ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- _private.Mathlib.Analysis.SpecialFunctions.Gamma.Basic.0.Complex.Gamma_integrand_deriv_integrable_B
private theorem Gamma_integrand_deriv_integrable_B {s : ℂ} (hs : 0 < s.re) {Y : ℝ} (hY : 0 ≤ Y) :
    IntervalIntegrable (fun x : ℝ => (-x).exp * (s * x ^ (s - 1)) : ℝ → ℂ) volume 0 Y 
Arguments: {s : ℂ} (hs : 0 < s.re) {Y : ℝ} (hY : 0 ≤ Y)
Goal: IntervalIntegrable (fun x => ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) MeasureTheory.volume 0 Y
-- Complex.partialGamma_add_one
theorem partialGamma_add_one {s : ℂ} (hs : 0 < s.re) {X : ℝ} (hX : 0 ≤ X) :
    partialGamma (s + 1) X = s * partialGamma s X - (-X).exp * X ^ s 
Arguments: {s : ℂ} (hs : 0 < s.re) {X : ℝ} (hX : 0 ≤ X)
Goal: (s + 1).partialGamma X = s * s.partialGamma X - ↑(rexp (-X)) * ↑X ^ s
-- Complex.GammaIntegral_add_one
theorem GammaIntegral_add_one {s : ℂ} (hs : 0 < s.re) :
    GammaIntegral (s + 1) = s * GammaIntegral s 
Arguments: {s : ℂ} (hs : 0 < s.re)
Goal: (s + 1).GammaIntegral = s * s.GammaIntegral
-- Complex.GammaAux_recurrence1
theorem GammaAux_recurrence1 (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n) :
    GammaAux n s = GammaAux n (s + 1) / s 
Arguments: (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n)
Goal: Complex.GammaAux n s = Complex.GammaAux n (s + 1) / s
-- Complex.GammaAux_recurrence2
theorem GammaAux_recurrence2 (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n) :
    GammaAux n s = GammaAux (n + 1) s 
Arguments: (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n)
Goal: Complex.GammaAux n s = Complex.GammaAux (n + 1) s
-- Complex.Gamma_def
@[pp_nodot]
irreducible_def Gamma (s : ℂ) : ℂ 
Arguments: (s : ℂ)
Goal: Complex.Gamma s = Complex.GammaAux ⌊1 - s.re⌋₊ s
-- Complex.Gamma_eq_GammaAux
theorem Gamma_eq_GammaAux (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n) : Gamma s = GammaAux n s 
Arguments: (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n)
Goal: Complex.Gamma s = Complex.GammaAux n s
-- Complex.Gamma_add_one
theorem Gamma_add_one (s : ℂ) (h2 : s ≠ 0) : Gamma (s + 1) = s * Gamma s 
Arguments: (s : ℂ) (h2 : s ≠ 0)
Goal: Complex.Gamma (s + 1) = s * Complex.Gamma s
-- Complex.Gamma_eq_integral
theorem Gamma_eq_integral {s : ℂ} (hs : 0 < s.re) : Gamma s = GammaIntegral s 
Arguments: {s : ℂ} (hs : 0 < s.re)
Goal: Complex.Gamma s = s.GammaIntegral
-- Complex.Gamma_one
@[simp]
theorem Gamma_one : Gamma 1 = 1 
Goal: Complex.Gamma 1 = 1
-- Complex.Gamma_nat_eq_factorial
theorem Gamma_nat_eq_factorial (n : ℕ) : Gamma (n + 1) = n ! 
Arguments: (n : ℕ)
Goal: Complex.Gamma (↑n + 1) = ↑n !
-- Complex.Gamma_ofNat_eq_factorial
@[simp]
theorem Gamma_ofNat_eq_factorial (n : ℕ) [(n + 1).AtLeastTwo] :
    Gamma (no_index (OfNat.ofNat (n + 1) : ℂ)) = n ! 
Arguments: (n : ℕ) [(n + 1).AtLeastTwo]
Goal: Complex.Gamma (OfNat.ofNat (n + 1)) = ↑n !
