ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.IsEquivalent.tendsto_const
theorem IsEquivalent.tendsto_const {c : β} (hu : u ~[l] const _ c) : Tendsto u l (𝓝 c) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u : α → β} {l : Filter α} {c : β} (hu : u ~[l] Function.const α c)
Goal: Filter.Tendsto u l (nhds c)
-- Asymptotics.IsEquivalent.tendsto_nhds
theorem IsEquivalent.tendsto_nhds {c : β} (huv : u ~[l] v) (hu : Tendsto u l (𝓝 c)) :
    Tendsto v l (𝓝 c) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} {c : β} (huv : u ~[l] v) (hu : Filter.Tendsto u l (nhds c))
Goal: Filter.Tendsto v l (nhds c)
-- Asymptotics.IsEquivalent.tendsto_nhds_iff
theorem IsEquivalent.tendsto_nhds_iff {c : β} (huv : u ~[l] v) :
    Tendsto u l (𝓝 c) ↔ Tendsto v l (𝓝 c) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} {c : β} (huv : u ~[l] v)
Goal: Filter.Tendsto u l (nhds c) ↔ Filter.Tendsto v l (nhds c)
-- Asymptotics.IsEquivalent.add_isLittleO
theorem IsEquivalent.add_isLittleO (huv : u ~[l] v) (hwv : w =o[l] v) : u + w ~[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v w : α → β} {l : Filter α} (huv : u ~[l] v) (hwv : w =o[l] v)
Goal: u + w ~[l] v
-- Asymptotics.IsEquivalent.sub_isLittleO
theorem IsEquivalent.sub_isLittleO (huv : u ~[l] v) (hwv : w =o[l] v) : u - w ~[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v w : α → β} {l : Filter α} (huv : u ~[l] v) (hwv : w =o[l] v)
Goal: u - w ~[l] v
-- Asymptotics.IsLittleO.add_isEquivalent
theorem IsLittleO.add_isEquivalent (hu : u =o[l] w) (hv : v ~[l] w) : u + v ~[l] w 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v w : α → β} {l : Filter α} (hu : u =o[l] w) (hv : v ~[l] w)
Goal: u + v ~[l] w
-- Asymptotics.IsLittleO.isEquivalent
theorem IsLittleO.isEquivalent (huv : (u - v) =o[l] v) : u ~[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} (huv : (u - v) =o[l] v)
Goal: u ~[l] v
-- Asymptotics.IsEquivalent.neg
theorem IsEquivalent.neg (huv : u ~[l] v) : (fun x ↦ -u x) ~[l] fun x ↦ -v x 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} (huv : u ~[l] v)
Goal: (fun x => -u x) ~[l] fun x => -v x
-- Asymptotics.isEquivalent_iff_exists_eq_mul
theorem isEquivalent_iff_exists_eq_mul :
    u ~[l] v ↔ ∃ (φ : α → β) (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField β] {u v : α → β} {l : Filter α}
Goal: u ~[l] v ↔ ∃ φ, ∃ (_ : Filter.Tendsto φ l (nhds 1)), u =ᶠ[l] φ * v
-- Asymptotics.IsEquivalent.exists_eq_mul
theorem IsEquivalent.exists_eq_mul (huv : u ~[l] v) :
    ∃ (φ : α → β) (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField β] {u v : α → β} {l : Filter α} (huv : u ~[l] v)
Goal: ∃ φ, ∃ (_ : Filter.Tendsto φ l (nhds 1)), u =ᶠ[l] φ * v
-- Asymptotics.isEquivalent_of_tendsto_one
theorem isEquivalent_of_tendsto_one (hz : ∀ᶠ x in l, v x = 0 → u x = 0)
    (huv : Tendsto (u / v) l (𝓝 1)) : u ~[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField β] {u v : α → β} {l : Filter α} (hz : ∀ᶠ (x : α) in l, v x = 0 → u x = 0) (huv : Filter.Tendsto (u / v) l (nhds 1))
