint_mem_segment
theorem midpoint_mem_segment [Invertible (2 : 𝕜)] (x y : E) : midpoint 𝕜 x y ∈ [x -[𝕜] y] 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [Invertible 2] (x y : E)
Goal: midpoint 𝕜 x y ∈ [x-[𝕜]y]
-- mem_segment_sub_add
theorem mem_segment_sub_add [Invertible (2 : 𝕜)] (x y : E) : x ∈ [x - y -[𝕜] x + y] 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [Invertible 2] (x y : E)
Goal: x ∈ [x - y-[𝕜]x + y]
-- mem_segment_add_sub
theorem mem_segment_add_sub [Invertible (2 : 𝕜)] (x y : E) : x ∈ [x + y -[𝕜] x - y] 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [Invertible 2] (x y : E)
Goal: x ∈ [x + y-[𝕜]x - y]
-- left_mem_openSegment_iff
@[simp]
theorem left_mem_openSegment_iff [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E] :
    x ∈ openSegment 𝕜 x y ↔ x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y : E} [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E]
Goal: x ∈ openSegment 𝕜 x y ↔ x = y
-- right_mem_openSegment_iff
@[simp]
theorem right_mem_openSegment_iff [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E] :
    y ∈ openSegment 𝕜 x y ↔ x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y : E} [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E]
Goal: y ∈ openSegment 𝕜 x y ↔ x = y
-- mem_segment_iff_div
theorem mem_segment_iff_div :
    x ∈ [y -[𝕜] z] ↔
      ∃ a b : 𝕜, 0 ≤ a ∧ 0 ≤ b ∧ 0 < a + b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedSemifield 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y z : E}
Goal: x ∈ [y-[𝕜]z] ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ 0 < a + b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x
-- mem_openSegment_iff_div
theorem mem_openSegment_iff_div : x ∈ openSegment 𝕜 y z ↔
    ∃ a b : 𝕜, 0 < a ∧ 0 < b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedSemifield 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y z : E}
Goal: x ∈ openSegment 𝕜 y z ↔ ∃ a b, 0 < a ∧ 0 < b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x
-- mem_segment_iff_sameRay
theorem mem_segment_iff_sameRay : x ∈ [y -[𝕜] z] ↔ SameRay 𝕜 (x - y) (z - x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y z : E}
Goal: x ∈ [y-[𝕜]z] ↔ SameRay 𝕜 (x - y) (z - x)
-- openSegment_subset_union
theorem openSegment_subset_union (x y : E) {z : E} (hz : z ∈ range (lineMap x y : 𝕜 → E)) :
    openSegment 𝕜 x y ⊆ insert z (openSegment 𝕜 x z ∪ openSegment 𝕜 z y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (x y : E) {z : E} (hz : z ∈ Set.range ⇑(AffineMap.lineMap x y))
Goal: openSegment 𝕜 x y ⊆ insert z (openSegment 𝕜 x z ∪ openSegment 𝕜 z y)
-- segment_subset_Icc
theorem segment_subset_Icc (h : x ≤ y) : [x -[𝕜] y] ⊆ Icc x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [OrderedAddCommMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {x y : E} (h : x ≤ y)
Goal: [x-[𝕜]y] ⊆ Set.Icc x y
-- openSegment_subset_Ioo
theorem openSegment_subset_Ioo (h : x < y) : openSegment 𝕜 x y ⊆ Ioo x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [OrderedCancelAddCommMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {x y : E} (h : x < y)
Goal: openSegment 𝕜 x y ⊆ Set.Ioo x y
-- segment_subset_uIcc
theorem segment_subset_uIcc (x y : E) : [x -[𝕜] y] ⊆ uIcc x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E] (x y : E)
