l3lab/ntp-mathlib huggingface dataset:
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
⊢ u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v

Next tactic:
have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by 

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
⊢ 0 < (v + 1).re

Next tactic:
rw [add_re, one_re]
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; 

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
⊢ 0 < v.re + 1

Next tactic:
positivity
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
⊢ u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v

Next tactic:
have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      exact hx.2
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
⊢ ContinuousOn F (Icc 0 1)

Next tactic:
refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · 

Proof state:
case refine'_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
x : ℝ
hx : x ∈ Icc 0 1
⊢ ContinuousAt (fun x => ↑x ^ u) x

Next tactic:
refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      

Proof state:
case refine'_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
x : ℝ
hx : x ∈ Icc 0 1
⊢ 0 ≤ (↑x).re

Next tactic:
rw [ofReal_re]
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; 

Proof state:
case refine'_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
x : ℝ
hx : x ∈ Icc 0 1
⊢ 0 ≤ x

Next tactic:
exact hx.1
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · 

Proof state:
case refine'_2
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
x : ℝ
hx : x ∈ Icc 0 1
⊢ ContinuousAt (fun x => (1 - ↑x) ^ v) x

Next tactic:
refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      

Proof state:
case refine'_2
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
x : ℝ
hx : x ∈ Icc 0 1
⊢ 0 ≤ (1 - ↑x).re

Next tactic:
rw [sub_re, one_re, ofReal_re, sub_nonneg]
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      

Proof state:
case refine'_2
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
x : ℝ
hx : x ∈ Icc 0 1
⊢ x ≤ 1

Next tactic:
exact hx.2
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      exact hx.2
  

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
⊢ u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v

Next tactic:
have hder : ∀ x : ℝ, x ∈ Ioo (0 : ℝ) 1 →
      HasDerivAt F (u * ((x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ v) -
        v * ((x : ℂ) ^ u * (1 - (x : ℂ)) ^ (v - 1))) x := by
    intro x hx
    have U : HasDerivAt (fun y : ℂ => y ^ u) (u * (x : ℂ) ^ (u - 1)) ↑x := by
      have := @HasDerivAt.cpow_const _ _ _ u (hasDerivAt_id (x : ℂ)) (Or.inl ?_)
      simp only [id_eq, mul_one] at this
      · exact this
      · rw [id_eq, ofReal_re]; exact hx.1
    have V : HasDerivAt (fun y : ℂ => (1 - y) ^ v) (-v * (1 - (x : ℂ)) ^ (v - 1)) ↑x := by
      have A := @HasDerivAt.cpow_const _ _ _ v (hasDerivAt_id (1 - (x : ℂ))) (Or.inl ?_)
      swap; · rw [id.def, sub_re, one_re, ofReal_re, sub_pos]; exact hx.2
      simp_rw [id.def] at A
      have B : HasDerivAt (fun y : ℂ => 1 - y) (-1) ↑x := by
        apply HasDerivAt.const_sub; apply hasDerivAt_id
      convert HasDerivAt.comp (↑x) A B using 1
      ring
    convert (U.mul V).comp_ofReal using 1
    ring
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      exact hx.2
  have hder : ∀ x : ℝ, x ∈ Ioo (0 : ℝ) 1 →
      HasDerivAt F (u * ((x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ v) -
        v * ((x : ℂ) ^ u * (1 - (x : ℂ)) ^ (v - 1))) x := by
    

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
⊢ ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x

Next tactic:
intro x hx
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      exact hx.2
  have hder : ∀ x : ℝ, x ∈ Ioo (0 : ℝ) 1 →
      HasDerivAt F (u * ((x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ v) -
        v * ((x : ℂ) ^ u * (1 - (x : ℂ)) ^ (v - 1))) x := by
    intro x hx
    

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
⊢ HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x

Next tactic:
have U : HasDerivAt (fun y : ℂ => y ^ u) (u * (x : ℂ) ^ (u - 1)) ↑x := by
      have := @HasDerivAt.cpow_const _ _ _ u (hasDerivAt_id (x : ℂ)) (Or.inl ?_)
      simp only [id_eq, mul_one] at this
      · exact this
      · rw [id_eq, ofReal_re]; exact hx.1
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      exact hx.2
  have hder : ∀ x : ℝ, x ∈ Ioo (0 : ℝ) 1 →
      HasDerivAt F (u * ((x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ v) -
        v * ((x : ℂ) ^ u * (1 - (x : ℂ)) ^ (v - 1))) x := by
    intro x hx
    have U : HasDerivAt (fun y : ℂ => y ^ u) (u * (x : ℂ) ^ (u - 1)) ↑x := by
      

Proof state:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
⊢ HasDerivAt (fun y => y ^ u) (u * ↑x ^ (u - 1)) ↑x

Next tactic:
have := @HasDerivAt.cpow_const _ _ _ u (hasDerivAt_id (x : ℂ)) (Or.inl ?_)
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      exact hx.2
  have hder : ∀ x : ℝ, x ∈ Ioo (0 : ℝ) 1 →
      HasDerivAt F (u * ((x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ v) -
        v * ((x : ℂ) ^ u * (1 - (x : ℂ)) ^ (v - 1))) x := by
    intro x hx
    have U : HasDerivAt (fun y : ℂ => y ^ u) (u * (x : ℂ) ^ (u - 1)) ↑x := by
      have := @HasDerivAt.cpow_const _ _ _ u (hasDerivAt_id (x : ℂ)) (Or.inl ?_)
      

Proof state:
case refine_2
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
this : HasDerivAt (fun x => id x ^ u) (u * id ↑x ^ (u - 1) * 1) ↑x
⊢ HasDerivAt (fun y => y ^ u) (u * ↑x ^ (u - 1)) ↑x
case refine_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
⊢ 0 < (id ↑x).re

Next tactic:
simp only [id_eq, mul_one] at this
/-- Recurrence formula for the Beta function. -/
theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
    u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by
  -- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
  -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but
  -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
  let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v
  have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity
  have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity
  have hc : ContinuousOn F (Icc 0 1) := by
    refine' (ContinuousAt.continuousOn fun x hx => _).mul (ContinuousAt.continuousOn fun x hx => _)
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hu).comp continuous_ofReal.continuousAt
      rw [ofReal_re]; exact hx.1
    · refine' (continuousAt_cpow_const_of_re_pos (Or.inl _) hv).comp
        (continuous_const.sub continuous_ofReal).continuousAt
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      exact hx.2
  have hder : ∀ x : ℝ, x ∈ Ioo (0 : ℝ) 1 →
      HasDerivAt F (u * ((x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ v) -
        v * ((x : ℂ) ^ u * (1 - (x : ℂ)) ^ (v - 1))) x := by
    intro x hx
    have U : HasDerivAt (fun y : ℂ => y ^ u) (u * (x : ℂ) ^ (u - 1)) ↑x := by
      have := @HasDerivAt.cpow_const _ _ _ u (hasDerivAt_id (x : ℂ)) (Or.inl ?_)
      simp only [id_eq, mul_one] at this
      · 

Proof state:
case refine_2
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
this : HasDerivAt (fun x => x ^ u) (u * ↑x ^ (u - 1)) ↑x
⊢ HasDerivAt (fun y => y ^ u) (u * ↑x ^ (u - 1)) ↑x

Next tactic:
exact this