Goal: u ~[l] v
-- Asymptotics.isEquivalent_of_tendsto_one'
theorem isEquivalent_of_tendsto_one' (hz : ∀ x, v x = 0 → u x = 0) (huv : Tendsto (u / v) l (𝓝 1)) :
    u ~[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField β] {u v : α → β} {l : Filter α} (hz : ∀ (x : α), v x = 0 → u x = 0) (huv : Filter.Tendsto (u / v) l (nhds 1))
Goal: u ~[l] v
-- Asymptotics.isEquivalent_iff_tendsto_one
theorem isEquivalent_iff_tendsto_one (hz : ∀ᶠ x in l, v x ≠ 0) :
    u ~[l] v ↔ Tendsto (u / v) l (𝓝 1) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField β] {u v : α → β} {l : Filter α} (hz : ∀ᶠ (x : α) in l, v x ≠ 0)
Goal: u ~[l] v ↔ Filter.Tendsto (u / v) l (nhds 1)
-- Asymptotics.IsEquivalent.smul
theorem IsEquivalent.smul {α E 𝕜 : Type*} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    {a b : α → 𝕜} {u v : α → E} {l : Filter α} (hab : a ~[l] b) (huv : u ~[l] v) :
    (fun x ↦ a x • u x) ~[l] fun x ↦ b x • v x 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_3} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {a b : α → 𝕜} {u v : α → E} {l : Filter α} (hab : a ~[l] b) (huv : u ~[l] v)
Goal: (fun x => a x • u x) ~[l] fun x => b x • v x
-- Asymptotics.IsEquivalent.mul
theorem IsEquivalent.mul (htu : t ~[l] u) (hvw : v ~[l] w) : t * v ~[l] u * w 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField β] {t u v w : α → β} {l : Filter α} (htu : t ~[l] u) (hvw : v ~[l] w)
Goal: t * v ~[l] u * w
-- Asymptotics.IsEquivalent.inv
theorem IsEquivalent.inv (huv : u ~[l] v) : (fun x ↦ (u x)⁻¹) ~[l] fun x ↦ (v x)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField β] {u v : α → β} {l : Filter α} (huv : u ~[l] v)
Goal: (fun x => (u x)⁻¹) ~[l] fun x => (v x)⁻¹
-- Asymptotics.IsEquivalent.div
theorem IsEquivalent.div (htu : t ~[l] u) (hvw : v ~[l] w) :
    (fun x ↦ t x / v x) ~[l] fun x ↦ u x / w x 
Arguments: {α : Type u_1} {β : Type u_2} [NormedField β] {t u v w : α → β} {l : Filter α} (htu : t ~[l] u) (hvw : v ~[l] w)
Goal: (fun x => t x / v x) ~[l] fun x => u x / w x
-- Asymptotics.IsEquivalent.tendsto_atTop
theorem IsEquivalent.tendsto_atTop [OrderTopology β] (huv : u ~[l] v) (hu : Tendsto u l atTop) :
    Tendsto v l atTop 
Arguments: {α : Type u_1} {β : Type u_2} [NormedLinearOrderedField β] {u v : α → β} {l : Filter α} [OrderTopology β] (huv : u ~[l] v) (hu : Filter.Tendsto u l Filter.atTop)
Goal: Filter.Tendsto v l Filter.atTop
-- Asymptotics.IsEquivalent.tendsto_atTop_iff
theorem IsEquivalent.tendsto_atTop_iff [OrderTopology β] (huv : u ~[l] v) :
    Tendsto u l atTop ↔ Tendsto v l atTop 
Arguments: {α : Type u_1} {β : Type u_2} [NormedLinearOrderedField β] {u v : α → β} {l : Filter α} [OrderTopology β] (huv : u ~[l] v)
Goal: Filter.Tendsto u l Filter.atTop ↔ Filter.Tendsto v l Filter.atTop
-- Asymptotics.IsEquivalent.tendsto_atBot
