-- Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
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)

Proof state:
case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < (↑n.succ).re

Tactic:
rw [← ofReal_natCast, ofReal_re]

Result:
case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < ↑n.succ

Proof state:
case zero
u : ℂ
hu : 0 < u.re
⊢ u.betaIntegral (↑0 + 1) = ↑0! / ∏ j ∈ Finset.range (0 + 1), (u + ↑j)

Tactic:
rw [Nat.cast_zero, zero_add, betaIntegral_eval_one_right hu, Nat.factorial_zero, Nat.cast_one]

Result:
case zero
u : ℂ
hu : 0 < u.re
⊢ 1 / u = 1 / ∏ j ∈ Finset.range (0 + 1), (u + ↑j)

Proof state:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u * u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ
⊢ u.betaIntegral (↑(n + 1) + 1) = ↑(n + 1)! / ∏ j ∈ Finset.range (n + 1 + 1), (u + ↑j)

case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < (↑n.succ).re

Tactic:
swap

Result:
case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < (↑n.succ).re

case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u * u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ
⊢ u.betaIntegral (↑(n + 1) + 1) = ↑(n + 1)! / ∏ j ∈ Finset.range (n + 1 + 1), (u + ↑j)

Proof state:
case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < ↑n.succ

Tactic:
positivity

Result:
no goals

Proof state:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ (↑n + 1) * (↑n ! / ∏ j ∈ Finset.range (n + 1), (u + 1 + ↑j)) / u =
    ↑(n + 1)! / ((∏ k ∈ Finset.range (n + 1), (u + ↑(k + 1))) * (u + ↑0))

Tactic:
rw [Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, Nat.cast_zero, add_zero, ←
  mul_div_assoc, ← div_div]

Result:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ ((↑n + 1) * ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + 1 + ↑j)) / u =
    ((↑n + 1) * ↑n ! / ∏ k ∈ Finset.range (n + 1), (u + ↑(k + 1))) / u

Proof state:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ u.betaIntegral (↑(n + 1) + 1) = ↑(n + 1)! / ∏ j ∈ Finset.range (n + 1 + 1), (u + ↑j)

case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) * u = ↑n.succ * (u + 1).betaIntegral ↑n.succ
⊢ u ≠ 0

Tactic:
swap

Result:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) * u = ↑n.succ * (u + 1).betaIntegral ↑n.succ
⊢ u ≠ 0

case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ u.betaIntegral (↑(n + 1) + 1) = ↑(n + 1)! / ∏ j ∈ Finset.range (n + 1 + 1), (u + ↑j)

Proof state:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ 0 < u.re + 1

Tactic:
positivity

Result:
no goals

Proof state:
case succ.refine_2.e_a.e_a.e_f.h
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
j : ℕ
⊢ u + 1 + ↑j = u + (↑j + 1)

Tactic:
abel

Result:
no goals

Proof state:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ (↑n + 1) * (↑n ! / ∏ j ∈ Finset.range (n + 1), (u + 1 + ↑j)) / u =
    ↑(n + 1)! / ((∏ k ∈ Finset.range (n + 1), (u + ↑(k + 1))) * (u + ↑0))

case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ 0 < (u + 1).re

Tactic:
swap

Result:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ 0 < (u + 1).re

case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ (↑n + 1) * (↑n ! / ∏ j ∈ Finset.range (n + 1), (u + 1 + ↑j)) / u =
    ↑(n + 1)! / ((∏ k ∈ Finset.range (n + 1), (u + ↑(k + 1))) * (u + ↑0))

Proof state:
case succ.refine_2.e_a.e_a.e_f.h
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
j : ℕ
⊢ u + 1 + ↑j = u + ↑(j + 1)

Tactic:
push_cast

Result:
case succ.refine_2.e_a.e_a.e_f.h
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
j : ℕ
⊢ u + 1 + ↑j = u + (↑j + 1)

Proof state:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ 0 < (u + 1).re

Tactic:
rw [add_re, one_re]

Result:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ 0 < u.re + 1

Proof state:
case succ
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ u.betaIntegral (↑(n + 1) + 1) = ↑(n + 1)! / ∏ j ∈ Finset.range (n + 1 + 1), (u + ↑j)

Tactic:
have := betaIntegral_recurrence hu (?_ : 0 < re n.succ)

Result:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u * u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ
⊢ u.betaIntegral (↑(n + 1) + 1) = ↑(n + 1)! / ∏ j ∈ Finset.range (n + 1 + 1), (u + ↑j)

case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < (↑n.succ).re