ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- geometric_hahn_banach_closed_point
theorem geometric_hahn_banach_closed_point (hs₁ : Convex ℝ s) (hs₂ : IsClosed s) (disj : x ∉ s) :
    ∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ u < f x 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {s : Set E} {x : E} [LocallyConvexSpace ℝ E] (hs₁ : Convex ℝ s) (hs₂ : IsClosed s) (disj : x ∉ s)
Goal: ∃ f u, (∀ a ∈ s, f a < u) ∧ u < f x
-- geometric_hahn_banach_point_point
theorem geometric_hahn_banach_point_point [T1Space E] (hxy : x ≠ y) :
    ∃ f : E →L[ℝ] ℝ, f x < f y 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {x y : E} [LocallyConvexSpace ℝ E] [T1Space E] (hxy : x ≠ y)
Goal: ∃ f, f x < f y
-- iInter_halfspaces_eq
theorem iInter_halfspaces_eq (hs₁ : Convex ℝ s) (hs₂ : IsClosed s) :
    ⋂ l : E →L[ℝ] ℝ, { x | ∃ y ∈ s, l x ≤ l y } = s 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {s : Set E} [LocallyConvexSpace ℝ E] (hs₁ : Convex ℝ s) (hs₂ : IsClosed s)
Goal: ⋂ l, {x | ∃ y ∈ s, l x ≤ l y} = s
-- inv_norm_smul_mem_closed_unit_ball
theorem inv_norm_smul_mem_closed_unit_ball (x : E) :
    ‖x‖⁻¹ • x ∈ closedBall (0 : E) 1 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E)
Goal: ‖x‖⁻¹ • x ∈ Metric.closedBall 0 1
-- norm_smul_of_nonneg
theorem norm_smul_of_nonneg {t : ℝ} (ht : 0 ≤ t) (x : E) : ‖t • x‖ = t * ‖x‖ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {t : ℝ} (ht : 0 ≤ t) (x : E)
Goal: ‖t • x‖ = t * ‖x‖
-- dist_smul_add_one_sub_smul_le
theorem dist_smul_add_one_sub_smul_le {r : ℝ} {x y : E} (h : r ∈ Icc 0 1) :
    dist (r • x + (1 - r) • y) x ≤ dist y x 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ} {x y : E} (h : r ∈ Set.Icc 0 1)
Goal: dist (r • x + (1 - r) • y) x ≤ dist y x
-- closure_ball
theorem closure_ball (x : E) {r : ℝ} (hr : r ≠ 0) : closure (ball x r) = closedBall x r 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0)
Goal: closure (Metric.ball x r) = Metric.closedBall x r
-- frontier_ball
theorem frontier_ball (x : E) {r : ℝ} (hr : r ≠ 0) :
    frontier (ball x r) = sphere x r 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0)
Goal: frontier (Metric.ball x r) = Metric.sphere x r
-- interior_closedBall
theorem interior_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
    interior (closedBall x r) = ball x r 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0)
Goal: interior (Metric.closedBall x r) = Metric.ball x r
-- frontier_closedBall
theorem frontier_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
    frontier (closedBall x r) = sphere x r 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0)
Goal: frontier (Metric.closedBall x r) = Metric.sphere x r
-- interior_sphere
theorem interior_sphere (x : E) {r : ℝ} (hr : r ≠ 0) : interior (sphere x r) = ∅ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0)
Goal: interior (Metric.sphere x r) = ∅
-- frontier_sphere
theorem frontier_sphere (x : E) {r : ℝ} (hr : r ≠ 0) : frontier (sphere x r) = sphere x r 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0)
Goal: frontier (Metric.sphere x r) = Metric.sphere x r
-- exists_norm_eq
theorem exists_norm_eq {c : ℝ} (hc : 0 ≤ c) : ∃ x : E, ‖x‖ = c 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] {c : ℝ} (hc : 0 ≤ c)
Goal: ∃ x, ‖x‖ = c
-- range_norm
@[simp]
theorem range_norm : range (norm : E → ℝ) = Ici 0 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E]
Goal: Set.range norm = Set.Ici 0