theorem IsEquivalent.tendsto_atBot [OrderTopology β] (huv : u ~[l] v) (hu : Tendsto u l atBot) :
    Tendsto v l atBot 
Arguments: {α : Type u_1} {β : Type u_2} [NormedLinearOrderedField β] {u v : α → β} {l : Filter α} [OrderTopology β] (huv : u ~[l] v) (hu : Filter.Tendsto u l Filter.atBot)
Goal: Filter.Tendsto v l Filter.atBot
-- Asymptotics.IsEquivalent.tendsto_atBot_iff
theorem IsEquivalent.tendsto_atBot_iff [OrderTopology β] (huv : u ~[l] v) :
    Tendsto u l atBot ↔ Tendsto v l atBot 
Arguments: {α : Type u_1} {β : Type u_2} [NormedLinearOrderedField β] {u v : α → β} {l : Filter α} [OrderTopology β] (huv : u ~[l] v)
Goal: Filter.Tendsto u l Filter.atBot ↔ Filter.Tendsto v l Filter.atBot
-- Filter.EventuallyEq.isEquivalent
theorem Filter.EventuallyEq.isEquivalent {u v : α → β} (h : u =ᶠ[l] v) : u ~[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {l : Filter α} {u v : α → β} (h : u =ᶠ[l] v)
Goal: u ~[l] v
-- Filter.EventuallyEq.trans_isEquivalent
@[trans]
theorem Filter.EventuallyEq.trans_isEquivalent {f g₁ g₂ : α → β} (h : f =ᶠ[l] g₁)
    (h₂ : g₁ ~[l] g₂) : f ~[l] g₂ 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {l : Filter α} {f g₁ g₂ : α → β} (h : f =ᶠ[l] g₁) (h₂ : g₁ ~[l] g₂)
Goal: f ~[l] g₂
-- Asymptotics.IsEquivalent.trans_eventuallyEq
@[trans]
theorem IsEquivalent.trans_eventuallyEq {f g₁ g₂ : α → β} (h : f ~[l] g₁)
    (h₂ : g₁ =ᶠ[l] g₂) : f ~[l] g₂ 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {l : Filter α} {f g₁ g₂ : α → β} (h : f ~[l] g₁) (h₂ : g₁ =ᶠ[l] g₂)
Goal: f ~[l] g₂
-- Asymptotics.IsEquivalent.trans_isBigO
@[trans]
theorem IsEquivalent.trans_isBigO {f g₁ : α → β} {g₂ : α → β₂} (h : f ~[l] g₁) (h₂ : g₁ =O[l] g₂) :
    f =O[l] g₂ 
Arguments: {α : Type u_1} {β : Type u_2} {β₂ : Type u_3} [NormedAddCommGroup β] [Norm β₂] {l : Filter α} {f g₁ : α → β} {g₂ : α → β₂} (h : f ~[l] g₁) (h₂ : g₁ =O[l] g₂)
Goal: f =O[l] g₂
-- Asymptotics.IsBigO.trans_isEquivalent
@[trans]
theorem IsBigO.trans_isEquivalent {f : α → β₂} {g₁ g₂ : α → β} (h : f =O[l] g₁) (h₂ : g₁ ~[l] g₂) :
    f =O[l] g₂ 
Arguments: {α : Type u_1} {β : Type u_2} {β₂ : Type u_3} [NormedAddCommGroup β] [Norm β₂] {l : Filter α} {f : α → β₂} {g₁ g₂ : α → β} (h : f =O[l] g₁) (h₂ : g₁ ~[l] g₂)
Goal: f =O[l] g₂
-- Asymptotics.IsEquivalent.trans_isLittleO
@[trans]
theorem IsEquivalent.trans_isLittleO {f g₁ : α → β} {g₂ : α → β₂} (h : f ~[l] g₁)
    (h₂ : g₁ =o[l] g₂) : f =o[l] g₂ 
Arguments: {α : Type u_1} {β : Type u_2} {β₂ : Type u_3} [NormedAddCommGroup β] [Norm β₂] {l : Filter α} {f g₁ : α → β} {g₂ : α → β₂} (h : f ~[l] g₁) (h₂ : g₁ =o[l] g₂)
Goal: f =o[l] g₂
-- Asymptotics.IsLittleO.trans_isEquivalent
@[trans]
theorem IsLittleO.trans_isEquivalent {f : α → β₂} {g₁ g₂ : α → β} (h : f =o[l] g₁)
    (h₂ : g₁ ~[l] g₂) : f =o[l] g₂ 
Arguments: {α : Type u_1} {β : Type u_2} {β₂ : Type u_3} [NormedAddCommGroup β] [Norm β₂] {l : Filter α} {f : α → β₂} {g₁ g₂ : α → β} (h : f =o[l] g₁) (h₂ : g₁ ~[l] g₂)
