l3lab/ntp-mathlib huggingface dataset:
/-- Squashing at position `n + 1` and taking the tail is the same as squashing the tail of the
sequence at position `n`. -/
theorem squashSeq_succ_n_tail_eq_squashSeq_tail_n :
    (squashSeq s (n + 1)).tail = squashSeq s.tail n := by
  cases' s_succ_succ_nth_eq : s.get? (n + 2) with gp_succ_succ_n
  case none =>
    cases s_succ_nth_eq : s.get? (n + 1) <;>
      simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_eq]
  case some =>
    obtain ⟨gp_succ_n, s_succ_nth_eq⟩ : ∃ gp_succ_n, s.get? (n + 1) = some gp_succ_n;
    exact s.ge_stable (n + 1).le_succ s_succ_succ_nth_eq
    -- apply extensionality with `m` and continue by cases `m = n`.
    ext1 m
    cases' Decidable.em (m = n) with m_eq_n m_ne_n
    · simp [*, squashSeq]
    · have : s.tail.get? m = s.get? (m + 1) := s.get?_tail m
      cases s_succ_mth_eq : s.get? (m + 1)
      all_goals 

Proof state:
case intro.h.inr.some
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
gp_succ_succ_n : Pair K
s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n
m : ℕ
m_ne_n : ¬m = n
this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)
val✝ : Pair K
s_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val✝
⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m

Next tactic:
have _ := this.trans s_succ_mth_eq
/-- Squashing at position `n + 1` and taking the tail is the same as squashing the tail of the
sequence at position `n`. -/
theorem squashSeq_succ_n_tail_eq_squashSeq_tail_n :
    (squashSeq s (n + 1)).tail = squashSeq s.tail n := by
  cases' s_succ_succ_nth_eq : s.get? (n + 2) with gp_succ_succ_n
  case none =>
    cases s_succ_nth_eq : s.get? (n + 1) <;>
      simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_eq]
  case some =>
    obtain ⟨gp_succ_n, s_succ_nth_eq⟩ : ∃ gp_succ_n, s.get? (n + 1) = some gp_succ_n;
    exact s.ge_stable (n + 1).le_succ s_succ_succ_nth_eq
    -- apply extensionality with `m` and continue by cases `m = n`.
    ext1 m
    cases' Decidable.em (m = n) with m_eq_n m_ne_n
    · simp [*, squashSeq]
    · have : s.tail.get? m = s.get? (m + 1) := s.get?_tail m
      cases s_succ_mth_eq : s.get? (m + 1)
      all_goals have _ := this.trans s_succ_mth_eq
      · 

Proof state:
case intro.h.inr.none
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
gp_succ_succ_n : Pair K
s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n
m : ℕ
m_ne_n : ¬m = n
this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)
s_succ_mth_eq : Stream'.Seq.get? s (m + 1) = none
x✝ : Stream'.Seq.get? (Stream'.Seq.tail s) m = none
⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m

Next tactic:
simp only [*, squashSeq, Stream'.Seq.get?_tail, Stream'.Seq.get?_zipWith,
          Option.map₂_none_right]
/-- Squashing at position `n + 1` and taking the tail is the same as squashing the tail of the
sequence at position `n`. -/
theorem squashSeq_succ_n_tail_eq_squashSeq_tail_n :
    (squashSeq s (n + 1)).tail = squashSeq s.tail n := by
  cases' s_succ_succ_nth_eq : s.get? (n + 2) with gp_succ_succ_n
  case none =>
    cases s_succ_nth_eq : s.get? (n + 1) <;>
      simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_eq]
  case some =>
    obtain ⟨gp_succ_n, s_succ_nth_eq⟩ : ∃ gp_succ_n, s.get? (n + 1) = some gp_succ_n;
    exact s.ge_stable (n + 1).le_succ s_succ_succ_nth_eq
    -- apply extensionality with `m` and continue by cases `m = n`.
    ext1 m
    cases' Decidable.em (m = n) with m_eq_n m_ne_n
    · simp [*, squashSeq]
    · have : s.tail.get? m = s.get? (m + 1) := s.get?_tail m
      cases s_succ_mth_eq : s.get? (m + 1)
      all_goals have _ := this.trans s_succ_mth_eq
      · simp only [*, squashSeq, Stream'.Seq.get?_tail, Stream'.Seq.get?_zipWith,
          Option.map₂_none_right]
      · 

Proof state:
case intro.h.inr.some
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
gp_succ_succ_n : Pair K
s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n
m : ℕ
m_ne_n : ¬m = n
this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)
val✝ : Pair K
s_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val✝
x✝ : Stream'.Seq.get? (Stream'.Seq.tail s) m = some val✝
⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m

Next tactic:
simp [*, squashSeq]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  

Proof state:
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)

Next tactic:
cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  

Proof state:
case none
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = none
⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)
case some
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n
⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)

Next tactic:
case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  

Proof state:
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = none
⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)

Next tactic:
case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    

Proof state:
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = none
⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)

Next tactic:
rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  

Proof state:
case some
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n
⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)

