ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- strongConvexOn_iff_convex
lemma strongConvexOn_iff_convex :
    StrongConvexOn s m f ↔ ConvexOn ℝ s fun x ↦ f x - m / (2 : ℝ) * ‖x‖ ^ 2 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {s : Set E} {m : ℝ} {f : E → ℝ}
Goal: StrongConvexOn s m f ↔ ConvexOn ℝ s fun x => f x - m / 2 * ‖x‖ ^ 2
-- strongConcaveOn_iff_convex
lemma strongConcaveOn_iff_convex :
    StrongConcaveOn s m f ↔ ConcaveOn ℝ s fun x ↦ f x + m / (2 : ℝ) * ‖x‖ ^ 2 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {s : Set E} {m : ℝ} {f : E → ℝ}
Goal: StrongConcaveOn s m f ↔ ConcaveOn ℝ s fun x => f x + m / 2 * ‖x‖ ^ 2
-- ContinuousLinearMap.toExposed.isExposed
theorem ContinuousLinearMap.toExposed.isExposed : IsExposed 𝕜 A (l.toExposed A) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {l : E →L[𝕜] 𝕜} {A : Set E}
Goal: IsExposed 𝕜 A (l.toExposed A)
-- isExposed_empty
theorem isExposed_empty : IsExposed 𝕜 A ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A : Set E}
Goal: IsExposed 𝕜 A ∅
-- IsExposed.subset
protected theorem subset (hAB : IsExposed 𝕜 A B) : B ⊆ A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A B : Set E} (hAB : IsExposed 𝕜 A B)
Goal: B ⊆ A
-- IsExposed.refl
@[refl]
protected theorem refl (A : Set E) : IsExposed 𝕜 A A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (A : Set E)
Goal: IsExposed 𝕜 A A
-- IsExposed.antisymm
protected theorem antisymm (hB : IsExposed 𝕜 A B) (hA : IsExposed 𝕜 B A) : A = B 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A B : Set E} (hB : IsExposed 𝕜 A B) (hA : IsExposed 𝕜 B A)
Goal: A = B
-- IsExposed.mono
protected theorem mono (hC : IsExposed 𝕜 A C) (hBA : B ⊆ A) (hCB : C ⊆ B) : IsExposed 𝕜 B C 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A B C : Set E} (hC : IsExposed 𝕜 A C) (hBA : B ⊆ A) (hCB : C ⊆ B)
Goal: IsExposed 𝕜 B C
-- IsExposed.eq_inter_halfspace'
theorem eq_inter_halfspace' {A B : Set E} (hAB : IsExposed 𝕜 A B) (hB : B.Nonempty) :
    ∃ l : E →L[𝕜] 𝕜, ∃ a, B = { x ∈ A | a ≤ l x } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A B : Set E} (hAB : IsExposed 𝕜 A B) (hB : B.Nonempty)
Goal: ∃ l a, B = {x | x ∈ A ∧ a ≤ l x}
-- IsExposed.eq_inter_halfspace
theorem eq_inter_halfspace [Nontrivial 𝕜] {A B : Set E} (hAB : IsExposed 𝕜 A B) :
    ∃ l : E →L[𝕜] 𝕜, ∃ a, B = { x ∈ A | a ≤ l x } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] [Nontrivial 𝕜] {A B : Set E} (hAB : IsExposed 𝕜 A B)
Goal: ∃ l a, B = {x | x ∈ A ∧ a ≤ l x}
-- IsExposed.inter
protected theorem inter [ContinuousAdd 𝕜] {A B C : Set E} (hB : IsExposed 𝕜 A B)
    (hC : IsExposed 𝕜 A C) : IsExposed 𝕜 A (B ∩ C) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] [ContinuousAdd 𝕜] {A B C : Set E} (hB : IsExposed 𝕜 A B) (hC : IsExposed 𝕜 A C)
Goal: IsExposed 𝕜 A (B ∩ C)
-- IsExposed.sInter
theorem sInter [ContinuousAdd 𝕜] {F : Finset (Set E)} (hF : F.Nonempty)
    (hAF : ∀ B ∈ F, IsExposed 𝕜 A B) : IsExposed 𝕜 A (⋂₀ F) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A : Set E} [ContinuousAdd 𝕜] {F : Finset (Set E)} (hF : F.Nonempty) (hAF : ∀ B ∈ F, IsExposed 𝕜 A B)
Goal: IsExposed 𝕜 A (⋂₀ ↑F)
-- IsExposed.inter_left
theorem inter_left (hC : IsExposed 𝕜 A C) (hCB : C ⊆ B) : IsExposed 𝕜 (A ∩ B) C 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A B C : Set E} (hC : IsExposed 𝕜 A C) (hCB : C ⊆ B)
Goal: IsExposed 𝕜 (A ∩ B) C
-- IsExposed.inter_right
theorem inter_right (hC : IsExposed 𝕜 B C) (hCA : C ⊆ A) : IsExposed 𝕜 (A ∩ B) C 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A B C : Set E} (hC : IsExposed 𝕜 B C) (hCA : C ⊆ A)
Goal: IsExposed 𝕜 (A ∩ B) C
-- IsExposed.isClosed
protected theorem isClosed [OrderClosedTopology 𝕜] {A B : Set E} (hAB : IsExposed 𝕜 A B)
    (hA : IsClosed A) : IsClosed B 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] [OrderClosedTopology 𝕜] {A B : Set E} (hAB : IsExposed 𝕜 A B) (hA : IsClosed A)
Goal: IsClosed B
-- IsExposed.isCompact
protected theorem isCompact [OrderClosedTopology 𝕜] [T2Space E] {A B : Set E}
