-- Mathlib/Analysis/PSeries.lean
theorem sum_schlomilch_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : Monotone u) (n : ℕ) :
    (∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1))) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k

Proof state:
case succ
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k
⊢ (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k

Tactic:
convert sum_le_sum this

Result:
case h.e'_3
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k
⊢ (u (n + 1) - u n) • f (u (n + 1)) = ∑ i ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1))

Proof state:
case succ
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
⊢ (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k

Tactic:
have : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k := fun k hk =>
  hf (Nat.lt_of_le_of_lt (Nat.succ_le_of_lt (h_pos n)) <| (Nat.lt_succ_of_le le_rfl).trans_le
    (mem_Ico.mp hk).1) (Nat.le_of_lt_succ <| (mem_Ico.mp hk).2)

Result:
case succ
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k
⊢ (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k

Proof state:
case h.e'_3
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k
⊢ (u (n + 1) - u n) • f (u (n + 1)) = ∑ i ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1))

Tactic:
simp [pow_succ, mul_two]

Result:
no goals

Proof state:
case zero
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
⊢ ∑ k ∈ range 0, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u 0 + 1), f k

Tactic:
simp

Result:
no goals

Proof state:
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
⊢ ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k

Tactic:
induction n with
| zero => simp
| succ n ihn =>
  suffices (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k by
    rw [sum_range_succ, ← sum_Ico_consecutive]
    exacts [add_le_add ihn this,
      (add_le_add_right (hu n.zero_le) _ : u 0 + 1 ≤ u n + 1),
      add_le_add_right (hu n.le_succ) _]
  have : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k := fun k hk =>
    hf (Nat.lt_of_le_of_lt (Nat.succ_le_of_lt (h_pos n)) <| (Nat.lt_succ_of_le le_rfl).trans_le
      (mem_Ico.mp hk).1) (Nat.le_of_lt_succ <| (mem_Ico.mp hk).2)
  convert sum_le_sum this
  simp [pow_succ, mul_two]

Result:
no goals

Proof state:
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ ∑ x ∈ range n, (u (x + 1) - u x) • f (u (x + 1)) + (u (n + 1) - u n) • f (u (n + 1)) ≤
    ∑ i ∈ Ico (u 0 + 1) ?n, f i + ∑ i ∈ Ico ?n (u (n + 1) + 1), f i

case n
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ ℕ

case hmn
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ u 0 + 1 ≤ ?n

case hnk
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ ?n ≤ u (n + 1) + 1

Tactic:
exacts [add_le_add ihn this,
  (add_le_add_right (hu n.zero_le) _ : u 0 + 1 ≤ u n + 1),
  add_le_add_right (hu n.le_succ) _]

Result:
no goals

Proof state:
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ ∑ k ∈ range (n + 1), (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u (n + 1) + 1), f k

Tactic:
rw [sum_range_succ, ← sum_Ico_consecutive]

Result:
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ ∑ x ∈ range n, (u (x + 1) - u x) • f (u (x + 1)) + (u (n + 1) - u n) • f (u (n + 1)) ≤
    ∑ i ∈ Ico (u 0 + 1) ?n, f i + ∑ i ∈ Ico ?n (u (n + 1) + 1), f i

case n
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ ℕ

case hmn
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ u 0 + 1 ≤ ?n

case hnk
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
⊢ ?n ≤ u (n + 1) + 1