l3lab/ntp-mathlib huggingface dataset:
/-- Given a bad sequence `f`, this constructs a bad sequence that agrees with `f` on the first `n`
  terms and is minimal at `n`.
-/
noncomputable def minBadSeqOfBadSeq (r : α → α → Prop) (rk : α → ℕ) (s : Set α) (n : ℕ) (f : ℕ → α)
    (hf : IsBadSeq r s f) :
    { g : ℕ → α // (∀ m : ℕ, m < n → f m = g m) ∧ IsBadSeq r s g ∧ IsMinBadSeq r rk s n g } := by
  classical
    have h : ∃ (k : ℕ) (g : ℕ → α), (∀ m, m < n → f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = k :=
      ⟨_, f, fun _ _ => rfl, hf, rfl⟩
    obtain ⟨h1, h2, h3⟩ := Classical.choose_spec (Nat.find_spec h)
    refine' ⟨Classical.choose (Nat.find_spec h), h1, by convert h2, fun g hg1 hg2 con => _⟩
    

Proof state:
case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
n : ℕ
f : ℕ → α
hf : IsBadSeq r s f
h : ∃ k g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = k
h1 : ∀ m < n, f m = Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h) m
h2 : IsBadSeq r s (Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h))
h3 : rk (Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h) n) = Nat.find h
g : ℕ → α
hg1 : ∀ m < n, Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h) m = g m
hg2 : rk (g n) < rk (Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h) n)
con : IsBadSeq r s g
⊢ False

Next tactic:
refine' Nat.find_min h _ ⟨g, fun m mn => (h1 m mn).trans (hg1 m mn), con, rfl⟩
/-- Given a bad sequence `f`, this constructs a bad sequence that agrees with `f` on the first `n`
  terms and is minimal at `n`.
-/
noncomputable def minBadSeqOfBadSeq (r : α → α → Prop) (rk : α → ℕ) (s : Set α) (n : ℕ) (f : ℕ → α)
    (hf : IsBadSeq r s f) :
    { g : ℕ → α // (∀ m : ℕ, m < n → f m = g m) ∧ IsBadSeq r s g ∧ IsMinBadSeq r rk s n g } := by
  classical
    have h : ∃ (k : ℕ) (g : ℕ → α), (∀ m, m < n → f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = k :=
      ⟨_, f, fun _ _ => rfl, hf, rfl⟩
    obtain ⟨h1, h2, h3⟩ := Classical.choose_spec (Nat.find_spec h)
    refine' ⟨Classical.choose (Nat.find_spec h), h1, by convert h2, fun g hg1 hg2 con => _⟩
    refine' Nat.find_min h _ ⟨g, fun m mn => (h1 m mn).trans (hg1 m mn), con, rfl⟩
    

Proof state:
case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
n : ℕ
f : ℕ → α
hf : IsBadSeq r s f
h : ∃ k g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = k
h1 : ∀ m < n, f m = Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h) m
h2 : IsBadSeq r s (Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h))
h3 : rk (Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h) n) = Nat.find h
g : ℕ → α
hg1 : ∀ m < n, Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h) m = g m
hg2 : rk (g n) < rk (Classical.choose (_ : ∃ g, (∀ m < n, f m = g m) ∧ IsBadSeq r s g ∧ rk (g n) = Nat.find h) n)
con : IsBadSeq r s g
⊢ rk (g n) < Nat.find h

Next tactic:
rwa [← h3]
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  

Proof state:
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
⊢ (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ (n : ℕ), IsMinBadSeq r rk s n f

Next tactic:
rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  

Proof state:
case intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
⊢ ∃ f, IsBadSeq r s f ∧ ∀ (n : ℕ), IsMinBadSeq r rk s n f

Next tactic:
let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    

Proof state:
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
⊢ (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f }

Next tactic:
refine' Nat.rec _ fun n fn => _
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · 

Proof state:
case refine'_1
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
⊢ { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s Nat.zero f }

Next tactic:
exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · 

Proof state:
case refine'_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
n : ℕ
fn : { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f }
⊢ { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s (Nat.succ n) f }

Next tactic:
exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  

Proof state:
case intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
⊢ ∃ f, IsBadSeq r s f ∧ ∀ (n : ℕ), IsMinBadSeq r rk s n f

Next tactic:
have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    obtain ⟨k, rfl⟩ := exists_add_of_le mn; clear mn
    induction' k with k ih
    · rfl
    · rw [ih, (minBadSeqOfBadSeq r rk s (m + k + 1) (fs (m + k)).1 (fs (m + k)).2.1).2.1 m
        (Nat.lt_succ_iff.2 (Nat.add_le_add_left k.zero_le m))]
      rfl
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    

Proof state:
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
m n : ℕ
mn : m ≤ n
⊢ ↑(fs m) m = ↑(fs n) m

Next tactic:
obtain ⟨k, rfl⟩ := exists_add_of_le mn
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    obtain ⟨k, rfl⟩ := exists_add_of_le mn; 

