l3lab/ntp-mathlib huggingface dataset:
theorem Gamma_conj (s : ℂ) : Gamma (conj s) = conj (Gamma s) := by
  suffices ∀ (n : ℕ) (s : ℂ), GammaAux n (conj s) = conj (GammaAux n s) by
    simp [Gamma, this]
  intro n
  induction' n with n IH
  · rw [GammaAux]; exact GammaIntegral_conj
  · intro s
    rw [GammaAux]
    dsimp only
    

Proof state:
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)

Next tactic:
rw [div_eq_mul_inv _ s, RingHom.map_mul, conj_inv, ← div_eq_mul_inv]
theorem Gamma_conj (s : ℂ) : Gamma (conj s) = conj (Gamma s) := by
  suffices ∀ (n : ℕ) (s : ℂ), GammaAux n (conj s) = conj (GammaAux n s) by
    simp [Gamma, this]
  intro n
  induction' n with n IH
  · rw [GammaAux]; exact GammaIntegral_conj
  · intro s
    rw [GammaAux]
    dsimp only
    rw [div_eq_mul_inv _ s, RingHom.map_mul, conj_inv, ← div_eq_mul_inv]
    

Proof state:
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

Next tactic:
suffices conj s + 1 = conj (s + 1) by rw [this, IH]
theorem Gamma_conj (s : ℂ) : Gamma (conj s) = conj (Gamma s) := by
  suffices ∀ (n : ℕ) (s : ℂ), GammaAux n (conj s) = conj (GammaAux n s) by
    simp [Gamma, this]
  intro n
  induction' n with n IH
  · rw [GammaAux]; exact GammaIntegral_conj
  · intro s
    rw [GammaAux]
    dsimp only
    rw [div_eq_mul_inv _ s, RingHom.map_mul, conj_inv, ← div_eq_mul_inv]
    suffices conj s + 1 = conj (s + 1) by 

Proof state:
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

Next tactic:
rw [this, IH]
theorem Gamma_conj (s : ℂ) : Gamma (conj s) = conj (Gamma s) := by
  suffices ∀ (n : ℕ) (s : ℂ), GammaAux n (conj s) = conj (GammaAux n s) by
    simp [Gamma, this]
  intro n
  induction' n with n IH
  · rw [GammaAux]; exact GammaIntegral_conj
  · intro s
    rw [GammaAux]
    dsimp only
    rw [div_eq_mul_inv _ s, RingHom.map_mul, conj_inv, ← div_eq_mul_inv]
    suffices conj s + 1 = conj (s + 1) by rw [this, IH]
    

Proof state:
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)

Next tactic:
rw [RingHom.map_add, RingHom.map_one]
/-- Rewrite the Gamma integral as an example of a Mellin transform. -/
theorem GammaIntegral_eq_mellin : GammaIntegral = mellin fun x => ↑(Real.exp (-x)) :=
  funext fun s => by 

Proof state:
s : ℂ
⊢ GammaIntegral s = mellin (fun x => ↑(rexp (-x))) s

Next tactic:
simp only [mellin, GammaIntegral, smul_eq_mul, mul_comm]
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  

Proof state:
s : ℂ
hs : 0 < s.re
⊢ HasDerivAt GammaIntegral (∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) * (↑(Real.log t) * ↑(rexp (-t)))) s

Next tactic:
rw [GammaIntegral_eq_mellin]
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  

Proof state:
s : ℂ
hs : 0 < s.re
⊢ HasDerivAt (mellin fun x => ↑(rexp (-x))) (∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) * (↑(Real.log t) * ↑(rexp (-t)))) s

Next tactic:
convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · 

Proof state:
case convert_2
s : ℂ
hs : 0 < s.re
⊢ LocallyIntegrableOn (fun x => ↑(rexp (-x))) (Ioi 0)

Next tactic:
refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    

Proof state:
case convert_2
s : ℂ
hs : 0 < s.re
⊢ Continuous fun x => ↑(rexp (-x))

Next tactic:
exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
  · 

Proof state:
case convert_3
s : ℂ
hs : 0 < s.re
⊢ (fun x => ↑(rexp (-x))) =O[atTop] fun x => x ^ (-(s.re + 1))

Next tactic:
rw [← isBigO_norm_left]
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
  · rw [← isBigO_norm_left]
    

Proof state:
case convert_3
s : ℂ
hs : 0 < s.re
⊢ (fun x => ‖↑(rexp (-x))‖) =O[atTop] fun x => x ^ (-(s.re + 1))

Next tactic:
simp_rw [Complex.norm_eq_abs, abs_ofReal, ← Real.norm_eq_abs, isBigO_norm_left]
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
  · rw [← isBigO_norm_left]
    simp_rw [Complex.norm_eq_abs, abs_ofReal, ← Real.norm_eq_abs, isBigO_norm_left]
    

