-- Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean
theorem exists_nth_stream_eq_none_of_rat (q : ℚ) : ∃ n : ℕ, IntFractPair.stream q n = none

Proof state:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
this✝ : fract_q_num - ↑n = -1
this : 0 ≤ ifp.fr
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
have : 0 ≤ ifp.fr.num := Rat.num_nonneg.mpr this

Result:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
this✝¹ : fract_q_num - ↑n = -1
this✝ : 0 ≤ ifp.fr
this : 0 ≤ ifp.fr.num
⊢ ∃ n, IntFractPair.stream q n = none

Proof state:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
have : fract_q_num - n = -1 := by
  have : 0 ≤ fract_q_num := Rat.num_nonneg.mpr (Int.fract_nonneg q)
  simp only [n, Nat.cast_add, Int.natAbs_of_nonneg this, Nat.cast_one,
    sub_add_eq_sub_sub_swap, sub_right_comm, sub_self, zero_sub]

Result:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
this : fract_q_num - ↑n = -1
⊢ ∃ n, IntFractPair.stream q n = none

Proof state:
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
⊢ fract_q_num - ↑n = -1

Tactic:
have : 0 ≤ fract_q_num := Rat.num_nonneg.mpr (Int.fract_nonneg q)

Result:
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
this : 0 ≤ fract_q_num
⊢ fract_q_num - ↑n = -1

Proof state:
case none
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
stream_nth_eq : IntFractPair.stream q n = none
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
use n, stream_nth_eq

Result:
no goals

Proof state:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
have ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - n :=
  stream_nth_fr_num_le_fr_num_sub_n_rat stream_nth_eq

Result:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
⊢ ∃ n, IntFractPair.stream q n = none

Proof state:
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
rcases stream_nth_eq : IntFractPair.stream q n with ifp | ifp

Result:
case none
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
stream_nth_eq : IntFractPair.stream q n = none
⊢ ∃ n, IntFractPair.stream q n = none

case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
⊢ ∃ n, IntFractPair.stream q n = none

Proof state:
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
this : 0 ≤ fract_q_num
⊢ fract_q_num - ↑n = -1

Tactic:
simp only [n, Nat.cast_add, Int.natAbs_of_nonneg this, Nat.cast_one,
  sub_add_eq_sub_sub_swap, sub_right_comm, sub_self, zero_sub]

Result:
no goals

Proof state:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
this : fract_q_num - ↑n = -1
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
have : 0 ≤ ifp.fr := (nth_stream_fr_nonneg_lt_one stream_nth_eq).left

Result:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
this✝ : fract_q_num - ↑n = -1
this : 0 ≤ ifp.fr
⊢ ∃ n, IntFractPair.stream q n = none

Proof state:
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
let n := fract_q_num.natAbs + 1

Result:
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
⊢ ∃ n, IntFractPair.stream q n = none

Proof state:
case some
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
n : ℕ := fract_q_num.natAbs + 1
ifp : IntFractPair ℚ
stream_nth_eq : IntFractPair.stream q n = some ifp
ifp_fr_num_le_q_fr_num_sub_n : ifp.fr.num ≤ fract_q_num - ↑n
this✝¹ : fract_q_num - ↑n = -1
this✝ : 0 ≤ ifp.fr
this : 0 ≤ ifp.fr.num
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
omega

Result:
no goals

Proof state:
q : ℚ
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
let fract_q_num := (Int.fract q).num

Result:
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
⊢ ∃ n, IntFractPair.stream q n = none