-- Mathlib/Analysis/SpecialFunctions/OrdinaryHypergeometric.lean
theorem ordinaryHypergeometricSeries_norm_div_succ_norm (n : ℕ)
    (habc : ∀ kn < n, (↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c)) :
    ‖ordinaryHypergeometricCoefficient a b c n‖ / ‖ordinaryHypergeometricCoefficient a b c n.succ‖ =
      ‖a + n‖⁻¹ * ‖b + n‖⁻¹ * ‖c + n‖ * ‖1 + (n : 𝕂)‖

Proof state:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ↑n ! ≠ 0

Tactic:
exact cast_ne_zero.2 (factorial_ne_zero n)

Result:
no goals

Proof state:
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖↑n !‖⁻¹ * ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ *
        ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹ /
      (‖↑n !‖⁻¹ * ‖↑n + 1‖⁻¹ * (‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖a + ↑n‖) *
          (‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ * ‖b + ↑n‖) *
        (‖c + ↑n‖⁻¹ * ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹)) =
    ‖a + ↑n‖⁻¹ * ‖b + ↑n‖⁻¹ * ‖c + ↑n‖ * ‖1 + ↑n‖

Tactic:
calc
  _ = ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖⁻¹ *
      ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖⁻¹ *
      ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹⁻¹ * ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹ *
      ‖(n ! : 𝕂)‖⁻¹⁻¹ * ‖(n ! : 𝕂)‖⁻¹ * ‖a + n‖⁻¹ * ‖b + n‖⁻¹ * ‖c + n‖⁻¹⁻¹ *
      ‖1 + (n : 𝕂)‖⁻¹⁻¹ := by ring_nf
  _ = _ := by
    simp only [inv_inv]
    repeat rw [DivisionRing.mul_inv_cancel, one_mul]
    all_goals
      rw [norm_ne_zero_iff]
    any_goals
      apply (ascPochhammer_eval_eq_zero_iff n _).not.2
      push_neg
      exact fun kn hkn ↦ by simp [habc kn hkn]
    exact cast_ne_zero.2 (factorial_ne_zero n)

Result:
no goals

Proof state:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ Polynomial.eval a (ascPochhammer 𝕂 n) ≠ 0

Tactic:
apply (ascPochhammer_eval_eq_zero_iff n _).not.2

Result:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ¬∃ k < n, ↑k = -a

Proof state:
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖⁻¹ *
                        ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ *
                      ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖⁻¹ *
                    ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖ *
                  ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹ *
                ‖↑n !‖ *
              ‖↑n !‖⁻¹ *
            ‖a + ↑n‖⁻¹ *
          ‖b + ↑n‖⁻¹ *
        ‖c + ↑n‖ *
      ‖1 + ↑n‖ =
    ‖a + ↑n‖⁻¹ * ‖b + ↑n‖⁻¹ * ‖c + ↑n‖ * ‖1 + ↑n‖

Tactic:
repeat rw [DivisionRing.mul_inv_cancel, one_mul]

Result:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖↑n !‖ ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖ ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ ≠ 0

Proof state:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖↑n !‖ ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖ ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ ≠ 0

Tactic:
all_goals
  rw [norm_ne_zero_iff]

Result:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ↑n ! ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ Polynomial.eval c (ascPochhammer 𝕂 n) ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ Polynomial.eval b (ascPochhammer 𝕂 n) ≠ 0

case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ Polynomial.eval a (ascPochhammer 𝕂 n) ≠ 0

Proof state:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ¬∃ k < n, ↑k = -a

Tactic:
push_neg

Result:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ∀ k < n, ↑k ≠ -a

Proof state:
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖⁻¹ *
                        ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ *
                      ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖⁻¹ *
                    ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹⁻¹ *
                  ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹ *
                ‖↑n !‖⁻¹⁻¹ *
              ‖↑n !‖⁻¹ *
            ‖a + ↑n‖⁻¹ *
          ‖b + ↑n‖⁻¹ *
        ‖c + ↑n‖⁻¹⁻¹ *
      ‖1 + ↑n‖⁻¹⁻¹ =
    ‖a + ↑n‖⁻¹ * ‖b + ↑n‖⁻¹ * ‖c + ↑n‖ * ‖1 + ↑n‖

Tactic:
simp only [inv_inv]

Result:
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖⁻¹ *
                        ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ *
                      ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖⁻¹ *
                    ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖ *
                  ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹ *
                ‖↑n !‖ *
              ‖↑n !‖⁻¹ *
            ‖a + ↑n‖⁻¹ *
          ‖b + ↑n‖⁻¹ *
        ‖c + ↑n‖ *
      ‖1 + ↑n‖ =
    ‖a + ↑n‖⁻¹ * ‖b + ↑n‖⁻¹ * ‖c + ↑n‖ * ‖1 + ↑n‖

Proof state:
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
kn : ℕ
hkn : kn < n
⊢ ↑kn ≠ -a

Tactic:
simp [habc kn hkn]

Result:
no goals

Proof state:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ ≠ 0

Tactic:
rw [norm_ne_zero_iff]

Result:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ Polynomial.eval a (ascPochhammer 𝕂 n) ≠ 0

Proof state:
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖ordinaryHypergeometricCoefficient a b c n‖ / ‖ordinaryHypergeometricCoefficient a b c n.succ‖ =
    ‖a + ↑n‖⁻¹ * ‖b + ↑n‖⁻¹ * ‖c + ↑n‖ * ‖1 + ↑n‖

Tactic:
simp only [mul_inv_rev, factorial_succ, cast_mul, cast_add,
  cast_one, ascPochhammer_succ_eval, norm_mul, norm_inv]

Result:
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ‖↑n !‖⁻¹ * ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ *
        ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹ /
      (‖↑n !‖⁻¹ * ‖↑n + 1‖⁻¹ * (‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖a + ↑n‖) *
          (‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ * ‖b + ↑n‖) *
        (‖c + ↑n‖⁻¹ * ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹)) =
    ‖a + ↑n‖⁻¹ * ‖b + ↑n‖⁻¹ * ‖c + ↑n‖ * ‖1 + ↑n‖

Proof state:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ∀ k < n, ↑k ≠ -a

Tactic:
exact fun kn hkn ↦ by simp [habc kn hkn]

Result:
no goals