awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
r : Finset ℕ := range (n + 1)
hbc : LE.le (α := ℝ) (∑ i ∈ r, (1 / 2) ^ i) 2
⊢ (∑ i ∈ r, (1 / 2) ^ i) * ε ≤ 2 * ε

Tactic:
gcongr
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
r : Finset ℕ := range (n + 1)
⊢ LE.le (α := ℝ) (∑ i ∈ r, (1 / 2) ^ i) 2

Tactic:
apply sum_geometric_two_le

Premises used: sum_geometric_two_le
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
r : Finset ℕ := range (n + 1)
d : ∑ i ∈ r, ε / HPow.hPow (α := ℝ) 2 i = (∑ i ∈ r, (1 / 2) ^ i) * ε
d₁ : (∑ i ∈ r, (1 / 2) ^ i) * ε ≤ 2 * ε
⊢ d x (u (n + 1)) ≤ 2 * ε

Tactic:
calc
          d (u 0) (u (n + 1)) ≤ ∑ i ∈ r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
          _ ≤ ∑ i ∈ r, ε / 2 ^ i :=
            (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
          _ = (∑ i ∈ r, (1 / 2 : ℝ) ^ i) * ε := by
            rw [Finset.sum_mul]
            simp
            field_simp
          _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le

Premises used: dist_le_range_sum_dist, HDiv.hDiv, Finset.mem_range, HPow.hPow, Nat.lt_succ_iff, And.left, HAdd.hAdd, Iff.mp, Finset.sum_le_sum, Dist.dist, OfNat.ofNat, Finset.sum
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
⊢ d (u (n + 1)) x ≤ 2 * ε

Tactic:
have A : d (u (n + 1)) x ≤ 2 * ε := by
        rw [dist_comm]
        let r := range (n + 1) -- range (n+1) = {0, ..., n}
        calc
          d (u 0) (u (n + 1)) ≤ ∑ i ∈ r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
          _ ≤ ∑ i ∈ r, ε / 2 ^ i :=
            (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
          _ = (∑ i ∈ r, (1 / 2 : ℝ) ^ i) * ε := by
            rw [Finset.sum_mul]
            simp
            field_simp
          _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
A : d (u (n + 1)) x ≤ 2 * ε
⊢ HPow.hPow (α := ℝ) 2 (n + 1) * ϕ x ≤ ϕ (u (n + 1))

Tactic:
@geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => ?_

Premises used: geom_le, HAdd.hAdd, zero_le_two, OfNat.ofNat
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
A : d (u (n + 1)) x ≤ 2 * ε
refine : ∀ (m : ℕ) (hm : m < n + 1), 2 * (ϕ ∘ u) m ≤ (ϕ ∘ u) (m + 1)
⊢ HPow.hPow (α := ℝ) 2 (n + 1) * ϕ x ≤ ϕ (u (n + 1))

Tactic:
refine @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => ?_

Premises used: geom_le, HAdd.hAdd, zero_le_two, OfNat.ofNat
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
A : d (u (n + 1)) x ≤ 2 * ε
m : ℕ
hm : m < n + 1
⊢ 2 * (ϕ ∘ u) m ≤ (ϕ ∘ u) (m + 1)

Tactic:
(IH _ <| Nat.lt_add_one_iff.1 hm).2.le

Premises used: Nat.lt_add_one_iff, LT.lt.le, And.right, Iff.mp
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
A : d (u (n + 1)) x ≤ 2 * ε
m : ℕ
hm : m < n + 1
⊢ 2 * (ϕ ∘ u) m ≤ (ϕ ∘ u) (m + 1)

Tactic:
exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le

Premises used: Nat.lt_add_one_iff, LT.lt.le, And.right, Iff.mp
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
A : d (u (n + 1)) x ≤ 2 * ε
⊢ HPow.hPow (α := ℝ) 2 (n + 1) * ϕ x ≤ ϕ (u (n + 1))

Tactic:
have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
        refine @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => ?_
        exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
A : d (u (n + 1)) x ≤ 2 * ε
B : HPow.hPow (α := ℝ) 2 (n + 1) * ϕ x ≤ ϕ (u (n + 1))
⊢ d (u (n + 1)) (u (n + 1 + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 (n + 1) ∧ 2 * ϕ (u (n + 1)) < ϕ (u (n + 1 + 1))

