ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mem_convexHull_iff_exists_fintype
lemma mem_convexHull_iff_exists_fintype {s : Set E} {x : E} :
    x ∈ convexHull R s ↔ ∃ (ι : Type) (_ : Fintype ι) (w : ι → R) (z : ι → E), (∀ i, 0 ≤ w i) ∧
      ∑ i, w i = 1 ∧ (∀ i, z i ∈ s) ∧ ∑ i, w i • z i = x 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E} {x : E}
Goal: x ∈ (convexHull R) s ↔
  ∃ ι x_1 w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i : ι, w i = 1 ∧ (∀ (i : ι), z i ∈ s) ∧ ∑ i : ι, w i • z i = x
-- Finset.convexHull_eq
theorem Finset.convexHull_eq (s : Finset E) : convexHull R ↑s =
    { x : E | ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x } 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s : Finset E)
Goal: (convexHull R) ↑s = {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x}
-- Finset.mem_convexHull
theorem Finset.mem_convexHull {s : Finset E} {x : E} : x ∈ convexHull R (s : Set E) ↔
    ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Finset E} {x : E}
Goal: x ∈ (convexHull R) ↑s ↔ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x
-- Finset.mem_convexHull'
lemma Finset.mem_convexHull' {s : Finset E} {x : E} :
    x ∈ convexHull R (s : Set E) ↔
      ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ ∑ y ∈ s, w y • y = x 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Finset E} {x : E}
Goal: x ∈ (convexHull R) ↑s ↔ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ ∑ y ∈ s, w y • y = x
-- Set.Finite.convexHull_eq
theorem Set.Finite.convexHull_eq {s : Set E} (hs : s.Finite) : convexHull R s =
    { x : E | ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ hs.toFinset, w y = 1 ∧
      hs.toFinset.centerMass w id = x } 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E} (hs : s.Finite)
Goal: (convexHull R) s = {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ hs.toFinset, w y = 1 ∧ hs.toFinset.centerMass w id = x}
-- convexHull_eq_union_convexHull_finite_subsets
theorem convexHull_eq_union_convexHull_finite_subsets (s : Set E) :
    convexHull R s = ⋃ (t : Finset E) (w : ↑t ⊆ s), convexHull R ↑t 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s : Set E)
Goal: (convexHull R) s = ⋃ t, ⋃ (_ : ↑t ⊆ s), (convexHull R) ↑t
-- mk_mem_convexHull_prod
theorem mk_mem_convexHull_prod {t : Set F} {x : E} {y : F} (hx : x ∈ convexHull R s)
    (hy : y ∈ convexHull R t) : (x, y) ∈ convexHull R (s ×ˢ t) 
Arguments: {R : Type u_1} {E : Type u_3} {F : Type u_4} [LinearOrderedField R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] {s : Set E} {t : Set F} {x : E} {y : F} (hx : x ∈ (convexHull R) s) (hy : y ∈ (convexHull R) t)
Goal: (x, y) ∈ (convexHull R) (s ×ˢ t)
-- convexHull_prod
@[simp]
theorem convexHull_prod (s : Set E) (t : Set F) :
    convexHull R (s ×ˢ t) = convexHull R s ×ˢ convexHull R t 
Arguments: {R : Type u_1} {E : Type u_3} {F : Type u_4} [LinearOrderedField R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] (s : Set E) (t : Set F)
