Require Import Definitions.

# SRH to SR

Section SRH_SR.

Variables (R : SRS) (x0 : string) (a0 : symbol).
Notation Sigma := (a0 :: x0 ++ sym R).

Definition P :=
R ++ map (fun a => [a; a0] / [a0]) Sigma ++ map (fun a => [a0; a] / [a0]) Sigma.

Lemma sym_P :
sym P <<= Sigma.
Proof.
unfold P. rewrite !sym_app. eapply incl_app; [ | eapply incl_app].
- eauto.
- eapply sym_map. cbn. eauto.
- eapply sym_map. cbn. eauto.
Qed.

Lemma rewt_a0_L x :
x <<= Sigma -> rewt P (a0 :: x) [a0].
Proof.
intros. induction x.
- reflexivity.
- econstructor.
replace (a0 :: a :: x) with ([] ++ [a0;a] ++ x) by now simpl_list. econstructor.
unfold P. rewrite !in_app_iff, !in_map_iff. eauto 9. firstorder.
Qed.

Lemma rewt_a0_R x :
x <<= Sigma -> rewt P (x ++ [a0]) [a0].
Proof.
induction x using rev_ind.
- econstructor.
- econstructor. replace ((x1 ++ [x]) ++ [a0]) with (x1 ++ ([x] ++ [a0]) ++ []). econstructor.
cbn. unfold P. rewrite !in_app_iff, !in_map_iff. eauto 9. now simpl_list.
rewrite app_nil_r. firstorder.
Qed.

Lemma x_rewt_a0 x :
a0 el x -> x <<= Sigma -> rewt P x [a0].
Proof.
intros (y & z & ->) % in_split ?.
transitivity (y ++ [a0]).
eapply rewt_app_L, rewt_a0_L. eauto.
eapply rewt_a0_R. eauto.
Qed.

Lemma SR_SRH x :
x <<= Sigma ->
rewt P x [a0] -> exists y, rewt R x y /\ a0 el y.
Proof.
intros. pattern x; refine (rewt_induct _ _ H0).
+ exists [a0]. split. reflexivity. eauto.
+ clear H H0. intros. inv H. destruct H1 as [y []].
unfold P in H2. eapply in_app_iff in H2 as [ | [ (? & ? & ?) % in_map_iff | (? & ? & ?) % in_map_iff ] % in_app_iff].
* exists y. eauto using rewt, rew.
* inv H2. eauto 9 using rewt.
* inv H2. eauto 9 using rewt.
Qed.

Lemma equi :
SRH (R, x0, a0) <-> SR (P, x0, [a0]).
Proof.
split.
- intros (y & H & Hi).
unfold SR. transitivity y. eapply (rewt_subset H). unfold P. eauto.
eapply x_rewt_a0. firstorder. eapply rewt_sym with (x := x0); eauto.
- intros H. unfold SRH, SR in *.
eapply SR_SRH in H; eauto.
Qed.

End SRH_SR.

Theorem reduction :
SRH SR.
Proof.
exists (fun '(R, x, a) => (P R x a, x, [a])). intros [[R x0] a0].
now eapply equi.
Qed.