Goal: f =o[l] g₂
-- Asymptotics.IsEquivalent.trans_isTheta
@[trans]
theorem IsEquivalent.trans_isTheta {f g₁ : α → β} {g₂ : α → β₂} (h : f ~[l] g₁)
    (h₂ : g₁ =Θ[l] g₂) : f =Θ[l] g₂ 
Arguments: {α : Type u_1} {β : Type u_2} {β₂ : Type u_3} [NormedAddCommGroup β] [Norm β₂] {l : Filter α} {f g₁ : α → β} {g₂ : α → β₂} (h : f ~[l] g₁) (h₂ : g₁ =Θ[l] g₂)
Goal: f =Θ[l] g₂
-- Asymptotics.IsTheta.trans_isEquivalent
@[trans]
theorem IsTheta.trans_isEquivalent {f : α → β₂} {g₁ g₂ : α → β} (h : f =Θ[l] g₁)
    (h₂ : g₁ ~[l] g₂) : f =Θ[l] g₂ 
Arguments: {α : Type u_1} {β : Type u_2} {β₂ : Type u_3} [NormedAddCommGroup β] [Norm β₂] {l : Filter α} {f : α → β₂} {g₁ g₂ : α → β} (h : f =Θ[l] g₁) (h₂ : g₁ ~[l] g₂)
Goal: f =Θ[l] g₂
-- SameRay.norm_add
theorem norm_add (h : SameRay ℝ x y) : ‖x + y‖ = ‖x‖ + ‖y‖ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y : E} (h : SameRay ℝ x y)
Goal: ‖x + y‖ = ‖x‖ + ‖y‖
-- SameRay.norm_sub
theorem norm_sub (h : SameRay ℝ x y) : ‖x - y‖ = |‖x‖ - ‖y‖| 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y : E} (h : SameRay ℝ x y)
Goal: ‖x - y‖ = |‖x‖ - ‖y‖|
-- SameRay.norm_smul_eq
theorem norm_smul_eq (h : SameRay ℝ x y) : ‖x‖ • y = ‖y‖ • x 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y : E} (h : SameRay ℝ x y)
Goal: ‖x‖ • y = ‖y‖ • x
-- norm_injOn_ray_left
theorem norm_injOn_ray_left (hx : x ≠ 0) : { y | SameRay ℝ x y }.InjOn norm 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x : F} (hx : x ≠ 0)
Goal: Set.InjOn norm {y | SameRay ℝ x y}
-- norm_injOn_ray_right
theorem norm_injOn_ray_right (hy : y ≠ 0) : { x | SameRay ℝ x y }.InjOn norm 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {y : F} (hy : y ≠ 0)
Goal: Set.InjOn norm {x | SameRay ℝ x y}
-- sameRay_iff_norm_smul_eq
theorem sameRay_iff_norm_smul_eq : SameRay ℝ x y ↔ ‖x‖ • y = ‖y‖ • x 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x y : F}
Goal: SameRay ℝ x y ↔ ‖x‖ • y = ‖y‖ • x
-- sameRay_iff_inv_norm_smul_eq_of_ne
theorem sameRay_iff_inv_norm_smul_eq_of_ne (hx : x ≠ 0) (hy : y ≠ 0) :
    SameRay ℝ x y ↔ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x y : F} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: SameRay ℝ x y ↔ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y
-- sameRay_iff_inv_norm_smul_eq
theorem sameRay_iff_inv_norm_smul_eq : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x y : F}
Goal: SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y
-- sameRay_iff_of_norm_eq
theorem sameRay_iff_of_norm_eq (h : ‖x‖ = ‖y‖) : SameRay ℝ x y ↔ x = y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x y : F} (h : ‖x‖ = ‖y‖)
Goal: SameRay ℝ x y ↔ x = y
-- not_sameRay_iff_of_norm_eq
theorem not_sameRay_iff_of_norm_eq (h : ‖x‖ = ‖y‖) : ¬SameRay ℝ x y ↔ x ≠ y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x y : F} (h : ‖x‖ = ‖y‖)
Goal: ¬SameRay ℝ x y ↔ x ≠ y
-- SameRay.eq_of_norm_eq