Goal: [x-[𝕜]y] ⊆ Set.uIcc x y
-- Convex.min_le_combo
theorem Convex.min_le_combo (x y : E) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
    min x y ≤ a • x + b • y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {a b : 𝕜} (x y : E) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: min x y ≤ a • x + b • y
-- Convex.combo_le_max
theorem Convex.combo_le_max (x y : E) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
    a • x + b • y ≤ max x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {a b : 𝕜} (x y : E) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: a • x + b • y ≤ max x y
-- Icc_subset_segment
theorem Icc_subset_segment : Icc x y ⊆ [x -[𝕜] y] 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y : 𝕜}
Goal: Set.Icc x y ⊆ [x-[𝕜]y]
-- segment_eq_Icc
@[simp]
theorem segment_eq_Icc (h : x ≤ y) : [x -[𝕜] y] = Icc x y 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y : 𝕜} (h : x ≤ y)
Goal: [x-[𝕜]y] = Set.Icc x y
-- Ioo_subset_openSegment
theorem Ioo_subset_openSegment : Ioo x y ⊆ openSegment 𝕜 x y 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y : 𝕜}
Goal: Set.Ioo x y ⊆ openSegment 𝕜 x y
-- openSegment_eq_Ioo
@[simp]
theorem openSegment_eq_Ioo (h : x < y) : openSegment 𝕜 x y = Ioo x y 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y : 𝕜} (h : x < y)
Goal: openSegment 𝕜 x y = Set.Ioo x y
-- segment_eq_Icc'
theorem segment_eq_Icc' (x y : 𝕜) : [x -[𝕜] y] = Icc (min x y) (max x y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] (x y : 𝕜)
Goal: [x-[𝕜]y] = Set.Icc (min x y) (max x y)
-- openSegment_eq_Ioo'
theorem openSegment_eq_Ioo' (hxy : x ≠ y) : openSegment 𝕜 x y = Ioo (min x y) (max x y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y : 𝕜} (hxy : x ≠ y)
Goal: openSegment 𝕜 x y = Set.Ioo (min x y) (max x y)
-- segment_eq_uIcc
theorem segment_eq_uIcc (x y : 𝕜) : [x -[𝕜] y] = uIcc x y 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] (x y : 𝕜)
Goal: [x-[𝕜]y] = Set.uIcc x y
-- Convex.mem_Icc
theorem Convex.mem_Icc (h : x ≤ y) :
    z ∈ Icc x y ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y z : 𝕜} (h : x ≤ y)
Goal: z ∈ Set.Icc x y ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z
-- Convex.mem_Ioo
theorem Convex.mem_Ioo (h : x < y) :
    z ∈ Ioo x y ↔ ∃ a b, 0 < a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y z : 𝕜} (h : x < y)
Goal: z ∈ Set.Ioo x y ↔ ∃ a b, 0 < a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z
-- Convex.mem_Ioc
theorem Convex.mem_Ioc (h : x < y) :
    z ∈ Ioc x y ↔ ∃ a b, 0 ≤ a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y z : 𝕜} (h : x < y)
Goal: z ∈ Set.Ioc x y ↔ ∃ a b, 0 ≤ a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z
-- Convex.mem_Ico
theorem Convex.mem_Ico (h : x < y) :
    z ∈ Ico x y ↔ ∃ a b, 0 < a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {x y z : 𝕜} (h : x < y)
Goal: z ∈ Set.Ico x y ↔ ∃ a b, 0 < a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z
-- Prod.segment_subset
theorem segment_subset (x y : E × F) : segment 𝕜 x y ⊆ segment 𝕜 x.1 y.1 ×ˢ segment 𝕜 x.2 y.2 
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 y : E × F)
Goal: [x-[𝕜]y] ⊆ [x.1-[𝕜]y.1] ×ˢ [x.2-[𝕜]y.2]
-- Prod.openSegment_subset
theorem openSegment_subset (x y : E × F) :
