l3lab/ntp-mathlib huggingface dataset:
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · 

Proof state:
case pos
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
he : Finset.Nonempty (Set.Finite.toFinset hbad)
⊢ ∃ m, ∀ (n : ℕ), m ≤ n → n ∉ bad

Next tactic:
refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · 

Proof state:
case neg
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
he : ¬Finset.Nonempty (Set.Finite.toFinset hbad)
⊢ ∃ m, ∀ (n : ℕ), m ≤ n → n ∉ bad

Next tactic:
exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      

Proof state:
case neg.intro
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
⊢ ∃ g, (∀ (n : ℕ), r (f (g n)) (f (g (n + 1)))) ∨ ∀ (m n : ℕ), m < n → ¬r (f (g m)) (f (g n))

Next tactic:
have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
        obtain ⟨n', hn1, hn2⟩ := h
        obtain ⟨x, hpos, rfl⟩ := exists_pos_add_of_lt hn1
        refine' ⟨n + x, add_lt_add_left hpos n, _⟩
        rw [add_assoc, add_comm x m, ← add_assoc]
        exact hn2
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        

Proof state:
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
⊢ ∀ (n : ℕ), ∃ n', n < n' ∧ r (f (n + m)) (f (n' + m))

Next tactic:
intro n
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        

Proof state:
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
n : ℕ
⊢ ∃ n', n < n' ∧ r (f (n + m)) (f (n' + m))

Next tactic:
have h := hm _ (le_add_of_nonneg_left n.zero_le)
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        

Proof state:
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
n : ℕ
h : n + m ∉ bad
⊢ ∃ n', n < n' ∧ r (f (n + m)) (f (n' + m))

Next tactic:
simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
        

Proof state:
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
n : ℕ
h : ∃ x, n + m < x ∧ r (f (n + m)) (f x)
⊢ ∃ n', n < n' ∧ r (f (n + m)) (f (n' + m))

Next tactic:
obtain ⟨n', hn1, hn2⟩ := h
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
        obtain ⟨n', hn1, hn2⟩ := h
        

Proof state:
case intro.intro
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
n n' : ℕ
hn1 : n + m < n'
hn2 : r (f (n + m)) (f n')
⊢ ∃ n', n < n' ∧ r (f (n + m)) (f (n' + m))

Next tactic:
obtain ⟨x, hpos, rfl⟩ := exists_pos_add_of_lt hn1
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
        obtain ⟨n', hn1, hn2⟩ := h
        obtain ⟨x, hpos, rfl⟩ := exists_pos_add_of_lt hn1
        

Proof state:
case intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
n x : ℕ
hpos : 0 < x
hn1 : n + m < n + m + x
hn2 : r (f (n + m)) (f (n + m + x))
⊢ ∃ n', n < n' ∧ r (f (n + m)) (f (n' + m))

Next tactic:
refine' ⟨n + x, add_lt_add_left hpos n, _⟩
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
        obtain ⟨n', hn1, hn2⟩ := h
        obtain ⟨x, hpos, rfl⟩ := exists_pos_add_of_lt hn1
        refine' ⟨n + x, add_lt_add_left hpos n, _⟩
        

Proof state:
case intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
n x : ℕ
hpos : 0 < x
hn1 : n + m < n + m + x
hn2 : r (f (n + m)) (f (n + m + x))
⊢ r (f (n + m)) (f (n + x + m))

Next tactic:
rw [add_assoc, add_comm x m, ← add_assoc]
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
        obtain ⟨n', hn1, hn2⟩ := h
        obtain ⟨x, hpos, rfl⟩ := exists_pos_add_of_lt hn1
        refine' ⟨n + x, add_lt_add_left hpos n, _⟩
        rw [add_assoc, add_comm x m, ← add_assoc]
        

Proof state:
case intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
n x : ℕ
hpos : 0 < x
hn1 : n + m < n + m + x
hn2 : r (f (n + m)) (f (n + m + x))
⊢ r (f (n + m)) (f (n + m + x))

Next tactic:
exact hn2
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
        obtain ⟨n', hn1, hn2⟩ := h
        obtain ⟨x, hpos, rfl⟩ := exists_pos_add_of_lt hn1
        refine' ⟨n + x, add_lt_add_left hpos n, _⟩
        rw [add_assoc, add_comm x m, ← add_assoc]
        exact hn2
      

