ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- balanced_ball_zero
theorem balanced_ball_zero : Balanced 𝕜 (Metric.ball (0 : E) r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
Goal: Balanced 𝕜 (Metric.ball 0 r)
-- rescale_to_shell_semi_normed_zpow
lemma rescale_to_shell_semi_normed_zpow {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E}
    (hx : ‖x‖ ≠ 0) :
    ∃ n : ℤ, c^n ≠ 0 ∧ ‖c^n • x‖ < ε ∧ (ε / ‖c‖ ≤ ‖c^n • x‖) ∧ (‖c^n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : ‖x‖ ≠ 0)
Goal: ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖
-- rescale_to_shell_semi_normed
lemma rescale_to_shell_semi_normed {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε)
    {x : E} (hx : ‖x‖ ≠ 0) :
    ∃d : 𝕜, d ≠ 0 ∧ ‖d • x‖ < ε ∧ (ε/‖c‖ ≤ ‖d • x‖) ∧ (‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : ‖x‖ ≠ 0)
Goal: ∃ d, d ≠ 0 ∧ ‖d • x‖ < ε ∧ ε / ‖c‖ ≤ ‖d • x‖ ∧ ‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖
-- rescale_to_shell_zpow
lemma rescale_to_shell_zpow [NormedAddCommGroup F] [NormedSpace 𝕜 F] {c : 𝕜} (hc : 1 < ‖c‖)
    {ε : ℝ} (εpos : 0 < ε) {x : F} (hx : x ≠ 0) :
    ∃ n : ℤ, c^n ≠ 0 ∧ ‖c^n • x‖ < ε ∧ (ε / ‖c‖ ≤ ‖c^n • x‖) ∧ (‖c^n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖) 
Arguments: {𝕜 : Type u_3} {F : Type u_10} [NormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : F} (hx : x ≠ 0)
Goal: ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖
-- rescale_to_shell
lemma rescale_to_shell [NormedAddCommGroup F] [NormedSpace 𝕜 F] {c : 𝕜} (hc : 1 < ‖c‖)
    {ε : ℝ} (εpos : 0 < ε) {x : F} (hx : x ≠ 0) :
    ∃d : 𝕜, d ≠ 0 ∧ ‖d • x‖ < ε ∧ (ε/‖c‖ ≤ ‖d • x‖) ∧ (‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖) 
Arguments: {𝕜 : Type u_3} {F : Type u_10} [NormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : F} (hx : x ≠ 0)
Goal: ∃ d, d ≠ 0 ∧ ‖d • x‖ < ε ∧ ε / ‖c‖ ≤ ‖d • x‖ ∧ ‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖
-- QuasiconvexOn.dual
theorem QuasiconvexOn.dual : QuasiconvexOn 𝕜 s f → QuasiconcaveOn 𝕜 s (toDual ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LE β] [SMul 𝕜 E] {s : Set E} {f : E → β}
Goal: QuasiconvexOn 𝕜 s f → QuasiconcaveOn 𝕜 s (⇑OrderDual.toDual ∘ f)
-- QuasiconcaveOn.dual
theorem QuasiconcaveOn.dual : QuasiconcaveOn 𝕜 s f → QuasiconvexOn 𝕜 s (toDual ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LE β] [SMul 𝕜 E] {s : Set E} {f : E → β}
Goal: QuasiconcaveOn 𝕜 s f → QuasiconvexOn 𝕜 s (⇑OrderDual.toDual ∘ f)
-- QuasilinearOn.dual
theorem QuasilinearOn.dual : QuasilinearOn 𝕜 s f → QuasilinearOn 𝕜 s (toDual ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LE β] [SMul 𝕜 E] {s : Set E} {f : E → β}
Goal: QuasilinearOn 𝕜 s f → QuasilinearOn 𝕜 s (⇑OrderDual.toDual ∘ f)
-- Convex.quasiconvexOn_of_convex_le
theorem Convex.quasiconvexOn_of_convex_le (hs : Convex 𝕜 s) (h : ∀ r, Convex 𝕜 { x | f x ≤ r }) :
    QuasiconvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LE β] [SMul 𝕜 E] {s : Set E} {f : E → β} (hs : Convex 𝕜 s) (h : ∀ (r : β), Convex 𝕜 {x | f x ≤ r})