-- Complex.Gamma_zero
@[simp]
theorem Gamma_zero : Gamma 0 = 0 
Goal: Complex.Gamma 0 = 0
-- Complex.Gamma_neg_nat_eq_zero
theorem Gamma_neg_nat_eq_zero (n : ℕ) : Gamma (-n) = 0 
Arguments: (n : ℕ)
Goal: Complex.Gamma (-↑n) = 0
-- Complex.Gamma_conj
theorem Gamma_conj (s : ℂ) : Gamma (conj s) = conj (Gamma s) 
Arguments: (s : ℂ)
Goal: Complex.Gamma ((starRingEnd ℂ) s) = (starRingEnd ℂ) (Complex.Gamma s)
-- Complex.integral_cpow_mul_exp_neg_mul_Ioi
lemma integral_cpow_mul_exp_neg_mul_Ioi {a : ℂ} {r : ℝ} (ha : 0 < a.re) (hr : 0 < r) :
    ∫ (t : ℝ) in Ioi 0, t ^ (a - 1) * exp (-(r * t)) = (1 / r) ^ a * Gamma a 
Arguments: {a : ℂ} {r : ℝ} (ha : 0 < a.re) (hr : 0 < r)
Goal: ∫ (t : ℝ) in Set.Ioi 0, ↑t ^ (a - 1) * Complex.exp (-(↑r * ↑t)) = (1 / ↑r) ^ a * Complex.Gamma a
-- Complex.GammaIntegral_eq_mellin
theorem GammaIntegral_eq_mellin : GammaIntegral = mellin fun x => ↑(Real.exp (-x)) 
Goal: Complex.GammaIntegral = mellin fun x => ↑(rexp (-x))
-- Complex.hasDerivAt_GammaIntegral
theorem hasDerivAt_GammaIntegral {s : ℂ} (hs : 0 < s.re) :
    HasDerivAt GammaIntegral (∫ t : ℝ in Ioi 0, t ^ (s - 1) * (Real.log t * Real.exp (-t))) s 
Arguments: {s : ℂ} (hs : 0 < s.re)
Goal: HasDerivAt Complex.GammaIntegral (∫ (t : ℝ) in Set.Ioi 0, ↑t ^ (s - 1) * (↑(Real.log t) * ↑(rexp (-t)))) s
-- Complex.differentiableAt_GammaAux
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s 
Arguments: (s : ℂ) (n : ℕ) (h1 : 1 - s.re < ↑n) (h2 : ∀ (m : ℕ), s ≠ -↑m)
Goal: DifferentiableAt ℂ (Complex.GammaAux n) s
-- Complex.differentiableAt_Gamma
theorem differentiableAt_Gamma (s : ℂ) (hs : ∀ m : ℕ, s ≠ -m) : DifferentiableAt ℂ Gamma s 
Arguments: (s : ℂ) (hs : ∀ (m : ℕ), s ≠ -↑m)
Goal: DifferentiableAt ℂ Complex.Gamma s
-- Complex.tendsto_self_mul_Gamma_nhds_zero
theorem tendsto_self_mul_Gamma_nhds_zero : Tendsto (fun z : ℂ => z * Gamma z) (𝓝[≠] 0) (𝓝 1) 
Goal: Filter.Tendsto (fun z => z * Complex.Gamma z) (𝓝[≠] 0) (𝓝 1)
-- Real.Gamma_eq_integral
theorem Gamma_eq_integral {s : ℝ} (hs : 0 < s) :
    Gamma s = ∫ x in Ioi 0, exp (-x) * x ^ (s - 1) 
Arguments: {s : ℝ} (hs : 0 < s)
Goal: Real.Gamma s = ∫ (x : ℝ) in Set.Ioi 0, rexp (-x) * x ^ (s - 1)
-- Real.Gamma_add_one
theorem Gamma_add_one {s : ℝ} (hs : s ≠ 0) : Gamma (s + 1) = s * Gamma s 
Arguments: {s : ℝ} (hs : s ≠ 0)
Goal: Real.Gamma (s + 1) = s * Real.Gamma s
-- Real.Gamma_one
@[simp]
theorem Gamma_one : Gamma 1 = 1 
Goal: Real.Gamma 1 = 1
-- Complex.Gamma_ofReal
theorem _root_.Complex.Gamma_ofReal (s : ℝ) : Complex.Gamma (s : ℂ) = Gamma s 
Arguments: (s : ℝ)
Goal: Complex.Gamma ↑s = ↑(Real.Gamma s)
-- Real.Gamma_nat_eq_factorial
theorem Gamma_nat_eq_factorial (n : ℕ) : Gamma (n + 1) = n ! 