theorem SameRay.eq_of_norm_eq (h : SameRay ℝ x y) (hn : ‖x‖ = ‖y‖) : x = y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x y : F} (h : SameRay ℝ x y) (hn : ‖x‖ = ‖y‖)
Goal: x = y
-- SameRay.norm_eq_iff
theorem SameRay.norm_eq_iff (h : SameRay ℝ x y) : ‖x‖ = ‖y‖ ↔ x = y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {x y : F} (h : SameRay ℝ x y)
Goal: ‖x‖ = ‖y‖ ↔ x = y
-- LinearMap.toSpanSingleton_homothety
theorem toSpanSingleton_homothety (x : E) (c : 𝕜) :
    ‖LinearMap.toSpanSingleton 𝕜 E x c‖ = ‖x‖ * ‖c‖ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (x : E) (c : 𝕜)
Goal: ‖(LinearMap.toSpanSingleton 𝕜 E x) c‖ = ‖x‖ * ‖c‖
-- LinearEquiv.toSpanNonzeroSingleton_homothety
theorem _root_.LinearEquiv.toSpanNonzeroSingleton_homothety (x : E) (h : x ≠ 0) (c : 𝕜) :
    ‖LinearEquiv.toSpanNonzeroSingleton 𝕜 E x h c‖ = ‖x‖ * ‖c‖ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (x : E) (h : x ≠ 0) (c : 𝕜)
Goal: ‖(LinearEquiv.toSpanNonzeroSingleton 𝕜 E x h) c‖ = ‖x‖ * ‖c‖
-- ContinuousLinearEquiv.coe_toSpanNonzeroSingleton_symm
@[simp]
theorem coe_toSpanNonzeroSingleton_symm {x : E} (h : x ≠ 0) :
    ⇑(toSpanNonzeroSingleton 𝕜 x h).symm = coord 𝕜 x h 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} (h : x ≠ 0)
Goal: ⇑(ContinuousLinearEquiv.toSpanNonzeroSingleton 𝕜 x h).symm = ⇑(ContinuousLinearEquiv.coord 𝕜 x h)
-- ContinuousLinearEquiv.coord_toSpanNonzeroSingleton
@[simp]
theorem coord_toSpanNonzeroSingleton {x : E} (h : x ≠ 0) (c : 𝕜) :
    coord 𝕜 x h (toSpanNonzeroSingleton 𝕜 x h c) = c 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} (h : x ≠ 0) (c : 𝕜)
Goal: (ContinuousLinearEquiv.coord 𝕜 x h) ((ContinuousLinearEquiv.toSpanNonzeroSingleton 𝕜 x h) c) = c
-- ContinuousLinearEquiv.toSpanNonzeroSingleton_coord
@[simp]
theorem toSpanNonzeroSingleton_coord {x : E} (h : x ≠ 0) (y : 𝕜 ∙ x) :
    toSpanNonzeroSingleton 𝕜 x h (coord 𝕜 x h y) = y 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} (h : x ≠ 0) (y : ↥(Submodule.span 𝕜 {x}))
Goal: (ContinuousLinearEquiv.toSpanNonzeroSingleton 𝕜 x h) ((ContinuousLinearEquiv.coord 𝕜 x h) y) = y
-- ContinuousLinearEquiv.coord_self
@[simp]
theorem coord_self (x : E) (h : x ≠ 0) :
    (coord 𝕜 x h) (⟨x, Submodule.mem_span_singleton_self x⟩ : 𝕜 ∙ x) = 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) (h : x ≠ 0)
Goal: (ContinuousLinearEquiv.coord 𝕜 x h) ⟨x, ⋯⟩ = 1
-- LinearIsometryEquiv.toSpanUnitSingleton_apply
@[simp] theorem toSpanUnitSingleton_apply (x : E) (hx : ‖x‖ = 1) (r : 𝕜) :
    toSpanUnitSingleton x hx r = (⟨r • x, by aesop⟩ : 𝕜 ∙ x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (x : E) (hx : ‖x‖ = 1) (r : 𝕜)
Goal: (LinearIsometryEquiv.toSpanUnitSingleton x hx) r = ⟨r • x, ⋯⟩
-- UniformSpace.Completion.coe_toComplₗᵢ
@[simp]
theorem coe_toComplₗᵢ : ⇑(toComplₗᵢ : E →ₗᵢ[𝕜] Completion E) = ((↑) : E → Completion E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: ⇑UniformSpace.Completion.toComplₗᵢ = ↑E