Goal: QuasiconvexOn 𝕜 s f
-- Convex.quasiconcaveOn_of_convex_ge
theorem Convex.quasiconcaveOn_of_convex_ge (hs : Convex 𝕜 s) (h : ∀ r, Convex 𝕜 { x | r ≤ f x }) :
    QuasiconcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LE β] [SMul 𝕜 E] {s : Set E} {f : E → β} (hs : Convex 𝕜 s) (h : ∀ (r : β), Convex 𝕜 {x | r ≤ f x})
Goal: QuasiconcaveOn 𝕜 s f
-- QuasiconvexOn.convex
theorem QuasiconvexOn.convex [IsDirected β (· ≤ ·)] (hf : QuasiconvexOn 𝕜 s f) : Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LE β] [SMul 𝕜 E] {s : Set E} {f : E → β} [IsDirected β fun x x_1 => x ≤ x_1] (hf : QuasiconvexOn 𝕜 s f)
Goal: Convex 𝕜 s
-- QuasiconcaveOn.convex
theorem QuasiconcaveOn.convex [IsDirected β (· ≥ ·)] (hf : QuasiconcaveOn 𝕜 s f) : Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LE β] [SMul 𝕜 E] {s : Set E} {f : E → β} [IsDirected β fun x x_1 => x ≥ x_1] (hf : QuasiconcaveOn 𝕜 s f)
Goal: Convex 𝕜 s
-- QuasiconvexOn.sup
theorem QuasiconvexOn.sup [SemilatticeSup β] (hf : QuasiconvexOn 𝕜 s f)
    (hg : QuasiconvexOn 𝕜 s g) : QuasiconvexOn 𝕜 s (f ⊔ g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {f g : E → β} [SemilatticeSup β] (hf : QuasiconvexOn 𝕜 s f) (hg : QuasiconvexOn 𝕜 s g)
Goal: QuasiconvexOn 𝕜 s (f ⊔ g)
-- QuasiconcaveOn.inf
theorem QuasiconcaveOn.inf [SemilatticeInf β] (hf : QuasiconcaveOn 𝕜 s f)
    (hg : QuasiconcaveOn 𝕜 s g) : QuasiconcaveOn 𝕜 s (f ⊓ g) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {f g : E → β} [SemilatticeInf β] (hf : QuasiconcaveOn 𝕜 s f) (hg : QuasiconcaveOn 𝕜 s g)
Goal: QuasiconcaveOn 𝕜 s (f ⊓ g)
-- quasiconvexOn_iff_le_max
theorem quasiconvexOn_iff_le_max : QuasiconvexOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄,
    y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ max (f x) (f y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LinearOrder β] [SMul 𝕜 E] {s : Set E} {f : E → β}
Goal: QuasiconvexOn 𝕜 s f ↔
  Convex 𝕜 s ∧
    ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ max (f x) (f y)
-- quasiconcaveOn_iff_min_le
theorem quasiconcaveOn_iff_min_le : QuasiconcaveOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄,
    y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → min (f x) (f y) ≤ f (a • x + b • y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LinearOrder β] [SMul 𝕜 E] {s : Set E} {f : E → β}
Goal: QuasiconcaveOn 𝕜 s f ↔
  Convex 𝕜 s ∧
    ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → min (f x) (f y) ≤ f (a • x + b • y)
-- quasilinearOn_iff_mem_uIcc
theorem quasilinearOn_iff_mem_uIcc : QuasilinearOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄,
    y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ uIcc (f x) (f y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LinearOrder β] [SMul 𝕜 E] {s : Set E} {f : E → β}
Goal: QuasilinearOn 𝕜 s f ↔
  Convex 𝕜 s ∧
    ∀ ⦃x : E⦄,
      x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ Set.uIcc (f x) (f y)