Proof state:
case neg.intro
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
h : ∀ (n : ℕ), ∃ n', n < n' ∧ r (f (n + m)) (f (n' + m))
⊢ ∃ g, (∀ (n : ℕ), r (f (g n)) (f (g (n + 1)))) ∨ ∀ (m n : ℕ), m < n → ¬r (f (g m)) (f (g n))

Next tactic:
let g' : ℕ → ℕ := @Nat.rec (fun _ => ℕ) m fun n gn => Nat.find (h gn)
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  classical
    let bad : Set ℕ := { m | ∀ n, m < n → ¬r (f m) (f n) }
    by_cases hbad : Infinite bad
    · haveI := hbad
      refine' ⟨Nat.orderEmbeddingOfSet bad, Or.intro_right _ fun m n mn => _⟩
      have h := @Set.mem_range_self _ _ ↑(Nat.orderEmbeddingOfSet bad) m
      rw [Nat.orderEmbeddingOfSet_range bad] at h
      exact h _ ((OrderEmbedding.lt_iff_lt _).2 mn)
    · rw [Set.infinite_coe_iff, Set.Infinite, not_not] at hbad
      obtain ⟨m, hm⟩ : ∃ m, ∀ n, m ≤ n → ¬n ∈ bad := by
        by_cases he : hbad.toFinset.Nonempty
        · refine'
            ⟨(hbad.toFinset.max' he).succ, fun n hn nbad =>
              Nat.not_succ_le_self _
                (hn.trans (hbad.toFinset.le_max' n (hbad.mem_toFinset.2 nbad)))⟩
        · exact ⟨0, fun n _ nbad => he ⟨n, hbad.mem_toFinset.2 nbad⟩⟩
      have h : ∀ n : ℕ, ∃ n' : ℕ, n < n' ∧ r (f (n + m)) (f (n' + m)) := by
        intro n
        have h := hm _ (le_add_of_nonneg_left n.zero_le)
        simp only [exists_prop, not_not, Set.mem_setOf_eq, not_forall] at h
        obtain ⟨n', hn1, hn2⟩ := h
        obtain ⟨x, hpos, rfl⟩ := exists_pos_add_of_lt hn1
        refine' ⟨n + x, add_lt_add_left hpos n, _⟩
        rw [add_assoc, add_comm x m, ← add_assoc]
        exact hn2
      let g' : ℕ → ℕ := @Nat.rec (fun _ => ℕ) m fun n gn => Nat.find (h gn)
      

Proof state:
case neg.intro
α : Type u_1
r : α → α → Prop
f : ℕ → α
bad : Set ℕ := {m | ∀ (n : ℕ), m < n → ¬r (f m) (f n)}
hbad : Set.Finite bad
m : ℕ
hm : ∀ (n : ℕ), m ≤ n → n ∉ bad
h : ∀ (n : ℕ), ∃ n', n < n' ∧ r (f (n + m)) (f (n' + m))
g' : ℕ → ℕ := Nat.rec m fun n gn => Nat.find (_ : ∃ n', gn < n' ∧ r (f (gn + m)) (f (n' + m)))
⊢ ∃ g, (∀ (n : ℕ), r (f (g n)) (f (g (n + 1)))) ∨ ∀ (m n : ℕ), m < n → ¬r (f (g m)) (f (g n))

Next tactic:
exact
        ⟨(RelEmbedding.natLT (fun n => g' n + m) fun n =>
              Nat.add_lt_add_right (Nat.find_spec (h (g' n))).1 m).orderEmbeddingOfLTEmbedding,
          Or.intro_left _ fun n => (Nat.find_spec (h (g' n))).2⟩
/-- This is the infinitary Erdős–Szekeres theorem, and an important lemma in the usual proof of
    Bolzano-Weierstrass for `ℝ`. -/
theorem exists_increasing_or_nonincreasing_subseq (r : α → α → Prop) [IsTrans α r] (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ m n : ℕ, m < n → r (f (g m)) (f (g n))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  

Proof state:
α : Type u_1
r : α → α → Prop
inst✝ : IsTrans α r
f : ℕ → α
⊢ ∃ g, (∀ (m n : ℕ), m < n → r (f (g m)) (f (g n))) ∨ ∀ (m n : ℕ), m < n → ¬r (f (g m)) (f (g n))

