ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ConvexOn.map_sum_le
theorem ConvexOn.map_sum_le (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1)
    (hmem : ∀ i ∈ t, p i ∈ s) : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s)
Goal: f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i)
-- ConcaveOn.le_map_sum
theorem ConcaveOn.le_map_sum (hf : ConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
    (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
    (∑ i ∈ t, w i • f (p i)) ≤ f (∑ i ∈ t, w i • p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : ConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s)
Goal: ∑ i ∈ t, w i • f (p i) ≤ f (∑ i ∈ t, w i • p i)
-- ConvexOn.map_add_sum_le
lemma ConvexOn.map_add_sum_le (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
    (h₁ : v + ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hv : 0 ≤ v) (hq : q ∈ s) :
    f (v • q + ∑ i ∈ t, w i • p i) ≤ v • f q + ∑ i ∈ t, w i • f (p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} {v : 𝕜} {q : E} (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : v + ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hv : 0 ≤ v) (hq : q ∈ s)
Goal: f (v • q + ∑ i ∈ t, w i • p i) ≤ v • f q + ∑ i ∈ t, w i • f (p i)
-- ConcaveOn.map_add_sum_le
lemma ConcaveOn.map_add_sum_le (hf : ConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
    (h₁ : v + ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hv : 0 ≤ v) (hq : q ∈ s) :
    v • f q + ∑ i ∈ t, w i • f (p i) ≤ f (v • q + ∑ i ∈ t, w i • p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} {v : 𝕜} {q : E} (hf : ConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : v + ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hv : 0 ≤ v) (hq : q ∈ s)
Goal: v • f q + ∑ i ∈ t, w i • f (p i) ≤ f (v • q + ∑ i ∈ t, w i • p i)
-- StrictConvexOn.map_sum_lt
lemma StrictConvexOn.map_sum_lt (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i)
    (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hp : ∃ j ∈ t, ∃ k ∈ t, p j ≠ p k) :
    f (∑ i ∈ t, w i • p i) < ∑ i ∈ t, w i • f (p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hp : ∃ j ∈ t, ∃ k ∈ t, p j ≠ p k)
Goal: f (∑ i ∈ t, w i • p i) < ∑ i ∈ t, w i • f (p i)
-- StrictConcaveOn.lt_map_sum
lemma StrictConcaveOn.lt_map_sum (hf : StrictConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i)
    (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hp : ∃ j ∈ t, ∃ k ∈ t, p j ≠ p k) :
    ∑ i ∈ t, w i • f (p i) < f (∑ i ∈ t, w i • p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : StrictConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hp : ∃ j ∈ t, ∃ k ∈ t, p j ≠ p k)
Goal: ∑ i ∈ t, w i • f (p i) < f (∑ i ∈ t, w i • p i)
-- StrictConvexOn.eq_of_le_map_sum
lemma StrictConvexOn.eq_of_le_map_sum (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i)
    (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s)
    (h_eq : ∑ i ∈ t, w i • f (p i) ≤ f (∑ i ∈ t, w i • p i)) :
    ∀ ⦃j⦄, j ∈ t → ∀ ⦃k⦄, k ∈ t → p j = p k 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (h_eq : ∑ i ∈ t, w i • f (p i) ≤ f (∑ i ∈ t, w i • p i)) ⦃j : ι⦄
Goal: j ∈ t → ∀ ⦃k : ι⦄, k ∈ t → p j = p k
-- StrictConcaveOn.eq_of_map_sum_eq
lemma StrictConcaveOn.eq_of_map_sum_eq (hf : StrictConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i)
    (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s)
    (h_eq : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i)) :
    ∀ ⦃j⦄, j ∈ t → ∀ ⦃k⦄, k ∈ t → p j = p k 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : StrictConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (h_eq : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i)) ⦃j : ι⦄
Goal: j ∈ t → ∀ ⦃k : ι⦄, k ∈ t → p j = p k
-- StrictConvexOn.map_sum_eq_iff
lemma StrictConvexOn.map_sum_eq_iff {w : ι → 𝕜} {p : ι → E} (hf : StrictConvexOn 𝕜 s f)
    (h₀ : ∀ i ∈ t, 0 < w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
