ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProperCone.mem_map
@[simp]
theorem mem_map {f : E →L[ℝ] F} {K : ProperCone ℝ E} {y : F} :
    y ∈ K.map f ↔ y ∈ (PointedCone.map (f : E →ₗ[ℝ] F) ↑K).closure 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {f : E →L[ℝ] F} {K : ProperCone ℝ E} {y : F}
Goal: y ∈ ProperCone.map f K ↔ y ∈ (PointedCone.map ↑f ↑K).closure
-- ProperCone.map_id
@[simp]
theorem map_id (K : ProperCone ℝ E) : K.map (ContinuousLinearMap.id ℝ E) = K 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (K : ProperCone ℝ E)
Goal: ProperCone.map (ContinuousLinearMap.id ℝ E) K = K
-- ProperCone.coe_dual
@[simp, norm_cast]
theorem coe_dual (K : ProperCone ℝ E) : K.dual = (K : Set E).innerDualCone 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (K : ProperCone ℝ E)
Goal: ↑↑K.dual = (↑K).innerDualCone
-- ProperCone.mem_dual
@[simp]
theorem mem_dual {K : ProperCone ℝ E} {y : E} : y ∈ dual K ↔ ∀ ⦃x⦄, x ∈ K → 0 ≤ ⟪x, y⟫_ℝ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {K : ProperCone ℝ E} {y : E}
Goal: y ∈ K.dual ↔ ∀ ⦃x : E⦄, x ∈ K → 0 ≤ ⟪x, y⟫_ℝ
-- ProperCone.coe_comap
@[simp]
theorem coe_comap (f : E →L[ℝ] F) (S : ProperCone ℝ F) : (S.comap f : Set E) = f ⁻¹' S 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (f : E →L[ℝ] F) (S : ProperCone ℝ F)
Goal: ↑(ProperCone.comap f S) = ⇑f ⁻¹' ↑S
-- ProperCone.comap_id
@[simp]
theorem comap_id (S : ConvexCone ℝ E) : S.comap LinearMap.id = S 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (S : ConvexCone ℝ E)
Goal: ConvexCone.comap LinearMap.id S = S
-- ProperCone.comap_comap
theorem comap_comap (g : F →L[ℝ] G) (f : E →L[ℝ] F) (S : ProperCone ℝ G) :
    (S.comap g).comap f = S.comap (g.comp f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {G : Type u_3} [NormedAddCommGroup G] [InnerProductSpace ℝ G] (g : F →L[ℝ] G) (f : E →L[ℝ] F) (S : ProperCone ℝ G)
Goal: ProperCone.comap f (ProperCone.comap g S) = ProperCone.comap (g.comp f) S
-- ProperCone.mem_comap
@[simp]
theorem mem_comap {f : E →L[ℝ] F} {S : ProperCone ℝ F} {x : E} : x ∈ S.comap f ↔ f x ∈ S 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {f : E →L[ℝ] F} {S : ProperCone ℝ F} {x : E}
Goal: x ∈ ProperCone.comap f S ↔ f x ∈ S
-- ProperCone.dual_dual
@[simp]
theorem dual_dual (K : ProperCone ℝ E) : K.dual.dual = K 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] (K : ProperCone ℝ E)
Goal: K.dual.dual = K
-- ProperCone.hyperplane_separation
theorem hyperplane_separation (K : ProperCone ℝ E) {f : E →L[ℝ] F} {b : F} :
    b ∈ K.map f ↔ ∀ y : F, adjoint f y ∈ K.dual → 0 ≤ ⟪y, b⟫_ℝ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F] (K : ProperCone ℝ E) {f : E →L[ℝ] F} {b : F}
Goal: b ∈ ProperCone.map f K ↔ ∀ (y : F), (ContinuousLinearMap.adjoint f) y ∈ K.dual → 0 ≤ ⟪y, b⟫_ℝ
-- ProperCone.hyperplane_separation_of_nmem
theorem hyperplane_separation_of_nmem (K : ProperCone ℝ E) {f : E →L[ℝ] F} {b : F}