Arguments: (n : ℕ)
Goal: Real.Gamma (↑n + 1) = ↑n !
-- Real.Gamma_ofNat_eq_factorial
@[simp]
theorem Gamma_ofNat_eq_factorial (n : ℕ) [(n + 1).AtLeastTwo] :
    Gamma (no_index (OfNat.ofNat (n + 1) : ℝ)) = n ! 
Arguments: (n : ℕ) [(n + 1).AtLeastTwo]
Goal: Real.Gamma (OfNat.ofNat (n + 1)) = ↑n !
-- Real.Gamma_zero
@[simp]
theorem Gamma_zero : Gamma 0 = 0 
Goal: Real.Gamma 0 = 0
-- Real.Gamma_neg_nat_eq_zero
theorem Gamma_neg_nat_eq_zero (n : ℕ) : Gamma (-n) = 0 
Arguments: (n : ℕ)
Goal: Real.Gamma (-↑n) = 0
-- Real.Gamma_pos_of_pos
theorem Gamma_pos_of_pos {s : ℝ} (hs : 0 < s) : 0 < Gamma s 
Arguments: {s : ℝ} (hs : 0 < s)
Goal: 0 < Real.Gamma s
-- Real.Gamma_nonneg_of_nonneg
theorem Gamma_nonneg_of_nonneg {s : ℝ} (hs : 0 ≤ s) : 0 ≤ Gamma s 
Arguments: {s : ℝ} (hs : 0 ≤ s)
Goal: 0 ≤ Real.Gamma s
-- Real.integral_rpow_mul_exp_neg_mul_Ioi
lemma integral_rpow_mul_exp_neg_mul_Ioi {a r : ℝ} (ha : 0 < a) (hr : 0 < r) :
    ∫ t : ℝ in Ioi 0, t ^ (a - 1) * exp (-(r * t)) = (1 / r) ^ a * Gamma a 
Arguments: {a r : ℝ} (ha : 0 < a) (hr : 0 < r)
Goal: ∫ (t : ℝ) in Set.Ioi 0, t ^ (a - 1) * rexp (-(r * t)) = (1 / r) ^ a * Real.Gamma a
-- Real.Gamma_ne_zero
theorem Gamma_ne_zero {s : ℝ} (hs : ∀ m : ℕ, s ≠ -m) : Gamma s ≠ 0 
Arguments: {s : ℝ} (hs : ∀ (m : ℕ), s ≠ -↑m)
Goal: Real.Gamma s ≠ 0
-- Real.Gamma_eq_zero_iff
theorem Gamma_eq_zero_iff (s : ℝ) : Gamma s = 0 ↔ ∃ m : ℕ, s = -m 
Arguments: (s : ℝ)
Goal: Real.Gamma s = 0 ↔ ∃ m, s = -↑m
-- Real.differentiableAt_Gamma
theorem differentiableAt_Gamma {s : ℝ} (hs : ∀ m : ℕ, s ≠ -m) : DifferentiableAt ℝ Gamma s 
Arguments: {s : ℝ} (hs : ∀ (m : ℕ), s ≠ -↑m)
Goal: DifferentiableAt ℝ Real.Gamma s
-- isConformalMap_id
theorem isConformalMap_id : IsConformalMap (id R M) 
Arguments: {R : Type u_1} {M : Type u_2} [NormedField R] [SeminormedAddCommGroup M] [NormedSpace R M]
Goal: IsConformalMap (ContinuousLinearMap.id R M)
-- IsConformalMap.smul
theorem IsConformalMap.smul (hf : IsConformalMap f) {c : R} (hc : c ≠ 0) :
    IsConformalMap (c • f) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_3} [NormedField R] [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] [NormedSpace R M] [NormedSpace R N] {f : M →L[R] N} (hf : IsConformalMap f) {c : R} (hc : c ≠ 0)
Goal: IsConformalMap (c • f)
-- isConformalMap_const_smul
theorem isConformalMap_const_smul (hc : c ≠ 0) : IsConformalMap (c • id R M) 
Arguments: {R : Type u_1} {M : Type u_2} [NormedField R] [SeminormedAddCommGroup M] [NormedSpace R M] {c : R} (hc : c ≠ 0)
Goal: IsConformalMap (c • ContinuousLinearMap.id R M)
-- LinearIsometry.isConformalMap
protected theorem LinearIsometry.isConformalMap (f' : M →ₗᵢ[R] N) :
    IsConformalMap f'.toContinuousLinearMap 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_3} [NormedField R] [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] [NormedSpace R M] [NormedSpace R N] (f' : M →ₗᵢ[R] N)
