ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ConcaveOn.comp_convexOn
theorem ConcaveOn.comp_convexOn (hg : ConcaveOn 𝕜 (f '' s) g) (hf : ConvexOn 𝕜 s f)
    (hg' : AntitoneOn g (f '' s)) : ConcaveOn 𝕜 s (g ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 α] [SMul 𝕜 β] {s : Set E} {f : E → β} {g : β → α} (hg : ConcaveOn 𝕜 (f '' s) g) (hf : ConvexOn 𝕜 s f) (hg' : AntitoneOn g (f '' s))
Goal: ConcaveOn 𝕜 s (g ∘ f)
-- StrictConvexOn.comp
theorem StrictConvexOn.comp (hg : StrictConvexOn 𝕜 (f '' s) g) (hf : StrictConvexOn 𝕜 s f)
    (hg' : StrictMonoOn g (f '' s)) (hf' : s.InjOn f) : StrictConvexOn 𝕜 s (g ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 α] [SMul 𝕜 β] {s : Set E} {f : E → β} {g : β → α} (hg : StrictConvexOn 𝕜 (f '' s) g) (hf : StrictConvexOn 𝕜 s f) (hg' : StrictMonoOn g (f '' s)) (hf' : Set.InjOn f s)
Goal: StrictConvexOn 𝕜 s (g ∘ f)
-- StrictConcaveOn.comp
theorem StrictConcaveOn.comp (hg : StrictConcaveOn 𝕜 (f '' s) g) (hf : StrictConcaveOn 𝕜 s f)
    (hg' : StrictMonoOn g (f '' s)) (hf' : s.InjOn f) : StrictConcaveOn 𝕜 s (g ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 α] [SMul 𝕜 β] {s : Set E} {f : E → β} {g : β → α} (hg : StrictConcaveOn 𝕜 (f '' s) g) (hf : StrictConcaveOn 𝕜 s f) (hg' : StrictMonoOn g (f '' s)) (hf' : Set.InjOn f s)
Goal: StrictConcaveOn 𝕜 s (g ∘ f)
-- StrictConvexOn.comp_strictConcaveOn
theorem StrictConvexOn.comp_strictConcaveOn (hg : StrictConvexOn 𝕜 (f '' s) g)
    (hf : StrictConcaveOn 𝕜 s f) (hg' : StrictAntiOn g (f '' s)) (hf' : s.InjOn f) :
    StrictConvexOn 𝕜 s (g ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 α] [SMul 𝕜 β] {s : Set E} {f : E → β} {g : β → α} (hg : StrictConvexOn 𝕜 (f '' s) g) (hf : StrictConcaveOn 𝕜 s f) (hg' : StrictAntiOn g (f '' s)) (hf' : Set.InjOn f s)
Goal: StrictConvexOn 𝕜 s (g ∘ f)
-- StrictConcaveOn.comp_strictConvexOn
theorem StrictConcaveOn.comp_strictConvexOn (hg : StrictConcaveOn 𝕜 (f '' s) g)
    (hf : StrictConvexOn 𝕜 s f) (hg' : StrictAntiOn g (f '' s)) (hf' : s.InjOn f) :
    StrictConcaveOn 𝕜 s (g ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 α] [SMul 𝕜 β] {s : Set E} {f : E → β} {g : β → α} (hg : StrictConcaveOn 𝕜 (f '' s) g) (hf : StrictConvexOn 𝕜 s f) (hg' : StrictAntiOn g (f '' s)) (hf' : Set.InjOn f s)
Goal: StrictConcaveOn 𝕜 s (g ∘ f)
-- ConvexOn.add
theorem ConvexOn.add (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) : ConvexOn 𝕜 s (f + g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [DistribMulAction 𝕜 β] {s : Set E} {f g : E → β} (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
Goal: ConvexOn 𝕜 s (f + g)
-- ConcaveOn.add
theorem ConcaveOn.add (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) : ConcaveOn 𝕜 s (f + g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [DistribMulAction 𝕜 β] {s : Set E} {f g : E → β} (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
Goal: ConcaveOn 𝕜 s (f + g)
-- convexOn_const
theorem convexOn_const (c : β) (hs : Convex 𝕜 s) : ConvexOn 𝕜 s fun _ : E => c 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] {s : Set E} (c : β) (hs : Convex 𝕜 s)
Goal: ConvexOn 𝕜 s fun x => c
-- concaveOn_const
theorem concaveOn_const (c : β) (hs : Convex 𝕜 s) : ConcaveOn 𝕜 s fun _ => c 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] {s : Set E} (c : β) (hs : Convex 𝕜 s)
Goal: ConcaveOn 𝕜 s fun x => c
-- ConvexOn.add_const
theorem ConvexOn.add_const (hf : ConvexOn 𝕜 s f) (b : β) :
    ConvexOn 𝕜 s (f + fun _ => b) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f) (b : β)