    (hAB : IsExposed 𝕜 A B) (hA : IsCompact A) : IsCompact B 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] [OrderClosedTopology 𝕜] [T2Space E] {A B : Set E} (hAB : IsExposed 𝕜 A B) (hA : IsCompact A)
Goal: IsCompact B
-- exposed_point_def
theorem exposed_point_def :
    x ∈ A.exposedPoints 𝕜 ↔ x ∈ A ∧ ∃ l : E →L[𝕜] 𝕜, ∀ y ∈ A, l y ≤ l x ∧ (l x ≤ l y → y = x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A : Set E} {x : E}
Goal: x ∈ Set.exposedPoints 𝕜 A ↔ x ∈ A ∧ ∃ l, ∀ y ∈ A, l y ≤ l x ∧ (l x ≤ l y → y = x)
-- exposedPoints_subset
theorem exposedPoints_subset : A.exposedPoints 𝕜 ⊆ A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A : Set E}
Goal: Set.exposedPoints 𝕜 A ⊆ A
-- exposedPoints_empty
@[simp]
theorem exposedPoints_empty : (∅ : Set E).exposedPoints 𝕜 = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E]
Goal: Set.exposedPoints 𝕜 ∅ = ∅
-- mem_exposedPoints_iff_exposed_singleton
theorem mem_exposedPoints_iff_exposed_singleton : x ∈ A.exposedPoints 𝕜 ↔ IsExposed 𝕜 A {x} 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [OrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A : Set E} {x : E}
Goal: x ∈ Set.exposedPoints 𝕜 A ↔ IsExposed 𝕜 A {x}
-- IsExposed.convex
protected theorem convex (hAB : IsExposed 𝕜 A B) (hA : Convex 𝕜 A) : Convex 𝕜 B 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [LinearOrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A B : Set E} (hAB : IsExposed 𝕜 A B) (hA : Convex 𝕜 A)
Goal: Convex 𝕜 B
-- IsExposed.isExtreme
protected theorem isExtreme (hAB : IsExposed 𝕜 A B) : IsExtreme 𝕜 A B 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [LinearOrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A B : Set E} (hAB : IsExposed 𝕜 A B)
Goal: IsExtreme 𝕜 A B
-- exposedPoints_subset_extremePoints
theorem exposedPoints_subset_extremePoints : A.exposedPoints 𝕜 ⊆ A.extremePoints 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace 𝕜] [LinearOrderedRing 𝕜] [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] {A : Set E}
Goal: Set.exposedPoints 𝕜 A ⊆ Set.extremePoints 𝕜 A
-- gauge_def
theorem gauge_def : gauge s x = sInf ({ r ∈ Set.Ioi (0 : ℝ) | x ∈ r • s }) 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E}
Goal: gauge s x = sInf {r | r ∈ Set.Ioi 0 ∧ x ∈ r • s}
-- gauge_def'
theorem gauge_def' : gauge s x = sInf {r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s} 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E}
Goal: gauge s x = sInf {r | r ∈ Set.Ioi 0 ∧ r⁻¹ • x ∈ s}
-- _private.Mathlib.Analysis.Convex.Gauge.0.gauge_set_bddBelow
private theorem gauge_set_bddBelow : BddBelow { r : ℝ | 0 < r ∧ x ∈ r • s } 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E}
Goal: BddBelow {r | 0 < r ∧ x ∈ r • s}
-- Absorbent.gauge_set_nonempty
theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
    { r : ℝ | 0 < r ∧ x ∈ r • s }.Nonempty 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} (absorbs : Absorbent ℝ s)
Goal: {r | 0 < r ∧ x ∈ r • s}.Nonempty
-- gauge_mono
theorem gauge_mono (hs : Absorbent ℝ s) (h : s ⊆ t) : gauge t ≤ gauge s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s t : Set E} (hs : Absorbent ℝ s) (h : s ⊆ t)
Goal: gauge t ≤ gauge s
-- exists_lt_of_gauge_lt
theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
    ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} {a : ℝ} (absorbs : Absorbent ℝ s) (h : gauge s x < a)
Goal: ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s
-- gauge_zero
@[simp]
theorem gauge_zero : gauge s 0 = 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E}
Goal: gauge s 0 = 0
-- gauge_zero'
@[simp]
theorem gauge_zero' : gauge (0 : Set E) = 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E]
Goal: gauge 0 = 0
-- gauge_empty
@[simp]
theorem gauge_empty : gauge (∅ : Set E) = 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E]
Goal: gauge ∅ = 0
-- gauge_of_subset_zero
theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (h : s ⊆ 0)
Goal: gauge s = 0
-- gauge_nonneg
theorem gauge_nonneg (x : E) : 0 ≤ gauge s x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (x : E)
Goal: 0 ≤ gauge s x