Tactic:
hu (n + 1) ⟨A, B⟩

Premises used: HAdd.hAdd, And.intro, OfNat.ofNat
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))
A : d (u (n + 1)) x ≤ 2 * ε
B : HPow.hPow (α := ℝ) 2 (n + 1) * ϕ x ≤ ϕ (u (n + 1))
⊢ d (u (n + 1)) (u (n + 1 + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 (n + 1) ∧ 2 * ϕ (u (n + 1)) < ϕ (u (n + 1 + 1))

Tactic:
exact hu (n + 1) ⟨A, B⟩

Premises used: HAdd.hAdd, And.intro, OfNat.ofNat
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
n : ℕ
hz :
  d (u 0) (u (HAdd.hAdd (α := ℕ) 0 1)) ≤ ε / HPow.hPow (α := ℝ) (β := ℕ) 2 0 ∧
    2 * ϕ (u 0) < ϕ (u (HAdd.hAdd (α := ℕ) 0 1))
hi :
  ∀ (n : ℕ) (IH : ∀ m ≤ n, d (u m) (u (m + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 m ∧ 2 * ϕ (u m) < ϕ (u (m + 1))),
    d (u (n + 1)) (u (n + 1 + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 (n + 1) ∧ 2 * ϕ (u (n + 1)) < ϕ (u (n + 1 + 1))
⊢ d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))

Tactic:
induction n using Nat.case_strong_induction_on with
    | hz => simpa [u, ε_pos.le] using hu 0
    | hi n IH =>
      have A : d (u (n + 1)) x ≤ 2 * ε := by
        rw [dist_comm]
        let r := range (n + 1) -- range (n+1) = {0, ..., n}
        calc
          d (u 0) (u (n + 1)) ≤ ∑ i ∈ r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
          _ ≤ ∑ i ∈ r, ε / 2 ^ i :=
            (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
          _ = (∑ i ∈ r, (1 / 2 : ℝ) ^ i) * ε := by
            rw [Finset.sum_mul]
            simp
            field_simp
          _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
      have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
        refine @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => ?_
        exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
      exact hu (n + 1) ⟨A, B⟩

Premises used: Nat.case_strong_induction_on, LT.lt.le, OfNat.ofNat
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
⊢ ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))

Tactic:
have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
    intro n
    induction n using Nat.case_strong_induction_on with
    | hz => simpa [u, ε_pos.le] using hu 0
    | hi n IH =>
      have A : d (u (n + 1)) x ≤ 2 * ε := by
        rw [dist_comm]
        let r := range (n + 1) -- range (n+1) = {0, ..., n}
        calc
          d (u 0) (u (n + 1)) ≤ ∑ i ∈ r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
          _ ≤ ∑ i ∈ r, ε / 2 ^ i :=
            (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
          _ = (∑ i ∈ r, (1 / 2 : ℝ) ^ i) * ε := by
            rw [Finset.sum_mul]
            simp
            field_simp
          _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
      have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
        refine @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => ?_
        exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
      exact hu (n + 1) ⟨A, B⟩
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
⊢ ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 x

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

Premises used: Iff.mp, forall_and
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 x
⊢ ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))

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

Premises used: Iff.mp, forall_and
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
⊢ CauchySeq u

Tactic:
cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => ?_

Premises used: cauchySeq_of_le_geometric, one_half_lt_one
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
refine : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε * (1 / 2) ^ n
⊢ CauchySeq u

Tactic:
refine cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => ?_

Premises used: cauchySeq_of_le_geometric, one_half_lt_one
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
n : ℕ
⊢ d (u n) (u (n + 1)) ≤ ε * (1 / 2) ^ n

Tactic:
simpa only [one_div, inv_pow] using key₁ n

Premises used: inv_pow, one_div
-- Module: Mathlib.Analysis.Hofer
Goal:
X : Type u_1
inst : MetricSpace X
inst_1 : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / HPow.hPow (α := ℝ) 2 k * ϕ x' ↔ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x'
this : Nonempty X
F : (k : ℕ) → (x' : X) → X
hF :
  ∀ (k : ℕ) (x' : X),
    d x' x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 k * ϕ x ≤ ϕ x' →
      d x' (F k x') ≤ ε / HPow.hPow (α := ℝ) 2 k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun (n : ℕ) ↦ Nat.recOn (motive := fun (x : ℕ) ↦ X) n x F
hu :
  ∀ (n : ℕ),
    d (u n) x ≤ 2 * ε ∧ HPow.hPow (α := ℝ) 2 n * ϕ x ≤ ϕ (u n) →
      d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / HPow.hPow (α := ℝ) 2 x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
⊢ CauchySeq u

Tactic:
have cauchy_u : CauchySeq u := by
    refine cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => ?_
    simpa only [one_div, inv_pow] using key₁ n