Goal: IsConformalMap f'.toContinuousLinearMap
-- isConformalMap_of_subsingleton
@[nontriviality]
theorem isConformalMap_of_subsingleton [Subsingleton M] (f' : M →L[R] N) : IsConformalMap f' 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_3} [NormedField R] [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] [NormedSpace R M] [NormedSpace R N] [Subsingleton M] (f' : M →L[R] N)
Goal: IsConformalMap f'
-- IsConformalMap.comp
theorem comp (hg : IsConformalMap g) (hf : IsConformalMap f) : IsConformalMap (g.comp f) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_3} {G : Type u_4} [NormedField R] [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] [SeminormedAddCommGroup G] [NormedSpace R M] [NormedSpace R N] [NormedSpace R G] {f : M →L[R] N} {g : N →L[R] G} (hg : IsConformalMap g) (hf : IsConformalMap f)
Goal: IsConformalMap (g.comp f)
-- IsConformalMap.injective
protected theorem injective {f : M' →L[R] N} (h : IsConformalMap f) : Function.Injective f 
Arguments: {R : Type u_1} {N : Type u_3} {M' : Type u_5} [NormedField R] [SeminormedAddCommGroup N] [NormedSpace R N] [NormedAddCommGroup M'] [NormedSpace R M'] {f : M' →L[R] N} (h : IsConformalMap f)
Goal: Function.Injective ⇑f
-- IsConformalMap.ne_zero
theorem ne_zero [Nontrivial M'] {f' : M' →L[R] N} (hf' : IsConformalMap f') : f' ≠ 0 
Arguments: {R : Type u_1} {N : Type u_3} {M' : Type u_5} [NormedField R] [SeminormedAddCommGroup N] [NormedSpace R N] [NormedAddCommGroup M'] [NormedSpace R M'] [Nontrivial M'] {f' : M' →L[R] N} (hf' : IsConformalMap f')
Goal: f' ≠ 0
-- Finset.centerMass_empty
theorem Finset.centerMass_empty : (∅ : Finset ι).centerMass w z = 0 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (w : ι → R) (z : ι → E)
Goal: ∅.centerMass w z = 0
-- Finset.centerMass_pair
theorem Finset.centerMass_pair (hne : i ≠ j) :
    ({i, j} : Finset ι).centerMass w z = (w i / (w i + w j)) • z i + (w j / (w i + w j)) • z j 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (i j : ι) (w : ι → R) (z : ι → E) (hne : i ≠ j)
Goal: {i, j}.centerMass w z = (w i / (w i + w j)) • z i + (w j / (w i + w j)) • z j
-- Finset.centerMass_insert
theorem Finset.centerMass_insert (ha : i ∉ t) (hw : ∑ j ∈ t, w j ≠ 0) :
    (insert i t).centerMass w z =
      (w i / (w i + ∑ j ∈ t, w j)) • z i +
        ((∑ j ∈ t, w j) / (w i + ∑ j ∈ t, w j)) • t.centerMass w z 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (i : ι) (t : Finset ι) {w : ι → R} (z : ι → E) (ha : i ∉ t) (hw : ∑ j ∈ t, w j ≠ 0)
Goal: (insert i t).centerMass w z =
  (w i / (w i + ∑ j ∈ t, w j)) • z i + ((∑ j ∈ t, w j) / (w i + ∑ j ∈ t, w j)) • t.centerMass w z
-- Finset.centerMass_singleton
theorem Finset.centerMass_singleton (hw : w i ≠ 0) : ({i} : Finset ι).centerMass w z = z i 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (i : ι) {w : ι → R} (z : ι → E) (hw : w i ≠ 0)
Goal: {i}.centerMass w z = z i
