ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousOn.rpow_const
theorem ContinuousOn.rpow_const (hf : ContinuousOn f s) (h : ∀ x ∈ s, f x ≠ 0 ∨ 0 ≤ p) :
    ContinuousOn (fun x => f x ^ p) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {s : Set α} {p : ℝ} (hf : ContinuousOn f s) (h : ∀ x ∈ s, f x ≠ 0 ∨ 0 ≤ p)
Goal: ContinuousOn (fun x => f x ^ p) s
-- Continuous.rpow_const
theorem Continuous.rpow_const (hf : Continuous f) (h : ∀ x, f x ≠ 0 ∨ 0 ≤ p) :
    Continuous fun x => f x ^ p 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {p : ℝ} (hf : Continuous f) (h : ∀ (x : α), f x ≠ 0 ∨ 0 ≤ p)
Goal: Continuous fun x => f x ^ p
-- Complex.continuousAt_cpow_zero_of_re_pos
theorem continuousAt_cpow_zero_of_re_pos {z : ℂ} (hz : 0 < z.re) :
    ContinuousAt (fun x : ℂ × ℂ => x.1 ^ x.2) (0, z) 
Arguments: {z : ℂ} (hz : 0 < z.re)
Goal: ContinuousAt (fun x => x.1 ^ x.2) (0, z)
-- Complex.continuousAt_cpow_of_re_pos
theorem continuousAt_cpow_of_re_pos {p : ℂ × ℂ} (h₁ : 0 ≤ p.1.re ∨ p.1.im ≠ 0) (h₂ : 0 < p.2.re) :
    ContinuousAt (fun x : ℂ × ℂ => x.1 ^ x.2) p 
Arguments: {p : ℂ × ℂ} (h₁ : 0 ≤ p.1.re ∨ p.1.im ≠ 0) (h₂ : 0 < p.2.re)
Goal: ContinuousAt (fun x => x.1 ^ x.2) p
-- Complex.continuousAt_cpow_const_of_re_pos
theorem continuousAt_cpow_const_of_re_pos {z w : ℂ} (hz : 0 ≤ re z ∨ im z ≠ 0) (hw : 0 < re w) :
    ContinuousAt (fun x => x ^ w) z 
Arguments: {z w : ℂ} (hz : 0 ≤ z.re ∨ z.im ≠ 0) (hw : 0 < w.re)
Goal: ContinuousAt (fun x => x ^ w) z
-- Complex.continuousAt_ofReal_cpow
theorem continuousAt_ofReal_cpow (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0) :
    ContinuousAt (fun p => (p.1 : ℂ) ^ p.2 : ℝ × ℂ → ℂ) (x, y) 
Arguments: (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0)
Goal: ContinuousAt (fun p => ↑p.1 ^ p.2) (x, y)
-- Complex.continuousAt_ofReal_cpow_const
theorem continuousAt_ofReal_cpow_const (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0) :
    ContinuousAt (fun a => (a : ℂ) ^ y : ℝ → ℂ) x 
Arguments: (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0)
Goal: ContinuousAt (fun a => ↑a ^ y) x
-- Complex.continuous_ofReal_cpow_const
theorem continuous_ofReal_cpow_const {y : ℂ} (hs : 0 < y.re) :
    Continuous (fun x => (x : ℂ) ^ y : ℝ → ℂ) 
Arguments: {y : ℂ} (hs : 0 < y.re)
Goal: Continuous fun x => ↑x ^ y
-- NNReal.continuousAt_rpow
theorem continuousAt_rpow {x : ℝ≥0} {y : ℝ} (h : x ≠ 0 ∨ 0 < y) :
    ContinuousAt (fun p : ℝ≥0 × ℝ => p.1 ^ p.2) (x, y) 
Arguments: {x : ℝ≥0} {y : ℝ} (h : x ≠ 0 ∨ 0 < y)
Goal: ContinuousAt (fun p => p.1 ^ p.2) (x, y)
-- NNReal.eventually_pow_one_div_le
theorem eventually_pow_one_div_le (x : ℝ≥0) {y : ℝ≥0} (hy : 1 < y) :
    ∀ᶠ n : ℕ in atTop, x ^ (1 / n : ℝ) ≤ y 
Arguments: (x : ℝ≥0) {y : ℝ≥0} (hy : 1 < y)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, x ^ (1 / ↑n) ≤ y
-- Filter.Tendsto.nnrpow
theorem Filter.Tendsto.nnrpow {α : Type*} {f : Filter α} {u : α → ℝ≥0} {v : α → ℝ} {x : ℝ≥0}
    {y : ℝ} (hx : Tendsto u f (𝓝 x)) (hy : Tendsto v f (𝓝 y)) (h : x ≠ 0 ∨ 0 < y) :
    Tendsto (fun a => u a ^ v a) f (𝓝 (x ^ y)) 