-- QuasiconvexOn.convex_lt
theorem QuasiconvexOn.convex_lt (hf : QuasiconvexOn 𝕜 s f) (r : β) :
    Convex 𝕜 ({ x ∈ s | f x < r }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LinearOrder β] [SMul 𝕜 E] {s : Set E} {f : E → β} (hf : QuasiconvexOn 𝕜 s f) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ f x < r}
-- QuasiconcaveOn.convex_gt
theorem QuasiconcaveOn.convex_gt (hf : QuasiconcaveOn 𝕜 s f) (r : β) :
    Convex 𝕜 ({ x ∈ s | r < f x }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [LinearOrder β] [SMul 𝕜 E] {s : Set E} {f : E → β} (hf : QuasiconcaveOn 𝕜 s f) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ r < f x}
-- ConvexOn.quasiconvexOn
theorem ConvexOn.quasiconvexOn (hf : ConvexOn 𝕜 s f) : QuasiconvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f)
Goal: QuasiconvexOn 𝕜 s f
-- ConcaveOn.quasiconcaveOn
theorem ConcaveOn.quasiconcaveOn (hf : ConcaveOn 𝕜 s f) : QuasiconcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f)
Goal: QuasiconcaveOn 𝕜 s f
-- MonotoneOn.quasiconvexOn
theorem MonotoneOn.quasiconvexOn (hf : MonotoneOn f s) (hs : Convex 𝕜 s) : QuasiconvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : MonotoneOn f s) (hs : Convex 𝕜 s)
Goal: QuasiconvexOn 𝕜 s f
-- MonotoneOn.quasiconcaveOn
theorem MonotoneOn.quasiconcaveOn (hf : MonotoneOn f s) (hs : Convex 𝕜 s) : QuasiconcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : MonotoneOn f s) (hs : Convex 𝕜 s)
Goal: QuasiconcaveOn 𝕜 s f
-- MonotoneOn.quasilinearOn
theorem MonotoneOn.quasilinearOn (hf : MonotoneOn f s) (hs : Convex 𝕜 s) : QuasilinearOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : MonotoneOn f s) (hs : Convex 𝕜 s)
Goal: QuasilinearOn 𝕜 s f
-- AntitoneOn.quasiconvexOn
theorem AntitoneOn.quasiconvexOn (hf : AntitoneOn f s) (hs : Convex 𝕜 s) : QuasiconvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : AntitoneOn f s) (hs : Convex 𝕜 s)
Goal: QuasiconvexOn 𝕜 s f
-- AntitoneOn.quasiconcaveOn
theorem AntitoneOn.quasiconcaveOn (hf : AntitoneOn f s) (hs : Convex 𝕜 s) : QuasiconcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : AntitoneOn f s) (hs : Convex 𝕜 s)
Goal: QuasiconcaveOn 𝕜 s f
-- AntitoneOn.quasilinearOn
theorem AntitoneOn.quasilinearOn (hf : AntitoneOn f s) (hs : Convex 𝕜 s) : QuasilinearOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : AntitoneOn f s) (hs : Convex 𝕜 s)
Goal: QuasilinearOn 𝕜 s f
-- Monotone.quasiconvexOn
theorem Monotone.quasiconvexOn (hf : Monotone f) : QuasiconvexOn 𝕜 univ f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Monotone f)
Goal: QuasiconvexOn 𝕜 Set.univ f
-- Monotone.quasiconcaveOn
theorem Monotone.quasiconcaveOn (hf : Monotone f) : QuasiconcaveOn 𝕜 univ f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Monotone f)
Goal: QuasiconcaveOn 𝕜 Set.univ f
-- Monotone.quasilinearOn
theorem Monotone.quasilinearOn (hf : Monotone f) : QuasilinearOn 𝕜 univ f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Monotone f)