    (disj : b ∉ K.map f) : ∃ y : F, adjoint f y ∈ K.dual ∧ ⟪y, b⟫_ℝ < 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F] (K : ProperCone ℝ E) {f : E →L[ℝ] F} {b : F} (disj : b ∉ ProperCone.map f K)
Goal: ∃ y, (ContinuousLinearMap.adjoint f) y ∈ K.dual ∧ ⟪y, b⟫_ℝ < 0
-- ConvexCone.add_mem'
structure ConvexCone [AddCommMonoid E] [SMul 𝕜 E] where
    carrier : Set E
  smul_mem' : ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : E⦄, x ∈ carrier → c • x ∈ carrier
  add_mem' : ∀ ⦃x⦄ (_ : x ∈ carrier) ⦃y⦄ (_ : y ∈ carrier), x + y ∈ carrier
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (self : ConvexCone 𝕜 E) ⦃x : E⦄
Goal: x ∈ self.carrier → ∀ ⦃y : E⦄, y ∈ self.carrier → x + y ∈ self.carrier
-- ConvexCone.smul_mem'
structure ConvexCone [AddCommMonoid E] [SMul 𝕜 E] where
    carrier : Set E
  smul_mem' : ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : E⦄, x ∈ carrier → c • x ∈ carrier
  add_mem' : ∀ ⦃x⦄ (_ : x ∈ carrier) ⦃y⦄ (_ : y ∈ carrier), x + y ∈ carrier
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (self : ConvexCone 𝕜 E) ⦃c : 𝕜⦄
Goal: 0 < c → ∀ ⦃x : E⦄, x ∈ self.carrier → c • x ∈ self.carrier
-- ConvexCone.coe_mk
@[simp]
theorem coe_mk {s : Set E} {h₁ h₂} : ↑(@mk 𝕜 _ _ _ _ s h₁ h₂) = s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {h₁ : ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : E⦄, x ∈ s → c • x ∈ s} {h₂ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x + y ∈ s}
Goal: ↑{ carrier := s, smul_mem' := h₁, add_mem' := h₂ } = s
-- ConvexCone.mem_mk
@[simp]
theorem mem_mk {s : Set E} {h₁ h₂ x} : x ∈ @mk 𝕜 _ _ _ _ s h₁ h₂ ↔ x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {h₁ : ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : E⦄, x ∈ s → c • x ∈ s} {h₂ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x + y ∈ s} {x : E}
Goal: x ∈ { carrier := s, smul_mem' := h₁, add_mem' := h₂ } ↔ x ∈ s
-- ConvexCone.ext
@[ext]
theorem ext {S T : ConvexCone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {S T : ConvexCone 𝕜 E} (h : ∀ (x : E), x ∈ S ↔ x ∈ T)
Goal: S = T
-- ConvexCone.smul_mem
@[aesop safe apply (rule_sets := [SetLike])]
theorem smul_mem {c : 𝕜} {x : E} (hc : 0 < c) (hx : x ∈ S) : c • x ∈ S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (S : ConvexCone 𝕜 E) {c : 𝕜} {x : E} (hc : 0 < c) (hx : x ∈ S)
Goal: c • x ∈ S
-- ConvexCone.add_mem
theorem add_mem ⦃x⦄ (hx : x ∈ S) ⦃y⦄ (hy : y ∈ S) : x + y ∈ S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (S : ConvexCone 𝕜 E) ⦃x : E⦄ (hx : x ∈ S) ⦃y : E⦄ (hy : y ∈ S)
Goal: x + y ∈ S
-- ConvexCone.coe_inf
@[simp]
theorem coe_inf : ((S ⊓ T : ConvexCone 𝕜 E) : Set E) = ↑S ∩ ↑T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (S T : ConvexCone 𝕜 E)
Goal: ↑(S ⊓ T) = ↑S ∩ ↑T
-- ConvexCone.mem_inf
theorem mem_inf {x} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (S T : ConvexCone 𝕜 E) {x : E}