Goal: ConvexOn 𝕜 s (f + fun x => b)
-- ConcaveOn.add_const
theorem ConcaveOn.add_const (hf : ConcaveOn 𝕜 s f) (b : β) :
    ConcaveOn 𝕜 s (f + fun _ => b) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f) (b : β)
Goal: ConcaveOn 𝕜 s (f + fun x => b)
-- convexOn_of_convex_epigraph
theorem convexOn_of_convex_epigraph (h : Convex 𝕜 { p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2 }) :
    ConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β} (h : Convex 𝕜 {p | p.1 ∈ s ∧ f p.1 ≤ p.2})
Goal: ConvexOn 𝕜 s f
-- concaveOn_of_convex_hypograph
theorem concaveOn_of_convex_hypograph (h : Convex 𝕜 { p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1 }) :
    ConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β} (h : Convex 𝕜 {p | p.1 ∈ s ∧ p.2 ≤ f p.1})
Goal: ConcaveOn 𝕜 s f
-- ConvexOn.convex_le
theorem ConvexOn.convex_le (hf : ConvexOn 𝕜 s f) (r : β) : Convex 𝕜 ({ x ∈ s | f x ≤ r }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ f x ≤ r}
-- ConcaveOn.convex_ge
theorem ConcaveOn.convex_ge (hf : ConcaveOn 𝕜 s f) (r : β) : Convex 𝕜 ({ x ∈ s | r ≤ f x }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ r ≤ f x}
-- ConvexOn.convex_epigraph
theorem ConvexOn.convex_epigraph (hf : ConvexOn 𝕜 s f) :
    Convex 𝕜 { p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f)
Goal: Convex 𝕜 {p | p.1 ∈ s ∧ f p.1 ≤ p.2}
-- ConcaveOn.convex_hypograph
theorem ConcaveOn.convex_hypograph (hf : ConcaveOn 𝕜 s f) :
    Convex 𝕜 { p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f)
Goal: Convex 𝕜 {p | p.1 ∈ s ∧ p.2 ≤ f p.1}
-- convexOn_iff_convex_epigraph
theorem convexOn_iff_convex_epigraph :
    ConvexOn 𝕜 s f ↔ Convex 𝕜 { p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β}
Goal: ConvexOn 𝕜 s f ↔ Convex 𝕜 {p | p.1 ∈ s ∧ f p.1 ≤ p.2}
-- concaveOn_iff_convex_hypograph
theorem concaveOn_iff_convex_hypograph :
    ConcaveOn 𝕜 s f ↔ Convex 𝕜 { p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β}
Goal: ConcaveOn 𝕜 s f ↔ Convex 𝕜 {p | p.1 ∈ s ∧ p.2 ≤ f p.1}
-- ConvexOn.translate_right
theorem ConvexOn.translate_right (hf : ConvexOn 𝕜 s f) (c : E) :
    ConvexOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => c + z) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f) (c : E)
Goal: ConvexOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => c + z)
-- ConcaveOn.translate_right
theorem ConcaveOn.translate_right (hf : ConcaveOn 𝕜 s f) (c : E) :
    ConcaveOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => c + z) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f) (c : E)
Goal: ConcaveOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => c + z)
-- ConvexOn.translate_left
theorem ConvexOn.translate_left (hf : ConvexOn 𝕜 s f) (c : E) :
    ConvexOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => z + c) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f) (c : E)
Goal: ConvexOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => z + c)
-- ConcaveOn.translate_left
theorem ConcaveOn.translate_left (hf : ConcaveOn 𝕜 s f) (c : E) :
    ConcaveOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => z + c) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f) (c : E)
Goal: ConcaveOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => z + c)
-- convexOn_iff_forall_pos
theorem convexOn_iff_forall_pos {s : Set E} {f : E → β} :
    ConvexOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b →
      a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β}
