Theorem: exists_monotone_Icc_subset_open_cover_unitInterval_prod_self
File: Mathlib/Topology/UnitInterval.lean

Proof steps:
state_before: ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
⊢ ∃ t,
    t 0 = 0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
state_after: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
⊢ ∃ t,
    t 0 = 0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
tactic: obtain ⟨δ, δ_pos, ball_subset⟩ := lebesgue_number_lemma_of_metric isCompact_univ hc₁ hc₂
state_before: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
⊢ ∃ t,
    t 0 = 0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
state_after: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
⊢ ∃ t,
    t 0 = 0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
tactic: have hδ := half_pos δ_pos
state_before: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
⊢ ∃ t,
    t 0 = 0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
state_after: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
⊢ ∃ t,
    ↑(t 0) = ↑0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, ↑(t m) = ↑1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
tactic: simp_rw [Subtype.ext_iff]
state_before: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
⊢ ∃ t,
    ↑(t 0) = ↑0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, ↑(t m) = ↑1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
state_after: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
h : 0 ≤ 1
⊢ ∃ t,
    ↑(t 0) = ↑0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, ↑(t m) = ↑1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
tactic: have h : (0 : ℝ) ≤ 1 := zero_le_one
state_before: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
h : 0 ≤ 1
⊢ ∃ t,
    ↑(t 0) = ↑0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, ↑(t m) = ↑1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
state_after: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
h : 0 ≤ 1
n m : ℕ
⊢ ∃ i,
    Icc (addNSMul h (δ / 2) n) (addNSMul h (δ / 2) (n + 1)) ×ˢ Icc (addNSMul h (δ / 2) m) (addNSMul h (δ / 2) (m + 1)) ⊆
      c i
tactic: refine ⟨addNSMul h (δ/2), addNSMul_zero h,
    monotone_addNSMul h hδ.le, addNSMul_eq_right h hδ, fun n m ↦ ?_⟩
state_before: case intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
h : 0 ≤ 1
n m : ℕ
⊢ ∃ i,
    Icc (addNSMul h (δ / 2) n) (addNSMul h (δ / 2) (n + 1)) ×ˢ Icc (addNSMul h (δ / 2) m) (addNSMul h (δ / 2) (m + 1)) ⊆
      c i
state_after: case intro.intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
h : 0 ≤ 1
n m : ℕ
i : ι
hsub : Metric.ball (addNSMul h (δ / 2) n, addNSMul h (δ / 2) m) δ ⊆ c i
⊢ ∃ i,
    Icc (addNSMul h (δ / 2) n) (addNSMul h (δ / 2) (n + 1)) ×ˢ Icc (addNSMul h (δ / 2) m) (addNSMul h (δ / 2) (m + 1)) ⊆
      c i
tactic: obtain ⟨i, hsub⟩ := ball_subset (addNSMul h (δ/2) n, addNSMul h (δ/2) m) trivial
state_before: case intro.intro.intro
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
h : 0 ≤ 1
n m : ℕ
i : ι
hsub : Metric.ball (addNSMul h (δ / 2) n, addNSMul h (δ / 2) m) δ ⊆ c i
⊢ ∃ i,
    Icc (addNSMul h (δ / 2) n) (addNSMul h (δ / 2) (n + 1)) ×ˢ Icc (addNSMul h (δ / 2) m) (addNSMul h (δ / 2) (m + 1)) ⊆
      c i
state_after: no goals
tactic: exact ⟨i, fun t ht ↦ hsub (Metric.mem_ball.mpr <| (max_le (abs_sub_addNSMul_le h hδ.le n ht.1) <|
    abs_sub_addNSMul_le h hδ.le m ht.2).trans_lt <| half_lt_self δ_pos)⟩