l3lab/ntp-mathlib huggingface dataset:
/-- The exponent of product of two monoids is the `lcm` of the exponents of the
individuaul monoids. -/
@[to_additive AddMonoid.exponent_prod "The exponent of product of two additive monoids is the `lcm`
of the exponents of the individuaul additive monoids."]
theorem Monoid.exponent_prod {M₁ M₂ : Type*} [Monoid M₁] [Monoid M₂] :
    exponent (M₁ × M₂) = lcm (exponent M₁) (exponent M₂) := by
  refine dvd_antisymm ?_ (lcm_dvd ?_ ?_)
  · refine exponent_dvd_of_forall_pow_eq_one _ _ fun g ↦ ?_
    ext1
    · rw [Prod.pow_fst, Prod.fst_one, ← orderOf_dvd_iff_pow_eq_one]
      exact dvd_trans (Monoid.order_dvd_exponent (g.1)) <| dvd_lcm_left _ _
    · 

Proof state:
case refine_1.a
G : Type u
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
g : M₁ × M₂
⊢ (g ^ GCDMonoid.lcm (exponent M₁) (exponent M₂)).2 = 1.2

Next tactic:
rw [Prod.pow_snd, Prod.snd_one, ← orderOf_dvd_iff_pow_eq_one]
/-- The exponent of product of two monoids is the `lcm` of the exponents of the
individuaul monoids. -/
@[to_additive AddMonoid.exponent_prod "The exponent of product of two additive monoids is the `lcm`
of the exponents of the individuaul additive monoids."]
theorem Monoid.exponent_prod {M₁ M₂ : Type*} [Monoid M₁] [Monoid M₂] :
    exponent (M₁ × M₂) = lcm (exponent M₁) (exponent M₂) := by
  refine dvd_antisymm ?_ (lcm_dvd ?_ ?_)
  · refine exponent_dvd_of_forall_pow_eq_one _ _ fun g ↦ ?_
    ext1
    · rw [Prod.pow_fst, Prod.fst_one, ← orderOf_dvd_iff_pow_eq_one]
      exact dvd_trans (Monoid.order_dvd_exponent (g.1)) <| dvd_lcm_left _ _
    · rw [Prod.pow_snd, Prod.snd_one, ← orderOf_dvd_iff_pow_eq_one]
      

Proof state:
case refine_1.a
G : Type u
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
g : M₁ × M₂
⊢ orderOf g.2 ∣ GCDMonoid.lcm (exponent M₁) (exponent M₂)

Next tactic:
exact dvd_trans (Monoid.order_dvd_exponent (g.2)) <| dvd_lcm_right _ _
/-- The exponent of product of two monoids is the `lcm` of the exponents of the
individuaul monoids. -/
@[to_additive AddMonoid.exponent_prod "The exponent of product of two additive monoids is the `lcm`
of the exponents of the individuaul additive monoids."]
theorem Monoid.exponent_prod {M₁ M₂ : Type*} [Monoid M₁] [Monoid M₂] :
    exponent (M₁ × M₂) = lcm (exponent M₁) (exponent M₂) := by
  refine dvd_antisymm ?_ (lcm_dvd ?_ ?_)
  · refine exponent_dvd_of_forall_pow_eq_one _ _ fun g ↦ ?_
    ext1
    · rw [Prod.pow_fst, Prod.fst_one, ← orderOf_dvd_iff_pow_eq_one]
      exact dvd_trans (Monoid.order_dvd_exponent (g.1)) <| dvd_lcm_left _ _
    · rw [Prod.pow_snd, Prod.snd_one, ← orderOf_dvd_iff_pow_eq_one]
      exact dvd_trans (Monoid.order_dvd_exponent (g.2)) <| dvd_lcm_right _ _
  · 

Proof state:
case refine_2
G : Type u
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
⊢ exponent M₁ ∣ exponent (M₁ × M₂)