-- Finset.centerMass_neg_left
@[simp] lemma Finset.centerMass_neg_left : t.centerMass (-w) z = t.centerMass w z 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (t : Finset ι) {w : ι → R} (z : ι → E)
Goal: t.centerMass (-w) z = t.centerMass w z
-- Finset.centerMass_smul_left
lemma Finset.centerMass_smul_left {c : R'} [Module R' R] [Module R' E] [SMulCommClass R' R R]
    [IsScalarTower R' R R] [SMulCommClass R R' E] [IsScalarTower R' R E] (hc : c ≠ 0) :
    t.centerMass (c • w) z = t.centerMass w z 
Arguments: {R : Type u_1} {R' : Type u_2} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [LinearOrderedField R'] [AddCommGroup E] [Module R E] (t : Finset ι) {w : ι → R} (z : ι → E) {c : R'} [Module R' R] [Module R' E] [SMulCommClass R' R R] [IsScalarTower R' R R] [SMulCommClass R R' E] [IsScalarTower R' R E] (hc : c ≠ 0)
Goal: t.centerMass (c • w) z = t.centerMass w z
-- Finset.centerMass_eq_of_sum_1
theorem Finset.centerMass_eq_of_sum_1 (hw : ∑ i ∈ t, w i = 1) :
    t.centerMass w z = ∑ i ∈ t, w i • z i 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (t : Finset ι) {w : ι → R} (z : ι → E) (hw : ∑ i ∈ t, w i = 1)
Goal: t.centerMass w z = ∑ i ∈ t, w i • z i
-- Finset.centerMass_smul
theorem Finset.centerMass_smul : (t.centerMass w fun i => c • z i) = c • t.centerMass w z 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (c : R) (t : Finset ι) {w : ι → R} (z : ι → E)
Goal: (t.centerMass w fun i => c • z i) = c • t.centerMass w z
-- Finset.centerMass_segment'
theorem Finset.centerMass_segment' (s : Finset ι) (t : Finset ι') (ws : ι → R) (zs : ι → E)
    (wt : ι' → R) (zt : ι' → E) (hws : ∑ i ∈ s, ws i = 1) (hwt : ∑ i ∈ t, wt i = 1) (a b : R)
    (hab : a + b = 1) : a • s.centerMass ws zs + b • t.centerMass wt zt = (s.disjSum t).centerMass
    (Sum.elim (fun i => a * ws i) fun j => b * wt j) (Sum.elim zs zt) 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} {ι' : Type u_6} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s : Finset ι) (t : Finset ι') (ws : ι → R) (zs : ι → E) (wt : ι' → R) (zt : ι' → E) (hws : ∑ i ∈ s, ws i = 1) (hwt : ∑ i ∈ t, wt i = 1) (a b : R) (hab : a + b = 1)
Goal: a • s.centerMass ws zs + b • t.centerMass wt zt =
  (s.disjSum t).centerMass (Sum.elim (fun i => a * ws i) fun j => b * wt j) (Sum.elim zs zt)
-- Finset.centerMass_segment
theorem Finset.centerMass_segment (s : Finset ι) (w₁ w₂ : ι → R) (z : ι → E)
    (hw₁ : ∑ i ∈ s, w₁ i = 1) (hw₂ : ∑ i ∈ s, w₂ i = 1) (a b : R) (hab : a + b = 1) :
    a • s.centerMass w₁ z + b • s.centerMass w₂ z =
    s.centerMass (fun i => a * w₁ i + b * w₂ i) z 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s : Finset ι) (w₁ w₂ : ι → R) (z : ι → E) (hw₁ : ∑ i ∈ s, w₁ i = 1) (hw₂ : ∑ i ∈ s, w₂ i = 1) (a b : R) (hab : a + b = 1)
Goal: a • s.centerMass w₁ z + b • s.centerMass w₂ z = s.centerMass (fun i => a * w₁ i + b * w₂ i) z
