Require Import Arith Nat Omega List Bool.
From Undecidability.Shared.Libs.DLW.Utils Require Import utils_tac utils_list rel_iter sums.
From Undecidability.H10.Dio Require Import dio_logic dio_expo dio_bounded.
Set Implicit Arguments.
Local Notation power := (mscal mult 1).
Section df_seq.
If R is a diophantine binary relation then the predicate
fun c q n => is_seq R c q n is also diophantine. It states
that the first (n+1) digits of c in base q say x0,...,xn
form a R-sequence, ie x0 R x1 R ... R xn
Variable (R : nat -> nat -> Prop) (HR : 𝔻R (fun ν => R (ν 1) (ν 0))).
Theorem dio_rel_is_seq c q n : 𝔻P c -> 𝔻P q -> 𝔻P n
-> 𝔻R (fun ν => is_seq R (c ν) (q ν) (n ν)).
Proof.
intros H1 H2 H3.
unfold is_seq.
apply dio_rel_fall_lt; dio_rel_auto; auto.
Defined.
End df_seq.
Hint Resolve dio_rel_is_seq.
Fact dio_rel_power_subst a b (R : nat -> (nat -> nat) -> Prop) :
𝔻P a -> 𝔻P b
-> 𝔻R (fun ν => R (ν 0) (fun n => ν (S n)))
-> 𝔻R (fun ν => R (power (a ν) (b ν)) ν).
Proof.
intros Ha Hb HR.
apply dio_rel_equiv with (fun v => exists p, p = power (a v) (b v) /\ R p v).
+ intros v; split; eauto.
intros (? & ? & ?); subst; auto.
+ dio_rel_auto.
Defined.
Section df_rel_iter.
we show that for a diophantine binary relation R,
the iterator fun n x y => rel_iter R n x y is also diophantine
using the rel_iter_bounded characterization as:
rel_iter R n x y <-> exists q c, is_seq R c q n /\ is_digit c q 0 x /\ is_digit c q n y.
Variable (R : nat -> nat -> Prop) (HR : dio_rel (fun ν => R (ν 1) (ν 0))).
Lemma dio_rel_rel_iter n x y :
𝔻P n -> 𝔻P x -> 𝔻P y
-> 𝔻R (fun ν => rel_iter R (n ν) (x ν) (y ν)).
Proof.
intros Hn Hx Hy.
apply dio_rel_equiv with (1 := fun v => rel_iter_seq_equiv R (n v) (x v) (y v)).
dio_rel_auto.
Defined.
Hint Resolve dio_rel_rel_iter.
Corollary dio_rel_rt x y : 𝔻P x -> 𝔻P y ->
𝔻R (fun ν => exists i, rel_iter R i (x ν) (y ν)).
Proof. intros; dio_rel_auto. Qed.
End df_rel_iter.
Hint Resolve dio_rel_rel_iter.
Check dio_rel_rt.
Print Assumptions dio_rel_rt.