-- Mathlib/Analysis/Hofer.lean
theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x'

Proof state:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
v : ℕ → ℝ := fun n => (ϕ ∘ u) (n + 1)
⊢ Tendsto (ϕ ∘ u) atTop atTop

Tactic:
suffices Tendsto v atTop atTop by rwa [tendsto_add_atTop_iff_nat] at this

Result:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
v : ℕ → ℝ := fun n => (ϕ ∘ u) (n + 1)
⊢ Tendsto v atTop atTop

Proof state:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
H : ¬∃ ε' > 0, ∃ x', ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ (y : X), d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x'
⊢ ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'

Tactic:
intro x' k

Result:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
H : ¬∃ ε' > 0, ∃ x', ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ (y : X), d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x'
x' : X
k : ℕ
⊢ ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'

Proof state:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
⊢ False

Tactic:
obtain ⟨key₁, key₂⟩ := forall_and.mp key

Result:
case intro
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
⊢ False

Proof state:
case intro
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
⊢ False

Tactic:
obtain ⟨y, limy⟩ : ∃ y, Tendsto u atTop (𝓝 y) := CompleteSpace.complete cauchy_u

Result:
case intro.intro
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
⊢ False

Proof state:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
v : ℕ → ℝ := fun n => (ϕ ∘ u) (n + 1)
hv₀ : 0 < v 0
⊢ Tendsto v atTop atTop

Tactic:
apply tendsto_atTop_of_geom_le hv₀ one_lt_two

Result:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
v : ℕ → ℝ := fun n => (ϕ ∘ u) (n + 1)
hv₀ : 0 < v 0
⊢ ∀ (n : ℕ), 2 * v n ≤ v (n + 1)