Goal: QuasilinearOn 𝕜 Set.univ f
-- Antitone.quasiconvexOn
theorem Antitone.quasiconvexOn (hf : Antitone f) : QuasiconvexOn 𝕜 univ f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Antitone f)
Goal: QuasiconvexOn 𝕜 Set.univ f
-- Antitone.quasiconcaveOn
theorem Antitone.quasiconcaveOn (hf : Antitone f) : QuasiconcaveOn 𝕜 univ f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Antitone f)
Goal: QuasiconcaveOn 𝕜 Set.univ f
-- Antitone.quasilinearOn
theorem Antitone.quasilinearOn (hf : Antitone f) : QuasilinearOn 𝕜 univ f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Antitone f)
Goal: QuasilinearOn 𝕜 Set.univ f
-- QuasilinearOn.monotoneOn_or_antitoneOn
theorem QuasilinearOn.monotoneOn_or_antitoneOn [LinearOrder β] (hf : QuasilinearOn 𝕜 s f) :
    MonotoneOn f s ∨ AntitoneOn f s 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → β} [LinearOrder β] (hf : QuasilinearOn 𝕜 s f)
Goal: MonotoneOn f s ∨ AntitoneOn f s
-- quasilinearOn_iff_monotoneOn_or_antitoneOn
theorem quasilinearOn_iff_monotoneOn_or_antitoneOn [LinearOrderedAddCommMonoid β]
    (hs : Convex 𝕜 s) : QuasilinearOn 𝕜 s f ↔ MonotoneOn f s ∨ AntitoneOn f s 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → β} [LinearOrderedAddCommMonoid β] (hs : Convex 𝕜 s)
Goal: QuasilinearOn 𝕜 s f ↔ MonotoneOn f s ∨ AntitoneOn f s
-- RCLike.uniqueContinuousFunctionalCalculus_of_compactSpace_spectrum
theorem RCLike.uniqueContinuousFunctionalCalculus_of_compactSpace_spectrum [TopologicalSpace A]
    [T2Space A] [Ring A] [StarRing A] [Algebra 𝕜 A] [h : ∀ a : A, CompactSpace (spectrum 𝕜 a)] :
    UniqueContinuousFunctionalCalculus 𝕜 A 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [TopologicalSpace A] [T2Space A] [Ring A] [StarRing A] [Algebra 𝕜 A] [h : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)]
Goal: UniqueContinuousFunctionalCalculus 𝕜 A
-- ContinuousMap.continuous_toNNReal
@[fun_prop]
lemma continuous_toNNReal : Continuous (toNNReal (X := X)) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Continuous ContinuousMap.toNNReal
-- ContinuousMap.toNNReal_apply
@[simp]
lemma toNNReal_apply (f : C(X, ℝ)) (x : X) : f.toNNReal x = (f x).toNNReal 
Arguments: {X : Type u_1} [TopologicalSpace X] (f : C(X, ℝ)) (x : X)
Goal: f.toNNReal x = (f x).toNNReal
-- ContinuousMap.toNNReal_add_add_neg_add_neg_eq
lemma toNNReal_add_add_neg_add_neg_eq (f g : C(X, ℝ)) :
    (f + g).toNNReal + (-f).toNNReal + (-g).toNNReal =
      (-(f + g)).toNNReal + f.toNNReal + g.toNNReal 
Arguments: {X : Type u_1} [TopologicalSpace X] (f g : C(X, ℝ))
Goal: (f + g).toNNReal + (-f).toNNReal + (-g).toNNReal = (-(f + g)).toNNReal + f.toNNReal + g.toNNReal
-- ContinuousMap.toNNReal_mul_add_neg_mul_add_mul_neg_eq
lemma toNNReal_mul_add_neg_mul_add_mul_neg_eq (f g : C(X, ℝ)) :
    (f * g).toNNReal + (-f).toNNReal * g.toNNReal + f.toNNReal * (-g).toNNReal =
      (-(f * g)).toNNReal + f.toNNReal * g.toNNReal + (-f).toNNReal * (-g).toNNReal 
Arguments: {X : Type u_1} [TopologicalSpace X] (f g : C(X, ℝ))