Goal: ConvexOn 𝕜 s f ↔
  Convex 𝕜 s ∧
    ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y
-- concaveOn_iff_forall_pos
theorem concaveOn_iff_forall_pos {s : Set E} {f : E → β} :
    ConcaveOn 𝕜 s f ↔
      Convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
        a • f x + b • f y ≤ f (a • x + b • y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β}
Goal: ConcaveOn 𝕜 s f ↔
  Convex 𝕜 s ∧
    ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)
-- convexOn_iff_pairwise_pos
theorem convexOn_iff_pairwise_pos {s : Set E} {f : E → β} :
    ConvexOn 𝕜 s f ↔
      Convex 𝕜 s ∧
        s.Pairwise fun x y =>
          ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β}
Goal: ConvexOn 𝕜 s f ↔
  Convex 𝕜 s ∧ s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y
-- concaveOn_iff_pairwise_pos
theorem concaveOn_iff_pairwise_pos {s : Set E} {f : E → β} :
    ConcaveOn 𝕜 s f ↔
      Convex 𝕜 s ∧
        s.Pairwise fun x y =>
          ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β}
Goal: ConcaveOn 𝕜 s f ↔
  Convex 𝕜 s ∧ s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)
-- LinearMap.convexOn
theorem LinearMap.convexOn (f : E →ₗ[𝕜] β) {s : Set E} (hs : Convex 𝕜 s) : ConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] (f : E →ₗ[𝕜] β) {s : Set E} (hs : Convex 𝕜 s)
Goal: ConvexOn 𝕜 s ⇑f
-- LinearMap.concaveOn
theorem LinearMap.concaveOn (f : E →ₗ[𝕜] β) {s : Set E} (hs : Convex 𝕜 s) : ConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] (f : E →ₗ[𝕜] β) {s : Set E} (hs : Convex 𝕜 s)
Goal: ConcaveOn 𝕜 s ⇑f
-- StrictConvexOn.convexOn
theorem StrictConvexOn.convexOn {s : Set E} {f : E → β} (hf : StrictConvexOn 𝕜 s f) :
    ConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β} (hf : StrictConvexOn 𝕜 s f)
Goal: ConvexOn 𝕜 s f
-- StrictConcaveOn.concaveOn
theorem StrictConcaveOn.concaveOn {s : Set E} {f : E → β} (hf : StrictConcaveOn 𝕜 s f) :
    ConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] {s : Set E} {f : E → β} (hf : StrictConcaveOn 𝕜 s f)
Goal: ConcaveOn 𝕜 s f
-- StrictConvexOn.convex_lt
theorem StrictConvexOn.convex_lt (hf : StrictConvexOn 𝕜 s f) (r : β) :
    Convex 𝕜 ({ x ∈ s | f x < r }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : StrictConvexOn 𝕜 s f) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ f x < r}
-- StrictConcaveOn.convex_gt
theorem StrictConcaveOn.convex_gt (hf : StrictConcaveOn 𝕜 s f) (r : β) :
    Convex 𝕜 ({ x ∈ s | r < f x }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : StrictConcaveOn 𝕜 s f) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ r < f x}
-- LinearOrder.convexOn_of_lt
theorem LinearOrder.convexOn_of_lt (hs : Convex 𝕜 s)
    (hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
      f (a • x + b • y) ≤ a • f x + b • f y) :
    ConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [LinearOrder E] {s : Set E} {f : E → β} (hs : Convex 𝕜 s) (hf :
  ∀ ⦃x : E⦄,
    x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y)
Goal: ConvexOn 𝕜 s f
-- LinearOrder.concaveOn_of_lt
theorem LinearOrder.concaveOn_of_lt (hs : Convex 𝕜 s)
    (hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
      a • f x + b • f y ≤ f (a • x + b • y)) :
    ConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [LinearOrder E] {s : Set E} {f : E → β} (hs : Convex 𝕜 s) (hf :
  ∀ ⦃x : E⦄,
    x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y))
Goal: ConcaveOn 𝕜 s f
-- LinearOrder.strictConvexOn_of_lt
