ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ConcaveOn.le_map_average
theorem ConcaveOn.le_map_average [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, g (f x) ∂μ) ≤ g (⨍ x, 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 : 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), g (f x) ∂μ ≤ g (⨍ (x : α), f x ∂μ)
-- ConvexOn.set_average_mem_epigraph
theorem ConvexOn.set_average_mem_epigraph (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g 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) (hgi : IntegrableOn (g ∘ f) t μ) :
    (⨍ x in t, f x ∂μ, ⨍ x in t, 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} {t : Set α} {f : α → E} {g : E → ℝ} (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g 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) (hgi : MeasureTheory.IntegrableOn (g ∘ f) t μ)
Goal: (⨍ (x : α) in t, f x ∂μ, ⨍ (x : α) in t, g (f x) ∂μ) ∈ {p | p.1 ∈ s ∧ g p.1 ≤ p.2}
-- ConcaveOn.set_average_mem_hypograph
theorem ConcaveOn.set_average_mem_hypograph (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g 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) (hgi : IntegrableOn (g ∘ f) t μ) :
    (⨍ x in t, f x ∂μ, ⨍ x in t, 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} {t : Set α} {f : α → E} {g : E → ℝ} (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g 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) (hgi : MeasureTheory.IntegrableOn (g ∘ f) t μ)
Goal: (⨍ (x : α) in t, f x ∂μ, ⨍ (x : α) in t, g (f x) ∂μ) ∈ {p | p.1 ∈ s ∧ p.2 ≤ g p.1}
-- ConvexOn.map_set_average_le
theorem ConvexOn.map_set_average_le (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g 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) (hgi : IntegrableOn (g ∘ f) t μ) :
    g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, 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} {t : Set α} {f : α → E} {g : E → ℝ} (hg : ConvexOn ℝ s g) (hgc : ContinuousOn g 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) (hgi : MeasureTheory.IntegrableOn (g ∘ f) t μ)
Goal: g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
-- ConcaveOn.le_map_set_average
theorem ConcaveOn.le_map_set_average (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g 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) (hgi : IntegrableOn (g ∘ f) t μ) :
    (⨍ x in t, g (f x) ∂μ) ≤ g (⨍ x in t, 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} {t : Set α} {f : α → E} {g : E → ℝ} (hg : ConcaveOn ℝ s g) (hgc : ContinuousOn g 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) (hgi : MeasureTheory.IntegrableOn (g ∘ f) t μ)
Goal: ⨍ (x : α) in t, g (f x) ∂μ ≤ g (⨍ (x : α) in t, f x ∂μ)
-- ConvexOn.map_integral_le
theorem ConvexOn.map_integral_le [IsProbabilityMeasure 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.IsProbabilityMeasure 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
-- ConcaveOn.le_map_integral
theorem ConcaveOn.le_map_integral [IsProbabilityMeasure 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, g (f x) ∂μ) ≤ g (∫ x, 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.IsProbabilityMeasure 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), g (f x) ∂μ ≤ g (∫ (x : α), f x ∂μ)
-- ae_eq_const_or_exists_average_ne_compl
theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integrable f μ) :
    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
      ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ tᶜ ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, 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} {f : α → E} [MeasureTheory.IsFiniteMeasure μ] (hfi : MeasureTheory.Integrable f μ)
Goal: f =ᵐ[μ] Function.const α (⨍ (x : α), f x ∂μ) ∨
  ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ tᶜ ≠ 0 ∧ ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
-- Convex.average_mem_interior_of_set
theorem Convex.average_mem_interior_of_set [IsFiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0)
    (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) (ht : (⨍ x in t, f x ∂μ) ∈ interior s) :
    (⨍ x, f x ∂μ) ∈ interior 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} [MeasureTheory.IsFiniteMeasure μ] (hs : Convex ℝ s) (h0 : μ t ≠ 0) (hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s) (hfi : MeasureTheory.Integrable f μ) (ht : ⨍ (x : α) in t, f x ∂μ ∈ interior s)