-- nnnorm_surjective
theorem nnnorm_surjective : Surjective (nnnorm : E → ℝ≥0) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E]
Goal: Function.Surjective nnnorm
-- range_nnnorm
@[simp]
theorem range_nnnorm : range (nnnorm : E → ℝ≥0) = univ 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E]
Goal: Set.range nnnorm = Set.univ
-- interior_closedBall'
theorem interior_closedBall' (x : E) (r : ℝ) : interior (closedBall x r) = ball x r 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] (x : E) (r : ℝ)
Goal: interior (Metric.closedBall x r) = Metric.ball x r
-- frontier_closedBall'
theorem frontier_closedBall' (x : E) (r : ℝ) : frontier (closedBall x r) = sphere x r 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] (x : E) (r : ℝ)
Goal: frontier (Metric.closedBall x r) = Metric.sphere x r
-- interior_sphere'
@[simp]
theorem interior_sphere' (x : E) (r : ℝ) : interior (sphere x r) = ∅ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] (x : E) (r : ℝ)
Goal: interior (Metric.sphere x r) = ∅
-- frontier_sphere'
@[simp]
theorem frontier_sphere' (x : E) (r : ℝ) : frontier (sphere x r) = sphere x r 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] (x : E) (r : ℝ)
Goal: frontier (Metric.sphere x r) = Metric.sphere x r
-- separatingDual_def
@[mk_iff separatingDual_def]
class SeparatingDual (R V : Type*) [Ring R] [AddCommGroup V] [TopologicalSpace V]
    [TopologicalSpace R] [Module R V] : Prop :=
    exists_ne_zero' : ∀ (x : V), x ≠ 0 → ∃ f : V →L[R] R, f x ≠ 0
Arguments: (R : Type u_1) (V : Type u_2) [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V]
Goal: SeparatingDual R V ↔ ∀ (x : V), x ≠ 0 → ∃ f, f x ≠ 0
-- SeparatingDual.exists_ne_zero'
@[mk_iff separatingDual_def]
class SeparatingDual (R V : Type*) [Ring R] [AddCommGroup V] [TopologicalSpace V]
    [TopologicalSpace R] [Module R V] : Prop :=
    exists_ne_zero' : ∀ (x : V), x ≠ 0 → ∃ f : V →L[R] R, f x ≠ 0
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [self : SeparatingDual R V] (x : V)
Goal: x ≠ 0 → ∃ f, f x ≠ 0
-- SeparatingDual.exists_ne_zero
lemma exists_ne_zero {x : V} (hx : x ≠ 0) :
    ∃ f : V →L[R] R, f x ≠ 0 
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [SeparatingDual R V] {x : V} (hx : x ≠ 0)
Goal: ∃ f, f x ≠ 0
-- SeparatingDual.exists_separating_of_ne
theorem exists_separating_of_ne {x y : V} (h : x ≠ y) :
    ∃ f : V →L[R] R, f x ≠ f y 
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [SeparatingDual R V] {x y : V} (h : x ≠ y)
Goal: ∃ f, f x ≠ f y
-- SeparatingDual.t1Space
protected theorem t1Space [T1Space R] : T1Space V 
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [SeparatingDual R V] [T1Space R]
Goal: T1Space V
-- SeparatingDual.t2Space
protected theorem t2Space [T2Space R] : T2Space V 
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [SeparatingDual R V] [T2Space R]
Goal: T2Space V
-- separatingDual_iff_injective
theorem _root_.separatingDual_iff_injective : SeparatingDual R V ↔
    Function.Injective (ContinuousLinearMap.coeLM (R := R) R (M := V) (N₃ := R)).flip 
Arguments: {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [TopologicalRing R] [Module R V]
Goal: SeparatingDual R V ↔ Function.Injective ⇑(ContinuousLinearMap.coeLM R).flip
-- SeparatingDual.dualMap_surjective_iff
theorem dualMap_surjective_iff {W} [AddCommGroup W] [Module R W] [FiniteDimensional R W]
    {f : W →ₗ[R] V} : Surjective (f.dualMap ∘ ContinuousLinearMap.toLinearMap) ↔ Injective f 
