ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Convex.openSegment_interior_closure_subset_interior
theorem Convex.openSegment_interior_closure_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
    (hx : x ∈ interior s) (hy : y ∈ closure s) : openSegment 𝕜 x y ⊆ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ closure s)
Goal: openSegment 𝕜 x y ⊆ interior s
-- Convex.openSegment_interior_self_subset_interior
theorem Convex.openSegment_interior_self_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
    (hx : x ∈ interior s) (hy : y ∈ s) : openSegment 𝕜 x y ⊆ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ s)
Goal: openSegment 𝕜 x y ⊆ interior s
-- Convex.openSegment_closure_interior_subset_interior
theorem Convex.openSegment_closure_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
    (hx : x ∈ closure s) (hy : y ∈ interior s) : openSegment 𝕜 x y ⊆ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s) (hy : y ∈ interior s)
Goal: openSegment 𝕜 x y ⊆ interior s
-- Convex.openSegment_self_interior_subset_interior
theorem Convex.openSegment_self_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
    (hx : x ∈ s) (hy : y ∈ interior s) : openSegment 𝕜 x y ⊆ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s)
Goal: openSegment 𝕜 x y ⊆ interior s
-- Convex.add_smul_sub_mem_interior'
theorem Convex.add_smul_sub_mem_interior' {s : Set E} (hs : Convex 𝕜 s) {x y : E}
    (hx : x ∈ closure s) (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) :
    x + t • (y - x) ∈ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s) (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Set.Ioc 0 1)
Goal: x + t • (y - x) ∈ interior s
-- Convex.add_smul_sub_mem_interior
theorem Convex.add_smul_sub_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s)
    (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • (y - x) ∈ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Set.Ioc 0 1)
Goal: x + t • (y - x) ∈ interior s
-- Convex.add_smul_mem_interior'
theorem Convex.add_smul_mem_interior' {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s)
    (hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • y ∈ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s) (hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Set.Ioc 0 1)
Goal: x + t • y ∈ interior s
-- Convex.add_smul_mem_interior
theorem Convex.add_smul_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s)
    (hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • y ∈ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Set.Ioc 0 1)
Goal: x + t • y ∈ interior s
-- Convex.interior
protected theorem Convex.interior {s : Set E} (hs : Convex 𝕜 s) : Convex 𝕜 (interior s) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s)
Goal: Convex 𝕜 (interior s)
-- Convex.closure
protected theorem Convex.closure {s : Set E} (hs : Convex 𝕜 s) : Convex 𝕜 (closure s) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s)
Goal: Convex 𝕜 (closure s)
-- Convex.strictConvex'
protected theorem Convex.strictConvex' {s : Set E} (hs : Convex 𝕜 s)
    (h : (s \ interior s).Pairwise fun x y => ∃ c : 𝕜, lineMap x y c ∈ interior s) :
    StrictConvex 𝕜 s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (h : (s \ interior s).Pairwise fun x y => ∃ c, (AffineMap.lineMap x y) c ∈ interior s)
Goal: StrictConvex 𝕜 s
-- Convex.strictConvex
protected theorem Convex.strictConvex {s : Set E} (hs : Convex 𝕜 s)
    (h : (s \ interior s).Pairwise fun x y => ([x -[𝕜] y] \ frontier s).Nonempty) :
    StrictConvex 𝕜 s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (h : (s \ interior s).Pairwise fun x y => ([x-[𝕜]y] \ frontier s).Nonempty)
Goal: StrictConvex 𝕜 s
-- Set.Finite.isCompact_convexHull
theorem Set.Finite.isCompact_convexHull {s : Set E} (hs : s.Finite) :
    IsCompact (convexHull ℝ s) 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s : Set E} (hs : s.Finite)