Goal: x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T
-- ConvexCone.coe_sInf
@[simp]
theorem coe_sInf (S : Set (ConvexCone 𝕜 E)) : ↑(sInf S) = ⋂ s ∈ S, (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (S : Set (ConvexCone 𝕜 E))
Goal: ↑(sInf S) = ⋂ s ∈ S, ↑s
-- ConvexCone.mem_sInf
theorem mem_sInf {x : E} {S : Set (ConvexCone 𝕜 E)} : x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {x : E} {S : Set (ConvexCone 𝕜 E)}
Goal: x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s
-- ConvexCone.coe_iInf
@[simp]
theorem coe_iInf {ι : Sort*} (f : ι → ConvexCone 𝕜 E) : ↑(iInf f) = ⋂ i, (f i : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {ι : Sort u_5} (f : ι → ConvexCone 𝕜 E)
Goal: ↑(iInf f) = ⋂ i, ↑(f i)
-- ConvexCone.mem_iInf
theorem mem_iInf {ι : Sort*} {x : E} {f : ι → ConvexCone 𝕜 E} : x ∈ iInf f ↔ ∀ i, x ∈ f i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {ι : Sort u_5} {x : E} {f : ι → ConvexCone 𝕜 E}
Goal: x ∈ iInf f ↔ ∀ (i : ι), x ∈ f i
-- ConvexCone.mem_bot
theorem mem_bot (x : E) : (x ∈ (⊥ : ConvexCone 𝕜 E)) = False 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x : E)
Goal: (x ∈ ⊥) = False
-- ConvexCone.coe_bot
@[simp]
theorem coe_bot : ↑(⊥ : ConvexCone 𝕜 E) = (∅ : Set E) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E]
Goal: ↑⊥ = ∅
-- ConvexCone.mem_top
theorem mem_top (x : E) : x ∈ (⊤ : ConvexCone 𝕜 E) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x : E)
Goal: x ∈ ⊤
-- ConvexCone.coe_top
@[simp]
theorem coe_top : ↑(⊤ : ConvexCone 𝕜 E) = (univ : Set E) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E]
Goal: ↑⊤ = Set.univ
-- ConvexCone.convex
protected theorem convex : Convex 𝕜 (S : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S : ConvexCone 𝕜 E)
Goal: Convex 𝕜 ↑S
-- ConvexCone.coe_map
@[simp, norm_cast]
theorem coe_map (S : ConvexCone 𝕜 E) (f : E →ₗ[𝕜] F) : (S.map f : Set F) = f '' S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (S : ConvexCone 𝕜 E) (f : E →ₗ[𝕜] F)
Goal: ↑(ConvexCone.map f S) = ⇑f '' ↑S
-- ConvexCone.mem_map
@[simp]
theorem mem_map {f : E →ₗ[𝕜] F} {S : ConvexCone 𝕜 E} {y : F} : y ∈ S.map f ↔ ∃ x ∈ S, f x = 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] {f : E →ₗ[𝕜] F} {S : ConvexCone 𝕜 E} {y : F}
Goal: y ∈ ConvexCone.map f S ↔ ∃ x ∈ S, f x = y
-- ConvexCone.map_map
theorem map_map (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : ConvexCone 𝕜 E) :
    (S.map f).map g = S.map (g.comp f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [AddCommMonoid G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : ConvexCone 𝕜 E)
Goal: ConvexCone.map g (ConvexCone.map f S) = ConvexCone.map (g ∘ₗ f) S
-- ConvexCone.map_id
@[simp]
theorem map_id (S : ConvexCone 𝕜 E) : S.map LinearMap.id = S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S : ConvexCone 𝕜 E)
Goal: ConvexCone.map LinearMap.id S = S
-- ConvexCone.coe_comap