Goal: ⨍ (x : α), f x ∂μ ∈ interior s
-- StrictConvex.ae_eq_const_or_average_mem_interior
theorem StrictConvex.ae_eq_const_or_average_mem_interior [IsFiniteMeasure μ] (hs : StrictConvex ℝ s)
    (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) :
    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ (⨍ x, f x ∂μ) ∈ interior 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] (hs : StrictConvex ℝ s) (hsc : IsClosed s) (hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s) (hfi : MeasureTheory.Integrable f μ)
Goal: f =ᵐ[μ] Function.const α (⨍ (x : α), f x ∂μ) ∨ ⨍ (x : α), f x ∂μ ∈ interior s
-- StrictConvexOn.ae_eq_const_or_map_average_lt
theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg : StrictConvexOn 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) :
    f =ᵐ[μ] const α (⨍ x, f x ∂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] (hg : StrictConvexOn 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: f =ᵐ[μ] Function.const α (⨍ (x : α), f x ∂μ) ∨ g (⨍ (x : α), f x ∂μ) < ⨍ (x : α), g (f x) ∂μ
-- StrictConcaveOn.ae_eq_const_or_lt_map_average
theorem StrictConcaveOn.ae_eq_const_or_lt_map_average [IsFiniteMeasure μ]
    (hg : StrictConcaveOn 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) :
    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ (⨍ x, g (f x) ∂μ) < g (⨍ x, 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] (hg : StrictConcaveOn 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: f =ᵐ[μ] Function.const α (⨍ (x : α), f x ∂μ) ∨ ⨍ (x : α), g (f x) ∂μ < g (⨍ (x : α), f x ∂μ)
-- ae_eq_const_or_norm_average_lt_of_norm_le_const
theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E]
    (h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖⨍ x, f x ∂μ‖ < C 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} {C : ℝ} [StrictConvexSpace ℝ E] (h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C)
Goal: f =ᵐ[μ] Function.const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C
-- ae_eq_const_or_norm_integral_lt_of_norm_le_const
theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ E] [IsFiniteMeasure μ]
    (h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖∫ x, f x ∂μ‖ < (μ univ).toReal * C 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} {C : ℝ} [StrictConvexSpace ℝ E] [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C)
Goal: f =ᵐ[μ] Function.const α (⨍ (x : α), f x ∂μ) ∨ ‖∫ (x : α), f x ∂μ‖ < (μ Set.univ).toReal * C
-- ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const
theorem ae_eq_const_or_norm_setIntegral_lt_of_norm_le_const [StrictConvexSpace ℝ E] (ht : μ t ≠ ∞)
    (h_le : ∀ᵐ x ∂μ.restrict t, ‖f x‖ ≤ C) :
    f =ᵐ[μ.restrict t] const α (⨍ x in t, f x ∂μ) ∨ ‖∫ x in t, f x ∂μ‖ < (μ t).toReal * C 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {t : Set α} {f : α → E} {C : ℝ} [StrictConvexSpace ℝ E] (ht : μ t ≠ ⊤) (h_le : ∀ᵐ (x : α) ∂μ.restrict t, ‖f x‖ ≤ C)
Goal: f =ᵐ[μ.restrict t] Function.const α (⨍ (x : α) in t, f x ∂μ) ∨ ‖∫ (x : α) in t, f x ∂μ‖ < (μ t).toReal * C
-- Convex.addHaar_frontier
theorem addHaar_frontier (hs : Convex ℝ s) : μ (frontier s) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {s : Set E} (hs : Convex ℝ s)
Goal: μ (frontier s) = 0
-- Convex.nullMeasurableSet
protected theorem nullMeasurableSet (hs : Convex ℝ s) : NullMeasurableSet s μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {s : Set E} (hs : Convex ℝ s)
Goal: MeasureTheory.NullMeasurableSet s μ
-- ConvexOn.smul'
lemma ConvexOn.smul' (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x)
    (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : MonovaryOn f g s) : ConvexOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [LinearOrderedAddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [Module E F] [IsScalarTower 𝕜 E F] [SMulCommClass 𝕜 E F] [OrderedSMul 𝕜 F] [OrderedSMul E F] {s : Set 𝕜} {f : 𝕜 → E} {g : 𝕜 → F} (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x) (hfg : MonovaryOn f g s)
Goal: ConvexOn 𝕜 s (f • g)
-- ConcaveOn.smul'
lemma ConcaveOn.smul' (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x)