-- gauge_neg
theorem gauge_neg (symmetric : ∀ x ∈ s, -x ∈ s) (x : E) : gauge s (-x) = gauge s x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (x : E)
Goal: gauge s (-x) = gauge s x
-- gauge_neg_set_neg
theorem gauge_neg_set_neg (x : E) : gauge (-s) (-x) = gauge s x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (x : E)
Goal: gauge (-s) (-x) = gauge s x
-- gauge_neg_set_eq_gauge_neg
theorem gauge_neg_set_eq_gauge_neg (x : E) : gauge (-s) x = gauge s (-x) 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (x : E)
Goal: gauge (-s) x = gauge s (-x)
-- gauge_le_of_mem
theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} {a : ℝ} (ha : 0 ≤ a) (hx : x ∈ a • s)
Goal: gauge s x ≤ a
-- gauge_le_eq
theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
    { x | gauge s x ≤ a } = ⋂ (r : ℝ) (_ : a < r), r • s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {a : ℝ} (hs₁ : Convex ℝ s) (hs₀ : 0 ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a)
Goal: {x | gauge s x ≤ a} = ⋂ r, ⋂ (_ : a < r), r • s
-- gauge_lt_eq'
theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
    { x | gauge s x < a } = ⋃ (r : ℝ) (_ : 0 < r) (_ : r < a), r • s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (absorbs : Absorbent ℝ s) (a : ℝ)
Goal: {x | gauge s x < a} = ⋃ r, ⋃ (_ : 0 < r), ⋃ (_ : r < a), r • s
-- gauge_lt_eq
theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
    { x | gauge s x < a } = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (absorbs : Absorbent ℝ s) (a : ℝ)
Goal: {x | gauge s x < a} = ⋃ r ∈ Set.Ioo 0 a, r • s
-- mem_openSegment_of_gauge_lt_one
theorem mem_openSegment_of_gauge_lt_one (absorbs : Absorbent ℝ s) (hgauge : gauge s x < 1) :
    ∃ y ∈ s, x ∈ openSegment ℝ 0 y 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} (absorbs : Absorbent ℝ s) (hgauge : gauge s x < 1)
Goal: ∃ y ∈ s, x ∈ openSegment ℝ 0 y
-- gauge_lt_one_subset_self
theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) :
    { x | gauge s x < 1 } ⊆ s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (hs : Convex ℝ s) (h₀ : 0 ∈ s) (absorbs : Absorbent ℝ s)
Goal: {x | gauge s x < 1} ⊆ s
-- gauge_le_one_of_mem
theorem gauge_le_one_of_mem {x : E} (hx : x ∈ s) : gauge s x ≤ 1 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} (hx : x ∈ s)
Goal: gauge s x ≤ 1
-- gauge_add_le
theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
    gauge s (x + y) ≤ gauge s x + gauge s y 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E)
Goal: gauge s (x + y) ≤ gauge s x + gauge s y
-- self_subset_gauge_le_one
theorem self_subset_gauge_le_one : s ⊆ { x | gauge s x ≤ 1 } 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E}
Goal: s ⊆ {x | gauge s x ≤ 1}
-- Convex.gauge_le
theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) (a : ℝ) :
    Convex ℝ { x | gauge s x ≤ a } 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (hs : Convex ℝ s) (h₀ : 0 ∈ s) (absorbs : Absorbent ℝ s) (a : ℝ)
Goal: Convex ℝ {x | gauge s x ≤ a}
-- Balanced.starConvex
theorem Balanced.starConvex (hs : Balanced ℝ s) : StarConvex ℝ 0 s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} (hs : Balanced ℝ s)
Goal: StarConvex ℝ 0 s
-- le_gauge_of_not_mem
theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s) :
    a ≤ gauge s x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} {a : ℝ} (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s)
Goal: a ≤ gauge s x
-- one_le_gauge_of_not_mem
theorem one_le_gauge_of_not_mem (hs₁ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ s) :
    1 ≤ gauge s x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} (hs₁ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ s)
Goal: 1 ≤ gauge s x
-- gauge_smul_of_nonneg
theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
    (ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {α : Type u_4} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ] [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α} (ha : 0 ≤ a) (x : E)
Goal: gauge s (a • x) = a • gauge s x
-- gauge_smul_left_of_nonneg
theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ ℝ]
    [IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} {a : α} (ha : 0 ≤ a) :
    gauge (a • s) = a⁻¹ • gauge s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {α : Type u_4} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ] [MulActionWithZero α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} {a : α} (ha : 0 ≤ a)
Goal: gauge (a • s) = a⁻¹ • gauge s
-- gauge_smul_left
theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ]
    [IsScalarTower α ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α) :
    gauge (a • s) = |a|⁻¹ • gauge s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {α : Type u_4} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ] [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α)
Goal: gauge (a • s) = |a|⁻¹ • gauge s
-- gauge_norm_smul
theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) :
    gauge s (‖r‖ • x) = gauge s (r • x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E] (hs : Balanced 𝕜 s) (r : 𝕜) (x : E)
Goal: gauge s (‖r‖ • x) = gauge s (r • x)
-- gauge_smul
theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E] (hs : Balanced 𝕜 s) (r : 𝕜) (x : E)
Goal: gauge s (r • x) = ‖r‖ * gauge s x
-- comap_gauge_nhds_zero_le
theorem comap_gauge_nhds_zero_le (ha : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
    comap (gauge s) (𝓝 0) ≤ 𝓝 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] (ha : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s)
Goal: Filter.comap (gauge s) (𝓝 0) ≤ 𝓝 0
-- gauge_eq_zero
theorem gauge_eq_zero (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
    gauge s x = 0 ↔ x = 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} [TopologicalSpace E] [T1Space E] (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s)
Goal: gauge s x = 0 ↔ x = 0
-- gauge_pos
theorem gauge_pos (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
    0 < gauge s x ↔ x ≠ 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} [TopologicalSpace E] [T1Space E] (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s)
Goal: 0 < gauge s x ↔ x ≠ 0
-- interior_subset_gauge_lt_one
theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s x < 1 } 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] (s : Set E)
Goal: interior s ⊆ {x | gauge s x < 1}
-- gauge_lt_one_eq_self_of_isOpen
theorem gauge_lt_one_eq_self_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
    { x | gauge s x < 1 } = s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [ContinuousSMul ℝ E] (hs₁ : Convex ℝ s) (hs₀ : 0 ∈ s) (hs₂ : IsOpen s)
Goal: {x | gauge s x < 1} = s
-- gauge_lt_one_of_mem_of_isOpen
theorem gauge_lt_one_of_mem_of_isOpen (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
    gauge s x < 1 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [ContinuousSMul ℝ E] (hs₂ : IsOpen s) {x : E} (hx : x ∈ s)
Goal: gauge s x < 1
-- gauge_lt_of_mem_smul
theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s) (hx : x ∈ ε • s) :
    gauge s x < ε 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [ContinuousSMul ℝ E] (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s) (hx : x ∈ ε • s)
Goal: gauge s x < ε
-- mem_closure_of_gauge_le_one
theorem mem_closure_of_gauge_le_one (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s)
    (h : gauge s x ≤ 1) : x ∈ closure s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} [TopologicalSpace E] [ContinuousSMul ℝ E] (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s) (h : gauge s x ≤ 1)
Goal: x ∈ closure s
-- mem_frontier_of_gauge_eq_one
theorem mem_frontier_of_gauge_eq_one (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s)
    (h : gauge s x = 1) : x ∈ frontier s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} [TopologicalSpace E] [ContinuousSMul ℝ E] (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s) (h : gauge s x = 1)
Goal: x ∈ frontier s
-- tendsto_gauge_nhds_zero'
theorem tendsto_gauge_nhds_zero' (hs : s ∈ 𝓝 0) : Tendsto (gauge s) (𝓝 0) (𝓝[≥] 0) 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [ContinuousSMul ℝ E] (hs : s ∈ 𝓝 0)