Arguments: {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [TopologicalRing R] [Module R V] [SeparatingDual R V] {W : Type u_3} [AddCommGroup W] [Module R W] [FiniteDimensional R W] {f : W →ₗ[R] V}
Goal: Function.Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap) ↔ Function.Injective ⇑f
-- SeparatingDual.exists_eq_one
lemma exists_eq_one {x : V} (hx : x ≠ 0) :
    ∃ f : V →L[R] R, f x = 1 
Arguments: {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [TopologicalRing R] [Module R V] [SeparatingDual R V] {x : V} (hx : x ≠ 0)
Goal: ∃ f, f x = 1
-- SeparatingDual.exists_eq_one_ne_zero_of_ne_zero_pair
theorem exists_eq_one_ne_zero_of_ne_zero_pair {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    ∃ f : V →L[R] R, f x = 1 ∧ f y ≠ 0 
Arguments: {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [TopologicalRing R] [Module R V] [SeparatingDual R V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ∃ f, f x = 1 ∧ f y ≠ 0
-- SeparatingDual.exists_continuousLinearEquiv_apply_eq
theorem exists_continuousLinearEquiv_apply_eq [ContinuousSMul R V]
    {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    ∃ A : V ≃L[R] V, A x = y 
Arguments: {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [TopologicalRing R] [TopologicalAddGroup V] [Module R V] [SeparatingDual R V] [ContinuousSMul R V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ∃ A, A x = y
-- SeparatingDual.completeSpace_of_completeSpace_continuousLinearMap
lemma completeSpace_of_completeSpace_continuousLinearMap [CompleteSpace (E →L[𝕜] F)] :
    CompleteSpace F 
Arguments: (𝕜 : Type u_3) (E : Type u_4) (F : Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [SeparatingDual 𝕜 E] [Nontrivial E] [CompleteSpace (E →L[𝕜] F)]
Goal: CompleteSpace F
-- SeparatingDual.completeSpace_continuousLinearMap_iff
lemma completeSpace_continuousLinearMap_iff :
    CompleteSpace (E →L[𝕜] F) ↔ CompleteSpace F 
Arguments: (𝕜 : Type u_3) (E : Type u_4) (F : Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [SeparatingDual 𝕜 E] [Nontrivial E]
Goal: CompleteSpace (E →L[𝕜] F) ↔ CompleteSpace F
-- SeparatingDual.completeSpace_of_completeSpace_continuousMultilinearMap
lemma completeSpace_of_completeSpace_continuousMultilinearMap
    [CompleteSpace (ContinuousMultilinearMap 𝕜 M F)]
    {m : ∀ i, M i} (hm : ∀ i, m i ≠ 0) : CompleteSpace F 
Arguments: (𝕜 : Type u_3) (F : Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_6} [Finite ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] [∀ (i : ι), SeparatingDual 𝕜 (M i)] [CompleteSpace (ContinuousMultilinearMap 𝕜 M F)] {m : (i : ι) → M i} (hm : ∀ (i : ι), m i ≠ 0)
Goal: CompleteSpace F
-- SeparatingDual.completeSpace_continuousMultilinearMap_iff
lemma completeSpace_continuousMultilinearMap_iff {m : ∀ i, M i} (hm : ∀ i, m i ≠ 0) :
    CompleteSpace (ContinuousMultilinearMap 𝕜 M F) ↔ CompleteSpace F 
Arguments: (𝕜 : Type u_3) (F : Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_6} [Finite ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] [∀ (i : ι), SeparatingDual 𝕜 (M i)] {m : (i : ι) → M i} (hm : ∀ (i : ι), m i ≠ 0)
Goal: CompleteSpace (ContinuousMultilinearMap 𝕜 M F) ↔ CompleteSpace F
-- homeomorphUnitSphereProd_apply_snd_coe
@[simps apply_fst_coe apply_snd_coe symm_apply_coe]
noncomputable def homeomorphUnitSphereProd :
    ({0}ᶜ : Set E) ≃ₜ (sphere (0 : E) 1 × Ioi (0 : ℝ)) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] (x : ↑{0}ᶜ)