    openSegment 𝕜 x y ⊆ openSegment 𝕜 x.1 y.1 ×ˢ openSegment 𝕜 x.2 y.2 
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 y : E × F)
Goal: openSegment 𝕜 x y ⊆ openSegment 𝕜 x.1 y.1 ×ˢ openSegment 𝕜 x.2 y.2
-- Prod.image_mk_segment_left
theorem image_mk_segment_left (x₁ x₂ : E) (y : F) :
    (fun x => (x, y)) '' [x₁ -[𝕜] x₂] = [(x₁, y) -[𝕜] (x₂, y)] 
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] (x₁ x₂ : E) (y : F)
Goal: (fun x => (x, y)) '' [x₁-[𝕜]x₂] = [(x₁, y)-[𝕜](x₂, y)]
-- Prod.image_mk_segment_right
theorem image_mk_segment_right (x : E) (y₁ y₂ : F) :
    (fun y => (x, y)) '' [y₁ -[𝕜] y₂] = [(x, y₁) -[𝕜] (x, y₂)] 
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) (y₁ y₂ : F)
Goal: (fun y => (x, y)) '' [y₁-[𝕜]y₂] = [(x, y₁)-[𝕜](x, y₂)]
-- Prod.image_mk_openSegment_left
theorem image_mk_openSegment_left (x₁ x₂ : E) (y : F) :
    (fun x => (x, y)) '' openSegment 𝕜 x₁ x₂ = openSegment 𝕜 (x₁, y) (x₂, y) 
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] (x₁ x₂ : E) (y : F)
Goal: (fun x => (x, y)) '' openSegment 𝕜 x₁ x₂ = openSegment 𝕜 (x₁, y) (x₂, y)
-- Prod.image_mk_openSegment_right
@[simp]
theorem image_mk_openSegment_right (x : E) (y₁ y₂ : F) :
    (fun y => (x, y)) '' openSegment 𝕜 y₁ y₂ = openSegment 𝕜 (x, y₁) (x, y₂) 
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) (y₁ y₂ : F)
Goal: (fun y => (x, y)) '' openSegment 𝕜 y₁ y₂ = openSegment 𝕜 (x, y₁) (x, y₂)
-- Pi.segment_subset
theorem segment_subset (x y : ∀ i, π i) : segment 𝕜 x y ⊆ s.pi fun i => segment 𝕜 (x i) (y i) 
Arguments: {𝕜 : Type u_1} {ι : Type u_5} {π : ι → Type u_6} [OrderedSemiring 𝕜] [(i : ι) → AddCommMonoid (π i)] [(i : ι) → Module 𝕜 (π i)] {s : Set ι} (x y : (i : ι) → π i)
Goal: [x-[𝕜]y] ⊆ s.pi fun i => [x i-[𝕜]y i]
-- Pi.openSegment_subset
theorem openSegment_subset (x y : ∀ i, π i) :
    openSegment 𝕜 x y ⊆ s.pi fun i => openSegment 𝕜 (x i) (y i) 
Arguments: {𝕜 : Type u_1} {ι : Type u_5} {π : ι → Type u_6} [OrderedSemiring 𝕜] [(i : ι) → AddCommMonoid (π i)] [(i : ι) → Module 𝕜 (π i)] {s : Set ι} (x y : (i : ι) → π i)
Goal: openSegment 𝕜 x y ⊆ s.pi fun i => openSegment 𝕜 (x i) (y i)
-- Pi.image_update_segment
theorem image_update_segment (i : ι) (x₁ x₂ : π i) (y : ∀ i, π i) :
    update y i '' [x₁ -[𝕜] x₂] = [update y i x₁ -[𝕜] update y i x₂] 
Arguments: {𝕜 : Type u_1} {ι : Type u_5} {π : ι → Type u_6} [OrderedSemiring 𝕜] [(i : ι) → AddCommMonoid (π i)] [(i : ι) → Module 𝕜 (π i)] [DecidableEq ι] (i : ι) (x₁ x₂ : π i) (y : (i : ι) → π i)
Goal: Function.update y i '' [x₁-[𝕜]x₂] = [Function.update y i x₁-[𝕜]Function.update y i x₂]
-- Pi.image_update_openSegment
theorem image_update_openSegment (i : ι) (x₁ x₂ : π i) (y : ∀ i, π i) :
    update y i '' openSegment 𝕜 x₁ x₂ = openSegment 𝕜 (update y i x₁) (update y i x₂) 
Arguments: {𝕜 : Type u_1} {ι : Type u_5} {π : ι → Type u_6} [OrderedSemiring 𝕜] [(i : ι) → AddCommMonoid (π i)] [(i : ι) → Module 𝕜 (π i)] [DecidableEq ι] (i : ι) (x₁ x₂ : π i) (y : (i : ι) → π i)
Goal: Function.update y i '' openSegment 𝕜 x₁ x₂ = openSegment 𝕜 (Function.update y i x₁) (Function.update y i x₂)
-- convexOn_norm
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hs : Convex ℝ s)
Goal: ConvexOn ℝ s norm
-- convexOn_univ_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: ConvexOn ℝ Set.univ norm
-- convexOn_dist