    (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s) : ConcaveOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [LinearOrderedAddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [Module E F] [IsScalarTower 𝕜 E F] [SMulCommClass 𝕜 E F] [OrderedSMul 𝕜 E] [OrderedSMul 𝕜 F] [OrderedSMul E F] {s : Set 𝕜} {f : 𝕜 → E} {g : 𝕜 → F} (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s)
Goal: ConcaveOn 𝕜 s (f • g)
-- ConvexOn.smul''
lemma ConvexOn.smul'' (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0)
    (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s) : ConcaveOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [LinearOrderedAddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [Module E F] [IsScalarTower 𝕜 E F] [SMulCommClass 𝕜 E F] [OrderedSMul 𝕜 E] [OrderedSMul 𝕜 F] [OrderedSMul E F] {s : Set 𝕜} {f : 𝕜 → E} {g : 𝕜 → F} (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s)
Goal: ConcaveOn 𝕜 s (f • g)
-- ConcaveOn.smul''
lemma ConcaveOn.smul'' (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0)
    (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s) : ConvexOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [LinearOrderedAddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [Module E F] [IsScalarTower 𝕜 E F] [SMulCommClass 𝕜 E F] [OrderedSMul 𝕜 F] [OrderedSMul E F] {s : Set 𝕜} {f : 𝕜 → E} {g : 𝕜 → F} (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s)
Goal: ConvexOn 𝕜 s (f • g)
-- ConvexOn.smul_concaveOn
lemma ConvexOn.smul_concaveOn (hf : ConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s) :
    ConcaveOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [LinearOrderedAddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [Module E F] [IsScalarTower 𝕜 E F] [SMulCommClass 𝕜 E F] [OrderedSMul 𝕜 F] [OrderedSMul E F] {s : Set 𝕜} {f : 𝕜 → E} {g : 𝕜 → F} (hf : ConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s)
Goal: ConcaveOn 𝕜 s (f • g)
-- ConcaveOn.smul_convexOn
lemma ConcaveOn.smul_convexOn (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s) :
    ConvexOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [LinearOrderedAddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [Module E F] [IsScalarTower 𝕜 E F] [SMulCommClass 𝕜 E F] [OrderedSMul 𝕜 E] [OrderedSMul 𝕜 F] [OrderedSMul E F] {s : Set 𝕜} {f : 𝕜 → E} {g : 𝕜 → F} (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s)
Goal: ConvexOn 𝕜 s (f • g)
-- ConvexOn.smul_concaveOn'
lemma ConvexOn.smul_concaveOn' (hf : ConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : MonovaryOn f g s) :
    ConvexOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [LinearOrderedAddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [Module E F] [IsScalarTower 𝕜 E F] [SMulCommClass 𝕜 E F] [OrderedSMul 𝕜 E] [OrderedSMul 𝕜 F] [OrderedSMul E F] {s : Set 𝕜} {f : 𝕜 → E} {g : 𝕜 → F} (hf : ConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x) (hfg : MonovaryOn f g s)
Goal: ConvexOn 𝕜 s (f • g)
-- ConcaveOn.smul_convexOn'
lemma ConcaveOn.smul_convexOn' (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s) :
    ConcaveOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [LinearOrderedAddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [Module E F] [IsScalarTower 𝕜 E F] [SMulCommClass 𝕜 E F] [OrderedSMul 𝕜 F] [OrderedSMul E F] {s : Set 𝕜} {f : 𝕜 → E} {g : 𝕜 → F} (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s)
Goal: ConcaveOn 𝕜 s (f • g)
-- ConvexOn.mul
lemma ConvexOn.mul (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x)
    (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : MonovaryOn f g s) :
    ConvexOn 𝕜 s (f * g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : 𝕜 → E} (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x) (hfg : MonovaryOn f g s)
Goal: ConvexOn 𝕜 s (f * g)
-- ConcaveOn.mul
lemma ConcaveOn.mul (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s) :
    ConcaveOn 𝕜 s (f * g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : 𝕜 → E} (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s)
Goal: ConcaveOn 𝕜 s (f * g)
-- ConvexOn.mul'
lemma ConvexOn.mul' (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0)
    (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s) :
    ConcaveOn 𝕜 s (f * g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : 𝕜 → E} (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s)