Arguments: {α : Type u_1} {f : Filter α} {u : α → ℝ≥0} {v : α → ℝ} {x : ℝ≥0} {y : ℝ} (hx : Filter.Tendsto u f (𝓝 x)) (hy : Filter.Tendsto v f (𝓝 y)) (h : x ≠ 0 ∨ 0 < y)
Goal: Filter.Tendsto (fun a => u a ^ v a) f (𝓝 (x ^ y))
-- NNReal.continuousAt_rpow_const
theorem continuousAt_rpow_const {x : ℝ≥0} {y : ℝ} (h : x ≠ 0 ∨ 0 ≤ y) :
    ContinuousAt (fun z => z ^ y) x 
Arguments: {x : ℝ≥0} {y : ℝ} (h : x ≠ 0 ∨ 0 ≤ y)
Goal: ContinuousAt (fun z => z ^ y) x
-- NNReal.continuous_rpow_const
@[fun_prop]
theorem continuous_rpow_const {y : ℝ} (h : 0 ≤ y) : Continuous fun x : ℝ≥0 => x ^ y 
Arguments: {y : ℝ} (h : 0 ≤ y)
Goal: Continuous fun x => x ^ y
-- ENNReal.eventually_pow_one_div_le
theorem eventually_pow_one_div_le {x : ℝ≥0∞} (hx : x ≠ ∞) {y : ℝ≥0∞} (hy : 1 < y) :
    ∀ᶠ n : ℕ in atTop, x ^ (1 / n : ℝ) ≤ y 
Arguments: {x : ℝ≥0∞} (hx : x ≠ ⊤) {y : ℝ≥0∞} (hy : 1 < y)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, x ^ (1 / ↑n) ≤ y
-- _private.Mathlib.Analysis.SpecialFunctions.Pow.Continuity.0.ENNReal.continuousAt_rpow_const_of_pos
private theorem continuousAt_rpow_const_of_pos {x : ℝ≥0∞} {y : ℝ} (h : 0 < y) :
    ContinuousAt (fun a : ℝ≥0∞ => a ^ y) x 
Arguments: {x : ℝ≥0∞} {y : ℝ} (h : 0 < y)
Goal: ContinuousAt (fun a => a ^ y) x
-- ENNReal.continuous_rpow_const
@[continuity, fun_prop]
theorem continuous_rpow_const {y : ℝ} : Continuous fun a : ℝ≥0∞ => a ^ y 
Arguments: {y : ℝ}
Goal: Continuous fun a => a ^ y
-- ENNReal.tendsto_const_mul_rpow_nhds_zero_of_pos
theorem tendsto_const_mul_rpow_nhds_zero_of_pos {c : ℝ≥0∞} (hc : c ≠ ∞) {y : ℝ} (hy : 0 < y) :
    Tendsto (fun x : ℝ≥0∞ => c * x ^ y) (𝓝 0) (𝓝 0) 
Arguments: {c : ℝ≥0∞} (hc : c ≠ ⊤) {y : ℝ} (hy : 0 < y)
Goal: Filter.Tendsto (fun x => c * x ^ y) (𝓝 0) (𝓝 0)
-- Filter.Tendsto.ennrpow_const
theorem Filter.Tendsto.ennrpow_const {α : Type*} {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} (r : ℝ)
    (hm : Tendsto m f (𝓝 a)) : Tendsto (fun x => m x ^ r) f (𝓝 (a ^ r)) 
Arguments: {α : Type u_1} {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} (r : ℝ) (hm : Filter.Tendsto m f (𝓝 a))
Goal: Filter.Tendsto (fun x => m x ^ r) f (𝓝 (a ^ r))
-- Real.Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma
theorem Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma {s t a b : ℝ} (hs : 0 < s) (ht : 0 < t)
    (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
    Gamma (a * s + b * t) ≤ Gamma s ^ a * Gamma t ^ b 
Arguments: {s t a b : ℝ} (hs : 0 < s) (ht : 0 < t) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1)
Goal: Real.Gamma (a * s + b * t) ≤ Real.Gamma s ^ a * Real.Gamma t ^ b
-- Real.convexOn_log_Gamma
theorem convexOn_log_Gamma : ConvexOn ℝ (Ioi 0) (log ∘ Gamma) 
Goal: ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ Real.Gamma)
-- Real.convexOn_Gamma
theorem convexOn_Gamma : ConvexOn ℝ (Ioi 0) Gamma 
Goal: ConvexOn ℝ (Set.Ioi 0) Real.Gamma
-- Real.BohrMollerup.f_nat_eq
theorem f_nat_eq (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hn : n ≠ 0) :
    f n = f 1 + log (n - 1)! 
