-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
theorem f_add_nat_ge (hf_conv : ConvexOn ℝ (Ioi 0) f)
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hn : 2 ≤ n) (hx : 0 < x) :
    f n + x * log (n - 1) ≤ f (n + x)

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : f ↑n - f (↑n - 1) ≤ (f (↑n + x) - f ↑n) / x
⊢ f (↑n - 1) = f ↑n - log (↑n - 1)

Tactic:
rw [eq_sub_iff_add_eq, ← hf_feq npos, sub_add_cancel]

Result:
no goals

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
⊢ 0 < ↑n - 1

Tactic:
rw [← Nat.cast_one, sub_pos, Nat.cast_lt]

Result:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
⊢ 1 < n

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : f ↑n - f (↑n - 1) ≤ (f (↑n + x) - f ↑n) / x
this : f (↑n - 1) = f ↑n - log (↑n - 1)
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Tactic:
rwa [this, le_div_iff₀ hx, sub_sub_cancel, le_sub_iff_add_le, mul_comm _ x, add_comm] at c

Result:
no goals

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
⊢ ↑n < ↑n + x

Tactic:
linarith

Result:
no goals

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
⊢ 0 < ↑n + x

Tactic:
linarith

Result:
no goals

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : f ↑n - f (↑n - 1) ≤ (f (↑n + x) - f ↑n) / x
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Tactic:
have : f (↑n - 1) = f n - log (↑n - 1) := by
  rw [eq_sub_iff_add_eq, ← hf_feq npos, sub_add_cancel]

Result:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : f ↑n - f (↑n - 1) ≤ (f (↑n + x) - f ↑n) / x
this : f (↑n - 1) = f ↑n - log (↑n - 1)
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
⊢ ↑n - 1 < ↑n

Tactic:
linarith

Result:
no goals

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Tactic:
have c :=
  (convexOn_iff_slope_mono_adjacent.mp <| hf_conv).2 npos (by linarith : 0 < (n : ℝ) + x)
    (by linarith : (n : ℝ) - 1 < (n : ℝ)) (by linarith)

Result:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : (f ↑n - f (↑n - 1)) / (↑n - (↑n - 1)) ≤ (f (↑n + x) - f ↑n) / (↑n + x - ↑n)
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Tactic:
have npos : 0 < (n : ℝ) - 1 := by rw [← Nat.cast_one, sub_pos, Nat.cast_lt]; omega

Result:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : (f ↑n - f (↑n - 1)) / (↑n - (↑n - 1)) ≤ (f (↑n + x) - f ↑n) / (↑n + x - ↑n)
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Tactic:
rw [add_sub_cancel_left, sub_sub_cancel, div_one] at c

Result:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : f ↑n - f (↑n - 1) ≤ (f (↑n + x) - f ↑n) / x
⊢ f ↑n + x * log (↑n - 1) ≤ f (↑n + x)

Proof state:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
⊢ 1 < n

Tactic:
omega

Result:
no goals