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/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import algebra.continued_fractions.translations
/-!
# Stabilisation of gcf Computations Under Termination
## Summary
We show that the continuants and convergents of a gcf stabilise once the gcf terminates.
-/
namespace generalized_continued_fraction
variables {K : Type*} {g : generalized_continued_fraction K} {n m : ℕ}
/-- If a gcf terminated at position `n`, it also terminated at `m ≥ n`.-/
lemma terminated_stable (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.terminated_at m :=
g.s.terminated_stable n_le_m terminated_at_n
variable [division_ring K]
lemma continuants_aux_stable_step_of_terminated (terminated_at_n : g.terminated_at n) :
g.continuants_aux (n + 2) = g.continuants_aux (n + 1) :=
by { rw [terminated_at_iff_s_none] at terminated_at_n,
simp only [terminated_at_n, continuants_aux] }
lemma continuants_aux_stable_of_terminated (succ_n_le_m : (n + 1) ≤ m)
(terminated_at_n : g.terminated_at n) :
g.continuants_aux m = g.continuants_aux (n + 1) :=
begin
induction succ_n_le_m with m succ_n_le_m IH,
{ refl },
{ have : g.continuants_aux (m + 1) = g.continuants_aux m, by
{ have : n ≤ m - 1, from nat.le_pred_of_lt succ_n_le_m,
have : g.terminated_at (m - 1), from terminated_stable this terminated_at_n,
have stable_step : g.continuants_aux (m - 1 + 2) = g.continuants_aux (m - 1 + 1), from
continuants_aux_stable_step_of_terminated this,
have one_le_m : 1 ≤ m, from nat.one_le_of_lt succ_n_le_m,
have : m - 1 + 2 = m + 2 - 1, from tsub_add_eq_add_tsub one_le_m,
have : m - 1 + 1 = m + 1 - 1, from tsub_add_eq_add_tsub one_le_m,
simpa [*] using stable_step },
exact (eq.trans this IH) }
end
lemma convergents'_aux_stable_step_of_terminated {s : seq $ pair K}
(terminated_at_n : s.terminated_at n) :
convergents'_aux s (n + 1) = convergents'_aux s n :=
begin
change s.nth n = none at terminated_at_n,
induction n with n IH generalizing s,
case nat.zero
{ simp only [convergents'_aux, terminated_at_n, seq.head] },
case nat.succ
{ cases s_head_eq : s.head with gp_head,
case option.none { simp only [convergents'_aux, s_head_eq] },
case option.some
{ have : s.tail.terminated_at n, by simp only [seq.terminated_at, s.nth_tail, terminated_at_n],
simp only [convergents'_aux, s_head_eq, (IH this)] } }
end
lemma convergents'_aux_stable_of_terminated
{s : seq $ pair K} (n_le_m : n ≤ m)
(terminated_at_n : s.terminated_at n) :
convergents'_aux s m = convergents'_aux s n :=
begin
induction n_le_m with m n_le_m IH generalizing s,
{ refl },
{ cases s_head_eq : s.head with gp_head,
case option.none { cases n; simp only [convergents'_aux, s_head_eq] },
case option.some
{ have : convergents'_aux s (n + 1) = convergents'_aux s n, from
convergents'_aux_stable_step_of_terminated terminated_at_n,
rw [←this],
have : s.tail.terminated_at n, by
simpa only [seq.terminated_at, seq.nth_tail] using (s.le_stable n.le_succ terminated_at_n),
have : convergents'_aux s.tail m = convergents'_aux s.tail n, from IH this,
simp only [convergents'_aux, s_head_eq, this] } }
end
lemma continuants_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.continuants m = g.continuants n :=
by simp only [nth_cont_eq_succ_nth_cont_aux,
(continuants_aux_stable_of_terminated (nat.pred_le_iff.elim_left n_le_m) terminated_at_n)]
lemma numerators_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.numerators m = g.numerators n :=
by simp only [num_eq_conts_a, (continuants_stable_of_terminated n_le_m terminated_at_n)]
lemma denominators_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.denominators m = g.denominators n :=
by simp only [denom_eq_conts_b, (continuants_stable_of_terminated n_le_m terminated_at_n)]
lemma convergents_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.convergents m = g.convergents n :=
by simp only [convergents, (denominators_stable_of_terminated n_le_m terminated_at_n),
(numerators_stable_of_terminated n_le_m terminated_at_n)]
lemma convergents'_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.convergents' m = g.convergents' n :=
by simp only [convergents', (convergents'_aux_stable_of_terminated n_le_m terminated_at_n)]
end generalized_continued_fraction
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