Theorem: Complex.Gamma_conj
File: Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean

tactic: suffices ∀ (n : ℕ) (s : ℂ), GammaAux n (conj s) = conj (GammaAux n s) by
  simp [Gamma, this]
state_before: s : ℂ
⊢ Gamma ((starRingEnd ℂ) s) = (starRingEnd ℂ) (Gamma s)
state_after: s : ℂ
⊢ ∀ (n : ℕ) (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)

tactic: intro n
state_before: s : ℂ
⊢ ∀ (n : ℕ) (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
state_after: s : ℂ
n : ℕ
⊢ ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)

tactic: induction' n with n IH
state_before: s : ℂ
n : ℕ
⊢ ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
state_after: case zero
s : ℂ
⊢ ∀ (s : ℂ), GammaAux 0 ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux 0 s)

case succ
s : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
⊢ ∀ (s : ℂ), GammaAux (n + 1) ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux (n + 1) s)

tactic: simp [Gamma, this]
state_before: s : ℂ
this : ∀ (n : ℕ) (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
⊢ Gamma ((starRingEnd ℂ) s) = (starRingEnd ℂ) (Gamma s)
state_after: no goals

tactic: rw [GammaAux]
state_before: case zero
s : ℂ
⊢ ∀ (s : ℂ), GammaAux 0 ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux 0 s)
state_after: case zero
s : ℂ
⊢ ∀ (s : ℂ), ((starRingEnd ℂ) s).GammaIntegral = (starRingEnd ℂ) s.GammaIntegral

tactic: exact GammaIntegral_conj
state_before: case zero
s : ℂ
⊢ ∀ (s : ℂ), ((starRingEnd ℂ) s).GammaIntegral = (starRingEnd ℂ) s.GammaIntegral
state_after: no goals

tactic: intro s
state_before: case succ
s : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
⊢ ∀ (s : ℂ), GammaAux (n + 1) ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux (n + 1) s)
state_after: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ GammaAux (n + 1) ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux (n + 1) s)

tactic: rw [GammaAux]
state_before: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ GammaAux (n + 1) ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux (n + 1) s)
state_after: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (fun s => GammaAux n (s + 1) / s) ((starRingEnd ℂ) s) = (starRingEnd ℂ) ((fun s => GammaAux n (s + 1) / s) s)

tactic: dsimp only
state_before: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (fun s => GammaAux n (s + 1) / s) ((starRingEnd ℂ) s) = (starRingEnd ℂ) ((fun s => GammaAux n (s + 1) / s) s)
state_after: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ GammaAux n ((starRingEnd ℂ) s + 1) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1) / s)

tactic: rw [div_eq_mul_inv _ s, RingHom.map_mul, conj_inv, ← div_eq_mul_inv]
state_before: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ GammaAux n ((starRingEnd ℂ) s + 1) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1) / s)
state_after: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ GammaAux n ((starRingEnd ℂ) s + 1) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s

tactic: suffices conj s + 1 = conj (s + 1) by rw [this, IH]
state_before: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ GammaAux n ((starRingEnd ℂ) s + 1) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s
state_after: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) (s + 1)

tactic: rw [RingHom.map_add, RingHom.map_one]
state_before: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) (s + 1)
state_after: no goals

tactic: rw [this, IH]
state_before: s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
this : (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) (s + 1)
⊢ GammaAux n ((starRingEnd ℂ) s + 1) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s
state_after: no goals