-- Finset.centerMass_ite_eq
theorem Finset.centerMass_ite_eq (hi : i ∈ t) :
    t.centerMass (fun j => if i = j then (1 : R) else 0) z = z i 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (i : ι) (t : Finset ι) (z : ι → E) (hi : i ∈ t)
Goal: t.centerMass (fun j => if i = j then 1 else 0) z = z i
-- Finset.centerMass_subset
theorem Finset.centerMass_subset {t' : Finset ι} (ht : t ⊆ t') (h : ∀ i ∈ t', i ∉ t → w i = 0) :
    t.centerMass w z = t'.centerMass w z 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {t : Finset ι} {w : ι → R} (z : ι → E) {t' : Finset ι} (ht : t ⊆ t') (h : ∀ i ∈ t', i ∉ t → w i = 0)
Goal: t.centerMass w z = t'.centerMass w z
-- Finset.centerMass_filter_ne_zero
theorem Finset.centerMass_filter_ne_zero :
    (t.filter fun i => w i ≠ 0).centerMass w z = t.centerMass w z 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {t : Finset ι} {w : ι → R} (z : ι → E)
Goal: (Finset.filter (fun i => w i ≠ 0) t).centerMass w z = t.centerMass w z
-- Finset.centerMass_le_sup
theorem centerMass_le_sup {s : Finset ι} {f : ι → α} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i)
    (hw₁ : 0 < ∑ i ∈ s, w i) :
    s.centerMass w f ≤ s.sup' (nonempty_of_ne_empty <| by rintro rfl; simp at hw₁) f 
Arguments: {R : Type u_1} {ι : Type u_5} {α : Type u_7} [LinearOrderedField R] [LinearOrderedAddCommGroup α] [Module R α] [OrderedSMul R α] {s : Finset ι} {f : ι → α} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : 0 < ∑ i ∈ s, w i)
Goal: s.centerMass w f ≤ s.sup' ⋯ f
-- Finset.inf_le_centerMass
theorem inf_le_centerMass {s : Finset ι} {f : ι → α} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i)
    (hw₁ : 0 < ∑ i ∈ s, w i) :
    s.inf' (nonempty_of_ne_empty <| by rintro rfl; simp at hw₁) f ≤ s.centerMass w f 
Arguments: {R : Type u_1} {ι : Type u_5} {α : Type u_7} [LinearOrderedField R] [LinearOrderedAddCommGroup α] [Module R α] [OrderedSMul R α] {s : Finset ι} {f : ι → α} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : 0 < ∑ i ∈ s, w i)
Goal: s.inf' ⋯ f ≤ s.centerMass w f
-- Finset.centerMass_of_sum_add_sum_eq_zero
lemma Finset.centerMass_of_sum_add_sum_eq_zero {s t : Finset ι}
    (hw : ∑ i ∈ s, w i + ∑ i ∈ t, w i = 0) (hz : ∑ i ∈ s, w i • z i + ∑ i ∈ t, w i • z i = 0) :
    s.centerMass w z = t.centerMass w z 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {w : ι → R} {z : ι → E} {s t : Finset ι} (hw : ∑ i ∈ s, w i + ∑ i ∈ t, w i = 0) (hz : ∑ i ∈ s, w i • z i + ∑ i ∈ t, w i • z i = 0)
Goal: s.centerMass w z = t.centerMass w z
-- Convex.centerMass_mem
theorem Convex.centerMass_mem (hs : Convex R s) :
    (∀ i ∈ t, 0 ≤ w i) → (0 < ∑ i ∈ t, w i) → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E} {t : Finset ι} {w : ι → R} {z : ι → E} (hs : Convex R s)
Goal: (∀ i ∈ t, 0 ≤ w i) → 0 < ∑ i ∈ t, w i → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s
-- Convex.sum_mem