Arguments: {f : ℝ → ℝ} {n : ℕ} (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + Real.log y) (hn : n ≠ 0)
Goal: f ↑n = f 1 + Real.log ↑(n - 1)!
-- Real.BohrMollerup.f_add_nat_eq
theorem f_add_nat_eq (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) (n : ℕ) :
    f (x + n) = f x + ∑ m ∈ Finset.range n, log (x + m) 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + Real.log y) (hx : 0 < x) (n : ℕ)
Goal: f (x + ↑n) = f x + ∑ m ∈ Finset.range n, Real.log (x + ↑m)
-- Real.BohrMollerup.f_add_nat_le
theorem f_add_nat_le (hf_conv : ConvexOn ℝ (Ioi 0) f)
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hn : n ≠ 0) (hx : 0 < x) (hx' : x ≤ 1) :
    f (n + x) ≤ f n + x * log n 
Arguments: {f : ℝ → ℝ} {x : ℝ} {n : ℕ} (hf_conv : ConvexOn ℝ (Set.Ioi 0) f) (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + Real.log y) (hn : n ≠ 0) (hx : 0 < x) (hx' : x ≤ 1)
Goal: f (↑n + x) ≤ f ↑n + x * Real.log ↑n
-- Real.BohrMollerup.f_add_nat_ge
theorem f_add_nat_ge (hf_conv : ConvexOn ℝ (Ioi 0) f)
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hn : 2 ≤ n) (hx : 0 < x) :
    f n + x * log (n - 1) ≤ f (n + x) 
Arguments: {f : ℝ → ℝ} {x : ℝ} {n : ℕ} (hf_conv : ConvexOn ℝ (Set.Ioi 0) f) (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + Real.log y) (hn : 2 ≤ n) (hx : 0 < x)
Goal: f ↑n + x * Real.log (↑n - 1) ≤ f (↑n + x)
-- Real.BohrMollerup.logGammaSeq_add_one
theorem logGammaSeq_add_one (x : ℝ) (n : ℕ) :
    logGammaSeq (x + 1) n = logGammaSeq x (n + 1) + log x - (x + 1) * (log (n + 1) - log n) 
Arguments: (x : ℝ) (n : ℕ)
Goal: Real.BohrMollerup.logGammaSeq (x + 1) n =
  Real.BohrMollerup.logGammaSeq x (n + 1) + Real.log x - (x + 1) * (Real.log (↑n + 1) - Real.log ↑n)
-- Real.BohrMollerup.le_logGammaSeq
theorem le_logGammaSeq (hf_conv : ConvexOn ℝ (Ioi 0) f)
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) (hx' : x ≤ 1) (n : ℕ) :
    f x ≤ f 1 + x * log (n + 1) - x * log n + logGammaSeq x n 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hf_conv : ConvexOn ℝ (Set.Ioi 0) f) (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + Real.log y) (hx : 0 < x) (hx' : x ≤ 1) (n : ℕ)
Goal: f x ≤ f 1 + x * Real.log (↑n + 1) - x * Real.log ↑n + Real.BohrMollerup.logGammaSeq x n
-- Real.BohrMollerup.ge_logGammaSeq
theorem ge_logGammaSeq (hf_conv : ConvexOn ℝ (Ioi 0) f)
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) (hn : n ≠ 0) :
    f 1 + logGammaSeq x n ≤ f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} {n : ℕ} (hf_conv : ConvexOn ℝ (Set.Ioi 0) f) (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + Real.log y) (hx : 0 < x) (hn : n ≠ 0)
Goal: f 1 + Real.BohrMollerup.logGammaSeq x n ≤ f x
-- Real.BohrMollerup.tendsto_logGammaSeq_of_le_one
theorem tendsto_logGammaSeq_of_le_one (hf_conv : ConvexOn ℝ (Ioi 0) f)
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) (hx' : x ≤ 1) :
    Tendsto (logGammaSeq x) atTop (𝓝 <| f x - f 1) 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hf_conv : ConvexOn ℝ (Set.Ioi 0) f) (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + Real.log y) (hx : 0 < x) (hx' : x ≤ 1)