Goal: ↑((homeomorphUnitSphereProd E) x).2 = ‖↑x‖
-- homeomorphUnitSphereProd_apply_fst_coe
@[simps apply_fst_coe apply_snd_coe symm_apply_coe]
noncomputable def homeomorphUnitSphereProd :
    ({0}ᶜ : Set E) ≃ₜ (sphere (0 : E) 1 × Ioi (0 : ℝ)) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] (x : ↑{0}ᶜ)
Goal: ↑((homeomorphUnitSphereProd E) x).1 = ‖↑x‖⁻¹ • ↑x
-- homeomorphUnitSphereProd_symm_apply_coe
@[simps apply_fst_coe apply_snd_coe symm_apply_coe]
noncomputable def homeomorphUnitSphereProd :
    ({0}ᶜ : Set E) ≃ₜ (sphere (0 : E) 1 × Ioi (0 : ℝ)) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] (x : ↑(Metric.sphere 0 1) × ↑(Set.Ioi 0))
Goal: ↑((homeomorphUnitSphereProd E).symm x) = ↑x.2 • ↑x.1
-- Complex.betaIntegral_convergent_left
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
    IntervalIntegrable (fun x =>
      (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) 
Arguments: {u : ℂ} (hu : 0 < u.re) (v : ℂ)
Goal: IntervalIntegrable (fun x => ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)) MeasureTheory.volume 0 (1 / 2)
-- Complex.betaIntegral_convergent
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    IntervalIntegrable (fun x =>
      (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 
Arguments: {u v : ℂ} (hu : 0 < u.re) (hv : 0 < v.re)
Goal: IntervalIntegrable (fun x => ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)) MeasureTheory.volume 0 1
-- Complex.betaIntegral_symm
theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v 
Arguments: (u v : ℂ)
Goal: v.betaIntegral u = u.betaIntegral v
-- Complex.betaIntegral_eval_one_right
theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u 
Arguments: {u : ℂ} (hu : 0 < u.re)
Goal: u.betaIntegral 1 = 1 / u
-- Complex.betaIntegral_scaled
theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
    ∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
    (a : ℂ) ^ (s + t - 1) * betaIntegral s t 
Arguments: (s t : ℂ) {a : ℝ} (ha : 0 < a)
Goal: ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t
-- Complex.Gamma_mul_Gamma_eq_betaIntegral
theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
    Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t 
Arguments: {s t : ℂ} (hs : 0 < s.re) (ht : 0 < t.re)
Goal: Complex.Gamma s * Complex.Gamma t = Complex.Gamma (s + t) * s.betaIntegral t
-- Complex.betaIntegral_recurrence
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v 
Arguments: {u v : ℂ} (hu : 0 < u.re) (hv : 0 < v.re)
Goal: u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v
-- Complex.betaIntegral_eval_nat_add_one_right
theorem betaIntegral_eval_nat_add_one_right {u : ℂ} (hu : 0 < re u) (n : ℕ) :
    betaIntegral u (n + 1) = n ! / ∏ j ∈ Finset.range (n + 1), (u + j) 
Arguments: {u : ℂ} (hu : 0 < u.re) (n : ℕ)
Goal: u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
-- Complex.GammaSeq_eq_betaIntegral_of_re_pos
theorem GammaSeq_eq_betaIntegral_of_re_pos {s : ℂ} (hs : 0 < re s) (n : ℕ) :
    GammaSeq s n = (n : ℂ) ^ s * betaIntegral s (n + 1) 
Arguments: {s : ℂ} (hs : 0 < s.re) (n : ℕ)
Goal: s.GammaSeq n = ↑n ^ s * s.betaIntegral (↑n + 1)
-- Complex.GammaSeq_add_one_left
theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
    GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n 
Arguments: (s : ℂ) {n : ℕ} (hn : n ≠ 0)
Goal: (s + 1).GammaSeq n / s = ↑n / (↑n + 1 + s) * s.GammaSeq n