@[simp]
theorem coe_comap (f : E →ₗ[𝕜] F) (S : ConvexCone 𝕜 F) : (S.comap f : Set E) = f ⁻¹' S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (f : E →ₗ[𝕜] F) (S : ConvexCone 𝕜 F)
Goal: ↑(ConvexCone.comap f S) = ⇑f ⁻¹' ↑S
-- ConvexCone.comap_id
@[simp] -- Porting note: was not a `dsimp` lemma
theorem comap_id (S : ConvexCone 𝕜 E) : S.comap LinearMap.id = S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S : ConvexCone 𝕜 E)
Goal: ConvexCone.comap LinearMap.id S = S
-- ConvexCone.comap_comap
theorem comap_comap (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : ConvexCone 𝕜 G) :
    (S.comap g).comap f = S.comap (g.comp f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [AddCommMonoid G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : ConvexCone 𝕜 G)
Goal: ConvexCone.comap f (ConvexCone.comap g S) = ConvexCone.comap (g ∘ₗ f) S
-- ConvexCone.mem_comap
@[simp]
theorem mem_comap {f : E →ₗ[𝕜] F} {S : ConvexCone 𝕜 F} {x : E} : x ∈ S.comap f ↔ f x ∈ S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {f : E →ₗ[𝕜] F} {S : ConvexCone 𝕜 F} {x : E}
Goal: x ∈ ConvexCone.comap f S ↔ f x ∈ S
-- ConvexCone.smul_mem_iff
theorem smul_mem_iff {c : 𝕜} (hc : 0 < c) {x : E} : c • x ∈ S ↔ x ∈ S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommMonoid E] [MulAction 𝕜 E] (S : ConvexCone 𝕜 E) {c : 𝕜} (hc : 0 < c) {x : E}
Goal: c • x ∈ S ↔ x ∈ S
-- ConvexCone.to_orderedSMul
theorem to_orderedSMul (S : ConvexCone 𝕜 E) (h : ∀ x y : E, x ≤ y ↔ y - x ∈ S) : OrderedSMul 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] (S : ConvexCone 𝕜 E) (h : ∀ (x y : E), x ≤ y ↔ y - x ∈ S)
Goal: OrderedSMul 𝕜 E
-- ConvexCone.pointed_iff_not_blunt
theorem pointed_iff_not_blunt (S : ConvexCone 𝕜 E) : S.Pointed ↔ ¬S.Blunt 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (S : ConvexCone 𝕜 E)
Goal: S.Pointed ↔ ¬S.Blunt
-- ConvexCone.blunt_iff_not_pointed
theorem blunt_iff_not_pointed (S : ConvexCone 𝕜 E) : S.Blunt ↔ ¬S.Pointed 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (S : ConvexCone 𝕜 E)
Goal: S.Blunt ↔ ¬S.Pointed
-- ConvexCone.Pointed.mono
theorem Pointed.mono {S T : ConvexCone 𝕜 E} (h : S ≤ T) : S.Pointed → T.Pointed 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {S T : ConvexCone 𝕜 E} (h : S ≤ T)
Goal: S.Pointed → T.Pointed
-- ConvexCone.Blunt.anti
theorem Blunt.anti {S T : ConvexCone 𝕜 E} (h : T ≤ S) : S.Blunt → T.Blunt 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {S T : ConvexCone 𝕜 E} (h : T ≤ S)
Goal: S.Blunt → T.Blunt
-- ConvexCone.salient_iff_not_flat
theorem salient_iff_not_flat (S : ConvexCone 𝕜 E) : S.Salient ↔ ¬S.Flat 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [SMul 𝕜 E] (S : ConvexCone 𝕜 E)
Goal: S.Salient ↔ ¬S.Flat
-- ConvexCone.Flat.mono
theorem Flat.mono {S T : ConvexCone 𝕜 E} (h : S ≤ T) : S.Flat → T.Flat
  
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [SMul 𝕜 E] {S T : ConvexCone 𝕜 E} (h : S ≤ T)