Goal: (convexHull R) (s ×ˢ t) = (convexHull R) s ×ˢ (convexHull R) t
-- convexHull_add
theorem convexHull_add (s t : Set E) : convexHull R (s + t) = convexHull R s + convexHull R t 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s t : Set E)
Goal: (convexHull R) (s + t) = (convexHull R) s + (convexHull R) t
-- convexHullAddMonoidHom_apply
@[simps]
def convexHullAddMonoidHom : Set E →+ Set E 
Arguments: (R : Type u_1) (E : Type u_3) [LinearOrderedField R] [AddCommGroup E] [Module R E] (a : Set E)
Goal: (convexHullAddMonoidHom R E) a = (convexHull R) a
-- convexHull_sub
theorem convexHull_sub (s t : Set E) : convexHull R (s - t) = convexHull R s - convexHull R t 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s t : Set E)
Goal: (convexHull R) (s - t) = (convexHull R) s - (convexHull R) t
-- convexHull_list_sum
theorem convexHull_list_sum (l : List (Set E)) : convexHull R l.sum = (l.map <| convexHull R).sum 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (l : List (Set E))
Goal: (convexHull R) l.sum = (List.map (⇑(convexHull R)) l).sum
-- convexHull_multiset_sum
theorem convexHull_multiset_sum (s : Multiset (Set E)) :
    convexHull R s.sum = (s.map <| convexHull R).sum 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s : Multiset (Set E))
Goal: (convexHull R) s.sum = (Multiset.map (⇑(convexHull R)) s).sum
-- convexHull_sum
theorem convexHull_sum {ι} (s : Finset ι) (t : ι → Set E) :
    convexHull R (∑ i ∈ s, t i) = ∑ i ∈ s, convexHull R (t i) 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {ι : Type u_8} (s : Finset ι) (t : ι → Set E)
Goal: (convexHull R) (∑ i ∈ s, t i) = ∑ i ∈ s, (convexHull R) (t i)
-- convexHull_basis_eq_stdSimplex
theorem convexHull_basis_eq_stdSimplex :
    convexHull R (range fun i j : ι => if i = j then (1 : R) else 0) = stdSimplex R ι 
Arguments: {R : Type u_1} (ι : Type u_5) [LinearOrderedField R] [Fintype ι]
Goal: (convexHull R) (Set.range fun i j => if i = j then 1 else 0) = stdSimplex R ι
-- Set.Finite.convexHull_eq_image
theorem Set.Finite.convexHull_eq_image {s : Set E} (hs : s.Finite) : convexHull R s =
    haveI := hs.fintype
    (⇑(∑ x : s, (@LinearMap.proj R s _ (fun _ => R) _ _ x).smulRight x.1)) '' stdSimplex R s 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E} (hs : s.Finite)
Goal: (convexHull R) s = ⇑(∑ x : ↑s, (LinearMap.proj x).smulRight ↑x) '' stdSimplex R ↑s
-- mem_Icc_of_mem_stdSimplex
theorem mem_Icc_of_mem_stdSimplex (hf : f ∈ stdSimplex R ι) (x) : f x ∈ Icc (0 : R) 1 
Arguments: {R : Type u_1} {ι : Type u_5} [LinearOrderedField R] [Fintype ι] {f : ι → R} (hf : f ∈ stdSimplex R ι) (x : ι)
Goal: f x ∈ Set.Icc 0 1
-- AffineBasis.convexHull_eq_nonneg_coord
theorem AffineBasis.convexHull_eq_nonneg_coord {ι : Type*} (b : AffineBasis ι R E) :
    convexHull R (range b) = { x | ∀ i, 0 ≤ b.coord i x } 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {ι : Type u_8} (b : AffineBasis ι R E)