Goal: IsCompact ((convexHull ℝ) s)
-- Set.Finite.isClosed_convexHull
theorem Set.Finite.isClosed_convexHull [T2Space E] {s : Set E} (hs : s.Finite) :
    IsClosed (convexHull ℝ s) 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] [T2Space E] {s : Set E} (hs : s.Finite)
Goal: IsClosed ((convexHull ℝ) s)
-- Convex.closure_subset_image_homothety_interior_of_one_lt
theorem Convex.closure_subset_image_homothety_interior_of_one_lt {s : Set E} (hs : Convex ℝ s)
    {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) :
    closure s ⊆ homothety x t '' interior s 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s : Set E} (hs : Convex ℝ s) {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t)
Goal: closure s ⊆ ⇑(AffineMap.homothety x t) '' interior s
-- Convex.closure_subset_interior_image_homothety_of_one_lt
theorem Convex.closure_subset_interior_image_homothety_of_one_lt {s : Set E} (hs : Convex ℝ s)
    {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) :
    closure s ⊆ interior (homothety x t '' s) 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s : Set E} (hs : Convex ℝ s) {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t)
Goal: closure s ⊆ interior (⇑(AffineMap.homothety x t) '' s)
-- Convex.subset_interior_image_homothety_of_one_lt
theorem Convex.subset_interior_image_homothety_of_one_lt {s : Set E} (hs : Convex ℝ s) {x : E}
    (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) : s ⊆ interior (homothety x t '' s) 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s : Set E} (hs : Convex ℝ s) {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t)
Goal: s ⊆ interior (⇑(AffineMap.homothety x t) '' s)
-- JoinedIn.of_segment_subset
theorem JoinedIn.of_segment_subset {E : Type*} [AddCommGroup E] [Module ℝ E]
    [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E]
    {x y : E} {s : Set E} (h : [x -[ℝ] y] ⊆ s) : JoinedIn s x y 
Arguments: {E : Type u_4} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {x y : E} {s : Set E} (h : [x-[ℝ]y] ⊆ s)
Goal: JoinedIn s x y
-- Convex.isPathConnected
protected theorem Convex.isPathConnected {s : Set E} (hconv : Convex ℝ s) (hne : s.Nonempty) :
    IsPathConnected s 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s : Set E} (hconv : Convex ℝ s) (hne : s.Nonempty)
Goal: IsPathConnected s
-- Convex.isConnected
protected theorem Convex.isConnected {s : Set E} (h : Convex ℝ s) (hne : s.Nonempty) :
    IsConnected s 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s : Set E} (h : Convex ℝ s) (hne : s.Nonempty)
Goal: IsConnected s
-- Convex.isPreconnected
protected theorem Convex.isPreconnected {s : Set E} (h : Convex ℝ s) : IsPreconnected s 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s : Set E} (h : Convex ℝ s)
Goal: IsPreconnected s
-- TopologicalAddGroup.pathConnectedSpace
protected theorem TopologicalAddGroup.pathConnectedSpace : PathConnectedSpace E 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E]
Goal: PathConnectedSpace E
-- isPathConnected_compl_of_isPathConnected_compl_zero
theorem isPathConnected_compl_of_isPathConnected_compl_zero [ContinuousSMul ℝ E]
    {p q : Submodule ℝ E} (hpq : IsCompl p q) (hpc : IsPathConnected ({0}ᶜ : Set p)) :
    IsPathConnected (qᶜ : Set E) 
Arguments: {E : Type u_4} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {p q : Submodule ℝ E} (hpq : IsCompl p q) (hpc : IsPathConnected {0}ᶜ)
Goal: IsPathConnected (↑q)ᶜ
-- PointedCone.toConvexCone_injective
theorem toConvexCone_injective : Injective ((↑) : PointedCone 𝕜 E → ConvexCone 𝕜 E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
Goal: Function.Injective PointedCone.toConvexCone
-- PointedCone.toConvexCone_pointed
@[simp]
theorem toConvexCone_pointed (S : PointedCone 𝕜 E) : (S : ConvexCone 𝕜 E).Pointed 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S : PointedCone 𝕜 E)