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (z : E) (hs : Convex ℝ s)
Goal: ConvexOn ℝ s fun z' => dist z' z
-- convexOn_univ_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (z : E)
Goal: ConvexOn ℝ Set.univ fun z' => dist z' z
-- convex_ball
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (a : E) (r : ℝ)
Goal: Convex ℝ (Metric.ball a r)
-- convex_closedBall
theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (a : E) (r : ℝ)
Goal: Convex ℝ (Metric.closedBall a r)
-- Convex.thickening
theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hs : Convex ℝ s) (δ : ℝ)
Goal: Convex ℝ (Metric.thickening δ s)
-- Convex.cthickening
theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hs : Convex ℝ s) (δ : ℝ)
Goal: Convex ℝ (Metric.cthickening δ s)
-- convexHull_exists_dist_ge
theorem convexHull_exists_dist_ge {s : Set E} {x : E} (hx : x ∈ convexHull ℝ s) (y : E) :
    ∃ x' ∈ s, dist x y ≤ dist x' y 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {x : E} (hx : x ∈ (convexHull ℝ) s) (y : E)
Goal: ∃ x' ∈ s, dist x y ≤ dist x' y
-- convexHull_exists_dist_ge2
theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
    (hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s t : Set E} {x y : E} (hx : x ∈ (convexHull ℝ) s) (hy : y ∈ (convexHull ℝ) t)
Goal: ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y'
-- convexHull_ediam
@[simp]
theorem convexHull_ediam (s : Set E) : EMetric.diam (convexHull ℝ s) = EMetric.diam s 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (s : Set E)
Goal: EMetric.diam ((convexHull ℝ) s) = EMetric.diam s
-- convexHull_diam
@[simp]
theorem convexHull_diam (s : Set E) : Metric.diam (convexHull ℝ s) = Metric.diam s 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (s : Set E)
Goal: Metric.diam ((convexHull ℝ) s) = Metric.diam s
-- isBounded_convexHull
@[simp]
theorem isBounded_convexHull {s : Set E} :
    Bornology.IsBounded (convexHull ℝ s) ↔ Bornology.IsBounded s 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E}
Goal: Bornology.IsBounded ((convexHull ℝ) s) ↔ Bornology.IsBounded s
-- Wbtw.dist_add_dist
theorem Wbtw.dist_add_dist {x y z : P} (h : Wbtw ℝ x y z) :
    dist x y + dist y z = dist x z 
Arguments: {E : Type u_2} {P : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P] {x y z : P} (h : Wbtw ℝ x y z)
Goal: dist x y + dist y z = dist x z
-- dist_add_dist_of_mem_segment
theorem dist_add_dist_of_mem_segment {x y z : E} (h : y ∈ [x -[ℝ] z]) :
    dist x y + dist y z = dist x z 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y z : E} (h : y ∈ [x-[ℝ]z])
Goal: dist x y + dist y z = dist x z
-- isConnected_setOf_sameRay
theorem isConnected_setOf_sameRay (x : E) : IsConnected { y | SameRay ℝ x y } 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E)
Goal: IsConnected {y | SameRay ℝ x y}
-- isConnected_setOf_sameRay_and_ne_zero
theorem isConnected_setOf_sameRay_and_ne_zero {x : E} (hx : x ≠ 0) :
    IsConnected { y | SameRay ℝ x y ∧ y ≠ 0 } 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x : E} (hx : x ≠ 0)
Goal: IsConnected {y | SameRay ℝ x y ∧ y ≠ 0}
-- uniformConvexOn_zero
@[simp] lemma uniformConvexOn_zero : UniformConvexOn s 0 f ↔ ConvexOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {f : E → ℝ}