    f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔ ∀ j ∈ t, p j = ∑ i ∈ t, w i • p i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s)
Goal: f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔ ∀ j ∈ t, p j = ∑ i ∈ t, w i • p i
-- StrictConcaveOn.map_sum_eq_iff
lemma StrictConcaveOn.map_sum_eq_iff (hf : StrictConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i)
    (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
    f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔ ∀ j ∈ t, p j = ∑ i ∈ t, w i • p i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : StrictConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 < w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s)
Goal: f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔ ∀ j ∈ t, p j = ∑ i ∈ t, w i • p i
-- StrictConvexOn.map_sum_eq_iff'
lemma StrictConvexOn.map_sum_eq_iff' (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
    (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
    f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔
      ∀ j ∈ t, w j ≠ 0 → p j = ∑ i ∈ t, w i • p i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s)
Goal: f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔ ∀ j ∈ t, w j ≠ 0 → p j = ∑ i ∈ t, w i • p i
-- StrictConcaveOn.map_sum_eq_iff'
lemma StrictConcaveOn.map_sum_eq_iff' (hf : StrictConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
    (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
    f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔
      ∀ j ∈ t, w j ≠ 0 → p j = ∑ i ∈ t, w i • p i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : StrictConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s)
Goal: f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔ ∀ j ∈ t, w j ≠ 0 → p j = ∑ i ∈ t, w i • p i
-- le_sup_of_mem_convexHull
theorem le_sup_of_mem_convexHull {s : Finset E} (hf : ConvexOn 𝕜 (convexHull 𝕜 (s : Set E)) f)
    (hx : x ∈ convexHull 𝕜 (s : Set E)) :
    f x ≤ s.sup' (coe_nonempty.1 <| convexHull_nonempty_iff.1 ⟨x, hx⟩) f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [LinearOrderedField 𝕜] [AddCommGroup E] [LinearOrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} {x : E} {s : Finset E} (hf : ConvexOn 𝕜 ((convexHull 𝕜) ↑s) f) (hx : x ∈ (convexHull 𝕜) ↑s)
Goal: f x ≤ s.sup' ⋯ f
-- inf_le_of_mem_convexHull
theorem inf_le_of_mem_convexHull {s : Finset E} (hf : ConcaveOn 𝕜 (convexHull 𝕜 (s : Set E)) f)
    (hx : x ∈ convexHull 𝕜 (s : Set E)) :
    s.inf' (coe_nonempty.1 <| convexHull_nonempty_iff.1 ⟨x, hx⟩) f ≤ f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [LinearOrderedField 𝕜] [AddCommGroup E] [LinearOrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} {x : E} {s : Finset E} (hf : ConcaveOn 𝕜 ((convexHull 𝕜) ↑s) f) (hx : x ∈ (convexHull 𝕜) ↑s)
Goal: s.inf' ⋯ f ≤ f x
-- ConvexOn.exists_ge_of_centerMass
theorem ConvexOn.exists_ge_of_centerMass (h : ConvexOn 𝕜 s f) (hw₀ : ∀ i ∈ t, 0 ≤ w i)
    (hw₁ : 0 < ∑ i ∈ t, w i) (hp : ∀ i ∈ t, p i ∈ s) :
    ∃ i ∈ t, f (t.centerMass w p) ≤ f (p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [LinearOrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (h : ConvexOn 𝕜 s f) (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : 0 < ∑ i ∈ t, w i) (hp : ∀ i ∈ t, p i ∈ s)
Goal: ∃ i ∈ t, f (t.centerMass w p) ≤ f (p i)
-- ConcaveOn.exists_le_of_centerMass
theorem ConcaveOn.exists_le_of_centerMass (h : ConcaveOn 𝕜 s f) (hw₀ : ∀ i ∈ t, 0 ≤ w i)
    (hw₁ : 0 < ∑ i ∈ t, w i) (hp : ∀ i ∈ t, p i ∈ s) : ∃ i ∈ t, f (p i) ≤ f (t.centerMass w p) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [LinearOrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (h : ConcaveOn 𝕜 s f) (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : 0 < ∑ i ∈ t, w i) (hp : ∀ i ∈ t, p i ∈ s)
Goal: ∃ i ∈ t, f (p i) ≤ f (t.centerMass w p)
-- ConvexOn.exists_ge_of_mem_convexHull