Proof state:
case convert_3
s : ℂ
hs : 0 < s.re
⊢ (fun x => rexp (-x)) =O[atTop] fun x => x ^ (-(s.re + 1))

Next tactic:
simpa only [neg_one_mul] using (isLittleO_exp_neg_mul_rpow_atTop zero_lt_one _).isBigO
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
  · rw [← isBigO_norm_left]
    simp_rw [Complex.norm_eq_abs, abs_ofReal, ← Real.norm_eq_abs, isBigO_norm_left]
    simpa only [neg_one_mul] using (isLittleO_exp_neg_mul_rpow_atTop zero_lt_one _).isBigO
  · 

Proof state:
case convert_4
s : ℂ
hs : 0 < s.re
⊢ (fun x => ↑(rexp (-x))) =O[𝓝[>] 0] fun x => x ^ (-0)

Next tactic:
simp_rw [neg_zero, rpow_zero]
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
  · rw [← isBigO_norm_left]
    simp_rw [Complex.norm_eq_abs, abs_ofReal, ← Real.norm_eq_abs, isBigO_norm_left]
    simpa only [neg_one_mul] using (isLittleO_exp_neg_mul_rpow_atTop zero_lt_one _).isBigO
  · simp_rw [neg_zero, rpow_zero]
    

Proof state:
case convert_4
s : ℂ
hs : 0 < s.re
⊢ (fun x => ↑(rexp (-x))) =O[𝓝[>] 0] fun x => 1

Next tactic:
refine' isBigO_const_of_tendsto (_ : Tendsto _ _ (𝓝 (1 : ℂ))) one_ne_zero
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
  · rw [← isBigO_norm_left]
    simp_rw [Complex.norm_eq_abs, abs_ofReal, ← Real.norm_eq_abs, isBigO_norm_left]
    simpa only [neg_one_mul] using (isLittleO_exp_neg_mul_rpow_atTop zero_lt_one _).isBigO
  · simp_rw [neg_zero, rpow_zero]
    refine' isBigO_const_of_tendsto (_ : Tendsto _ _ (𝓝 (1 : ℂ))) one_ne_zero
    

Proof state:
case convert_4
s : ℂ
hs : 0 < s.re
⊢ Tendsto (fun x => ↑(rexp (-x))) (𝓝[>] 0) (𝓝 1)

Next tactic:
rw [(by simp : (1 : ℂ) = Real.exp (-0))]
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
  · rw [← isBigO_norm_left]
    simp_rw [Complex.norm_eq_abs, abs_ofReal, ← Real.norm_eq_abs, isBigO_norm_left]
    simpa only [neg_one_mul] using (isLittleO_exp_neg_mul_rpow_atTop zero_lt_one _).isBigO
  · simp_rw [neg_zero, rpow_zero]
    refine' isBigO_const_of_tendsto (_ : Tendsto _ _ (𝓝 (1 : ℂ))) one_ne_zero
    rw [(by 

Proof state:
s : ℂ
hs : 0 < s.re
⊢ 1 = ↑(rexp (-0))

Next tactic:
simp
/-- The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. -/
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 := by
  rw [GammaIntegral_eq_mellin]
  convert (mellin_hasDerivAt_of_isBigO_rpow (E := ℂ) _ _ (lt_add_one _) _ hs).2
  · refine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi
    exact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)
  · rw [← isBigO_norm_left]
    simp_rw [Complex.norm_eq_abs, abs_ofReal, ← Real.norm_eq_abs, isBigO_norm_left]
    simpa only [neg_one_mul] using (isLittleO_exp_neg_mul_rpow_atTop zero_lt_one _).isBigO
  · simp_rw [neg_zero, rpow_zero]
    refine' isBigO_const_of_tendsto (_ : Tendsto _ _ (𝓝 (1 : ℂ))) one_ne_zero
    rw [(by simp : (1 : ℂ) = Real.exp (-0))]
    

Proof state:
case convert_4
s : ℂ
hs : 0 < s.re
⊢ Tendsto (fun x => ↑(rexp (-x))) (𝓝[>] 0) (𝓝 ↑(rexp (-0)))

Next tactic:
exact (continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)).continuousWithinAt
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  

Proof state:
s : ℂ
n : ℕ
h1 : 1 - s.re < ↑n
h2 : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ (GammaAux n) s

Next tactic:
induction' n with n hn generalizing s
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · 

Proof state:
case zero
s : ℂ
h1 : 1 - s.re < ↑Nat.zero
h2 : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ (GammaAux Nat.zero) s

Next tactic:
refine' (hasDerivAt_GammaIntegral _).differentiableAt
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    