theorem Convex.sum_mem (hs : Convex R s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1)
    (hz : ∀ i ∈ t, z i ∈ s) : (∑ i ∈ t, w i • z i) ∈ s 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E} {t : Finset ι} {w : ι → R} {z : ι → E} (hs : Convex R s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hz : ∀ i ∈ t, z i ∈ s)
Goal: ∑ i ∈ t, w i • z i ∈ s
-- Convex.finsum_mem
theorem Convex.finsum_mem {ι : Sort*} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s)
    (h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) :
    (∑ᶠ i, w i • z i) ∈ s 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {ι : Sort u_8} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s) (h₀ : ∀ (i : ι), 0 ≤ w i) (h₁ : ∑ᶠ (i : ι), w i = 1) (hz : ∀ (i : ι), w i ≠ 0 → z i ∈ s)
Goal: ∑ᶠ (i : ι), w i • z i ∈ s
-- convex_iff_sum_mem
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x ∈ t, w x • x) ∈ s 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E}
Goal: Convex R s ↔
  ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s
-- Finset.centerMass_mem_convexHull
theorem Finset.centerMass_mem_convexHull (t : Finset ι) {w : ι → R} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
    (hws : 0 < ∑ i ∈ t, w i) {z : ι → E} (hz : ∀ i ∈ t, z i ∈ s) :
    t.centerMass w z ∈ convexHull R s 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E} (t : Finset ι) {w : ι → R} (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hws : 0 < ∑ i ∈ t, w i) {z : ι → E} (hz : ∀ i ∈ t, z i ∈ s)
Goal: t.centerMass w z ∈ (convexHull R) s
-- Finset.centerMass_mem_convexHull_of_nonpos
lemma Finset.centerMass_mem_convexHull_of_nonpos (t : Finset ι) (hw₀ : ∀ i ∈ t, w i ≤ 0)
    (hws : ∑ i ∈ t, w i < 0) (hz : ∀ i ∈ t, z i ∈ s) : t.centerMass w z ∈ convexHull R s 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E} {w : ι → R} {z : ι → E} (t : Finset ι) (hw₀ : ∀ i ∈ t, w i ≤ 0) (hws : ∑ i ∈ t, w i < 0) (hz : ∀ i ∈ t, z i ∈ s)
Goal: t.centerMass w z ∈ (convexHull R) s
-- Finset.centerMass_id_mem_convexHull
theorem Finset.centerMass_id_mem_convexHull (t : Finset E) {w : E → R} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
    (hws : 0 < ∑ i ∈ t, w i) : t.centerMass w id ∈ convexHull R (t : Set E) 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (t : Finset E) {w : E → R} (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hws : 0 < ∑ i ∈ t, w i)
Goal: t.centerMass w id ∈ (convexHull R) ↑t
-- Finset.centerMass_id_mem_convexHull_of_nonpos
lemma Finset.centerMass_id_mem_convexHull_of_nonpos (t : Finset E) {w : E → R}
    (hw₀ : ∀ i ∈ t, w i ≤ 0) (hws : ∑ i ∈ t, w i < 0) :
    t.centerMass w id ∈ convexHull R (t : Set E) 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (t : Finset E) {w : E → R} (hw₀ : ∀ i ∈ t, w i ≤ 0) (hws : ∑ i ∈ t, w i < 0)
Goal: t.centerMass w id ∈ (convexHull R) ↑t
-- affineCombination_eq_centerMass