Goal: (convexHull R) (Set.range ⇑b) = {x | ∀ (i : ι), 0 ≤ (b.coord i) x}
-- AffineIndependent.convexHull_inter
lemma AffineIndependent.convexHull_inter (hs : AffineIndependent R ((↑) : s → E))
    (ht₁ : t₁ ⊆ s) (ht₂ : t₂ ⊆ s) :
    convexHull R (t₁ ∩ t₂ : Set E) = convexHull R t₁ ∩ convexHull R t₂ 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s t₁ t₂ : Finset E} (hs : AffineIndependent R Subtype.val) (ht₁ : t₁ ⊆ s) (ht₂ : t₂ ⊆ s)
Goal: (convexHull R) (↑t₁ ∩ ↑t₂) = (convexHull R) ↑t₁ ∩ (convexHull R) ↑t₂
-- AffineIndependent.convexHull_inter'
lemma AffineIndependent.convexHull_inter' (hs : AffineIndependent R ((↑) : ↑(t₁ ∪ t₂) → E)) :
    convexHull R (t₁ ∩ t₂ : Set E) = convexHull R t₁ ∩ convexHull R t₂ 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {t₁ t₂ : Finset E} (hs : AffineIndependent R Subtype.val)
Goal: (convexHull R) (↑t₁ ∩ ↑t₂) = (convexHull R) ↑t₁ ∩ (convexHull R) ↑t₂
-- mem_convexHull_pi
lemma mem_convexHull_pi (h : ∀ i ∈ s, x i ∈ convexHull 𝕜 (t i)) : x ∈ convexHull 𝕜 (s.pi t) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} [Finite ι] [LinearOrderedField 𝕜] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] {s : Set ι} {t : (i : ι) → Set (E i)} {x : (i : ι) → E i} (h : ∀ i ∈ s, x i ∈ (convexHull 𝕜) (t i))
Goal: x ∈ (convexHull 𝕜) (s.pi t)
-- convexHull_pi
@[simp] lemma convexHull_pi (s : Set ι) (t : Π i, Set (E i)) :
    convexHull 𝕜 (s.pi t) = s.pi (fun i ↦ convexHull 𝕜 (t i)) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} [Finite ι] [LinearOrderedField 𝕜] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] (s : Set ι) (t : (i : ι) → Set (E i))
Goal: (convexHull 𝕜) (s.pi t) = s.pi fun i => (convexHull 𝕜) (t i)
-- ContinuousLinearMap.nnnorm_def
theorem nnnorm_def (f : E →SL[σ₁₂] F) : ‖f‖₊ = sInf { c | ∀ x, ‖f x‖₊ ≤ c * ‖x‖₊ } 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: ‖f‖₊ = sInf {c | ∀ (x : E), ‖f x‖₊ ≤ c * ‖x‖₊}
-- ContinuousLinearMap.opNNNorm_le_bound
theorem opNNNorm_le_bound (f : E →SL[σ₁₂] F) (M : ℝ≥0) (hM : ∀ x, ‖f x‖₊ ≤ M * ‖x‖₊) : ‖f‖₊ ≤ M 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (M : ℝ≥0) (hM : ∀ (x : E), ‖f x‖₊ ≤ M * ‖x‖₊)
Goal: ‖f‖₊ ≤ M
-- ContinuousLinearMap.opNNNorm_le_bound'
theorem opNNNorm_le_bound' (f : E →SL[σ₁₂] F) (M : ℝ≥0) (hM : ∀ x, ‖x‖₊ ≠ 0 → ‖f x‖₊ ≤ M * ‖x‖₊) :
    ‖f‖₊ ≤ M 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (M : ℝ≥0) (hM : ∀ (x : E), ‖x‖₊ ≠ 0 → ‖f x‖₊ ≤ M * ‖x‖₊)
Goal: ‖f‖₊ ≤ M
-- ContinuousLinearMap.opNNNorm_le_of_unit_nnnorm
theorem opNNNorm_le_of_unit_nnnorm [NormedSpace ℝ E] [NormedSpace ℝ F] {f : E →L[ℝ] F} {C : ℝ≥0}
    (hf : ∀ x, ‖x‖₊ = 1 → ‖f x‖₊ ≤ C) : ‖f‖₊ ≤ C 
Arguments: {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] {f : E →L[ℝ] F} {C : ℝ≥0} (hf : ∀ (x : E), ‖x‖₊ = 1 → ‖f x‖₊ ≤ C)
Goal: ‖f‖₊ ≤ C
-- ContinuousLinearMap.opNNNorm_le_of_lipschitz
theorem opNNNorm_le_of_lipschitz {f : E →SL[σ₁₂] F} {K : ℝ≥0} (hf : LipschitzWith K f) :