theorem ConvexOn.exists_ge_of_mem_convexHull (hf : ConvexOn 𝕜 (convexHull 𝕜 s) f) {x}
    (hx : x ∈ convexHull 𝕜 s) : ∃ y ∈ s, f x ≤ f y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [LinearOrderedField 𝕜] [AddCommGroup E] [LinearOrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 ((convexHull 𝕜) s) f) {x : E} (hx : x ∈ (convexHull 𝕜) s)
Goal: ∃ y ∈ s, f x ≤ f y
-- ConcaveOn.exists_le_of_mem_convexHull
theorem ConcaveOn.exists_le_of_mem_convexHull (hf : ConcaveOn 𝕜 (convexHull 𝕜 s) f) {x}
    (hx : x ∈ convexHull 𝕜 s) : ∃ y ∈ s, f y ≤ f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [LinearOrderedField 𝕜] [AddCommGroup E] [LinearOrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 ((convexHull 𝕜) s) f) {x : E} (hx : x ∈ (convexHull 𝕜) s)
Goal: ∃ y ∈ s, f y ≤ f x
-- ConvexOn.le_max_of_mem_segment
lemma ConvexOn.le_max_of_mem_segment (hf : ConvexOn 𝕜 [x -[𝕜] y] f) (hz : z ∈ [x -[𝕜] y]) :
    f z ≤ max (f x) (f y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [LinearOrderedField 𝕜] [AddCommGroup E] [LinearOrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} {x y z : E} (hf : ConvexOn 𝕜 [x-[𝕜]y] f) (hz : z ∈ [x-[𝕜]y])
Goal: f z ≤ max (f x) (f y)
-- ConcaveOn.min_le_of_mem_segment
lemma ConcaveOn.min_le_of_mem_segment (hf : ConcaveOn 𝕜 [x -[𝕜] y] f) (hz : z ∈ [x -[𝕜] y]) :
    min (f x) (f y) ≤ f z 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [LinearOrderedField 𝕜] [AddCommGroup E] [LinearOrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} {x y z : E} (hf : ConcaveOn 𝕜 [x-[𝕜]y] f) (hz : z ∈ [x-[𝕜]y])
Goal: min (f x) (f y) ≤ f z
-- ConvexOn.le_max_of_mem_Icc
lemma ConvexOn.le_max_of_mem_Icc {f : 𝕜 → β} {x y z : 𝕜} (hf : ConvexOn 𝕜 (Icc x y) f)
    (hz : z ∈ Icc x y) : f z ≤ max (f x) (f y) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [LinearOrderedField 𝕜] [LinearOrderedAddCommGroup β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : 𝕜 → β} {x y z : 𝕜} (hf : ConvexOn 𝕜 (Set.Icc x y) f) (hz : z ∈ Set.Icc x y)
Goal: f z ≤ max (f x) (f y)
-- ConcaveOn.min_le_of_mem_Icc
lemma ConcaveOn.min_le_of_mem_Icc {f : 𝕜 → β} {x y z : 𝕜} (hf : ConcaveOn 𝕜 (Icc x y) f)
    (hz : z ∈ Icc x y) : min (f x) (f y) ≤ f z 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [LinearOrderedField 𝕜] [LinearOrderedAddCommGroup β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : 𝕜 → β} {x y z : 𝕜} (hf : ConcaveOn 𝕜 (Set.Icc x y) f) (hz : z ∈ Set.Icc x y)
Goal: min (f x) (f y) ≤ f z
-- ConvexBody.isCompact'
structure ConvexBody (V : Type*) [TopologicalSpace V] [AddCommMonoid V] [SMul ℝ V] where
    carrier : Set V
    convex' : Convex ℝ carrier
    isCompact' : IsCompact carrier
    nonempty' : carrier.Nonempty
Arguments: {V : Type u_2} [TopologicalSpace V] [AddCommMonoid V] [SMul ℝ V] (self : ConvexBody V)
Goal: IsCompact self.carrier
-- ConvexBody.convex'
structure ConvexBody (V : Type*) [TopologicalSpace V] [AddCommMonoid V] [SMul ℝ V] where
    carrier : Set V
    convex' : Convex ℝ carrier
    isCompact' : IsCompact carrier
    nonempty' : carrier.Nonempty
Arguments: {V : Type u_2} [TopologicalSpace V] [AddCommMonoid V] [SMul ℝ V] (self : ConvexBody V)
Goal: Convex ℝ self.carrier
-- ConvexBody.nonempty'
structure ConvexBody (V : Type*) [TopologicalSpace V] [AddCommMonoid V] [SMul ℝ V] where
    carrier : Set V
    convex' : Convex ℝ carrier
    isCompact' : IsCompact carrier
    nonempty' : carrier.Nonempty
Arguments: {V : Type u_2} [TopologicalSpace V] [AddCommMonoid V] [SMul ℝ V] (self : ConvexBody V)
Goal: self.carrier.Nonempty
-- ConvexBody.convex
protected theorem convex (K : ConvexBody V) : Convex ℝ (K : Set V) 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] (K : ConvexBody V)
Goal: Convex ℝ ↑K
-- ConvexBody.isCompact
protected theorem isCompact (K : ConvexBody V) : IsCompact (K : Set V) 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] (K : ConvexBody V)
Goal: IsCompact ↑K
-- ConvexBody.isClosed