Goal: Filter.Tendsto (Real.BohrMollerup.logGammaSeq x) Filter.atTop (𝓝 (f x - f 1))
-- Real.BohrMollerup.tendsto_logGammaSeq
theorem tendsto_logGammaSeq (hf_conv : ConvexOn ℝ (Ioi 0) f)
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) :
    Tendsto (logGammaSeq x) atTop (𝓝 <| f x - f 1) 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hf_conv : ConvexOn ℝ (Set.Ioi 0) f) (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + Real.log y) (hx : 0 < x)
Goal: Filter.Tendsto (Real.BohrMollerup.logGammaSeq x) Filter.atTop (𝓝 (f x - f 1))
-- Real.BohrMollerup.tendsto_log_gamma
theorem tendsto_log_gamma {x : ℝ} (hx : 0 < x) :
    Tendsto (logGammaSeq x) atTop (𝓝 <| log (Gamma x)) 
Arguments: {x : ℝ} (hx : 0 < x)
Goal: Filter.Tendsto (Real.BohrMollerup.logGammaSeq x) Filter.atTop (𝓝 (Real.log (Real.Gamma x)))
-- Real.eq_Gamma_of_log_convex
theorem eq_Gamma_of_log_convex {f : ℝ → ℝ} (hf_conv : ConvexOn ℝ (Ioi 0) (log ∘ f))
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = y * f y) (hf_pos : ∀ {y : ℝ}, 0 < y → 0 < f y)
    (hf_one : f 1 = 1) : EqOn f Gamma (Ioi (0 : ℝ)) 
Arguments: {f : ℝ → ℝ} (hf_conv : ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ f)) (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = y * f y) (hf_pos : ∀ {y : ℝ}, 0 < y → 0 < f y) (hf_one : f 1 = 1)
Goal: Set.EqOn f Real.Gamma (Set.Ioi 0)
-- Real.Gamma_two
theorem Gamma_two : Gamma 2 = 1 
Goal: Real.Gamma 2 = 1
-- Real.Gamma_three_div_two_lt_one
theorem Gamma_three_div_two_lt_one : Gamma (3 / 2) < 1 
Goal: Real.Gamma (3 / 2) < 1
-- Real.Gamma_strictMonoOn_Ici
theorem Gamma_strictMonoOn_Ici : StrictMonoOn Gamma (Ici 2) 
Goal: StrictMonoOn Real.Gamma (Set.Ici 2)
-- Real.doublingGamma_add_one
theorem doublingGamma_add_one (s : ℝ) (hs : s ≠ 0) :
    doublingGamma (s + 1) = s * doublingGamma s 
Arguments: (s : ℝ) (hs : s ≠ 0)
Goal: (s + 1).doublingGamma = s * s.doublingGamma
-- Real.doublingGamma_one
theorem doublingGamma_one : doublingGamma 1 = 1 
Goal: Real.doublingGamma 1 = 1
-- Real.log_doublingGamma_eq
theorem log_doublingGamma_eq :
    EqOn (log ∘ doublingGamma)
      (fun s => log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + s * log 2 - log (2 * √π))
      (Ioi 0) 
Goal: Set.EqOn (Real.log ∘ Real.doublingGamma)
  (fun s => Real.log (Real.Gamma (s / 2)) + Real.log (Real.Gamma (s / 2 + 1 / 2)) + s * Real.log 2 - Real.log (2 * √π))
  (Set.Ioi 0)
-- Real.doublingGamma_log_convex_Ioi
theorem doublingGamma_log_convex_Ioi : ConvexOn ℝ (Ioi (0 : ℝ)) (log ∘ doublingGamma) 
Goal: ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ Real.doublingGamma)
-- Real.doublingGamma_eq_Gamma
theorem doublingGamma_eq_Gamma {s : ℝ} (hs : 0 < s) : doublingGamma s = Gamma s 
Arguments: {s : ℝ} (hs : 0 < s)
Goal: s.doublingGamma = Real.Gamma s
-- Real.Gamma_mul_Gamma_add_half_of_pos
theorem Gamma_mul_Gamma_add_half_of_pos {s : ℝ} (hs : 0 < s) :
    Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * √π 