Goal: ConcaveOn 𝕜 s (f * g)
-- ConcaveOn.mul'
lemma ConcaveOn.mul' (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0)
    (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s) :
    ConvexOn 𝕜 s (f * g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : 𝕜 → E} (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s)
Goal: ConvexOn 𝕜 s (f * g)
-- ConvexOn.mul_concaveOn
lemma ConvexOn.mul_concaveOn (hf : ConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s) :
    ConcaveOn 𝕜 s (f * g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : 𝕜 → E} (hf : ConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s)
Goal: ConcaveOn 𝕜 s (f * g)
-- ConcaveOn.mul_convexOn
lemma ConcaveOn.mul_convexOn (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s) :
    ConvexOn 𝕜 s (f * g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : 𝕜 → E} (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s)
Goal: ConvexOn 𝕜 s (f * g)
-- ConvexOn.mul_concaveOn'
lemma ConvexOn.mul_concaveOn' (hf : ConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : MonovaryOn f g s) :
    ConvexOn 𝕜 s (f * g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : 𝕜 → E} (hf : ConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x) (hfg : MonovaryOn f g s)
Goal: ConvexOn 𝕜 s (f * g)
-- ConcaveOn.mul_convexOn'
lemma ConcaveOn.mul_convexOn' (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
    (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s) :
    ConcaveOn 𝕜 s (f • g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : 𝕜 → E} (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s)
Goal: ConcaveOn 𝕜 s (f • g)
-- ConvexOn.pow
lemma ConvexOn.pow (hf : ConvexOn 𝕜 s f) (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) :
    ∀ n, ConvexOn 𝕜 s (f ^ n)
  
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedCommRing 𝕜] [LinearOrderedCommRing E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set 𝕜} [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f : 𝕜 → E} (hf : ConvexOn 𝕜 s f) (hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x) (n : ℕ)
Goal: ConvexOn 𝕜 s (f ^ n)
-- convexOn_pow
lemma convexOn_pow : ∀ n, ConvexOn 𝕜 (Ici 0) fun x : 𝕜 ↦ x ^ n 
Arguments: {𝕜 : Type u_1} [LinearOrderedCommRing 𝕜] (n : ℕ)
Goal: ConvexOn 𝕜 (Set.Ici 0) fun x => x ^ n
-- Even.convexOn_pow
protected lemma Even.convexOn_pow {n : ℕ} (hn : Even n) : ConvexOn 𝕜 univ fun x : 𝕜 ↦ x ^ n 
Arguments: {𝕜 : Type u_1} [LinearOrderedCommRing 𝕜] {n : ℕ} (hn : Even n)
Goal: ConvexOn 𝕜 Set.univ fun x => x ^ n
-- convexOn_zpow
lemma convexOn_zpow : ∀ n : ℤ, ConvexOn 𝕜 (Ioi 0) fun x : 𝕜 ↦ x ^ n
  
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] (n : ℤ)
Goal: ConvexOn 𝕜 (Set.Ioi 0) fun x => x ^ n
-- RieszExtension.step
theorem step (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
    (dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) (hdom : f.domain ≠ ⊤) :
    ∃ g, f < g ∧ ∀ x : g.domain, (x : E) ∈ s → 0 ≤ g x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] (s : ConvexCone ℝ E) (f : E →ₗ.[ℝ] ℝ) (nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x) (dense : ∀ (y : E), ∃ x, ↑x + y ∈ s) (hdom : f.domain ≠ ⊤)
Goal: ∃ g, f < g ∧ ∀ (x : ↥g.domain), ↑x ∈ s → 0 ≤ ↑g x
-- RieszExtension.exists_top
theorem exists_top (p : E →ₗ.[ℝ] ℝ) (hp_nonneg : ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x)
    (hp_dense : ∀ y, ∃ x : p.domain, (x : E) + y ∈ s) :
    ∃ q ≥ p, q.domain = ⊤ ∧ ∀ x : q.domain, (x : E) ∈ s → 0 ≤ q x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] (s : ConvexCone ℝ E) (p : E →ₗ.[ℝ] ℝ) (hp_nonneg : ∀ (x : ↥p.domain), ↑x ∈ s → 0 ≤ ↑p x) (hp_dense : ∀ (y : E), ∃ x, ↑x + y ∈ s)
Goal: ∃ q ≥ p, q.domain = ⊤ ∧ ∀ (x : ↥q.domain), ↑x ∈ s → 0 ≤ ↑q x
-- riesz_extension
theorem riesz_extension (s : ConvexCone ℝ E) (f : E →ₗ.[ℝ] ℝ)
    (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
    (dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) :
    ∃ g : E →ₗ[ℝ] ℝ, (∀ x : f.domain, g x = f x) ∧ ∀ x ∈ s, 0 ≤ g x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] (s : ConvexCone ℝ E) (f : E →ₗ.[ℝ] ℝ) (nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x) (dense : ∀ (y : E), ∃ x, ↑x + y ∈ s)
Goal: ∃ g, (∀ (x : ↥f.domain), g ↑x = ↑f x) ∧ ∀ x ∈ s, 0 ≤ g x