    ‖f‖₊ ≤ K 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F} {K : ℝ≥0} (hf : LipschitzWith K ⇑f)
Goal: ‖f‖₊ ≤ K
-- ContinuousLinearMap.opNNNorm_eq_of_bounds
theorem opNNNorm_eq_of_bounds {φ : E →SL[σ₁₂] F} (M : ℝ≥0) (h_above : ∀ x, ‖φ x‖₊ ≤ M * ‖x‖₊)
    (h_below : ∀ N, (∀ x, ‖φ x‖₊ ≤ N * ‖x‖₊) → M ≤ N) : ‖φ‖₊ = M 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {φ : E →SL[σ₁₂] F} (M : ℝ≥0) (h_above : ∀ (x : E), ‖φ x‖₊ ≤ M * ‖x‖₊) (h_below : ∀ (N : ℝ≥0), (∀ (x : E), ‖φ x‖₊ ≤ N * ‖x‖₊) → M ≤ N)
Goal: ‖φ‖₊ = M
-- ContinuousLinearMap.opNNNorm_le_iff
theorem opNNNorm_le_iff {f : E →SL[σ₁₂] F} {C : ℝ≥0} : ‖f‖₊ ≤ C ↔ ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F} {C : ℝ≥0}
Goal: ‖f‖₊ ≤ C ↔ ∀ (x : E), ‖f x‖₊ ≤ C * ‖x‖₊
-- ContinuousLinearMap.isLeast_opNNNorm
theorem isLeast_opNNNorm : IsLeast {C : ℝ≥0 | ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊} ‖f‖₊ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: IsLeast {C | ∀ (x : E), ‖f x‖₊ ≤ C * ‖x‖₊} ‖f‖₊
-- ContinuousLinearMap.opNNNorm_comp_le
theorem opNNNorm_comp_le [RingHomIsometric σ₁₃] (f : E →SL[σ₁₂] F) : ‖h.comp f‖₊ ≤ ‖h‖₊ * ‖f‖₊ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₁₂] [RingHomIsometric σ₂₃] (h : F →SL[σ₂₃] G) [RingHomIsometric σ₁₃] (f : E →SL[σ₁₂] F)
Goal: ‖h.comp f‖₊ ≤ ‖h‖₊ * ‖f‖₊
-- ContinuousLinearMap.le_opNNNorm
theorem le_opNNNorm : ‖f x‖₊ ≤ ‖f‖₊ * ‖x‖₊ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (x : E)
Goal: ‖f x‖₊ ≤ ‖f‖₊ * ‖x‖₊
-- ContinuousLinearMap.nndist_le_opNNNorm
theorem nndist_le_opNNNorm (x y : E) : nndist (f x) (f y) ≤ ‖f‖₊ * nndist x y 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (x y : E)
Goal: nndist (f x) (f y) ≤ ‖f‖₊ * nndist x y
-- ContinuousLinearMap.lipschitz
theorem lipschitz : LipschitzWith ‖f‖₊ f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: LipschitzWith ‖f‖₊ ⇑f
-- ContinuousLinearMap.lipschitz_apply
theorem lipschitz_apply (x : E) : LipschitzWith ‖x‖₊ fun f : E →SL[σ₁₂] F => f x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (x : E)
Goal: LipschitzWith ‖x‖₊ fun f => f x
-- ContinuousLinearMap.exists_mul_lt_apply_of_lt_opNNNorm
theorem exists_mul_lt_apply_of_lt_opNNNorm (f : E →SL[σ₁₂] F) {r : ℝ≥0} (hr : r < ‖f‖₊) :
    ∃ x, r * ‖x‖₊ < ‖f x‖₊ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ≥0} (hr : r < ‖f‖₊)
Goal: ∃ x, r * ‖x‖₊ < ‖f x‖₊
-- ContinuousLinearMap.exists_mul_lt_of_lt_opNorm
theorem exists_mul_lt_of_lt_opNorm (f : E →SL[σ₁₂] F) {r : ℝ} (hr₀ : 0 ≤ r) (hr : r < ‖f‖) :
    ∃ x, r * ‖x‖ < ‖f x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ} (hr₀ : 0 ≤ r) (hr : r < ‖f‖)
Goal: ∃ x, r * ‖x‖ < ‖f x‖
-- ContinuousLinearMap.exists_lt_apply_of_lt_opNNNorm
theorem exists_lt_apply_of_lt_opNNNorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ≥0}
    (hr : r < ‖f‖₊) : ∃ x : E, ‖x‖₊ < 1 ∧ r < ‖f x‖₊ 
Arguments: {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} [NormedAddCommGroup E] [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ≥0} (hr : r < ‖f‖₊)