Proof state:
case zero
s : ℂ
h1 : 1 - s.re < ↑Nat.zero
h2 : ∀ (m : ℕ), s ≠ -↑m
⊢ 0 < s.re

Next tactic:
rw [Nat.cast_zero] at h1
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; 

Proof state:
case zero
s : ℂ
h1 : 1 - s.re < 0
h2 : ∀ (m : ℕ), s ≠ -↑m
⊢ 0 < s.re

Next tactic:
linarith
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · 

Proof state:
case succ
n : ℕ
hn : ∀ (s : ℂ), 1 - s.re < ↑n → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) s
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ (GammaAux (Nat.succ n)) s

Next tactic:
dsimp only [GammaAux]
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    

Proof state:
case succ
n : ℕ
hn : ∀ (s : ℂ), 1 - s.re < ↑n → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) s
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ (fun s => GammaAux n (s + 1) / s) s

Next tactic:
specialize hn (s + 1)
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    

Proof state:
case succ
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
⊢ DifferentiableAt ℂ (fun s => GammaAux n (s + 1) / s) s

Next tactic:
have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; linarith
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      

Proof state:
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
⊢ 1 - (s + 1).re < ↑n

Next tactic:
rw [Nat.cast_succ] at h1
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; 

Proof state:
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑n + 1
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
⊢ 1 - (s + 1).re < ↑n

Next tactic:
rw [Complex.add_re, Complex.one_re]
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; 

Proof state:
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑n + 1
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
⊢ 1 - (s.re + 1) < ↑n

Next tactic:
linarith
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; linarith
    

Proof state:
case succ
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
a : 1 - (s + 1).re < ↑n
⊢ DifferentiableAt ℂ (fun s => GammaAux n (s + 1) / s) s

Next tactic:
have b : ∀ m : ℕ, s + 1 ≠ -m := by
      intro m; have := h2 (1 + m)
      contrapose! this
      rw [← eq_sub_iff_add_eq] at this
      simpa using this
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; linarith
    have b : ∀ m : ℕ, s + 1 ≠ -m := by
      

Proof state:
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
a : 1 - (s + 1).re < ↑n
⊢ ∀ (m : ℕ), s + 1 ≠ -↑m

Next tactic:
intro m
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; linarith
    have b : ∀ m : ℕ, s + 1 ≠ -m := by
      intro m; 

Proof state:
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
a : 1 - (s + 1).re < ↑n
m : ℕ
⊢ s + 1 ≠ -↑m

Next tactic:
have := h2 (1 + m)
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; linarith
    have b : ∀ m : ℕ, s + 1 ≠ -m := by
      intro m; have := h2 (1 + m)
      

Proof state:
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
a : 1 - (s + 1).re < ↑n
m : ℕ
this : s ≠ -↑(1 + m)
⊢ s + 1 ≠ -↑m

Next tactic:
contrapose! this
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; linarith
    have b : ∀ m : ℕ, s + 1 ≠ -m := by
      intro m; have := h2 (1 + m)
      contrapose! this
      

Proof state:
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
a : 1 - (s + 1).re < ↑n
m : ℕ
this : s + 1 = -↑m
⊢ s = -↑(1 + m)

Next tactic:
rw [← eq_sub_iff_add_eq] at this
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; linarith
    have b : ∀ m : ℕ, s + 1 ≠ -m := by
      intro m; have := h2 (1 + m)
      contrapose! this
      rw [← eq_sub_iff_add_eq] at this
      

Proof state:
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
a : 1 - (s + 1).re < ↑n
m : ℕ
this : s = -↑m - 1
⊢ s = -↑(1 + m)

Next tactic:
simpa using this
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
    DifferentiableAt ℂ (GammaAux n) s := by
  induction' n with n hn generalizing s
  · refine' (hasDerivAt_GammaIntegral _).differentiableAt
    rw [Nat.cast_zero] at h1; linarith
  · dsimp only [GammaAux]
    specialize hn (s + 1)
    have a : 1 - (s + 1).re < ↑n := by
      rw [Nat.cast_succ] at h1; rw [Complex.add_re, Complex.one_re]; linarith
    have b : ∀ m : ℕ, s + 1 ≠ -m := by
      intro m; have := h2 (1 + m)
      contrapose! this
      rw [← eq_sub_iff_add_eq] at this
      simpa using this
    

Proof state:
case succ
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑(Nat.succ n)
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
a : 1 - (s + 1).re < ↑n
b : ∀ (m : ℕ), s + 1 ≠ -↑m
⊢ DifferentiableAt ℂ (fun s => GammaAux n (s + 1) / s) s

Next tactic:
refine' DifferentiableAt.div (DifferentiableAt.comp _ (hn a b) _) _ _