Next tactic:
case some =>
    induction' n with m IH generalizing s gp_succ_n
    case zero =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable zero_le_one s_succ_nth_eq
      have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=
        squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq
      simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]
    case succ =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable (m + 2).zero_le s_succ_nth_eq
      suffices
        gp_head.a / (gp_head.b + convergents'Aux s.tail (m + 2)) =
          convergents'Aux (squashSeq s (m + 1)) (m + 2)
        by simpa only [convergents'Aux, s_head_eq]
      have : convergents'Aux s.tail (m + 2) = convergents'Aux (squashSeq s.tail m) (m + 1) := by
        refine' IH gp_succ_n _
        simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq
      have : (squashSeq s (m + 1)).head = some gp_head :=
        (squashSeq_nth_of_lt m.succ_pos).trans s_head_eq
      simp_all [convergents'Aux, squashSeq_succ_n_tail_eq_squashSeq_tail_n]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  

Proof state:
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n
⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)

Next tactic:
case some =>
    induction' n with m IH generalizing s gp_succ_n
    case zero =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable zero_le_one s_succ_nth_eq
      have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=
        squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq
      simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]
    case succ =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable (m + 2).zero_le s_succ_nth_eq
      suffices
        gp_head.a / (gp_head.b + convergents'Aux s.tail (m + 2)) =
          convergents'Aux (squashSeq s (m + 1)) (m + 2)
        by simpa only [convergents'Aux, s_head_eq]
      have : convergents'Aux s.tail (m + 2) = convergents'Aux (squashSeq s.tail m) (m + 1) := by
        refine' IH gp_succ_n _
        simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq
      have : (squashSeq s (m + 1)).head = some gp_head :=
        (squashSeq_nth_of_lt m.succ_pos).trans s_head_eq
      simp_all [convergents'Aux, squashSeq_succ_n_tail_eq_squashSeq_tail_n]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  case some =>
    

Proof state:
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n
⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)

Next tactic:
induction' n with m IH generalizing s gp_succ_n
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  case some =>
    induction' n with m IH generalizing s gp_succ_n
    

Proof state:
case zero
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n
⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)
case succ
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
m : ℕ
IH :
  ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),
    Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n
⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)

Next tactic:
case zero =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable zero_le_one s_succ_nth_eq
      have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=
        squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq
      simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  case some =>
    induction' n with m IH generalizing s gp_succ_n
    

Proof state:
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n
⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)

Next tactic:
case zero =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable zero_le_one s_succ_nth_eq
      have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=
        squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq
      simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  case some =>
    induction' n with m IH generalizing s gp_succ_n
    case zero =>
      

Proof state:
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n
⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)

Next tactic:
obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  case some =>
    induction' n with m IH generalizing s gp_succ_n
    case zero =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      

Proof state:
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n
⊢ ∃ gp_head, Stream'.Seq.head s = some gp_head
case intro
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n
gp_head : Pair K
s_head_eq : Stream'.Seq.head s = some gp_head
⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)

Next tactic:
exact s.ge_stable zero_le_one s_succ_nth_eq
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  case some =>
    induction' n with m IH generalizing s gp_succ_n
    case zero =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable zero_le_one s_succ_nth_eq
      

Proof state:
case intro
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n
gp_head : Pair K
s_head_eq : Stream'.Seq.head s = some gp_head
⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)

Next tactic:
have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=
        squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  case some =>
    induction' n with m IH generalizing s gp_succ_n
    case zero =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable zero_le_one s_succ_nth_eq
      have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=
        squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq
      

Proof state:
case intro
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n
gp_head : Pair K
s_head_eq : Stream'.Seq.head s = some gp_head
this : Stream'.Seq.head (squashSeq s 0) = some { a := gp_head.a, b := gp_head.b + gp_succ_n.a / gp_succ_n.b }
⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)

Next tactic:
simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]
/-- The auxiliary function `convergents'Aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. -/
theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
    convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by
  cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n
  case none =>
    rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
      convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
  case some =>
    induction' n with m IH generalizing s gp_succ_n
    case zero =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable zero_le_one s_succ_nth_eq
      have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=
        squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq
      simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]
    

Proof state:
case succ
K : Type u_1
g : GeneralizedContinuedFraction K
inst✝ : DivisionRing K
m : ℕ
IH :
  ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),
    Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)
s : Stream'.Seq (Pair K)
gp_succ_n : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n
⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)

Next tactic:
case succ =>
      obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head
      exact s.ge_stable (m + 2).zero_le s_succ_nth_eq
      suffices
        gp_head.a / (gp_head.b + convergents'Aux s.tail (m + 2)) =
          convergents'Aux (squashSeq s (m + 1)) (m + 2)
        by simpa only [convergents'Aux, s_head_eq]
      have : convergents'Aux s.tail (m + 2) = convergents'Aux (squashSeq s.tail m) (m + 1) := by
        refine' IH gp_succ_n _
        simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq
      have : (squashSeq s (m + 1)).head = some gp_head :=
        (squashSeq_nth_of_lt m.succ_pos).trans s_head_eq
      simp_all [convergents'Aux, squashSeq_succ_n_tail_eq_squashSeq_tail_n]