theorem LinearOrder.strictConvexOn_of_lt (hs : Convex 𝕜 s)
    (hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
      f (a • x + b • y) < a • f x + b • f y) :
    StrictConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [LinearOrder E] {s : Set E} {f : E → β} (hs : Convex 𝕜 s) (hf :
  ∀ ⦃x : E⦄,
    x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y)
Goal: StrictConvexOn 𝕜 s f
-- LinearOrder.strictConcaveOn_of_lt
theorem LinearOrder.strictConcaveOn_of_lt (hs : Convex 𝕜 s)
    (hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
      a • f x + b • f y < f (a • x + b • y)) :
    StrictConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [LinearOrder E] {s : Set E} {f : E → β} (hs : Convex 𝕜 s) (hf :
  ∀ ⦃x : E⦄,
    x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • f x + b • f y < f (a • x + b • y))
Goal: StrictConcaveOn 𝕜 s f
-- ConvexOn.comp_linearMap
theorem ConvexOn.comp_linearMap {f : F → β} {s : Set F} (hf : ConvexOn 𝕜 s f) (g : E →ₗ[𝕜] F) :
    ConvexOn 𝕜 (g ⁻¹' s) (f ∘ g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 F] [SMul 𝕜 β] {f : F → β} {s : Set F} (hf : ConvexOn 𝕜 s f) (g : E →ₗ[𝕜] F)
Goal: ConvexOn 𝕜 (⇑g ⁻¹' s) (f ∘ ⇑g)
-- ConcaveOn.comp_linearMap
theorem ConcaveOn.comp_linearMap {f : F → β} {s : Set F} (hf : ConcaveOn 𝕜 s f) (g : E →ₗ[𝕜] F) :
    ConcaveOn 𝕜 (g ⁻¹' s) (f ∘ g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 F] [SMul 𝕜 β] {f : F → β} {s : Set F} (hf : ConcaveOn 𝕜 s f) (g : E →ₗ[𝕜] F)
Goal: ConcaveOn 𝕜 (⇑g ⁻¹' s) (f ∘ ⇑g)
-- StrictConvexOn.add_convexOn
theorem StrictConvexOn.add_convexOn (hf : StrictConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) :
    StrictConvexOn 𝕜 s (f + g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [SMul 𝕜 E] [DistribMulAction 𝕜 β] {s : Set E} {f g : E → β} (hf : StrictConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
Goal: StrictConvexOn 𝕜 s (f + g)
-- ConvexOn.add_strictConvexOn
theorem ConvexOn.add_strictConvexOn (hf : ConvexOn 𝕜 s f) (hg : StrictConvexOn 𝕜 s g) :
    StrictConvexOn 𝕜 s (f + g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [SMul 𝕜 E] [DistribMulAction 𝕜 β] {s : Set E} {f g : E → β} (hf : ConvexOn 𝕜 s f) (hg : StrictConvexOn 𝕜 s g)
Goal: StrictConvexOn 𝕜 s (f + g)
-- StrictConvexOn.add
theorem StrictConvexOn.add (hf : StrictConvexOn 𝕜 s f) (hg : StrictConvexOn 𝕜 s g) :
    StrictConvexOn 𝕜 s (f + g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [SMul 𝕜 E] [DistribMulAction 𝕜 β] {s : Set E} {f g : E → β} (hf : StrictConvexOn 𝕜 s f) (hg : StrictConvexOn 𝕜 s g)
Goal: StrictConvexOn 𝕜 s (f + g)
-- StrictConcaveOn.add_concaveOn
theorem StrictConcaveOn.add_concaveOn (hf : StrictConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) :
    StrictConcaveOn 𝕜 s (f + g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [SMul 𝕜 E] [DistribMulAction 𝕜 β] {s : Set E} {f g : E → β} (hf : StrictConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
Goal: StrictConcaveOn 𝕜 s (f + g)
-- ConcaveOn.add_strictConcaveOn
theorem ConcaveOn.add_strictConcaveOn (hf : ConcaveOn 𝕜 s f) (hg : StrictConcaveOn 𝕜 s g) :
    StrictConcaveOn 𝕜 s (f + g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [SMul 𝕜 E] [DistribMulAction 𝕜 β] {s : Set E} {f g : E → β} (hf : ConcaveOn 𝕜 s f) (hg : StrictConcaveOn 𝕜 s g)
Goal: StrictConcaveOn 𝕜 s (f + g)
-- StrictConcaveOn.add
theorem StrictConcaveOn.add (hf : StrictConcaveOn 𝕜 s f) (hg : StrictConcaveOn 𝕜 s g) :