Goal: (↑S).Pointed
-- PointedCone.ext
@[ext]
theorem ext {S T : PointedCone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {S T : PointedCone 𝕜 E} (h : ∀ (x : E), x ∈ S ↔ x ∈ T)
Goal: S = T
-- ConvexCone.mem_toPointedCone
@[simp]
lemma _root_.ConvexCone.mem_toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) (x : E) :
    x ∈ S.toPointedCone hS ↔ x ∈ S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {S : ConvexCone 𝕜 E} (hS : S.Pointed) (x : E)
Goal: x ∈ ConvexCone.toPointedCone hS ↔ x ∈ S
-- ConvexCone.coe_toPointedCone
@[simp, norm_cast]
lemma _root_.ConvexCone.coe_toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) :
    S.toPointedCone hS = S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {S : ConvexCone 𝕜 E} (hS : S.Pointed)
Goal: ↑(ConvexCone.toPointedCone hS) = S
-- PointedCone.toConvexCone_map
@[simp, norm_cast]
theorem toConvexCone_map (S : PointedCone 𝕜 E) (f : E →ₗ[𝕜] F) :
    (S.map f : ConvexCone 𝕜 F) = (S : ConvexCone 𝕜 E).map f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (S : PointedCone 𝕜 E) (f : E →ₗ[𝕜] F)
Goal: ↑(PointedCone.map f S) = ConvexCone.map f ↑S
-- PointedCone.coe_map
@[simp, norm_cast]
theorem coe_map (S : PointedCone 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] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (S : PointedCone 𝕜 E) (f : E →ₗ[𝕜] F)
Goal: ↑(PointedCone.map f S) = ⇑f '' ↑S
-- PointedCone.mem_map
@[simp]
theorem mem_map {f : E →ₗ[𝕜] F} {S : PointedCone 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] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] {f : E →ₗ[𝕜] F} {S : PointedCone 𝕜 E} {y : F}
Goal: y ∈ PointedCone.map f S ↔ ∃ x ∈ S, f x = y
-- PointedCone.map_map
theorem map_map (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : PointedCone 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] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [AddCommMonoid G] [Module 𝕜 G] (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : PointedCone 𝕜 E)
Goal: PointedCone.map g (PointedCone.map f S) = PointedCone.map (g ∘ₗ f) S
-- PointedCone.map_id
@[simp]
theorem map_id (S : PointedCone 𝕜 E) : S.map LinearMap.id = S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S : PointedCone 𝕜 E)
Goal: PointedCone.map LinearMap.id S = S
-- PointedCone.coe_comap
@[simp, norm_cast]
theorem coe_comap (f : E →ₗ[𝕜] F) (S : PointedCone 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] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (f : E →ₗ[𝕜] F) (S : PointedCone 𝕜 F)
Goal: ↑(PointedCone.comap f S) = ⇑f ⁻¹' ↑S
-- PointedCone.comap_id
@[simp]
theorem comap_id (S : PointedCone 𝕜 E) : S.comap LinearMap.id = S 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (S : PointedCone 𝕜 E)
Goal: PointedCone.comap LinearMap.id S = S
-- PointedCone.comap_comap
theorem comap_comap (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : PointedCone 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] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [AddCommMonoid G] [Module 𝕜 G] (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : PointedCone 𝕜 G)
Goal: PointedCone.comap f (PointedCone.comap g S) = PointedCone.comap (g ∘ₗ f) S
-- PointedCone.mem_comap
@[simp]
theorem mem_comap {f : E →ₗ[𝕜] F} {S : PointedCone 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] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] {f : E →ₗ[𝕜] F} {S : PointedCone 𝕜 F} {x : E}
Goal: x ∈ PointedCone.comap f S ↔ f x ∈ S
-- PointedCone.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 ∈ PointedCone.positive 𝕜 E ↔ 0 ≤ x
-- PointedCone.toConvexCone_positive
@[simp, norm_cast]
theorem toConvexCone_positive : ↑(positive 𝕜 E) = ConvexCone.positive 𝕜 E 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
Goal: ↑(PointedCone.positive 𝕜 E) = ConvexCone.positive 𝕜 E
-- PointedCone.toConvexCone_dual