Goal: (f * g).toNNReal + (-f).toNNReal * g.toNNReal + f.toNNReal * (-g).toNNReal =
  (-(f * g)).toNNReal + f.toNNReal * g.toNNReal + (-f).toNNReal * (-g).toNNReal
-- ContinuousMap.toNNReal_algebraMap
@[simp]
lemma toNNReal_algebraMap (r : ℝ≥0) :
    (algebraMap ℝ C(X, ℝ) r).toNNReal = algebraMap ℝ≥0 C(X, ℝ≥0) r 
Arguments: {X : Type u_1} [TopologicalSpace X] (r : NNReal)
Goal: ((algebraMap ℝ C(X, ℝ)) ↑r).toNNReal = (algebraMap NNReal C(X, NNReal)) r
-- ContinuousMap.toNNReal_neg_algebraMap
@[simp]
lemma toNNReal_neg_algebraMap (r : ℝ≥0) : (- algebraMap ℝ C(X, ℝ) r).toNNReal = 0 
Arguments: {X : Type u_1} [TopologicalSpace X] (r : NNReal)
Goal: (-(algebraMap ℝ C(X, ℝ)) ↑r).toNNReal = 0
-- ContinuousMap.toNNReal_one
@[simp]
lemma toNNReal_one : (1 : C(X, ℝ)).toNNReal = 1 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: ContinuousMap.toNNReal 1 = 1
-- ContinuousMap.toNNReal_neg_one
@[simp]
lemma toNNReal_neg_one : (-1 : C(X, ℝ)).toNNReal = 0 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: (-1).toNNReal = 0
-- StarAlgHom.realContinuousMapOfNNReal_apply
@[simps]
noncomputable def realContinuousMapOfNNReal (φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A) :
    C(X, ℝ) →⋆ₐ[ℝ] A 
Arguments: {X : Type u_1} [TopologicalSpace X] {A : Type u_2} [Ring A] [StarRing A] [Algebra ℝ A] (φ : C(X, NNReal) →⋆ₐ[NNReal] A) (f : C(X, ℝ))
Goal: φ.realContinuousMapOfNNReal f = φ f.toNNReal - φ (-f).toNNReal
-- StarAlgHom.continuous_realContinuousMapOfNNReal
@[fun_prop]
lemma continuous_realContinuousMapOfNNReal (φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A)
    (hφ : Continuous φ) : Continuous φ.realContinuousMapOfNNReal 
Arguments: {X : Type u_1} [TopologicalSpace X] {A : Type u_2} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℝ A] [TopologicalRing A] (φ : C(X, NNReal) →⋆ₐ[NNReal] A) (hφ : Continuous ⇑φ)
Goal: Continuous ⇑φ.realContinuousMapOfNNReal
-- StarAlgHom.realContinuousMapOfNNReal_apply_comp_toReal
@[simp high]
lemma realContinuousMapOfNNReal_apply_comp_toReal (φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A)
    (f : C(X, ℝ≥0)) :
    φ.realContinuousMapOfNNReal ((ContinuousMap.mk toReal continuous_coe).comp f) = φ f 
Arguments: {X : Type u_1} [TopologicalSpace X] {A : Type u_2} [Ring A] [StarRing A] [Algebra ℝ A] (φ : C(X, NNReal) →⋆ₐ[NNReal] A) (f : C(X, NNReal))
Goal: φ.realContinuousMapOfNNReal ({ toFun := NNReal.toReal, continuous_toFun := NNReal.continuous_coe }.comp f) = φ f
-- StarAlgHom.realContinuousMapOfNNReal_injective
lemma realContinuousMapOfNNReal_injective :
    Function.Injective (realContinuousMapOfNNReal (X := X) (A := A)) 
Arguments: {X : Type u_1} [TopologicalSpace X] {A : Type u_2} [Ring A] [StarRing A] [Algebra ℝ A]
Goal: Function.Injective StarAlgHom.realContinuousMapOfNNReal
-- RCLike.uniqueNonUnitalContinuousFunctionalCalculus_of_compactSpace_quasispectrum
theorem RCLike.uniqueNonUnitalContinuousFunctionalCalculus_of_compactSpace_quasispectrum