Next tactic:
exact MonoidHom.exponent_dvd (f := MonoidHom.fst M₁ M₂) Prod.fst_surjective
/-- The exponent of product of two monoids is the `lcm` of the exponents of the
individuaul monoids. -/
@[to_additive AddMonoid.exponent_prod "The exponent of product of two additive monoids is the `lcm`
of the exponents of the individuaul additive monoids."]
theorem Monoid.exponent_prod {M₁ M₂ : Type*} [Monoid M₁] [Monoid M₂] :
    exponent (M₁ × M₂) = lcm (exponent M₁) (exponent M₂) := by
  refine dvd_antisymm ?_ (lcm_dvd ?_ ?_)
  · refine exponent_dvd_of_forall_pow_eq_one _ _ fun g ↦ ?_
    ext1
    · rw [Prod.pow_fst, Prod.fst_one, ← orderOf_dvd_iff_pow_eq_one]
      exact dvd_trans (Monoid.order_dvd_exponent (g.1)) <| dvd_lcm_left _ _
    · rw [Prod.pow_snd, Prod.snd_one, ← orderOf_dvd_iff_pow_eq_one]
      exact dvd_trans (Monoid.order_dvd_exponent (g.2)) <| dvd_lcm_right _ _
  · exact MonoidHom.exponent_dvd (f := MonoidHom.fst M₁ M₂) Prod.fst_surjective
  · 

Proof state:
case refine_3
G : Type u
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
⊢ exponent M₂ ∣ exponent (M₁ × M₂)

Next tactic:
exact MonoidHom.exponent_dvd (f := MonoidHom.snd M₁ M₂) Prod.snd_surjective
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      

Proof state:
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
⊢ ∀ (n : ℕ),
    (∀ m < n, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts) →
      ∃ conts, continuantsAux (of v) n = Pair.map Rat.cast conts

Next tactic:
clear n
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      

Proof state:
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
⊢ ∀ (n : ℕ),
    (∀ m < n, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts) →
      ∃ conts, continuantsAux (of v) n = Pair.map Rat.cast conts

Next tactic:
let g := of v
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      

Proof state:
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
⊢ ∀ (n : ℕ),
    (∀ m < n, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts) →
      ∃ conts, continuantsAux (of v) n = Pair.map Rat.cast conts

Next tactic:
intro n IH
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      

Proof state:
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < n, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
⊢ ∃ conts, continuantsAux (of v) n = Pair.map Rat.cast conts

Next tactic:
rcases n with (_ | _ | n)
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · 

Proof state:
case zero
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
IH : ∀ m < Nat.zero, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
⊢ ∃ conts, continuantsAux (of v) Nat.zero = Pair.map Rat.cast conts

Next tactic:
suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by 

Proof state:
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
IH : ∀ m < Nat.zero, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
this : ∃ gp, { a := 1, b := 0 } = Pair.map Rat.cast gp
⊢ ∃ conts, continuantsAux (of v) Nat.zero = Pair.map Rat.cast conts

Next tactic:
simpa [continuantsAux]
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        

Proof state:
case zero
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
IH : ∀ m < Nat.zero, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
⊢ ∃ gp, { a := 1, b := 0 } = Pair.map Rat.cast gp

Next tactic:
use Pair.mk 1 0
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        

Proof state:
case h
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
IH : ∀ m < Nat.zero, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
⊢ { a := 1, b := 0 } = Pair.map Rat.cast { a := 1, b := 0 }

Next tactic:
simp
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · 

Proof state:
case succ.zero
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
IH : ∀ m < Nat.succ Nat.zero, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
⊢ ∃ conts, continuantsAux (of v) (Nat.succ Nat.zero) = Pair.map Rat.cast conts

Next tactic:
suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by 

Proof state:
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
IH : ∀ m < Nat.succ Nat.zero, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
this : ∃ conts, { a := g.h, b := 1 } = Pair.map Rat.cast conts
⊢ ∃ conts, continuantsAux (of v) (Nat.succ Nat.zero) = Pair.map Rat.cast conts

Next tactic:
simpa [continuantsAux]
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        

Proof state:
case succ.zero
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
IH : ∀ m < Nat.succ Nat.zero, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
⊢ ∃ conts, { a := g.h, b := 1 } = Pair.map Rat.cast conts

Next tactic:
use Pair.mk ⌊v⌋ 1
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        

Proof state:
case h
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
IH : ∀ m < Nat.succ Nat.zero, ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
⊢ { a := g.h, b := 1 } = Pair.map Rat.cast { a := ↑⌊v⌋, b := 1 }

Next tactic:
simp
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        simp
      -- 2 ≤ n
      · 

Proof state:
case succ.succ
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
⊢ ∃ conts, continuantsAux (of v) (Nat.succ (Nat.succ n)) = Pair.map Rat.cast conts