Arguments: {s : ℝ} (hs : 0 < s)
Goal: Real.Gamma s * Real.Gamma (s + 1 / 2) = Real.Gamma (2 * s) * 2 ^ (1 - 2 * s) * √π
-- Complex.tan_arctan
theorem tan_arctan {z : ℂ} (h₁ : z ≠ I) (h₂ : z ≠ -I) : tan (arctan z) = z 
Arguments: {z : ℂ} (h₁ : z ≠ Complex.I) (h₂ : z ≠ -Complex.I)
Goal: Complex.tan z.arctan = z
-- Complex.cos_ne_zero_of_arctan_bounds
lemma cos_ne_zero_of_arctan_bounds {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re)
    (h₂ : z.re ≤ π / 2) : cos z ≠ 0 
Arguments: {z : ℂ} (h₀ : z ≠ ↑Real.pi / 2) (h₁ : -(Real.pi / 2) < z.re) (h₂ : z.re ≤ Real.pi / 2)
Goal: Complex.cos z ≠ 0
-- Complex.arctan_tan
theorem arctan_tan {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re) (h₂ : z.re ≤ π / 2) :
    arctan (tan z) = z 
Arguments: {z : ℂ} (h₀ : z ≠ ↑Real.pi / 2) (h₁ : -(Real.pi / 2) < z.re) (h₂ : z.re ≤ Real.pi / 2)
Goal: (Complex.tan z).arctan = z
-- Complex.ofReal_arctan
@[simp, norm_cast]
theorem ofReal_arctan (x : ℝ) : (Real.arctan x : ℂ) = arctan x 
Arguments: (x : ℝ)
Goal: ↑(Real.arctan x) = (↑x).arctan
-- Complex.arg_one_add_mem_Ioo
lemma arg_one_add_mem_Ioo {z : ℂ} (hz : ‖z‖ < 1) : (1 + z).arg ∈ Set.Ioo (-(π / 2)) (π / 2) 
Arguments: {z : ℂ} (hz : ‖z‖ < 1)
Goal: (1 + z).arg ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)
-- Complex.hasSum_arctan_aux
lemma hasSum_arctan_aux {z : ℂ} (hz : ‖z‖ < 1) :
    log (1 + z * I) + -log (1 - z * I) = log ((1 + z * I) / (1 - z * I)) 
Arguments: {z : ℂ} (hz : ‖z‖ < 1)
Goal: Complex.log (1 + z * Complex.I) + -Complex.log (1 - z * Complex.I) =
  Complex.log ((1 + z * Complex.I) / (1 - z * Complex.I))
-- Complex.hasSum_arctan
theorem hasSum_arctan {z : ℂ} (hz : ‖z‖ < 1) :
    HasSum (fun n : ℕ ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)) (arctan z) 
Arguments: {z : ℂ} (hz : ‖z‖ < 1)
Goal: HasSum (fun n => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)) z.arctan
-- Real.hasSum_arctan
theorem Real.hasSum_arctan {x : ℝ} (hx : ‖x‖ < 1) :
    HasSum (fun n : ℕ => (-1) ^ n * x ^ (2 * n + 1) / ↑(2 * n + 1)) (arctan x) 
Arguments: {x : ℝ} (hx : ‖x‖ < 1)
Goal: HasSum (fun n => (-1) ^ n * x ^ (2 * n + 1) / ↑(2 * n + 1)) (Real.arctan x)
-- PartialHomeomorph.univUnitBall_symm_apply
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (y : E)
Goal: ↑PartialHomeomorph.univUnitBall.symm y = (√(1 - ‖y‖ ^ 2))⁻¹ • y
-- PartialHomeomorph.univUnitBall_apply
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E)
Goal: ↑PartialHomeomorph.univUnitBall x = (√(1 + ‖x‖ ^ 2))⁻¹ • x
-- PartialHomeomorph.univUnitBall_target
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: PartialHomeomorph.univUnitBall.target = Metric.ball 0 1
-- PartialHomeomorph.univUnitBall_source
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: PartialHomeomorph.univUnitBall.source = Set.univ
-- PartialHomeomorph.univUnitBall_apply_zero
@[simp]
theorem PartialHomeomorph.univUnitBall_apply_zero : univUnitBall (0 : E) = 0 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: ↑PartialHomeomorph.univUnitBall 0 = 0