-- UniformSpace.Completion.coe_toComplL
@[simp]
theorem coe_toComplL : ⇑(toComplL : E →L[𝕜] Completion E) = ((↑) : E → Completion E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: ⇑UniformSpace.Completion.toComplL = ↑E
-- UniformSpace.Completion.norm_toComplL
@[simp]
theorem norm_toComplL {𝕜 E : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E]
    [NormedSpace 𝕜 E] [Nontrivial E] : ‖(toComplL : E →L[𝕜] Completion E)‖ = 1 
Arguments: {𝕜 : Type u_3} {E : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [Nontrivial E]
Goal: ‖UniformSpace.Completion.toComplL‖ = 1
-- WithLp.equiv_zero
@[simp]
theorem equiv_zero : WithLp.equiv p V 0 = 0 
Arguments: (p : ℝ≥0∞) {V : Type uV} [AddCommGroup V]
Goal: (WithLp.equiv p V) 0 = 0
-- WithLp.equiv_symm_zero
@[simp]
theorem equiv_symm_zero : (WithLp.equiv p V).symm 0 = 0 
Arguments: (p : ℝ≥0∞) {V : Type uV} [AddCommGroup V]
Goal: (WithLp.equiv p V).symm 0 = 0
-- WithLp.equiv_add
@[simp]
theorem equiv_add : WithLp.equiv p V (x + y) = WithLp.equiv p V x + WithLp.equiv p V y 
Arguments: (p : ℝ≥0∞) {V : Type uV} [AddCommGroup V] (x y : WithLp p V)
Goal: (WithLp.equiv p V) (x + y) = (WithLp.equiv p V) x + (WithLp.equiv p V) y
-- WithLp.equiv_symm_add
@[simp]
theorem equiv_symm_add :
    (WithLp.equiv p V).symm (x' + y') = (WithLp.equiv p V).symm x' + (WithLp.equiv p V).symm y' 
Arguments: (p : ℝ≥0∞) {V : Type uV} [AddCommGroup V] (x' y' : V)
Goal: (WithLp.equiv p V).symm (x' + y') = (WithLp.equiv p V).symm x' + (WithLp.equiv p V).symm y'
-- WithLp.equiv_sub
@[simp]
theorem equiv_sub : WithLp.equiv p V (x - y) = WithLp.equiv p V x - WithLp.equiv p V y 
Arguments: (p : ℝ≥0∞) {V : Type uV} [AddCommGroup V] (x y : WithLp p V)
Goal: (WithLp.equiv p V) (x - y) = (WithLp.equiv p V) x - (WithLp.equiv p V) y
-- WithLp.equiv_symm_sub
@[simp]
theorem equiv_symm_sub :
    (WithLp.equiv p V).symm (x' - y') = (WithLp.equiv p V).symm x' - (WithLp.equiv p V).symm y' 
Arguments: (p : ℝ≥0∞) {V : Type uV} [AddCommGroup V] (x' y' : V)
Goal: (WithLp.equiv p V).symm (x' - y') = (WithLp.equiv p V).symm x' - (WithLp.equiv p V).symm y'
-- WithLp.equiv_neg
@[simp]
theorem equiv_neg : WithLp.equiv p V (-x) = -WithLp.equiv p V x 
Arguments: (p : ℝ≥0∞) {V : Type uV} [AddCommGroup V] (x : WithLp p V)
Goal: (WithLp.equiv p V) (-x) = -(WithLp.equiv p V) x
-- WithLp.equiv_symm_neg
@[simp]
theorem equiv_symm_neg : (WithLp.equiv p V).symm (-x') = -(WithLp.equiv p V).symm x' 
Arguments: (p : ℝ≥0∞) {V : Type uV} [AddCommGroup V] (x' : V)
Goal: (WithLp.equiv p V).symm (-x') = -(WithLp.equiv p V).symm x'
-- WithLp.equiv_smul
@[simp]
theorem equiv_smul : WithLp.equiv p V (c • x) = c • WithLp.equiv p V x 
Arguments: (p : ℝ≥0∞) {K : Type uK} {V : Type uV} [Semiring K] [AddCommGroup V] [Module K V] (c : K) (x : WithLp p V)
Goal: (WithLp.equiv p V) (c • x) = c • (WithLp.equiv p V) x
-- WithLp.equiv_symm_smul
@[simp]