@[simp, norm_cast]
theorem toConvexCone_dual (S : PointedCone ℝ E) : ↑(dual S) = (S : Set E).innerDualCone 
Arguments: {E : Type u_5} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (S : PointedCone ℝ E)
Goal: ↑S.dual = (↑S).innerDualCone
-- PointedCone.mem_dual
@[simp]
theorem mem_dual {S : PointedCone ℝ E} {y : E} : y ∈ dual S ↔ ∀ ⦃x⦄, x ∈ S → 0 ≤ ⟪x, y⟫_ℝ 
Arguments: {E : Type u_5} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {S : PointedCone ℝ E} {y : E}
Goal: y ∈ S.dual ↔ ∀ ⦃x : E⦄, x ∈ S → 0 ≤ ⟪x, y⟫_ℝ
-- mem_innerDualCone
@[simp]
theorem mem_innerDualCone (y : H) (s : Set H) : y ∈ s.innerDualCone ↔ ∀ x ∈ s, 0 ≤ ⟪x, y⟫ 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (y : H) (s : Set H)
Goal: y ∈ s.innerDualCone ↔ ∀ x ∈ s, 0 ≤ ⟪x, y⟫_ℝ
-- innerDualCone_empty
@[simp]
theorem innerDualCone_empty : (∅ : Set H).innerDualCone = ⊤ 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H]
Goal: ∅.innerDualCone = ⊤
-- innerDualCone_zero
@[simp]
theorem innerDualCone_zero : (0 : Set H).innerDualCone = ⊤ 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H]
Goal: Set.innerDualCone 0 = ⊤
-- innerDualCone_univ
@[simp]
theorem innerDualCone_univ : (univ : Set H).innerDualCone = 0 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H]
Goal: Set.univ.innerDualCone = 0
-- innerDualCone_le_innerDualCone
theorem innerDualCone_le_innerDualCone (h : t ⊆ s) : s.innerDualCone ≤ t.innerDualCone 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s t : Set H) (h : t ⊆ s)
Goal: s.innerDualCone ≤ t.innerDualCone
-- pointed_innerDualCone
theorem pointed_innerDualCone : s.innerDualCone.Pointed 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H)
Goal: s.innerDualCone.Pointed
-- innerDualCone_singleton
theorem innerDualCone_singleton (x : H) :
    ({x} : Set H).innerDualCone = (ConvexCone.positive ℝ ℝ).comap (innerₛₗ ℝ x) 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (x : H)
Goal: {x}.innerDualCone = ConvexCone.comap ((innerₛₗ ℝ) x) (ConvexCone.positive ℝ ℝ)
-- innerDualCone_union
theorem innerDualCone_union (s t : Set H) :
    (s ∪ t).innerDualCone = s.innerDualCone ⊓ t.innerDualCone 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s t : Set H)
Goal: (s ∪ t).innerDualCone = s.innerDualCone ⊓ t.innerDualCone
-- innerDualCone_insert
theorem innerDualCone_insert (x : H) (s : Set H) :
    (insert x s).innerDualCone = Set.innerDualCone {x} ⊓ s.innerDualCone 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (x : H) (s : Set H)
Goal: (insert x s).innerDualCone = {x}.innerDualCone ⊓ s.innerDualCone
-- innerDualCone_iUnion
theorem innerDualCone_iUnion {ι : Sort*} (f : ι → Set H) :
    (⋃ i, f i).innerDualCone = ⨅ i, (f i).innerDualCone 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] {ι : Sort u_6} (f : ι → Set H)
Goal: (⋃ i, f i).innerDualCone = ⨅ i, (f i).innerDualCone
-- innerDualCone_sUnion
theorem innerDualCone_sUnion (S : Set (Set H)) :
    (⋃₀ S).innerDualCone = sInf (Set.innerDualCone '' S) 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (S : Set (Set H))
Goal: (⋃₀ S).innerDualCone = sInf (Set.innerDualCone '' S)
-- innerDualCone_eq_iInter_innerDualCone_singleton
theorem innerDualCone_eq_iInter_innerDualCone_singleton :
    (s.innerDualCone : Set H) = ⋂ i : s, (({↑i} : Set H).innerDualCone : Set H) 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H)
Goal: ↑s.innerDualCone = ⋂ i, ↑{↑i}.innerDualCone
-- isClosed_innerDualCone
theorem isClosed_innerDualCone : IsClosed (s.innerDualCone : Set H) 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H)
Goal: IsClosed ↑s.innerDualCone
-- ConvexCone.pointed_of_nonempty_of_isClosed
theorem ConvexCone.pointed_of_nonempty_of_isClosed (K : ConvexCone ℝ H) (ne : (K : Set H).Nonempty)
    (hc : IsClosed (K : Set H)) : K.Pointed 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (K : ConvexCone ℝ H) (ne : (↑K).Nonempty) (hc : IsClosed ↑K)