Goal: S.Flat → T.Flat
-- ConvexCone.Salient.anti
theorem Salient.anti {S T : ConvexCone 𝕜 E} (h : T ≤ S) : S.Salient → T.Salient 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [SMul 𝕜 E] {S T : ConvexCone 𝕜 E} (h : T ≤ S)
Goal: S.Salient → T.Salient
-- ConvexCone.Flat.pointed
theorem Flat.pointed {S : ConvexCone 𝕜 E} (hS : S.Flat) : S.Pointed 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [SMul 𝕜 E] {S : ConvexCone 𝕜 E} (hS : S.Flat)
Goal: S.Pointed
-- ConvexCone.Blunt.salient
theorem Blunt.salient {S : ConvexCone 𝕜 E} : S.Blunt → S.Salient 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [SMul 𝕜 E] {S : ConvexCone 𝕜 E}
Goal: S.Blunt → S.Salient
-- ConvexCone.mem_zero
@[simp]
theorem mem_zero (x : E) : x ∈ (0 : ConvexCone 𝕜 E) ↔ x = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (x : E)
Goal: x ∈ 0 ↔ x = 0
-- ConvexCone.coe_zero
@[simp]
theorem coe_zero : ((0 : ConvexCone 𝕜 E) : Set E) = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
Goal: ↑0 = 0
-- ConvexCone.pointed_zero
theorem pointed_zero : (0 : ConvexCone 𝕜 E).Pointed 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
Goal: ConvexCone.Pointed 0
-- ConvexCone.mem_add
@[simp]
theorem mem_add {K₁ K₂ : ConvexCone 𝕜 E} {a : E} :
    a ∈ K₁ + K₂ ↔ ∃ x ∈ K₁, ∃ y ∈ K₂, x + y = a 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {K₁ K₂ : ConvexCone 𝕜 E} {a : E}
Goal: a ∈ K₁ + K₂ ↔ ∃ x ∈ K₁, ∃ y ∈ K₂, x + y = a
-- Submodule.coe_toConvexCone
@[simp]
theorem coe_toConvexCone (S : Submodule 𝕜 E) : ↑S.toConvexCone = (S : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S : Submodule 𝕜 E)
Goal: ↑S.toConvexCone = ↑S
-- Submodule.mem_toConvexCone
@[simp]
theorem mem_toConvexCone {x : E} {S : Submodule 𝕜 E} : x ∈ S.toConvexCone ↔ x ∈ S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x : E} {S : Submodule 𝕜 E}
Goal: x ∈ S.toConvexCone ↔ x ∈ S
-- Submodule.toConvexCone_le_iff
@[simp]
theorem toConvexCone_le_iff {S T : Submodule 𝕜 E} : S.toConvexCone ≤ T.toConvexCone ↔ S ≤ T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {S T : Submodule 𝕜 E}
Goal: S.toConvexCone ≤ T.toConvexCone ↔ S ≤ T
-- Submodule.toConvexCone_bot
@[simp]
theorem toConvexCone_bot : (⊥ : Submodule 𝕜 E).toConvexCone = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
Goal: ⊥.toConvexCone = 0
-- Submodule.toConvexCone_top
@[simp]
theorem toConvexCone_top : (⊤ : Submodule 𝕜 E).toConvexCone = ⊤ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
Goal: ⊤.toConvexCone = ⊤
-- Submodule.toConvexCone_inf
@[simp]
theorem toConvexCone_inf (S T : Submodule 𝕜 E) :
    (S ⊓ T).toConvexCone = S.toConvexCone ⊓ T.toConvexCone 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S T : Submodule 𝕜 E)
Goal: (S ⊓ T).toConvexCone = S.toConvexCone ⊓ T.toConvexCone
-- Submodule.pointed_toConvexCone
@[simp]