Goal: ∃ x, ‖x‖₊ < 1 ∧ r < ‖f x‖₊
-- ContinuousLinearMap.exists_lt_apply_of_lt_opNorm
theorem exists_lt_apply_of_lt_opNorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ}
    (hr : r < ‖f‖) : ∃ x : E, ‖x‖ < 1 ∧ r < ‖f x‖ 
Arguments: {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} [NormedAddCommGroup E] [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ} (hr : r < ‖f‖)
Goal: ∃ x, ‖x‖ < 1 ∧ r < ‖f x‖
-- ContinuousLinearMap.sSup_unit_ball_eq_nnnorm
theorem sSup_unit_ball_eq_nnnorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    sSup ((fun x => ‖f x‖₊) '' ball 0 1) = ‖f‖₊ 
Arguments: {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} [NormedAddCommGroup E] [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: sSup ((fun x => ‖f x‖₊) '' Metric.ball 0 1) = ‖f‖₊
-- ContinuousLinearMap.sSup_unit_ball_eq_norm
theorem sSup_unit_ball_eq_norm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E] [SeminormedAddCommGroup F]
    [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedSpace 𝕜 E]
    [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    sSup ((fun x => ‖f x‖) '' ball 0 1) = ‖f‖ 
Arguments: {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} [NormedAddCommGroup E] [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: sSup ((fun x => ‖f x‖) '' Metric.ball 0 1) = ‖f‖
-- ContinuousLinearMap.sSup_closed_unit_ball_eq_nnnorm
theorem sSup_closed_unit_ball_eq_nnnorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    sSup ((fun x => ‖f x‖₊) '' closedBall 0 1) = ‖f‖₊ 
Arguments: {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} [NormedAddCommGroup E] [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: sSup ((fun x => ‖f x‖₊) '' Metric.closedBall 0 1) = ‖f‖₊
-- ContinuousLinearMap.sSup_closed_unit_ball_eq_norm
theorem sSup_closed_unit_ball_eq_norm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    sSup ((fun x => ‖f x‖) '' closedBall 0 1) = ‖f‖ 
Arguments: {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} [NormedAddCommGroup E] [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: sSup ((fun x => ‖f x‖) '' Metric.closedBall 0 1) = ‖f‖
-- ContinuousLinearEquiv.lipschitz
protected theorem lipschitz : LipschitzWith ‖(e : E →SL[σ₁₂] F)‖₊ e 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₁₂] (e : E ≃SL[σ₁₂] F)
Goal: LipschitzWith ‖↑e‖₊ ⇑e
-- ContinuousLinearMap.ofMemClosureImageCoeBounded_apply
@[simps! (config := .asFn) apply]
def ofMemClosureImageCoeBounded (f : E' → F) {s : Set (E' →SL[σ₁₂] F)} (hs : IsBounded s)
    (hf : f ∈ closure (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' s)) : E' →SL[σ₁₂] F 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [SeminormedAddCommGroup E'] [NormedSpace 𝕜 E'] [RingHomIsometric σ₁₂] (f : E' → F) {s : Set (E' →SL[σ₁₂] F)} (hs : Bornology.IsBounded s) (hf : f ∈ closure (DFunLike.coe '' s))
Goal: ⇑(ContinuousLinearMap.ofMemClosureImageCoeBounded f hs hf) = f