Goal: UniformConvexOn s 0 f ↔ ConvexOn ℝ s f
-- uniformConcaveOn_zero
@[simp] lemma uniformConcaveOn_zero : UniformConcaveOn s 0 f ↔ ConcaveOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {f : E → ℝ}
Goal: UniformConcaveOn s 0 f ↔ ConcaveOn ℝ s f
-- UniformConvexOn.mono
lemma UniformConvexOn.mono (hψφ : ψ ≤ φ) (hf : UniformConvexOn s φ f) : UniformConvexOn s ψ f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ ψ : ℝ → ℝ} {s : Set E} {f : E → ℝ} (hψφ : ψ ≤ φ) (hf : UniformConvexOn s φ f)
Goal: UniformConvexOn s ψ f
-- UniformConcaveOn.mono
lemma UniformConcaveOn.mono (hψφ : ψ ≤ φ) (hf : UniformConcaveOn s φ f) : UniformConcaveOn s ψ f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ ψ : ℝ → ℝ} {s : Set E} {f : E → ℝ} (hψφ : ψ ≤ φ) (hf : UniformConcaveOn s φ f)
Goal: UniformConcaveOn s ψ f
-- UniformConvexOn.convexOn
lemma UniformConvexOn.convexOn (hf : UniformConvexOn s φ f) (hφ : 0 ≤ φ) : ConvexOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ : ℝ → ℝ} {s : Set E} {f : E → ℝ} (hf : UniformConvexOn s φ f) (hφ : 0 ≤ φ)
Goal: ConvexOn ℝ s f
-- UniformConcaveOn.concaveOn
lemma UniformConcaveOn.concaveOn (hf : UniformConcaveOn s φ f) (hφ : 0 ≤ φ) : ConcaveOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ : ℝ → ℝ} {s : Set E} {f : E → ℝ} (hf : UniformConcaveOn s φ f) (hφ : 0 ≤ φ)
Goal: ConcaveOn ℝ s f
-- UniformConvexOn.strictConvexOn
lemma UniformConvexOn.strictConvexOn (hf : UniformConvexOn s φ f) (hφ : ∀ r, r ≠ 0 → 0 < φ r) :
    StrictConvexOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ : ℝ → ℝ} {s : Set E} {f : E → ℝ} (hf : UniformConvexOn s φ f) (hφ : ∀ (r : ℝ), r ≠ 0 → 0 < φ r)
Goal: StrictConvexOn ℝ s f
-- UniformConcaveOn.strictConcaveOn
lemma UniformConcaveOn.strictConcaveOn (hf : UniformConcaveOn s φ f) (hφ : ∀ r, r ≠ 0 → 0 < φ r) :
    StrictConcaveOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ : ℝ → ℝ} {s : Set E} {f : E → ℝ} (hf : UniformConcaveOn s φ f) (hφ : ∀ (r : ℝ), r ≠ 0 → 0 < φ r)
Goal: StrictConcaveOn ℝ s f
-- UniformConvexOn.add
lemma UniformConvexOn.add (hf : UniformConvexOn s φ f) (hg : UniformConvexOn s ψ g) :
    UniformConvexOn s (φ + ψ) (f + g) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ ψ : ℝ → ℝ} {s : Set E} {f g : E → ℝ} (hf : UniformConvexOn s φ f) (hg : UniformConvexOn s ψ g)
Goal: UniformConvexOn s (φ + ψ) (f + g)
-- UniformConcaveOn.add
lemma UniformConcaveOn.add (hf : UniformConcaveOn s φ f) (hg : UniformConcaveOn s ψ g) :
    UniformConcaveOn s (φ + ψ) (f + g) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ ψ : ℝ → ℝ} {s : Set E} {f g : E → ℝ} (hf : UniformConcaveOn s φ f) (hg : UniformConcaveOn s ψ g)
Goal: UniformConcaveOn s (φ + ψ) (f + g)
-- UniformConvexOn.neg
lemma UniformConvexOn.neg (hf : UniformConvexOn s φ f) : UniformConcaveOn s φ (-f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ : ℝ → ℝ} {s : Set E} {f : E → ℝ} (hf : UniformConvexOn s φ f)
Goal: UniformConcaveOn s φ (-f)
-- UniformConcaveOn.neg
lemma UniformConcaveOn.neg (hf : UniformConcaveOn s φ f) : UniformConvexOn s φ (-f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ : ℝ → ℝ} {s : Set E} {f : E → ℝ} (hf : UniformConcaveOn s φ f)
Goal: UniformConvexOn s φ (-f)
-- UniformConvexOn.sub
lemma UniformConvexOn.sub (hf : UniformConvexOn s φ f) (hg : UniformConcaveOn s ψ g) :