theorem pointed_toConvexCone (S : Submodule 𝕜 E) : S.toConvexCone.Pointed 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S : Submodule 𝕜 E)
Goal: S.toConvexCone.Pointed
-- ConvexCone.mem_positive
@[simp]
theorem mem_positive {x : E} : x ∈ positive 𝕜 E ↔ 0 ≤ x 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {x : E}
Goal: x ∈ ConvexCone.positive 𝕜 E ↔ 0 ≤ x
-- ConvexCone.coe_positive
@[simp]
theorem coe_positive : ↑(positive 𝕜 E) = Set.Ici (0 : E) 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
Goal: ↑(ConvexCone.positive 𝕜 E) = Set.Ici 0
-- ConvexCone.salient_positive
theorem salient_positive : Salient (positive 𝕜 E) 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
Goal: (ConvexCone.positive 𝕜 E).Salient
-- ConvexCone.pointed_positive
theorem pointed_positive : Pointed (positive 𝕜 E) 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
Goal: (ConvexCone.positive 𝕜 E).Pointed
-- ConvexCone.mem_strictlyPositive
@[simp]
theorem mem_strictlyPositive {x : E} : x ∈ strictlyPositive 𝕜 E ↔ 0 < x 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {x : E}
Goal: x ∈ ConvexCone.strictlyPositive 𝕜 E ↔ 0 < x
-- ConvexCone.coe_strictlyPositive
@[simp]
theorem coe_strictlyPositive : ↑(strictlyPositive 𝕜 E) = Set.Ioi (0 : E) 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
Goal: ↑(ConvexCone.strictlyPositive 𝕜 E) = Set.Ioi 0
-- ConvexCone.positive_le_strictlyPositive
theorem positive_le_strictlyPositive : strictlyPositive 𝕜 E ≤ positive 𝕜 E 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
Goal: ConvexCone.strictlyPositive 𝕜 E ≤ ConvexCone.positive 𝕜 E
-- ConvexCone.salient_strictlyPositive
theorem salient_strictlyPositive : Salient (strictlyPositive 𝕜 E) 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
Goal: (ConvexCone.strictlyPositive 𝕜 E).Salient
-- ConvexCone.blunt_strictlyPositive
theorem blunt_strictlyPositive : Blunt (strictlyPositive 𝕜 E) 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
Goal: (ConvexCone.strictlyPositive 𝕜 E).Blunt
-- Convex.mem_toCone
theorem mem_toCone : x ∈ hs.toCone s ↔ ∃ c : 𝕜, 0 < c ∧ ∃ y ∈ s, c • y = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x : E}
Goal: x ∈ Convex.toCone s hs ↔ ∃ c, 0 < c ∧ ∃ y ∈ s, c • y = x
-- Convex.mem_toCone'
theorem mem_toCone' : x ∈ hs.toCone s ↔ ∃ c : 𝕜, 0 < c ∧ c • x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x : E}
Goal: x ∈ Convex.toCone s hs ↔ ∃ c, 0 < c ∧ c • x ∈ s
-- Convex.subset_toCone
theorem subset_toCone : s ⊆ hs.toCone s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s)
Goal: s ⊆ ↑(Convex.toCone s hs)
-- Convex.toCone_isLeast
theorem toCone_isLeast : IsLeast { t : ConvexCone 𝕜 E | s ⊆ t } (hs.toCone s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s)
Goal: IsLeast {t | s ⊆ ↑t} (Convex.toCone s hs)
-- Convex.toCone_eq_sInf
theorem toCone_eq_sInf : hs.toCone s = sInf { t : ConvexCone 𝕜 E | s ⊆ t } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s)