Next tactic:
cases' IH (n + 1) <| lt_add_one (n + 1) with pred_conts pred_conts_eq
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        simp
      -- 2 ≤ n
      · cases' IH (n + 1) <| lt_add_one (n + 1) with pred_conts pred_conts_eq
        -- invoke the IH
        

Proof state:
case succ.succ.intro
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
pred_conts : Pair ℚ
pred_conts_eq : continuantsAux (of v) (n + 1) = Pair.map Rat.cast pred_conts
⊢ ∃ conts, continuantsAux (of v) (Nat.succ (Nat.succ n)) = Pair.map Rat.cast conts

Next tactic:
cases' s_ppred_nth_eq : g.s.get? n with gp_n
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        simp
      -- 2 ≤ n
      · cases' IH (n + 1) <| lt_add_one (n + 1) with pred_conts pred_conts_eq
        -- invoke the IH
        cases' s_ppred_nth_eq : g.s.get? n with gp_n
        -- option.none
        · 

Proof state:
case succ.succ.intro.none
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
pred_conts : Pair ℚ
pred_conts_eq : continuantsAux (of v) (n + 1) = Pair.map Rat.cast pred_conts
s_ppred_nth_eq : Stream'.Seq.get? g.s n = none
⊢ ∃ conts, continuantsAux (of v) (Nat.succ (Nat.succ n)) = Pair.map Rat.cast conts

Next tactic:
use pred_conts
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        simp
      -- 2 ≤ n
      · cases' IH (n + 1) <| lt_add_one (n + 1) with pred_conts pred_conts_eq
        -- invoke the IH
        cases' s_ppred_nth_eq : g.s.get? n with gp_n
        -- option.none
        · use pred_conts
          

Proof state:
case h
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
pred_conts : Pair ℚ
pred_conts_eq : continuantsAux (of v) (n + 1) = Pair.map Rat.cast pred_conts
s_ppred_nth_eq : Stream'.Seq.get? g.s n = none
⊢ continuantsAux (of v) (Nat.succ (Nat.succ n)) = Pair.map Rat.cast pred_conts

Next tactic:
have : g.continuantsAux (n + 2) = g.continuantsAux (n + 1) :=
            continuantsAux_stable_of_terminated (n + 1).le_succ s_ppred_nth_eq
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        simp
      -- 2 ≤ n
      · cases' IH (n + 1) <| lt_add_one (n + 1) with pred_conts pred_conts_eq
        -- invoke the IH
        cases' s_ppred_nth_eq : g.s.get? n with gp_n
        -- option.none
        · use pred_conts
          have : g.continuantsAux (n + 2) = g.continuantsAux (n + 1) :=
            continuantsAux_stable_of_terminated (n + 1).le_succ s_ppred_nth_eq
          

Proof state:
case h
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
pred_conts : Pair ℚ
pred_conts_eq : continuantsAux (of v) (n + 1) = Pair.map Rat.cast pred_conts
s_ppred_nth_eq : Stream'.Seq.get? g.s n = none
this : continuantsAux g (n + 2) = continuantsAux g (n + 1)
⊢ continuantsAux (of v) (Nat.succ (Nat.succ n)) = Pair.map Rat.cast pred_conts

Next tactic:
simp only [this, pred_conts_eq]
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        simp
      -- 2 ≤ n
      · cases' IH (n + 1) <| lt_add_one (n + 1) with pred_conts pred_conts_eq
        -- invoke the IH
        cases' s_ppred_nth_eq : g.s.get? n with gp_n
        -- option.none
        · use pred_conts
          have : g.continuantsAux (n + 2) = g.continuantsAux (n + 1) :=
            continuantsAux_stable_of_terminated (n + 1).le_succ s_ppred_nth_eq
          simp only [this, pred_conts_eq]
        -- option.some
        · -- invoke the IH a second time
          

Proof state:
case succ.succ.intro.some
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
pred_conts : Pair ℚ
pred_conts_eq : continuantsAux (of v) (n + 1) = Pair.map Rat.cast pred_conts
gp_n : Pair K
s_ppred_nth_eq : Stream'.Seq.get? g.s n = some gp_n
⊢ ∃ conts, continuantsAux (of v) (Nat.succ (Nat.succ n)) = Pair.map Rat.cast conts