theorem equiv_symm_smul : (WithLp.equiv p V).symm (c • x') = c • (WithLp.equiv p V).symm x' 
Arguments: (p : ℝ≥0∞) {K : Type uK} {V : Type uV} [Semiring K] [AddCommGroup V] [Module K V] (c : K) (x' : V)
Goal: (WithLp.equiv p V).symm (c • x') = c • (WithLp.equiv p V).symm x'
-- WithLp.linearEquiv_symm_apply
@[simps (config := .asFn)]
protected def linearEquiv : WithLp p V ≃ₗ[K] V 
Arguments: (p : ℝ≥0∞) (K : Type uK) (V : Type uV) [Semiring K] [AddCommGroup V] [Module K V]
Goal: ⇑(WithLp.linearEquiv p K V).symm = ⇑(WithLp.equiv p V).symm
-- WithLp.linearEquiv_apply
@[simps (config := .asFn)]
protected def linearEquiv : WithLp p V ≃ₗ[K] V 
Arguments: (p : ℝ≥0∞) (K : Type uK) (V : Type uV) [Semiring K] [AddCommGroup V] [Module K V]
Goal: ⇑(WithLp.linearEquiv p K V) = ⇑(WithLp.equiv p V)
-- IsUpperSet.thickening'
@[to_additive IsUpperSet.thickening]
protected theorem IsUpperSet.thickening' (hs : IsUpperSet s) (ε : ℝ) :
    IsUpperSet (thickening ε s) 
Arguments: {α : Type u_1} [NormedOrderedGroup α] {s : Set α} (hs : IsUpperSet s) (ε : ℝ)
Goal: IsUpperSet (Metric.thickening ε s)
-- IsUpperSet.thickening
@[to_additive IsUpperSet.thickening]
protected theorem IsUpperSet.thickening' (hs : IsUpperSet s) (ε : ℝ) :
    IsUpperSet (thickening ε s) 
Arguments: {α : Type u_1} [NormedOrderedAddGroup α] {s : Set α} (hs : IsUpperSet s) (ε : ℝ)
Goal: IsUpperSet (Metric.thickening ε s)
-- IsLowerSet.thickening'
@[to_additive IsLowerSet.thickening]
protected theorem IsLowerSet.thickening' (hs : IsLowerSet s) (ε : ℝ) :
    IsLowerSet (thickening ε s) 
Arguments: {α : Type u_1} [NormedOrderedGroup α] {s : Set α} (hs : IsLowerSet s) (ε : ℝ)
Goal: IsLowerSet (Metric.thickening ε s)
-- IsLowerSet.thickening
@[to_additive IsLowerSet.thickening]
protected theorem IsLowerSet.thickening' (hs : IsLowerSet s) (ε : ℝ) :
    IsLowerSet (thickening ε s) 
Arguments: {α : Type u_1} [NormedOrderedAddGroup α] {s : Set α} (hs : IsLowerSet s) (ε : ℝ)
Goal: IsLowerSet (Metric.thickening ε s)
-- IsUpperSet.cthickening
@[to_additive IsUpperSet.cthickening]
protected theorem IsUpperSet.cthickening' (hs : IsUpperSet s) (ε : ℝ) :
    IsUpperSet (cthickening ε s) 
Arguments: {α : Type u_1} [NormedOrderedAddGroup α] {s : Set α} (hs : IsUpperSet s) (ε : ℝ)
Goal: IsUpperSet (Metric.cthickening ε s)
-- IsUpperSet.cthickening'
@[to_additive IsUpperSet.cthickening]
protected theorem IsUpperSet.cthickening' (hs : IsUpperSet s) (ε : ℝ) :
    IsUpperSet (cthickening ε s) 
Arguments: {α : Type u_1} [NormedOrderedGroup α] {s : Set α} (hs : IsUpperSet s) (ε : ℝ)
Goal: IsUpperSet (Metric.cthickening ε s)
-- IsLowerSet.cthickening
@[to_additive IsLowerSet.cthickening]
protected theorem IsLowerSet.cthickening' (hs : IsLowerSet s) (ε : ℝ) :
    IsLowerSet (cthickening ε s) 
Arguments: {α : Type u_1} [NormedOrderedAddGroup α] {s : Set α} (hs : IsLowerSet s) (ε : ℝ)
Goal: IsLowerSet (Metric.cthickening ε s)
-- IsLowerSet.cthickening'