    UniformConvexOn s (φ + ψ) (f - g) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ ψ : ℝ → ℝ} {s : Set E} {f g : E → ℝ} (hf : UniformConvexOn s φ f) (hg : UniformConcaveOn s ψ g)
Goal: UniformConvexOn s (φ + ψ) (f - g)
-- UniformConcaveOn.sub
lemma UniformConcaveOn.sub (hf : UniformConcaveOn s φ f) (hg : UniformConvexOn s ψ g) :
    UniformConcaveOn s (φ + ψ) (f - g) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {φ ψ : ℝ → ℝ} {s : Set E} {f g : E → ℝ} (hf : UniformConcaveOn s φ f) (hg : UniformConvexOn s ψ g)
Goal: UniformConcaveOn s (φ + ψ) (f - g)
-- StrongConvexOn.mono
nonrec lemma StrongConvexOn.mono (hmn : m ≤ n) (hf : StrongConvexOn s n f) : StrongConvexOn s m f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {f : E → ℝ} {m n : ℝ} (hmn : m ≤ n) (hf : StrongConvexOn s n f)
Goal: StrongConvexOn s m f
-- StrongConcaveOn.mono
nonrec lemma StrongConcaveOn.mono (hmn : m ≤ n) (hf : StrongConcaveOn s n f) :
    StrongConcaveOn s m f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {f : E → ℝ} {m n : ℝ} (hmn : m ≤ n) (hf : StrongConcaveOn s n f)
Goal: StrongConcaveOn s m f
-- strongConvexOn_zero
@[simp] lemma strongConvexOn_zero : StrongConvexOn s 0 f ↔ ConvexOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {f : E → ℝ}
Goal: StrongConvexOn s 0 f ↔ ConvexOn ℝ s f
-- strongConcaveOn_zero
@[simp] lemma strongConcaveOn_zero : StrongConcaveOn s 0 f ↔ ConcaveOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {f : E → ℝ}
Goal: StrongConcaveOn s 0 f ↔ ConcaveOn ℝ s f
-- StrongConvexOn.strictConvexOn
nonrec lemma StrongConvexOn.strictConvexOn (hf : StrongConvexOn s m f) (hm : 0 < m) :
    StrictConvexOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {f : E → ℝ} {m : ℝ} (hf : StrongConvexOn s m f) (hm : 0 < m)
Goal: StrictConvexOn ℝ s f
-- StrongConcaveOn.strictConcaveOn
nonrec lemma StrongConcaveOn.strictConcaveOn (hf : StrongConcaveOn s m f) (hm : 0 < m) :
    StrictConcaveOn ℝ s f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {f : E → ℝ} {m : ℝ} (hf : StrongConcaveOn s m f) (hm : 0 < m)
Goal: StrictConcaveOn ℝ s f
-- _private.Mathlib.Analysis.Convex.Strong.0.aux_sub
private lemma aux_sub (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
    a * (f x - m / (2 : ℝ) * ‖x‖ ^ 2) + b * (f y - m / (2 : ℝ) * ‖y‖ ^ 2) +
      m / (2 : ℝ) * ‖a • x + b • y‖ ^ 2
      = a * f x + b * f y - m / (2 : ℝ) * a * b * ‖x - y‖ ^ 2 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {a b m : ℝ} {x y : E} {f : E → ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: a * (f x - m / 2 * ‖x‖ ^ 2) + b * (f y - m / 2 * ‖y‖ ^ 2) + m / 2 * ‖a • x + b • y‖ ^ 2 =
  a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2
-- _private.Mathlib.Analysis.Convex.Strong.0.aux_add
private lemma aux_add (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
    a * (f x + m / (2 : ℝ) * ‖x‖ ^ 2) + b * (f y + m / (2 : ℝ) * ‖y‖ ^ 2) -
      m / (2 : ℝ) * ‖a • x + b • y‖ ^ 2
      = a * f x + b * f y + m / (2 : ℝ) * a * b * ‖x - y‖ ^ 2 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {a b m : ℝ} {x y : E} {f : E → ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: a * (f x + m / 2 * ‖x‖ ^ 2) + b * (f y + m / 2 * ‖y‖ ^ 2) - m / 2 * ‖a • x + b • y‖ ^ 2 =
  a * f x + b * f y + m / 2 * a * b * ‖x - y‖ ^ 2