Next tactic:
obtain ⟨g, hr | hnr⟩ := exists_increasing_or_nonincreasing_subseq' r f
/-- This is the infinitary Erdős–Szekeres theorem, and an important lemma in the usual proof of
    Bolzano-Weierstrass for `ℝ`. -/
theorem exists_increasing_or_nonincreasing_subseq (r : α → α → Prop) [IsTrans α r] (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ m n : ℕ, m < n → r (f (g m)) (f (g n))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  obtain ⟨g, hr | hnr⟩ := exists_increasing_or_nonincreasing_subseq' r f
  · 

Proof state:
case intro.inl
α : Type u_1
r : α → α → Prop
inst✝ : IsTrans α r
f : ℕ → α
g : ℕ ↪o ℕ
hr : ∀ (n : ℕ), r (f (g n)) (f (g (n + 1)))
⊢ ∃ g, (∀ (m n : ℕ), m < n → r (f (g m)) (f (g n))) ∨ ∀ (m n : ℕ), m < n → ¬r (f (g m)) (f (g n))

Next tactic:
refine' ⟨g, Or.intro_left _ fun m n mn => _⟩
/-- This is the infinitary Erdős–Szekeres theorem, and an important lemma in the usual proof of
    Bolzano-Weierstrass for `ℝ`. -/
theorem exists_increasing_or_nonincreasing_subseq (r : α → α → Prop) [IsTrans α r] (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ m n : ℕ, m < n → r (f (g m)) (f (g n))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  obtain ⟨g, hr | hnr⟩ := exists_increasing_or_nonincreasing_subseq' r f
  · refine' ⟨g, Or.intro_left _ fun m n mn => _⟩
    

Proof state:
case intro.inl
α : Type u_1
r : α → α → Prop
inst✝ : IsTrans α r
f : ℕ → α
g : ℕ ↪o ℕ
hr : ∀ (n : ℕ), r (f (g n)) (f (g (n + 1)))
m n : ℕ
mn : m < n
⊢ r (f (g m)) (f (g n))

Next tactic:
obtain ⟨x, rfl⟩ := exists_add_of_le (Nat.succ_le_iff.2 mn)
/-- This is the infinitary Erdős–Szekeres theorem, and an important lemma in the usual proof of
    Bolzano-Weierstrass for `ℝ`. -/
theorem exists_increasing_or_nonincreasing_subseq (r : α → α → Prop) [IsTrans α r] (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ m n : ℕ, m < n → r (f (g m)) (f (g n))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  obtain ⟨g, hr | hnr⟩ := exists_increasing_or_nonincreasing_subseq' r f
  · refine' ⟨g, Or.intro_left _ fun m n mn => _⟩
    obtain ⟨x, rfl⟩ := exists_add_of_le (Nat.succ_le_iff.2 mn)
    

Proof state:
case intro.inl.intro
α : Type u_1
r : α → α → Prop
inst✝ : IsTrans α r
f : ℕ → α
g : ℕ ↪o ℕ
hr : ∀ (n : ℕ), r (f (g n)) (f (g (n + 1)))
m x : ℕ
mn : m < Nat.succ m + x
⊢ r (f (g m)) (f (g (Nat.succ m + x)))

Next tactic:
induction' x with x ih
/-- This is the infinitary Erdős–Szekeres theorem, and an important lemma in the usual proof of
    Bolzano-Weierstrass for `ℝ`. -/
theorem exists_increasing_or_nonincreasing_subseq (r : α → α → Prop) [IsTrans α r] (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ m n : ℕ, m < n → r (f (g m)) (f (g n))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) := by
  obtain ⟨g, hr | hnr⟩ := exists_increasing_or_nonincreasing_subseq' r f
  · refine' ⟨g, Or.intro_left _ fun m n mn => _⟩
    obtain ⟨x, rfl⟩ := exists_add_of_le (Nat.succ_le_iff.2 mn)
    induction' x with x ih
    · 

Proof state:
case intro.inl.intro.zero
α : Type u_1
r : α → α → Prop
inst✝ : IsTrans α r
f : ℕ → α
g : ℕ ↪o ℕ
hr : ∀ (n : ℕ), r (f (g n)) (f (g (n + 1)))
m : ℕ
mn : m < Nat.succ m + Nat.zero
⊢ r (f (g m)) (f (g (Nat.succ m + Nat.zero)))

Next tactic:
apply hr