Next tactic:
cases' IH n <| lt_of_le_of_lt n.le_succ <| lt_add_one <| n + 1 with ppred_conts
            ppred_conts_eq
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        simp
      -- 2 ≤ n
      · cases' IH (n + 1) <| lt_add_one (n + 1) with pred_conts pred_conts_eq
        -- invoke the IH
        cases' s_ppred_nth_eq : g.s.get? n with gp_n
        -- option.none
        · use pred_conts
          have : g.continuantsAux (n + 2) = g.continuantsAux (n + 1) :=
            continuantsAux_stable_of_terminated (n + 1).le_succ s_ppred_nth_eq
          simp only [this, pred_conts_eq]
        -- option.some
        · -- invoke the IH a second time
          cases' IH n <| lt_of_le_of_lt n.le_succ <| lt_add_one <| n + 1 with ppred_conts
            ppred_conts_eq
          

Proof state:
case succ.succ.intro.some.intro
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
pred_conts : Pair ℚ
pred_conts_eq : continuantsAux (of v) (n + 1) = Pair.map Rat.cast pred_conts
gp_n : Pair K
s_ppred_nth_eq : Stream'.Seq.get? g.s n = some gp_n
ppred_conts : Pair ℚ
ppred_conts_eq : continuantsAux (of v) n = Pair.map Rat.cast ppred_conts
⊢ ∃ conts, continuantsAux (of v) (Nat.succ (Nat.succ n)) = Pair.map Rat.cast conts

Next tactic:
obtain ⟨a_eq_one, z, b_eq_z⟩ : gp_n.a = 1 ∧ ∃ z : ℤ, gp_n.b = (z : K)
nonrec theorem exists_gcf_pair_rat_eq_of_nth_conts_aux :
    ∃ conts : Pair ℚ, (of v).continuantsAux n = (conts.map (↑) : Pair K) :=
  Nat.strong_induction_on n
    (by
      clear n
      let g := of v
      intro n IH
      rcases n with (_ | _ | n)
      -- n = 0
      · suffices ∃ gp : Pair ℚ, Pair.mk (1 : K) 0 = gp.map (↑) by simpa [continuantsAux]
        use Pair.mk 1 0
        simp
      -- n = 1
      · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [continuantsAux]
        use Pair.mk ⌊v⌋ 1
        simp
      -- 2 ≤ n
      · cases' IH (n + 1) <| lt_add_one (n + 1) with pred_conts pred_conts_eq
        -- invoke the IH
        cases' s_ppred_nth_eq : g.s.get? n with gp_n
        -- option.none
        · use pred_conts
          have : g.continuantsAux (n + 2) = g.continuantsAux (n + 1) :=
            continuantsAux_stable_of_terminated (n + 1).le_succ s_ppred_nth_eq
          simp only [this, pred_conts_eq]
        -- option.some
        · -- invoke the IH a second time
          cases' IH n <| lt_of_le_of_lt n.le_succ <| lt_add_one <| n + 1 with ppred_conts
            ppred_conts_eq
          obtain ⟨a_eq_one, z, b_eq_z⟩ : gp_n.a = 1 ∧ ∃ z : ℤ, gp_n.b = (z : K);
          

Proof state:
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
pred_conts : Pair ℚ
pred_conts_eq : continuantsAux (of v) (n + 1) = Pair.map Rat.cast pred_conts
gp_n : Pair K
s_ppred_nth_eq : Stream'.Seq.get? g.s n = some gp_n
ppred_conts : Pair ℚ
ppred_conts_eq : continuantsAux (of v) n = Pair.map Rat.cast ppred_conts
⊢ gp_n.a = 1 ∧ ∃ z, gp_n.b = ↑z
case succ.succ.intro.some.intro.intro.intro
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
g : GeneralizedContinuedFraction K := of v
n : ℕ
IH : ∀ m < Nat.succ (Nat.succ n), ∃ conts, continuantsAux (of v) m = Pair.map Rat.cast conts
pred_conts : Pair ℚ
pred_conts_eq : continuantsAux (of v) (n + 1) = Pair.map Rat.cast pred_conts
gp_n : Pair K
s_ppred_nth_eq : Stream'.Seq.get? g.s n = some gp_n
ppred_conts : Pair ℚ
ppred_conts_eq : continuantsAux (of v) n = Pair.map Rat.cast ppred_conts
a_eq_one : gp_n.a = 1
z : ℤ
b_eq_z : gp_n.b = ↑z
⊢ ∃ conts, continuantsAux (of v) (Nat.succ (Nat.succ n)) = Pair.map Rat.cast conts

Next tactic:
exact of_part_num_eq_one_and_exists_int_part_denom_eq s_ppred_nth_eq