-- Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
/-- Explicit formula for the derivative of the Gamma function at positive integers, in terms of
harmonic numbers and the Euler-Mascheroni constant `γ`. -/
lemma deriv_Gamma_nat (n : ℕ) :
    deriv Gamma (n + 1) = n ! * (-γ + harmonic n)

Proof state:
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
this : -deriv f 1 = γ
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)

Tactic:
rw [hder_nat n, ← this, neg_neg]

Result:
no goals

Proof state:
case a
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ γ ≤ -deriv f 1

Tactic:
apply le_of_tendsto tendsto_harmonic_sub_log_add_one

Result:
case a
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ ∀ᶠ (c : ℕ) in atTop, ↑(harmonic c) - log (↑c + 1) ≤ -deriv f 1

Proof state:
case succ
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
n : ℕ
hn : deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
⊢ deriv f 1 + ↑(harmonic n) + 1 / (↑n + 1) = deriv f 1 + ↑(harmonic n + (↑(n + 1))⁻¹)

Tactic:
push_cast

Result:
case succ
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
n : ℕ
hn : deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
⊢ deriv f 1 + ↑(harmonic n) + 1 / (↑n + 1) = deriv f 1 + (↑(harmonic n) + (↑n + 1)⁻¹)

Proof state:
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ -deriv f 1 = γ

Tactic:
apply le_antisymm

Result:
case a
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ -deriv f 1 ≤ γ

case a
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ γ ≤ -deriv f 1

Proof state:
case h
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
n : ℕ
hn : 0 < n
⊢ -deriv f 1 ≤ ↑(harmonic n) - log ↑n

Tactic:
rw [le_sub_iff_add_le', ← sub_eq_add_neg, sub_le_iff_le_add', ← hder_nat]

Result:
case h
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
n : ℕ
hn : 0 < n
⊢ log ↑n ≤ deriv f (↑n + 1)

Proof state:
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
n : ℕ
⊢ ↑n + 2 = ↑n + 1 + 1

Tactic:
ring

Result:
no goals

Proof state:
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
x : ℝ
hx : 0 < x
⊢ 0 < x

Tactic:
positivity

Result:
no goals

Proof state:
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
x : ℝ
hx : 0 < x
⊢ deriv f (x + 1) = deriv f x + 1 / x

Tactic:
rw [← deriv_comp_add_const, one_div, ← deriv_log,
  ← deriv_add (hder <| by positivity) (differentiableAt_log hx.ne')]

Result:
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
x : ℝ
hx : 0 < x
⊢ deriv (fun x => f (x + 1)) x = deriv (fun y => f y + log y) x

Proof state:
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
n : ℕ
hn : 0 < n
⊢ log ↑n = slope f (↑n) (↑n + 1)

Tactic:
rw [slope_def_field, show n + 1 - n = (1 : ℝ) by ring, div_one, h_rec n (by positivity),
  add_sub_cancel_left]

Result:
no goals

Proof state:
case h
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
n : ℕ
hn : 0 < n
⊢ log ↑n ≤ deriv f (↑n + 1)

Tactic:
exact derivLB n hn

Result:
no goals

Proof state:
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
this : deriv (fun x => log (Gamma x)) (↑n + 1) = -γ + ↑(harmonic n)
m : ℕ
⊢ ↑n + 1 ≠ -↑m

Tactic:
linarith

Result:
no goals

Proof state:
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
x : ℝ
hx : 0 < x
⊢ DifferentiableAt ℝ f x

Tactic:
refine ((differentiableAt_Gamma ?_).log (Gamma_ne_zero ?_)) <;>
exact fun m ↦ ne_of_gt (by linarith)

Result:
no goals

Proof state:
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
n : ℕ
⊢ deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)

Tactic:
induction n with
| zero => simp
| succ n hn =>
  rw [cast_succ, hder_rec (n + 1) (by positivity), hn, harmonic_succ]
  push_cast
  ring

Result:
no goals