-- ContinuousLinearMap.ofTendstoOfBoundedRange_apply
@[simps! (config := .asFn) apply]
def ofTendstoOfBoundedRange {α : Type*} {l : Filter α} [l.NeBot] (f : E' → F)
    (g : α → E' →SL[σ₁₂] F) (hf : Tendsto (fun a x => g a x) l (𝓝 f))
    (hg : IsBounded (Set.range g)) : E' →SL[σ₁₂] F 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [SeminormedAddCommGroup E'] [NormedSpace 𝕜 E'] [RingHomIsometric σ₁₂] {α : Type u_12} {l : Filter α} [l.NeBot] (f : E' → F) (g : α → E' →SL[σ₁₂] F) (hf : Filter.Tendsto (fun a x => (g a) x) l (𝓝 f)) (hg : Bornology.IsBounded (Set.range g))
Goal: ⇑(ContinuousLinearMap.ofTendstoOfBoundedRange f g hf hg) = f
-- ContinuousLinearMap.tendsto_of_tendsto_pointwise_of_cauchySeq
theorem tendsto_of_tendsto_pointwise_of_cauchySeq {f : ℕ → E' →SL[σ₁₂] F} {g : E' →SL[σ₁₂] F}
    (hg : Tendsto (fun n x => f n x) atTop (𝓝 g)) (hf : CauchySeq f) : Tendsto f atTop (𝓝 g) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [SeminormedAddCommGroup E'] [NormedSpace 𝕜 E'] [RingHomIsometric σ₁₂] {f : ℕ → E' →SL[σ₁₂] F} {g : E' →SL[σ₁₂] F} (hg : Filter.Tendsto (fun n x => (f n) x) Filter.atTop (𝓝 ⇑g)) (hf : CauchySeq f)
Goal: Filter.Tendsto f Filter.atTop (𝓝 g)
-- ContinuousLinearMap.isCompact_closure_image_coe_of_bounded
theorem isCompact_closure_image_coe_of_bounded [ProperSpace F] {s : Set (E' →SL[σ₁₂] F)}
    (hb : IsBounded s) : IsCompact (closure (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' s)) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [SeminormedAddCommGroup E'] [NormedSpace 𝕜 E'] [RingHomIsometric σ₁₂] [ProperSpace F] {s : Set (E' →SL[σ₁₂] F)} (hb : Bornology.IsBounded s)
Goal: IsCompact (closure (DFunLike.coe '' s))
-- ContinuousLinearMap.isCompact_image_coe_of_bounded_of_closed_image
theorem isCompact_image_coe_of_bounded_of_closed_image [ProperSpace F] {s : Set (E' →SL[σ₁₂] F)}
    (hb : IsBounded s) (hc : IsClosed (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' s)) :
    IsCompact (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' s) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [SeminormedAddCommGroup E'] [NormedSpace 𝕜 E'] [RingHomIsometric σ₁₂] [ProperSpace F] {s : Set (E' →SL[σ₁₂] F)} (hb : Bornology.IsBounded s) (hc : IsClosed (DFunLike.coe '' s))
Goal: IsCompact (DFunLike.coe '' s)
-- ContinuousLinearMap.isClosed_image_coe_of_bounded_of_weak_closed
theorem isClosed_image_coe_of_bounded_of_weak_closed {s : Set (E' →SL[σ₁₂] F)} (hb : IsBounded s)
    (hc : ∀ f : E' →SL[σ₁₂] F,
      (⇑f : E' → F) ∈ closure (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' s) → f ∈ s) :
    IsClosed (((↑) : (E' →SL[σ₁₂] F) → E' → F) '' s) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [SeminormedAddCommGroup E'] [NormedSpace 𝕜 E'] [RingHomIsometric σ₁₂] {s : Set (E' →SL[σ₁₂] F)} (hb : Bornology.IsBounded s) (hc : ∀ (f : E' →SL[σ₁₂] F), ⇑f ∈ closure (DFunLike.coe '' s) → f ∈ s)
Goal: IsClosed (DFunLike.coe '' s)