l3lab/ntp-mathlib huggingface dataset:
/-- The values before the squashed position stay the same. -/
theorem squashSeq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squashSeq s n).get? m = s.get? m := by
  cases s_succ_nth_eq : s.get? (n + 1)
  case none => rw [squashSeq_eq_self_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
m : ℕ
m_lt_n : m < n
val✝ : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some val✝
⊢ Stream'.Seq.get? (squashSeq s n) m = Stream'.Seq.get? s m

Next tactic:
obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n
/-- The values before the squashed position stay the same. -/
theorem squashSeq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squashSeq s n).get? m = s.get? m := by
  cases s_succ_nth_eq : s.get? (n + 1)
  case none => rw [squashSeq_eq_self_of_terminated s_succ_nth_eq]
  case some =>
    obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n
    

Proof state:
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
m : ℕ
m_lt_n : m < n
val✝ : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some val✝
⊢ ∃ gp_n, Stream'.Seq.get? s n = some gp_n
case intro
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
m : ℕ
m_lt_n : m < n
val✝ : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some val✝
gp_n : Pair K
s_nth_eq : Stream'.Seq.get? s n = some gp_n
⊢ Stream'.Seq.get? (squashSeq s n) m = Stream'.Seq.get? s m

Next tactic:
exact s.ge_stable n.le_succ s_succ_nth_eq
/-- The values before the squashed position stay the same. -/
theorem squashSeq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squashSeq s n).get? m = s.get? m := by
  cases s_succ_nth_eq : s.get? (n + 1)
  case none => rw [squashSeq_eq_self_of_terminated s_succ_nth_eq]
  case some =>
    obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n
    exact s.ge_stable n.le_succ s_succ_nth_eq
    

Proof state:
case intro
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
m : ℕ
m_lt_n : m < n
val✝ : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some val✝
gp_n : Pair K
s_nth_eq : Stream'.Seq.get? s n = some gp_n
⊢ Stream'.Seq.get? (squashSeq s n) m = Stream'.Seq.get? s m

Next tactic:
obtain ⟨gp_m, s_mth_eq⟩ : ∃ gp_m, s.get? m = some gp_m
/-- The values before the squashed position stay the same. -/
theorem squashSeq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squashSeq s n).get? m = s.get? m := by
  cases s_succ_nth_eq : s.get? (n + 1)
  case none => rw [squashSeq_eq_self_of_terminated s_succ_nth_eq]
  case some =>
    obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n
    exact s.ge_stable n.le_succ s_succ_nth_eq
    obtain ⟨gp_m, s_mth_eq⟩ : ∃ gp_m, s.get? m = some gp_m
    

Proof state:
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
m : ℕ
m_lt_n : m < n
val✝ : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some val✝
gp_n : Pair K
s_nth_eq : Stream'.Seq.get? s n = some gp_n
⊢ ∃ gp_m, Stream'.Seq.get? s m = some gp_m
case intro.intro
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
m : ℕ
m_lt_n : m < n
val✝ : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some val✝
gp_n : Pair K
s_nth_eq : Stream'.Seq.get? s n = some gp_n
gp_m : Pair K
s_mth_eq : Stream'.Seq.get? s m = some gp_m
⊢ Stream'.Seq.get? (squashSeq s n) m = Stream'.Seq.get? s m

Next tactic:
exact s.ge_stable (le_of_lt m_lt_n) s_nth_eq
/-- The values before the squashed position stay the same. -/
theorem squashSeq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squashSeq s n).get? m = s.get? m := by
  cases s_succ_nth_eq : s.get? (n + 1)
  case none => rw [squashSeq_eq_self_of_terminated s_succ_nth_eq]
  case some =>
    obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n
    exact s.ge_stable n.le_succ s_succ_nth_eq
    obtain ⟨gp_m, s_mth_eq⟩ : ∃ gp_m, s.get? m = some gp_m
    exact s.ge_stable (le_of_lt m_lt_n) s_nth_eq
    

Proof state:
case intro.intro
K : Type u_1
n : ℕ
g : GeneralizedContinuedFraction K
s : Stream'.Seq (Pair K)
inst✝ : DivisionRing K
m : ℕ
m_lt_n : m < n
val✝ : Pair K
s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some val✝
gp_n : Pair K
s_nth_eq : Stream'.Seq.get? s n = some gp_n
gp_m : Pair K
s_mth_eq : Stream'.Seq.get? s m = some gp_m
⊢ Stream'.Seq.get? (squashSeq s n) m = Stream'.Seq.get? s m

Next tactic:
simp [*, squashSeq, m_lt_n.ne]
/-- 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
  

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

Next tactic:
cases' s_succ_succ_nth_eq : s.get? (n + 2) with gp_succ_succ_n
/-- 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
  

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

Next tactic:
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]
/-- 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
  

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

Next tactic:
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]
/-- 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 =>
    

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

Next tactic:
cases s_succ_nth_eq : s.get? (n + 1)
/-- 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) <;>
      

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

Next tactic:
simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_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) <;>
      

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

Next tactic:
simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_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]
  

Proof state:
case 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
⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n

Next tactic:
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]
      · simp [*, squashSeq]
/-- 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]
  

Proof state:
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
⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n

Next tactic:
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]
      · simp [*, squashSeq]
/-- 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 =>
    

Proof state:
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
⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n

Next tactic:
obtain ⟨gp_succ_n, s_succ_nth_eq⟩ : ∃ gp_succ_n, s.get? (n + 1) = some gp_succ_n
/-- 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;
    

Proof state:
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, Stream'.Seq.get? s (n + 1) = some gp_succ_n
case intro
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
⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n

Next tactic:
exact s.ge_stable (n + 1).le_succ s_succ_succ_nth_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`.
    

Proof state:
case intro
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
⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n

Next tactic:
ext1 m
/-- 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
    

Proof state:
case intro.h
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
⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m

Next tactic:
cases' Decidable.em (m = n) with m_eq_n m_ne_n
/-- 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
    · 

Proof state:
case intro.h.inl
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_eq_n : m = n
⊢ 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]
/-- 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]
    · 

Proof state:
case intro.h.inr
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
⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m

Next tactic:
have : s.tail.get? m = s.get? (m + 1) := s.get?_tail m
/-- 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
      

Proof state:
case intro.h.inr
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)
⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m

Next tactic:
cases s_succ_mth_eq : s.get? (m + 1)
/-- 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)
      

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
⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m
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:
all_goals 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 

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
⊢ 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