-- Complex.GammaSeq_eq_approx_Gamma_integral
theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
    GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) 
Arguments: {s : ℂ} (hs : 0 < s.re) {n : ℕ} (hn : n ≠ 0)
Goal: s.GammaSeq n = ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
-- Complex.approx_Gamma_integral_tendsto_Gamma_integral
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
    Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 - x / n) ^ n : ℝ) * (x : ℂ) ^ (s - 1)) atTop
      (𝓝 <| Gamma s) 
Arguments: {s : ℂ} (hs : 0 < s.re)
Goal: Filter.Tendsto (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) Filter.atTop (𝓝 (Complex.Gamma s))
-- Complex.GammaSeq_tendsto_Gamma
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) 
Arguments: (s : ℂ)
Goal: Filter.Tendsto s.GammaSeq Filter.atTop (𝓝 (Complex.Gamma s))
-- Complex.GammaSeq_mul
theorem GammaSeq_mul (z : ℂ) {n : ℕ} (hn : n ≠ 0) :
    GammaSeq z n * GammaSeq (1 - z) n =
      n / (n + ↑1 - z) * (↑1 / (z * ∏ j ∈ Finset.range n, (↑1 - z ^ 2 / ((j : ℂ) + 1) ^ 2))) 
Arguments: (z : ℂ) {n : ℕ} (hn : n ≠ 0)
Goal: z.GammaSeq n * (1 - z).GammaSeq n = ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))
-- Complex.Gamma_mul_Gamma_one_sub
theorem Gamma_mul_Gamma_one_sub (z : ℂ) : Gamma z * Gamma (1 - z) = π / sin (π * z) 
Arguments: (z : ℂ)
Goal: Complex.Gamma z * Complex.Gamma (1 - z) = ↑π / Complex.sin (↑π * z)
-- Complex.Gamma_ne_zero
theorem Gamma_ne_zero {s : ℂ} (hs : ∀ m : ℕ, s ≠ -m) : Gamma s ≠ 0 
Arguments: {s : ℂ} (hs : ∀ (m : ℕ), s ≠ -↑m)
Goal: Complex.Gamma s ≠ 0
-- Complex.Gamma_eq_zero_iff
theorem Gamma_eq_zero_iff (s : ℂ) : Gamma s = 0 ↔ ∃ m : ℕ, s = -m 
Arguments: (s : ℂ)
Goal: Complex.Gamma s = 0 ↔ ∃ m, s = -↑m
-- Complex.Gamma_ne_zero_of_re_pos
theorem Gamma_ne_zero_of_re_pos {s : ℂ} (hs : 0 < re s) : Gamma s ≠ 0 
Arguments: {s : ℂ} (hs : 0 < s.re)
Goal: Complex.Gamma s ≠ 0
-- Real.GammaSeq_tendsto_Gamma
theorem GammaSeq_tendsto_Gamma (s : ℝ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) 
Arguments: (s : ℝ)
Goal: Filter.Tendsto s.GammaSeq Filter.atTop (𝓝 (Real.Gamma s))
-- Real.Gamma_mul_Gamma_one_sub
theorem Gamma_mul_Gamma_one_sub (s : ℝ) : Gamma s * Gamma (1 - s) = π / sin (π * s) 
Arguments: (s : ℝ)
Goal: Real.Gamma s * Real.Gamma (1 - s) = π / Real.sin (π * s)
-- Complex.one_div_Gamma_eq_self_mul_one_div_Gamma_add_one
theorem one_div_Gamma_eq_self_mul_one_div_Gamma_add_one (s : ℂ) :
    (Gamma s)⁻¹ = s * (Gamma (s + 1))⁻¹ 
Arguments: (s : ℂ)
Goal: (Complex.Gamma s)⁻¹ = s * (Complex.Gamma (s + 1))⁻¹
-- Complex.differentiable_one_div_Gamma
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ 
Goal: Differentiable ℂ fun s => (Complex.Gamma s)⁻¹
-- Complex.Gamma_mul_Gamma_add_half
theorem Gamma_mul_Gamma_add_half (s : ℂ) :
    Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(√π) 
Arguments: (s : ℂ)
Goal: Complex.Gamma s * Complex.Gamma (s + 1 / 2) = Complex.Gamma (2 * s) * 2 ^ (1 - 2 * s) * ↑√π
-- Real.Gamma_mul_Gamma_add_half
theorem Gamma_mul_Gamma_add_half (s : ℝ) :
    Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℝ) ^ (1 - 2 * s) * √π 
Arguments: (s : ℝ)
Goal: Real.Gamma s * Real.Gamma (s + 1 / 2) = Real.Gamma (2 * s) * 2 ^ (1 - 2 * s) * √π
-- AffineSubspace.isClosed_direction_iff
theorem AffineSubspace.isClosed_direction_iff (s : AffineSubspace 𝕜 Q) :
    IsClosed (s.direction : Set W) ↔ IsClosed (s : Set Q) 