    StrictConcaveOn 𝕜 s (f + g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [SMul 𝕜 E] [DistribMulAction 𝕜 β] {s : Set E} {f g : E → β} (hf : StrictConcaveOn 𝕜 s f) (hg : StrictConcaveOn 𝕜 s g)
Goal: StrictConcaveOn 𝕜 s (f + g)
-- StrictConvexOn.add_const
theorem StrictConvexOn.add_const {γ : Type*} {f : E → γ} [OrderedCancelAddCommMonoid γ]
    [Module 𝕜 γ] (hf : StrictConvexOn 𝕜 s f) (b : γ) : StrictConvexOn 𝕜 s (f + fun _ => b) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {γ : Type u_7} {f : E → γ} [OrderedCancelAddCommMonoid γ] [Module 𝕜 γ] (hf : StrictConvexOn 𝕜 s f) (b : γ)
Goal: StrictConvexOn 𝕜 s (f + fun x => b)
-- StrictConcaveOn.add_const
theorem StrictConcaveOn.add_const {γ : Type*} {f : E → γ} [OrderedCancelAddCommMonoid γ]
    [Module 𝕜 γ] (hf : StrictConcaveOn 𝕜 s f) (b : γ) : StrictConcaveOn 𝕜 s (f + fun _ => b) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {γ : Type u_7} {f : E → γ} [OrderedCancelAddCommMonoid γ] [Module 𝕜 γ] (hf : StrictConcaveOn 𝕜 s f) (b : γ)
Goal: StrictConcaveOn 𝕜 s (f + fun x => b)
-- ConvexOn.convex_lt
theorem ConvexOn.convex_lt (hf : ConvexOn 𝕜 s f) (r : β) : Convex 𝕜 ({ x ∈ s | f x < r }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ f x < r}
-- ConcaveOn.convex_gt
theorem ConcaveOn.convex_gt (hf : ConcaveOn 𝕜 s f) (r : β) : Convex 𝕜 ({ x ∈ s | r < f x }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ r < f x}
-- ConvexOn.openSegment_subset_strict_epigraph
theorem ConvexOn.openSegment_subset_strict_epigraph (hf : ConvexOn 𝕜 s f) (p q : E × β)
    (hp : p.1 ∈ s ∧ f p.1 < p.2) (hq : q.1 ∈ s ∧ f q.1 ≤ q.2) :
    openSegment 𝕜 p q ⊆ { p : E × β | p.1 ∈ s ∧ f p.1 < p.2 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f) (p q : E × β) (hp : p.1 ∈ s ∧ f p.1 < p.2) (hq : q.1 ∈ s ∧ f q.1 ≤ q.2)
Goal: openSegment 𝕜 p q ⊆ {p | p.1 ∈ s ∧ f p.1 < p.2}
-- ConcaveOn.openSegment_subset_strict_hypograph
theorem ConcaveOn.openSegment_subset_strict_hypograph (hf : ConcaveOn 𝕜 s f) (p q : E × β)
    (hp : p.1 ∈ s ∧ p.2 < f p.1) (hq : q.1 ∈ s ∧ q.2 ≤ f q.1) :
    openSegment 𝕜 p q ⊆ { p : E × β | p.1 ∈ s ∧ p.2 < f p.1 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f) (p q : E × β) (hp : p.1 ∈ s ∧ p.2 < f p.1) (hq : q.1 ∈ s ∧ q.2 ≤ f q.1)
Goal: openSegment 𝕜 p q ⊆ {p | p.1 ∈ s ∧ p.2 < f p.1}
-- ConvexOn.convex_strict_epigraph
theorem ConvexOn.convex_strict_epigraph (hf : ConvexOn 𝕜 s f) :
    Convex 𝕜 { p : E × β | p.1 ∈ s ∧ f p.1 < p.2 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f)
Goal: Convex 𝕜 {p | p.1 ∈ s ∧ f p.1 < p.2}
-- ConcaveOn.convex_strict_hypograph
theorem ConcaveOn.convex_strict_hypograph (hf : ConcaveOn 𝕜 s f) :
    Convex 𝕜 { p : E × β | p.1 ∈ s ∧ p.2 < f p.1 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedCancelAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f)
Goal: Convex 𝕜 {p | p.1 ∈ s ∧ p.2 < f p.1}
-- ConvexOn.sup
theorem ConvexOn.sup (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) : ConvexOn 𝕜 s (f ⊔ g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [LinearOrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f g : E → β} (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
Goal: ConvexOn 𝕜 s (f ⊔ g)
-- ConcaveOn.inf
theorem ConcaveOn.inf (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) : ConcaveOn 𝕜 s (f ⊓ g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [LinearOrderedAddCommMonoid β] [SMul 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f g : E → β} (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
Goal: ConcaveOn 𝕜 s (f ⊓ g)