-- PartialHomeomorph.univUnitBall_symm_apply_zero
@[simp]
theorem PartialHomeomorph.univUnitBall_symm_apply_zero : univUnitBall.symm (0 : E) = 0 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: ↑PartialHomeomorph.univUnitBall.symm 0 = 0
-- Homeomorph.unitBall_apply_coe
@[simps! (config := .lemmasOnly)]
def Homeomorph.unitBall : E ≃ₜ ball (0 : E) 1 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: ∀ (a : E), ↑(Homeomorph.unitBall a) = ↑PartialHomeomorph.univUnitBall a
-- Homeomorph.unitBall_symm_apply
@[simps! (config := .lemmasOnly)]
def Homeomorph.unitBall : E ≃ₜ ball (0 : E) 1 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: ∀ (a : ↑(Metric.ball 0 1)),
  Homeomorph.unitBall.symm a = ↑(PartialHomeomorph.univUnitBall.toHomeomorphSourceTarget.symm a)
-- Homeomorph.coe_unitBall_apply_zero
@[simp]
theorem Homeomorph.coe_unitBall_apply_zero :
    (Homeomorph.unitBall (0 : E) : E) = 0 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: ↑(Homeomorph.unitBall 0) = 0
-- PartialHomeomorph.unitBallBall_symm_apply
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ) (hr : 0 < r) (a : P)
Goal: ↑(PartialHomeomorph.unitBallBall c r hr).symm a = r⁻¹ • (a -ᵥ c)
-- PartialHomeomorph.unitBallBall_source
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ) (hr : 0 < r)
Goal: (PartialHomeomorph.unitBallBall c r hr).source = Metric.ball 0 1
-- PartialHomeomorph.unitBallBall_apply
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ) (hr : 0 < r) (a : E)
Goal: ↑(PartialHomeomorph.unitBallBall c r hr) a = r • a +ᵥ c
-- PartialHomeomorph.unitBallBall_target
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ) (hr : 0 < r)
Goal: (PartialHomeomorph.unitBallBall c r hr).target = Metric.ball c r
-- PartialHomeomorph.univBall_source
@[simp]
theorem univBall_source (c : P) (r : ℝ) : (univBall c r).source = univ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ)
Goal: (PartialHomeomorph.univBall c r).source = Set.univ
-- PartialHomeomorph.univBall_target
theorem univBall_target (c : P) {r : ℝ} (hr : 0 < r) : (univBall c r).target = ball c r 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) {r : ℝ} (hr : 0 < r)
Goal: (PartialHomeomorph.univBall c r).target = Metric.ball c r
-- PartialHomeomorph.ball_subset_univBall_target
theorem ball_subset_univBall_target (c : P) (r : ℝ) : ball c r ⊆ (univBall c r).target 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ)
Goal: Metric.ball c r ⊆ (PartialHomeomorph.univBall c r).target
-- PartialHomeomorph.univBall_apply_zero
@[simp]
theorem univBall_apply_zero (c : P) (r : ℝ) : univBall c r 0 = c 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ)
Goal: ↑(PartialHomeomorph.univBall c r) 0 = c
-- PartialHomeomorph.univBall_symm_apply_center
@[simp]
theorem univBall_symm_apply_center (c : P) (r : ℝ) : (univBall c r).symm c = 0 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ)
Goal: ↑(PartialHomeomorph.univBall c r).symm c = 0
-- PartialHomeomorph.continuous_univBall
@[continuity]