protected theorem isClosed [T2Space V] (K : ConvexBody V) : IsClosed (K : Set V) 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] [T2Space V] (K : ConvexBody V)
Goal: IsClosed ↑K
-- ConvexBody.nonempty
protected theorem nonempty (K : ConvexBody V) : (K : Set V).Nonempty 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] (K : ConvexBody V)
Goal: (↑K).Nonempty
-- ConvexBody.ext
@[ext]
protected theorem ext {K L : ConvexBody V} (h : (K : Set V) = L) : K = L 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] {K L : ConvexBody V} (h : ↑K = ↑L)
Goal: K = L
-- ConvexBody.coe_mk
@[simp]
theorem coe_mk (s : Set V) (h₁ h₂ h₃) : (mk s h₁ h₂ h₃ : Set V) = s 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] (s : Set V) (h₁ : Convex ℝ s) (h₂ : IsCompact s) (h₃ : s.Nonempty)
Goal: ↑{ carrier := s, convex' := h₁, isCompact' := h₂, nonempty' := h₃ } = s
-- ConvexBody.zero_mem_of_symmetric
theorem zero_mem_of_symmetric (K : ConvexBody V) (h_symm : ∀ x ∈ K, - x ∈ K) : 0 ∈ K 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] (K : ConvexBody V) (h_symm : ∀ x ∈ K, -x ∈ K)
Goal: 0 ∈ K
-- ConvexBody.coe_nsmul
theorem coe_nsmul : ∀ (n : ℕ) (K : ConvexBody V), ↑(n • K) = n • (K : Set V)
  
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] [ContinuousAdd V] (n : ℕ) (K : ConvexBody V)
Goal: ↑(n • K) = n • ↑K
-- ConvexBody.coe_add
@[simp] -- Porting note: add norm_cast; we leave it out for now to reproduce mathlib3 behavior.
theorem coe_add (K L : ConvexBody V) : (↑(K + L) : Set V) = (K : Set V) + L 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] [ContinuousAdd V] (K L : ConvexBody V)
Goal: ↑(K + L) = ↑K + ↑L
-- ConvexBody.coe_zero
@[simp] -- Porting note: add norm_cast; we leave it out for now to reproduce mathlib3 behavior.
theorem coe_zero : (↑(0 : ConvexBody V) : Set V) = 0 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V]
Goal: ↑0 = 0
-- ConvexBody.coe_smul
@[simp] -- Porting note: add norm_cast; we leave it out for now to reproduce mathlib3 behavior.
theorem coe_smul (c : ℝ) (K : ConvexBody V) : (↑(c • K) : Set V) = c • (K : Set V) 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] [ContinuousSMul ℝ V] (c : ℝ) (K : ConvexBody V)
Goal: ↑(c • K) = c • ↑K
-- ConvexBody.coe_smul'
@[simp] -- Porting note: add norm_cast; we leave it out for now to reproduce mathlib3 behavior.
theorem coe_smul' (c : ℝ≥0) (K : ConvexBody V) : (↑(c • K) : Set V) = c • (K : Set V) 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] [ContinuousSMul ℝ V] [ContinuousAdd V] (c : ℝ≥0) (K : ConvexBody V)
Goal: ↑(c • K) = c • ↑K
-- ConvexBody.smul_le_of_le
theorem smul_le_of_le (K : ConvexBody V) (h_zero : 0 ∈ K) {a b : ℝ≥0} (h : a ≤ b) :
    a • K ≤ b • K 
Arguments: {V : Type u_1} [TopologicalSpace V] [AddCommGroup V] [Module ℝ V] [ContinuousSMul ℝ V] [ContinuousAdd V] (K : ConvexBody V) (h_zero : 0 ∈ K) {a b : ℝ≥0} (h : a ≤ b)
Goal: a • K ≤ b • K
-- ConvexBody.isBounded
protected theorem isBounded : Bornology.IsBounded (K : Set V) 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V] [NormedSpace ℝ V] (K : ConvexBody V)
Goal: Bornology.IsBounded ↑K
-- ConvexBody.hausdorffEdist_ne_top
theorem hausdorffEdist_ne_top {K L : ConvexBody V} : EMetric.hausdorffEdist (K : Set V) L ≠ ⊤ 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V] [NormedSpace ℝ V] {K L : ConvexBody V}
Goal: EMetric.hausdorffEdist ↑K ↑L ≠ ⊤
-- ConvexBody.hausdorffDist_coe
@[simp, norm_cast]
theorem hausdorffDist_coe : Metric.hausdorffDist (K : Set V) L = dist K L 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V] [NormedSpace ℝ V] (K L : ConvexBody V)
Goal: Metric.hausdorffDist ↑K ↑L = dist K L
-- ConvexBody.hausdorffEdist_coe
@[simp, norm_cast]
theorem hausdorffEdist_coe : EMetric.hausdorffEdist (K : Set V) L = edist K L 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V] [NormedSpace ℝ V] (K L : ConvexBody V)
Goal: EMetric.hausdorffEdist ↑K ↑L = edist K L