@[to_additive IsLowerSet.cthickening]
protected theorem IsLowerSet.cthickening' (hs : IsLowerSet s) (ε : ℝ) :
    IsLowerSet (cthickening ε s) 
Arguments: {α : Type u_1} [NormedOrderedGroup α] {s : Set α} (hs : IsLowerSet s) (ε : ℝ)
Goal: IsLowerSet (Metric.cthickening ε s)
-- upperClosure_interior_subset
@[to_additive upperClosure_interior_subset] lemma upperClosure_interior_subset' (s : Set α) :
    (upperClosure (interior s) : Set α) ⊆ interior (upperClosure s) 
Arguments: {α : Type u_1} [NormedOrderedAddGroup α] (s : Set α)
Goal: ↑(upperClosure (interior s)) ⊆ interior ↑(upperClosure s)
-- upperClosure_interior_subset'
@[to_additive upperClosure_interior_subset] lemma upperClosure_interior_subset' (s : Set α) :
    (upperClosure (interior s) : Set α) ⊆ interior (upperClosure s) 
Arguments: {α : Type u_1} [NormedOrderedGroup α] (s : Set α)
Goal: ↑(upperClosure (interior s)) ⊆ interior ↑(upperClosure s)
-- lowerClosure_interior_subset
@[to_additive lowerClosure_interior_subset] lemma lowerClosure_interior_subset' (s : Set α) :
    (lowerClosure (interior s) : Set α) ⊆ interior (lowerClosure s) 
Arguments: {α : Type u_1} [NormedOrderedAddGroup α] (s : Set α)
Goal: ↑(lowerClosure (interior s)) ⊆ interior ↑(lowerClosure s)
-- lowerClosure_interior_subset'
@[to_additive lowerClosure_interior_subset] lemma lowerClosure_interior_subset' (s : Set α) :
    (lowerClosure (interior s) : Set α) ⊆ interior (lowerClosure s) 
Arguments: {α : Type u_1} [NormedOrderedGroup α] (s : Set α)
Goal: ↑(lowerClosure (interior s)) ⊆ interior ↑(lowerClosure s)
-- IsUpperSet.mem_interior_of_forall_lt
theorem IsUpperSet.mem_interior_of_forall_lt (hs : IsUpperSet s) (hx : x ∈ closure s)
    (h : ∀ i, x i < y i) : y ∈ interior s 
Arguments: {ι : Type u_2} [Finite ι] {s : Set (ι → ℝ)} {x y : ι → ℝ} (hs : IsUpperSet s) (hx : x ∈ closure s) (h : ∀ (i : ι), x i < y i)
Goal: y ∈ interior s
-- IsLowerSet.mem_interior_of_forall_lt
theorem IsLowerSet.mem_interior_of_forall_lt (hs : IsLowerSet s) (hx : x ∈ closure s)
    (h : ∀ i, y i < x i) : y ∈ interior s 
Arguments: {ι : Type u_2} [Finite ι] {s : Set (ι → ℝ)} {x y : ι → ℝ} (hs : IsLowerSet s) (hx : x ∈ closure s) (h : ∀ (i : ι), y i < x i)
Goal: y ∈ interior s
-- dist_inf_sup_pi
lemma dist_inf_sup_pi (x y : ι → ℝ) : dist (x ⊓ y) (x ⊔ y) = dist x y 
Arguments: {ι : Type u_2} [Fintype ι] (x y : ι → ℝ)
Goal: dist (x ⊓ y) (x ⊔ y) = dist x y
-- dist_mono_left_pi
lemma dist_mono_left_pi : MonotoneOn (dist · y) (Ici y) 
Arguments: {ι : Type u_2} [Fintype ι] {y : ι → ℝ}
Goal: MonotoneOn (fun x => dist x y) (Set.Ici y)
-- dist_mono_right_pi
lemma dist_mono_right_pi : MonotoneOn (dist x) (Ici x) 
Arguments: {ι : Type u_2} [Fintype ι] {x : ι → ℝ}
Goal: MonotoneOn (dist x) (Set.Ici x)
-- dist_anti_left_pi
lemma dist_anti_left_pi : AntitoneOn (dist · y) (Iic y) 
Arguments: {ι : Type u_2} [Fintype ι] {y : ι → ℝ}
Goal: AntitoneOn (fun x => dist x y) (Set.Iic y)