Arguments: {W : Type u_4} {Q : Type u_5} [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 W] (s : AffineSubspace 𝕜 Q)
Goal: IsClosed ↑s.direction ↔ IsClosed ↑s
-- dist_center_homothety
@[simp]
theorem dist_center_homothety (p₁ p₂ : P) (c : 𝕜) :
    dist p₁ (homothety p₁ c p₂) = ‖c‖ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: dist p₁ ((AffineMap.homothety p₁ c) p₂) = ‖c‖ * dist p₁ p₂
-- nndist_center_homothety
@[simp]
theorem nndist_center_homothety (p₁ p₂ : P) (c : 𝕜) :
    nndist p₁ (homothety p₁ c p₂) = ‖c‖₊ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: nndist p₁ ((AffineMap.homothety p₁ c) p₂) = ‖c‖₊ * nndist p₁ p₂
-- dist_homothety_center
@[simp]
theorem dist_homothety_center (p₁ p₂ : P) (c : 𝕜) :
    dist (homothety p₁ c p₂) p₁ = ‖c‖ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: dist ((AffineMap.homothety p₁ c) p₂) p₁ = ‖c‖ * dist p₁ p₂
-- nndist_homothety_center
@[simp]
theorem nndist_homothety_center (p₁ p₂ : P) (c : 𝕜) :
    nndist (homothety p₁ c p₂) p₁ = ‖c‖₊ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: nndist ((AffineMap.homothety p₁ c) p₂) p₁ = ‖c‖₊ * nndist p₁ p₂
-- dist_lineMap_lineMap
@[simp]
theorem dist_lineMap_lineMap (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
    dist (lineMap p₁ p₂ c₁) (lineMap p₁ p₂ c₂) = dist c₁ c₂ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c₁ c₂ : 𝕜)
Goal: dist ((AffineMap.lineMap p₁ p₂) c₁) ((AffineMap.lineMap p₁ p₂) c₂) = dist c₁ c₂ * dist p₁ p₂
-- nndist_lineMap_lineMap
@[simp]
theorem nndist_lineMap_lineMap (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
    nndist (lineMap p₁ p₂ c₁) (lineMap p₁ p₂ c₂) = nndist c₁ c₂ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c₁ c₂ : 𝕜)
Goal: nndist ((AffineMap.lineMap p₁ p₂) c₁) ((AffineMap.lineMap p₁ p₂) c₂) = nndist c₁ c₂ * nndist p₁ p₂
-- lipschitzWith_lineMap
theorem lipschitzWith_lineMap (p₁ p₂ : P) : LipschitzWith (nndist p₁ p₂) (lineMap p₁ p₂ : 𝕜 → P) 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P)
Goal: LipschitzWith (nndist p₁ p₂) ⇑(AffineMap.lineMap p₁ p₂)
-- dist_lineMap_left
@[simp]
theorem dist_lineMap_left (p₁ p₂ : P) (c : 𝕜) : dist (lineMap p₁ p₂ c) p₁ = ‖c‖ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: dist ((AffineMap.lineMap p₁ p₂) c) p₁ = ‖c‖ * dist p₁ p₂
-- nndist_lineMap_left
@[simp]
theorem nndist_lineMap_left (p₁ p₂ : P) (c : 𝕜) :
    nndist (lineMap p₁ p₂ c) p₁ = ‖c‖₊ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: nndist ((AffineMap.lineMap p₁ p₂) c) p₁ = ‖c‖₊ * nndist p₁ p₂
-- dist_left_lineMap
@[simp]
theorem dist_left_lineMap (p₁ p₂ : P) (c : 𝕜) : dist p₁ (lineMap p₁ p₂ c) = ‖c‖ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: dist p₁ ((AffineMap.lineMap p₁ p₂) c) = ‖c‖ * dist p₁ p₂
-- nndist_left_lineMap
@[simp]
theorem nndist_left_lineMap (p₁ p₂ : P) (c : 𝕜) :
    nndist p₁ (lineMap p₁ p₂ c) = ‖c‖₊ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: nndist p₁ ((AffineMap.lineMap p₁ p₂) c) = ‖c‖₊ * nndist p₁ p₂
-- dist_lineMap_right
@[simp]
theorem dist_lineMap_right (p₁ p₂ : P) (c : 𝕜) :
    dist (lineMap p₁ p₂ c) p₂ = ‖1 - c‖ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: dist ((AffineMap.lineMap p₁ p₂) c) p₂ = ‖1 - c‖ * dist p₁ p₂
-- nndist_lineMap_right
@[simp]
theorem nndist_lineMap_right (p₁ p₂ : P) (c : 𝕜) :
    nndist (lineMap p₁ p₂ c) p₂ = ‖1 - c‖₊ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: nndist ((AffineMap.lineMap p₁ p₂) c) p₂ = ‖1 - c‖₊ * nndist p₁ p₂