Goal: Filter.Tendsto (gauge s) (𝓝 0) (𝓝[≥] 0)
-- tendsto_gauge_nhds_zero
theorem tendsto_gauge_nhds_zero (hs : s ∈ 𝓝 0) : Tendsto (gauge s) (𝓝 0) (𝓝 0) 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [ContinuousSMul ℝ E] (hs : s ∈ 𝓝 0)
Goal: Filter.Tendsto (gauge s) (𝓝 0) (𝓝 0)
-- continuousAt_gauge_zero
theorem continuousAt_gauge_zero (hs : s ∈ 𝓝 0) : ContinuousAt (gauge s) 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [ContinuousSMul ℝ E] (hs : s ∈ 𝓝 0)
Goal: ContinuousAt (gauge s) 0
-- comap_gauge_nhds_zero
theorem comap_gauge_nhds_zero (hb : Bornology.IsVonNBounded ℝ s) (h₀ : s ∈ 𝓝 0) :
    comap (gauge s) (𝓝 0) = 𝓝 0 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [ContinuousSMul ℝ E] (hb : Bornology.IsVonNBounded ℝ s) (h₀ : s ∈ 𝓝 0)
Goal: Filter.comap (gauge s) (𝓝 0) = 𝓝 0
-- continuousAt_gauge
theorem continuousAt_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : ContinuousAt (gauge s) x 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
Goal: ContinuousAt (gauge s) x
-- continuous_gauge
@[continuity]
theorem continuous_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuous (gauge s) 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
Goal: Continuous (gauge s)
-- gauge_lt_one_eq_interior
theorem gauge_lt_one_eq_interior (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
    { x | gauge s x < 1 } = interior s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
Goal: {x | gauge s x < 1} = interior s
-- gauge_lt_one_iff_mem_interior
theorem gauge_lt_one_iff_mem_interior (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
    gauge s x < 1 ↔ x ∈ interior s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
Goal: gauge s x < 1 ↔ x ∈ interior s
-- gauge_le_one_iff_mem_closure
theorem gauge_le_one_iff_mem_closure (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
    gauge s x ≤ 1 ↔ x ∈ closure s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
Goal: gauge s x ≤ 1 ↔ x ∈ closure s
-- gauge_eq_one_iff_mem_frontier
theorem gauge_eq_one_iff_mem_frontier (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
    gauge s x = 1 ↔ x ∈ frontier s 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} {x : E} [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
Goal: gauge s x = 1 ↔ x ∈ frontier s
-- gaugeSeminorm_toFun
@[simps!]
def gaugeSeminorm (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Absorbent ℝ s) : Seminorm 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E] (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Absorbent ℝ s) (x : E)
Goal: (gaugeSeminorm hs₀ hs₁ hs₂) x = gauge s x
-- gaugeSeminorm_lt_one_of_isOpen
theorem gaugeSeminorm_lt_one_of_isOpen (hs : IsOpen s) {x : E} (hx : x ∈ s) :
    gaugeSeminorm hs₀ hs₁ hs₂ x < 1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E] {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [TopologicalSpace E] [ContinuousSMul ℝ E] (hs : IsOpen s) {x : E} (hx : x ∈ s)
Goal: (gaugeSeminorm hs₀ hs₁ hs₂) x < 1
-- gaugeSeminorm_ball_one
theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [AddCommGroup E] [Module ℝ E] {s : Set E} [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E] {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [TopologicalSpace E] [ContinuousSMul ℝ E] (hs : IsOpen s)
Goal: (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s
-- Seminorm.gauge_ball
@[simp]
protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1) = p 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] (p : Seminorm ℝ E)
Goal: gauge (p.ball 0 1) = ⇑p
-- Seminorm.gaugeSeminorm_ball
theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
    gaugeSeminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1) (p.absorbent_ball_zero zero_lt_one) =
      p 
Arguments: {E : Type u_2} [AddCommGroup E] [Module ℝ E] (p : Seminorm ℝ E)
Goal: gaugeSeminorm ⋯ ⋯ ⋯ = p
-- gauge_unit_ball
theorem gauge_unit_ball (x : E) : gauge (ball (0 : E) 1) x = ‖x‖ 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E)
Goal: gauge (Metric.ball 0 1) x = ‖x‖