theorem continuous_univBall (c : P) (r : ℝ) : Continuous (univBall c r) 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ)
Goal: Continuous ↑(PartialHomeomorph.univBall c r)
-- PartialHomeomorph.continuousOn_univBall_symm
theorem continuousOn_univBall_symm (c : P) (r : ℝ) : ContinuousOn (univBall c r).symm (ball c r) 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {P : Type u_2} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) (r : ℝ)
Goal: ContinuousOn (↑(PartialHomeomorph.univBall c r).symm) (Metric.ball c r)
-- ContinuousLinearMap.isBigOWith_id
theorem isBigOWith_id : IsBigOWith ‖f‖ l f fun x => x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (l : Filter E)
Goal: Asymptotics.IsBigOWith ‖f‖ l ⇑f fun x => x
-- ContinuousLinearMap.isBigO_id
theorem isBigO_id : f =O[l] fun x => x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (l : Filter E)
Goal: ⇑f =O[l] fun x => x
-- ContinuousLinearMap.isBigOWith_comp
theorem isBigOWith_comp [RingHomIsometric σ₂₃] {α : Type*} (g : F →SL[σ₂₃] G) (f : α → F)
    (l : Filter α) : IsBigOWith ‖g‖ l (fun x' => g (f x')) f 
Arguments: {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {F : Type u_5} {G : Type u_6} [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} [RingHomIsometric σ₂₃] {α : Type u_7} (g : F →SL[σ₂₃] G) (f : α → F) (l : Filter α)
Goal: Asymptotics.IsBigOWith ‖g‖ l (fun x' => g (f x')) f
-- ContinuousLinearMap.isBigO_comp
theorem isBigO_comp [RingHomIsometric σ₂₃] {α : Type*} (g : F →SL[σ₂₃] G) (f : α → F)
    (l : Filter α) : (fun x' => g (f x')) =O[l] f 
Arguments: {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {F : Type u_5} {G : Type u_6} [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} [RingHomIsometric σ₂₃] {α : Type u_7} (g : F →SL[σ₂₃] G) (f : α → F) (l : Filter α)
Goal: (fun x' => g (f x')) =O[l] f
-- ContinuousLinearMap.isBigOWith_sub
theorem isBigOWith_sub (x : E) :
    IsBigOWith ‖f‖ l (fun x' => f (x' - x)) fun x' => x' - x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (l : Filter E) (x : E)
Goal: Asymptotics.IsBigOWith ‖f‖ l (fun x' => f (x' - x)) fun x' => x' - x
-- ContinuousLinearMap.isBigO_sub
theorem isBigO_sub (x : E) :
    (fun x' => f (x' - x)) =O[l] fun x' => x' - x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (l : Filter E) (x : E)
Goal: (fun x' => f (x' - x)) =O[l] fun x' => x' - x
-- ContinuousLinearEquiv.isBigO_comp
theorem isBigO_comp {α : Type*} (f : α → E) (l : Filter α) : (fun x' => e (f x')) =O[l] f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [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] (e : E ≃SL[σ₁₂] F) [RingHomIsometric σ₁₂] {α : Type u_7} (f : α → E) (l : Filter α)
Goal: (fun x' => e (f x')) =O[l] f
-- ContinuousLinearEquiv.isBigO_sub
theorem isBigO_sub (l : Filter E) (x : E) : (fun x' => e (x' - x)) =O[l] fun x' => x' - x 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [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] (e : E ≃SL[σ₁₂] F) [RingHomIsometric σ₁₂] (l : Filter E) (x : E)
Goal: (fun x' => e (x' - x)) =O[l] fun x' => x' - x