    [TopologicalSpace A] [T2Space A] [NonUnitalRing A] [StarRing A] [Module 𝕜 A]
    [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [h : ∀ a : A, CompactSpace (quasispectrum 𝕜 a)] :
    UniqueNonUnitalContinuousFunctionalCalculus 𝕜 A 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [TopologicalSpace A] [T2Space A] [NonUnitalRing A] [StarRing A] [Module 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [h : ∀ (a : A), CompactSpace ↑(quasispectrum 𝕜 a)]
Goal: UniqueNonUnitalContinuousFunctionalCalculus 𝕜 A
-- ContinuousMapZero.toNNReal_apply
@[simp]
lemma toNNReal_apply (f : C(X, ℝ)₀) (x : X) : f.toNNReal x = Real.toNNReal (f x) 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] (f : ContinuousMapZero X ℝ) (x : X)
Goal: f.toNNReal x = (f x).toNNReal
-- ContinuousMapZero.continuous_toNNReal
@[fun_prop]
lemma continuous_toNNReal : Continuous (toNNReal (X := X)) 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X]
Goal: Continuous ContinuousMapZero.toNNReal
-- ContinuousMapZero.toContinuousMapHom_toNNReal
lemma toContinuousMapHom_toNNReal (f : C(X, ℝ)₀) :
    (toContinuousMapHom (X := X) (R := ℝ) f).toNNReal =
      toContinuousMapHom (X := X) (R := ℝ≥0) f.toNNReal 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] (f : ContinuousMapZero X ℝ)
Goal: (ContinuousMapZero.toContinuousMapHom f).toNNReal = ContinuousMapZero.toContinuousMapHom f.toNNReal
-- ContinuousMapZero.toNNReal_smul
@[simp]
lemma toNNReal_smul (r : ℝ≥0) (f : C(X, ℝ)₀) : (r • f).toNNReal = r • f.toNNReal 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] (r : NNReal) (f : ContinuousMapZero X ℝ)
Goal: (r • f).toNNReal = r • f.toNNReal
-- ContinuousMapZero.toNNReal_neg_smul
@[simp]
lemma toNNReal_neg_smul (r : ℝ≥0) (f : C(X, ℝ)₀) : (-(r • f)).toNNReal = r • (-f).toNNReal 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] (r : NNReal) (f : ContinuousMapZero X ℝ)
Goal: (-(r • f)).toNNReal = r • (-f).toNNReal
-- ContinuousMapZero.toNNReal_mul_add_neg_mul_add_mul_neg_eq
lemma toNNReal_mul_add_neg_mul_add_mul_neg_eq (f g : C(X, ℝ)₀) :
    ((f * g).toNNReal + (-f).toNNReal * g.toNNReal + f.toNNReal * (-g).toNNReal) =
    ((-(f * g)).toNNReal + f.toNNReal * g.toNNReal + (-f).toNNReal * (-g).toNNReal) 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] (f g : ContinuousMapZero X ℝ)
Goal: (f * g).toNNReal + (-f).toNNReal * g.toNNReal + f.toNNReal * (-g).toNNReal =
  (-(f * g)).toNNReal + f.toNNReal * g.toNNReal + (-f).toNNReal * (-g).toNNReal
-- ContinuousMapZero.toNNReal_add_add_neg_add_neg_eq
lemma toNNReal_add_add_neg_add_neg_eq (f g : C(X, ℝ)₀) :
    ((f + g).toNNReal + (-f).toNNReal + (-g).toNNReal) =
      ((-(f + g)).toNNReal + f.toNNReal + g.toNNReal) 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] (f g : ContinuousMapZero X ℝ)
Goal: (f + g).toNNReal + (-f).toNNReal + (-g).toNNReal = (-(f + g)).toNNReal + f.toNNReal + g.toNNReal
-- NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply
@[simps]
noncomputable def realContinuousMapZeroOfNNReal (φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A) :
    C(X, ℝ)₀ →⋆ₙₐ[ℝ] A 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] {A : Type u_2} [NonUnitalRing A] [StarRing A] [Module ℝ A] (φ : ContinuousMapZero X NNReal →⋆ₙₐ[NNReal] A) (f : ContinuousMapZero X ℝ)
Goal: φ.realContinuousMapZeroOfNNReal f = φ f.toNNReal - φ (-f).toNNReal
-- NonUnitalStarAlgHom.continuous_realContinuousMapZeroOfNNReal
@[fun_prop]
lemma continuous_realContinuousMapZeroOfNNReal (φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A)
    (hφ : Continuous φ) : Continuous φ.realContinuousMapZeroOfNNReal 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] {A : Type u_2} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℝ A] [TopologicalRing A] (φ : ContinuousMapZero X NNReal →⋆ₙₐ[NNReal] A) (hφ : Continuous ⇑φ)
Goal: Continuous ⇑φ.realContinuousMapZeroOfNNReal
-- NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply_comp_toReal
@[simp high]
lemma realContinuousMapZeroOfNNReal_apply_comp_toReal (φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A)
    (f : C(X, ℝ≥0)₀) :
    φ.realContinuousMapZeroOfNNReal ((ContinuousMapZero.mk ⟨toReal, continuous_coe⟩ rfl).comp f) =
      φ f 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] {A : Type u_2} [NonUnitalRing A] [StarRing A] [Module ℝ A] (φ : ContinuousMapZero X NNReal →⋆ₙₐ[NNReal] A) (f : ContinuousMapZero X NNReal)
Goal: φ.realContinuousMapZeroOfNNReal
    ({ toFun := NNReal.toReal, continuous_toFun := NNReal.continuous_coe, map_zero' := ⋯ }.comp f) =
  φ f
-- NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_injective
lemma realContinuousMapZeroOfNNReal_injective :
    Function.Injective (realContinuousMapZeroOfNNReal (X := X) (A := A)) 
Arguments: {X : Type u_1} [TopologicalSpace X] [Zero X] {A : Type u_2} [NonUnitalRing A] [StarRing A] [Module ℝ A]
Goal: Function.Injective NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal
-- ConvexOn.map_centerMass_le
theorem ConvexOn.map_centerMass_le (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
    (h₁ : 0 < ∑ i ∈ t, w i) (hmem : ∀ i ∈ t, p i ∈ s) :
    f (t.centerMass w p) ≤ t.centerMass w (f ∘ p) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : 0 < ∑ i ∈ t, w i) (hmem : ∀ i ∈ t, p i ∈ s)
Goal: f (t.centerMass w p) ≤ t.centerMass w (f ∘ p)
-- ConcaveOn.le_map_centerMass
theorem ConcaveOn.le_map_centerMass (hf : ConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
    (h₁ : 0 < ∑ i ∈ t, w i) (hmem : ∀ i ∈ t, p i ∈ s) :
    t.centerMass w (f ∘ p) ≤ f (t.centerMass w p) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [LinearOrderedField 𝕜] [AddCommGroup E] [OrderedAddCommGroup β] [Module 𝕜 E] [Module 𝕜 β] [OrderedSMul 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : ConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : 0 < ∑ i ∈ t, w i) (hmem : ∀ i ∈ t, p i ∈ s)
Goal: t.centerMass w (f ∘ p) ≤ f (t.centerMass w p)