-- ConvexBody.iInter_smul_eq_self
theorem iInter_smul_eq_self [T2Space V] {u : ℕ → ℝ≥0} (K : ConvexBody V) (h_zero : 0 ∈ K)
    (hu : Tendsto u atTop (𝓝 0)) :
    ⋂ n : ℕ, (1 + (u n : ℝ)) • (K : Set V) = K 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [T2Space V] {u : ℕ → ℝ≥0} (K : ConvexBody V) (h_zero : 0 ∈ K) (hu : Filter.Tendsto u Filter.atTop (𝓝 0))
Goal: ⋂ n, (1 + ↑(u n)) • ↑K = ↑K
-- Convex.integral_mem
theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
    (hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (∫ x, f x ∂μ) ∈ s 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set E} {f : α → E} [MeasureTheory.IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hf : ∀ᵐ (x : α) ∂μ, f x ∈ s) (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ ∈ s
-- Convex.average_mem
theorem Convex.average_mem [IsFiniteMeasure μ] [NeZero μ] (hs : Convex ℝ s) (hsc : IsClosed s)
    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (⨍ x, f x ∂μ) ∈ s 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set E} {f : α → E} [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s) (hfi : MeasureTheory.Integrable f μ)
Goal: ⨍ (x : α), f x ∂μ ∈ s
-- Convex.set_average_mem
theorem Convex.set_average_mem (hs : Convex ℝ s) (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞)
    (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) : (⨍ x in t, f x ∂μ) ∈ s 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set E} {t : Set α} {f : α → E} (hs : Convex ℝ s) (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ⊤) (hfs : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ s) (hfi : MeasureTheory.IntegrableOn f t μ)
Goal: ⨍ (x : α) in t, f x ∂μ ∈ s
-- Convex.set_average_mem_closure
theorem Convex.set_average_mem_closure (hs : Convex ℝ s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞)
    (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) :
    (⨍ x in t, f x ∂μ) ∈ closure s 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set E} {t : Set α} {f : α → E} (hs : Convex ℝ s) (h0 : μ t ≠ 0) (ht : μ t ≠ ⊤) (hfs : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ s) (hfi : MeasureTheory.IntegrableOn f t μ)
Goal: ⨍ (x : α) in t, f x ∂μ ∈ closure s
-- ConvexOn.average_mem_epigraph
theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] [NeZero μ] (hg : ConvexOn ℝ s g)
    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
    (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
    (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set E} {f : α → E} {g : E → ℝ} [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s) (hfi : MeasureTheory.Integrable f μ) (hgi : MeasureTheory.Integrable (g ∘ f) μ)
Goal: (⨍ (x : α), f x ∂μ, ⨍ (x : α), g (f x) ∂μ) ∈ {p | p.1 ∈ s ∧ g p.1 ≤ p.2}
-- ConcaveOn.average_mem_hypograph
theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure μ] [NeZero μ] (hg : ConcaveOn ℝ s g)
    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
    (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
    (⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1} 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set E} {f : α → E} {g : E → ℝ} [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s) (hfi : MeasureTheory.Integrable f μ) (hgi : MeasureTheory.Integrable (g ∘ f) μ)
Goal: (⨍ (x : α), f x ∂μ, ⨍ (x : α), g (f x) ∂μ) ∈ {p | p.1 ∈ s ∧ p.2 ≤ g p.1}
-- ConvexOn.map_average_le
theorem ConvexOn.map_average_le [IsFiniteMeasure μ] [NeZero μ]
    (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s)
    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
    g (⨍ x, f x ∂μ) ≤ ⨍ x, g (f x) ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set E} {f : α → E} {g : E → ℝ} [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s) (hfi : MeasureTheory.Integrable f μ) (hgi : MeasureTheory.Integrable (g ∘ f) μ)
Goal: g (⨍ (x : α), f x ∂μ) ≤ ⨍ (x : α), g (f x) ∂μ