Goal: K.Pointed
-- ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem
theorem ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem (K : ConvexCone ℝ H)
    (ne : (K : Set H).Nonempty) (hc : IsClosed (K : Set H)) {b : H} (disj : b ∉ K) :
    ∃ y : H, (∀ x : H, x ∈ K → 0 ≤ ⟪x, y⟫_ℝ) ∧ ⟪y, b⟫_ℝ < 0 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] (K : ConvexCone ℝ H) (ne : (↑K).Nonempty) (hc : IsClosed ↑K) {b : H} (disj : b ∉ K)
Goal: ∃ y, (∀ x ∈ K, 0 ≤ ⟪x, y⟫_ℝ) ∧ ⟪y, b⟫_ℝ < 0
-- ConvexCone.innerDualCone_of_innerDualCone_eq_self
theorem ConvexCone.innerDualCone_of_innerDualCone_eq_self (K : ConvexCone ℝ H)
    (ne : (K : Set H).Nonempty) (hc : IsClosed (K : Set H)) :
    ((K : Set H).innerDualCone : Set H).innerDualCone = K 
Arguments: {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] (K : ConvexCone ℝ H) (ne : (↑K).Nonempty) (hc : IsClosed ↑K)
Goal: (↑(↑K).innerDualCone).innerDualCone = K
-- ProperCone.isClosed'
structure ProperCone (𝕜 : Type*) (E : Type*) [OrderedSemiring 𝕜] [AddCommMonoid E]
    [TopologicalSpace E] [Module 𝕜 E] extends Submodule {c : 𝕜 // 0 ≤ c} E where
  isClosed' : IsClosed (carrier : Set E)
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (self : ProperCone 𝕜 E)
Goal: IsClosed self.carrier
-- ProperCone.toPointedCone_injective
theorem toPointedCone_injective : Function.Injective ((↑) : ProperCone 𝕜 E → PointedCone 𝕜 E) 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E]
Goal: Function.Injective ProperCone.toPointedCone
-- ProperCone.ext
@[ext]
theorem ext {S T : ProperCone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {S T : ProperCone 𝕜 E} (h : ∀ (x : E), x ∈ S ↔ x ∈ T)
Goal: S = T
-- ProperCone.mem_coe
@[simp]
theorem mem_coe {x : E} {K : ProperCone 𝕜 E} : x ∈ (K : PointedCone 𝕜 E) ↔ x ∈ K 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {x : E} {K : ProperCone 𝕜 E}
Goal: x ∈ ↑K ↔ x ∈ K
-- ProperCone.nonempty
protected theorem nonempty (K : ProperCone 𝕜 E) : (K : Set E).Nonempty 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (K : ProperCone 𝕜 E)
Goal: (↑K).Nonempty
-- ProperCone.isClosed
protected theorem isClosed (K : ProperCone 𝕜 E) : IsClosed (K : Set E) 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (K : ProperCone 𝕜 E)
Goal: IsClosed ↑K
-- ProperCone.mem_positive
@[simp]
theorem mem_positive {x : E} : x ∈ positive 𝕜 E ↔ 0 ≤ x 
Arguments: (𝕜 : Type u_2) (E : Type u_1) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] [TopologicalSpace E] [OrderClosedTopology E] {x : E}
Goal: x ∈ ProperCone.positive 𝕜 E ↔ 0 ≤ x
-- ProperCone.coe_positive
@[simp]
theorem coe_positive : ↑(positive 𝕜 E) = ConvexCone.positive 𝕜 E 
Arguments: (𝕜 : Type u_2) (E : Type u_1) [OrderedSemiring 𝕜] [OrderedAddCommGroup E] [Module 𝕜 E] [OrderedSMul 𝕜 E] [TopologicalSpace E] [OrderClosedTopology E]
Goal: ↑↑(ProperCone.positive 𝕜 E) = ConvexCone.positive 𝕜 E
-- ProperCone.mem_zero
@[simp]
theorem mem_zero (x : E) : x ∈ (0 : ProperCone 𝕜 E) ↔ x = 0 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E] (x : E)
Goal: x ∈ 0 ↔ x = 0
-- ProperCone.coe_zero
@[simp, norm_cast]
theorem coe_zero : ↑(0 : ProperCone 𝕜 E) = (0 : ConvexCone 𝕜 E) 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E]
Goal: ↑↑0 = 0
-- ProperCone.pointed_zero
theorem pointed_zero : ((0 : ProperCone 𝕜 E) : ConvexCone 𝕜 E).Pointed 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E]
Goal: (↑↑0).Pointed
-- ProperCone.pointed
protected theorem pointed (K : ProperCone ℝ E) : (K : ConvexCone ℝ E).Pointed 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (K : ProperCone ℝ E)
Goal: (↑↑K).Pointed
-- ProperCone.coe_map
@[simp, norm_cast]
theorem coe_map (f : E →L[ℝ] F) (K : ProperCone ℝ E) :
    ↑(K.map f) = (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)
Goal: ↑(ProperCone.map f K) = (PointedCone.map ↑f ↑K).closure