theorem affineCombination_eq_centerMass {ι : Type*} {t : Finset ι} {p : ι → E} {w : ι → R}
    (hw₂ : ∑ i ∈ t, w i = 1) : t.affineCombination R p w = centerMass t w p 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] {ι : Type u_8} {t : Finset ι} {p : ι → E} {w : ι → R} (hw₂ : ∑ i ∈ t, w i = 1)
Goal: (Finset.affineCombination R t p) w = t.centerMass w p
-- affineCombination_mem_convexHull
theorem affineCombination_mem_convexHull {s : Finset ι} {v : ι → E} {w : ι → R}
    (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : s.sum w = 1) :
    s.affineCombination R v w ∈ convexHull R (range v) 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Finset ι} {v : ι → E} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : s.sum w = 1)
Goal: (Finset.affineCombination R s v) w ∈ (convexHull R) (Set.range v)
-- Finset.centroid_eq_centerMass
@[simp]
theorem Finset.centroid_eq_centerMass (s : Finset ι) (hs : s.Nonempty) (p : ι → E) :
    s.centroid R p = s.centerMass (s.centroidWeights R) p 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s : Finset ι) (hs : s.Nonempty) (p : ι → E)
Goal: Finset.centroid R s p = s.centerMass (Finset.centroidWeights R s) p
-- Finset.centroid_mem_convexHull
theorem Finset.centroid_mem_convexHull (s : Finset E) (hs : s.Nonempty) :
    s.centroid R id ∈ convexHull R (s : Set E) 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s : Finset E) (hs : s.Nonempty)
Goal: Finset.centroid R s id ∈ (convexHull R) ↑s
-- convexHull_range_eq_exists_affineCombination
theorem convexHull_range_eq_exists_affineCombination (v : ι → E) : convexHull R (range v) =
    { x | ∃ (s : Finset ι) (w : ι → R), (∀ i ∈ s, 0 ≤ w i) ∧ s.sum w = 1 ∧
      s.affineCombination R v w = x } 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] (v : ι → E)
Goal: (convexHull R) (Set.range v) = {x | ∃ s w, (∀ i ∈ s, 0 ≤ w i) ∧ s.sum w = 1 ∧ (Finset.affineCombination R s v) w = x}
-- convexHull_eq
theorem convexHull_eq (s : Set E) : convexHull R s =
    { x : E | ∃ (ι : Type) (t : Finset ι) (w : ι → R) (z : ι → E), (∀ i ∈ t, 0 ≤ w i) ∧
      ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x } 
Arguments: {R : Type u_1} {E : Type u_3} [LinearOrderedField R] [AddCommGroup E] [Module R E] (s : Set E)
Goal: (convexHull R) s = {x | ∃ ι t w z, (∀ i ∈ t, 0 ≤ w i) ∧ ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x}
-- mem_convexHull_of_exists_fintype
lemma mem_convexHull_of_exists_fintype {s : Set E} {x : E} [Fintype ι] (w : ι → R) (z : ι → E)
    (hw₀ : ∀ i, 0 ≤ w i) (hw₁ : ∑ i, w i = 1) (hz : ∀ i, z i ∈ s) (hx : ∑ i, w i • z i = x) :
    x ∈ convexHull R s 
Arguments: {R : Type u_1} {E : Type u_3} {ι : Type u_5} [LinearOrderedField R] [AddCommGroup E] [Module R E] {s : Set E} {x : E} [Fintype ι] (w : ι → R) (z : ι → E) (hw₀ : ∀ (i : ι), 0 ≤ w i) (hw₁ : ∑ i : ι, w i = 1) (hz : ∀ (i : ι), z i ∈ s) (hx : ∑ i : ι, w i • z i = x)
Goal: x ∈ (convexHull R) s