Goal: Convex.toCone s hs = sInf {t | s ⊆ ↑t}
-- convexHull_toCone_isLeast
theorem convexHull_toCone_isLeast (s : Set E) :
    IsLeast { t : ConvexCone 𝕜 E | s ⊆ t } ((convex_convexHull 𝕜 s).toCone _) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (s : Set E)
Goal: IsLeast {t | s ⊆ ↑t} (Convex.toCone ((convexHull 𝕜) s) ⋯)
-- convexHull_toCone_eq_sInf
theorem convexHull_toCone_eq_sInf (s : Set E) :
    (convex_convexHull 𝕜 s).toCone _ = sInf { t : ConvexCone 𝕜 E | s ⊆ t } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (s : Set E)
Goal: Convex.toCone ((convexHull 𝕜) s) ⋯ = sInf {t | s ⊆ ↑t}
-- IsExtreme.refl
@[refl]
protected theorem IsExtreme.refl (A : Set E) : IsExtreme 𝕜 A A 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (A : Set E)
Goal: IsExtreme 𝕜 A A
-- IsExtreme.rfl
protected theorem IsExtreme.rfl : IsExtreme 𝕜 A A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A : Set E}
Goal: IsExtreme 𝕜 A A
-- IsExtreme.trans
@[trans]
protected theorem IsExtreme.trans (hAB : IsExtreme 𝕜 A B) (hBC : IsExtreme 𝕜 B C) :
    IsExtreme 𝕜 A C 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A B C : Set E} (hAB : IsExtreme 𝕜 A B) (hBC : IsExtreme 𝕜 B C)
Goal: IsExtreme 𝕜 A C
-- IsExtreme.antisymm
protected theorem IsExtreme.antisymm : AntiSymmetric (IsExtreme 𝕜 : Set E → Set E → Prop) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E]
Goal: AntiSymmetric (IsExtreme 𝕜)
-- IsExtreme.inter
theorem IsExtreme.inter (hAB : IsExtreme 𝕜 A B) (hAC : IsExtreme 𝕜 A C) :
    IsExtreme 𝕜 A (B ∩ C) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A B C : Set E} (hAB : IsExtreme 𝕜 A B) (hAC : IsExtreme 𝕜 A C)
Goal: IsExtreme 𝕜 A (B ∩ C)
-- IsExtreme.mono
protected theorem IsExtreme.mono (hAC : IsExtreme 𝕜 A C) (hBA : B ⊆ A) (hCB : C ⊆ B) :
    IsExtreme 𝕜 B C 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A B C : Set E} (hAC : IsExtreme 𝕜 A C) (hBA : B ⊆ A) (hCB : C ⊆ B)
Goal: IsExtreme 𝕜 B C
-- isExtreme_iInter
theorem isExtreme_iInter {ι : Sort*} [Nonempty ι] {F : ι → Set E}
    (hAF : ∀ i : ι, IsExtreme 𝕜 A (F i)) : IsExtreme 𝕜 A (⋂ i : ι, F i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A : Set E} {ι : Sort u_6} [Nonempty ι] {F : ι → Set E} (hAF : ∀ (i : ι), IsExtreme 𝕜 A (F i))
Goal: IsExtreme 𝕜 A (⋂ i, F i)
-- isExtreme_biInter
theorem isExtreme_biInter {F : Set (Set E)} (hF : F.Nonempty) (hA : ∀ B ∈ F, IsExtreme 𝕜 A B) :
    IsExtreme 𝕜 A (⋂ B ∈ F, B) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A : Set E} {F : Set (Set E)} (hF : F.Nonempty) (hA : ∀ B ∈ F, IsExtreme 𝕜 A B)
Goal: IsExtreme 𝕜 A (⋂ B ∈ F, B)
-- isExtreme_sInter
theorem isExtreme_sInter {F : Set (Set E)} (hF : F.Nonempty) (hAF : ∀ B ∈ F, IsExtreme 𝕜 A B) :
    IsExtreme 𝕜 A (⋂₀ F) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {A : Set E} {F : Set (Set E)} (hF : F.Nonempty) (hAF : ∀ B ∈ F, IsExtreme 𝕜 A B)
Goal: IsExtreme 𝕜 A (⋂₀ F)