Proof state:
case intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
m k : ℕ
mn : m ≤ m + k
⊢ ↑(fs m) m = ↑(fs (m + k)) m

Next tactic:
clear mn
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    obtain ⟨k, rfl⟩ := exists_add_of_le mn; clear mn
    

Proof state:
case intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
m k : ℕ
⊢ ↑(fs m) m = ↑(fs (m + k)) m

Next tactic:
induction' k with k ih
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    obtain ⟨k, rfl⟩ := exists_add_of_le mn; clear mn
    induction' k with k ih
    · 

Proof state:
case intro.zero
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
m : ℕ
⊢ ↑(fs m) m = ↑(fs (m + Nat.zero)) m

Next tactic:
rfl
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    obtain ⟨k, rfl⟩ := exists_add_of_le mn; clear mn
    induction' k with k ih
    · rfl
    · 

Proof state:
case intro.succ
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
m k : ℕ
ih : ↑(fs m) m = ↑(fs (m + k)) m
⊢ ↑(fs m) m = ↑(fs (m + Nat.succ k)) m

Next tactic:
rw [ih, (minBadSeqOfBadSeq r rk s (m + k + 1) (fs (m + k)).1 (fs (m + k)).2.1).2.1 m
        (Nat.lt_succ_iff.2 (Nat.add_le_add_left k.zero_le m))]
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    obtain ⟨k, rfl⟩ := exists_add_of_le mn; clear mn
    induction' k with k ih
    · rfl
    · rw [ih, (minBadSeqOfBadSeq r rk s (m + k + 1) (fs (m + k)).1 (fs (m + k)).2.1).2.1 m
        (Nat.lt_succ_iff.2 (Nat.add_le_add_left k.zero_le m))]
      

Proof state:
case intro.succ
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
m k : ℕ
ih : ↑(fs m) m = ↑(fs (m + k)) m
⊢ ↑(minBadSeqOfBadSeq r rk s (m + k + 1) ↑(fs (m + k)) (_ : IsBadSeq r s ↑(fs (m + k)))) m = ↑(fs (m + Nat.succ k)) m

Next tactic:
rfl
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    obtain ⟨k, rfl⟩ := exists_add_of_le mn; clear mn
    induction' k with k ih
    · rfl
    · rw [ih, (minBadSeqOfBadSeq r rk s (m + k + 1) (fs (m + k)).1 (fs (m + k)).2.1).2.1 m
        (Nat.lt_succ_iff.2 (Nat.add_le_add_left k.zero_le m))]
      rfl
  

Proof state:
case intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
h : ∀ (m n : ℕ), m ≤ n → ↑(fs m) m = ↑(fs n) m
⊢ ∃ f, IsBadSeq r s f ∧ ∀ (n : ℕ), IsMinBadSeq r rk s n f

Next tactic:
refine ⟨fun n => (fs n).1 n, ⟨fun n => (fs n).2.1.1 n, fun m n mn => ?_⟩, fun n g hg1 hg2 => ?_⟩
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
    (∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
  rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
  let fs : ∀ n : ℕ, { f : ℕ → α // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } := by
    refine' Nat.rec _ fun n fn => _
    · exact ⟨(minBadSeqOfBadSeq r rk s 0 f0 hf0).1, (minBadSeqOfBadSeq r rk s 0 f0 hf0).2.2⟩
    · exact ⟨(minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).1,
        (minBadSeqOfBadSeq r rk s (n + 1) fn.1 fn.2.1).2.2⟩
  have h : ∀ m n, m ≤ n → (fs m).1 m = (fs n).1 m := fun m n mn => by
    obtain ⟨k, rfl⟩ := exists_add_of_le mn; clear mn
    induction' k with k ih
    · rfl
    · rw [ih, (minBadSeqOfBadSeq r rk s (m + k + 1) (fs (m + k)).1 (fs (m + k)).2.1).2.1 m
        (Nat.lt_succ_iff.2 (Nat.add_le_add_left k.zero_le m))]
      rfl
  refine ⟨fun n => (fs n).1 n, ⟨fun n => (fs n).2.1.1 n, fun m n mn => ?_⟩, fun n g hg1 hg2 => ?_⟩
  · 

Proof state:
case intro.refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r✝ r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
  fun t =>
    Nat.rec
      { val := ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0),
        property :=
          (_ :
            IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0) ∧
              IsMinBadSeq r rk s 0 ↑(minBadSeqOfBadSeq r rk s 0 f0 hf0)) }
      (fun n fn =>
        { val := ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)),
          property :=
            (_ :
              IsBadSeq r s ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn)) ∧
                IsMinBadSeq r rk s (n + 1) ↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn (_ : IsBadSeq r s ↑fn))) })
      t
h : ∀ (m n : ℕ), m ≤ n → ↑(fs m) m = ↑(fs n) m
m n : ℕ
mn : m < n
⊢ ¬r ((fun n => ↑(fs n) n) m) ((fun n => ↑(fs n) n) n)

Next tactic:
dsimp