ARS
Abstract Reduction Systems, from Semantics Lecture at Programming Systems Lab, https://www.ps.uni-saarland.de/courses/sem-ws13/
Require Export Base.
Notation "p '<=1' q" := (forall x, p x -> q x) (at level 70).
Notation "p '=1' q" := (p <=1 q /\ q <=1 p) (at level 70).
Notation "R '<=2' S" := (forall x y, R x y -> S x y) (at level 70).
Notation "R '=2' S" := (R <=2 S /\ S <=2 R) (at level 70).
Relational composition
Definition rcomp X Y Z (R : X -> Y -> Prop) (S : Y -> Z -> Prop)
: X -> Z -> Prop :=
fun x z => exists y, R x y /\ S y z.
Power predicates
Require Import Arith.
Definition pow X R n : X -> X -> Prop := it (rcomp R) n eq.
Section FixX.
Variable X : Type.
Implicit Types R S : X -> X -> Prop.
Implicit Types x y z : X.
Definition reflexive R := forall x, R x x.
Definition symmetric R := forall x y, R x y -> R y x.
Definition transitive R := forall x y z, R x y -> R y z -> R x z.
Definition functional R := forall x y z, R x y -> R x z -> y = z.
Reflexive transitive closure
Inductive star R : X -> X -> Prop :=
| starR x : star R x x
| starC x y z : R x y -> star R y z -> star R x z.
(* Making first argument a non-uniform parameter doesn't simplify the induction principle. *)
Lemma star_simpl_ind R (p : X -> Prop) y :
p y ->
(forall x x', R x x' -> star R x' y -> p x' -> p x) ->
forall x, star R x y -> p x.
Proof.
intros A B. induction 1; eauto.
Qed.
Lemma star_trans R:
transitive (star R).
Proof.
induction 1; eauto using star.
Qed.
Power characterization
Lemma star_pow R x y :
star R x y <-> exists n, pow R n x y.
Proof.
split; intros A.
- induction A as [|x x' y B _ [n IH]].
+ exists 0. reflexivity.
+ exists (S n), x'. auto.
- destruct A as [n A].
revert x A. induction n; intros x A.
+ destruct A. constructor.
+ destruct A as [x' [A B]]. econstructor; eauto.
Qed.
Lemma pow_star R x y n:
pow R n x y -> star R x y.
Proof.
intros A. erewrite star_pow. eauto.
Qed.
Equivalence closure
Inductive ecl R : X -> X -> Prop :=
| eclR x : ecl R x x
| eclC x y z : R x y -> ecl R y z -> ecl R x z
| eclS x y z : R y x -> ecl R y z -> ecl R x z.
Lemma ecl_trans R :
transitive (ecl R).
Proof.
induction 1; eauto using ecl.
Qed.
Lemma ecl_sym R :
symmetric (ecl R).
Proof.
induction 1; eauto using ecl, (@ecl_trans R).
Qed.
Lemma star_ecl R :
star R <=2 ecl R.
Proof.
induction 1; eauto using ecl.
Qed.
Diamond, confluence, Church-Rosser
Definition joinable R x y :=
exists z, R x z /\ R y z.
Definition diamond R :=
forall x y z, R x y -> R x z -> joinable R y z.
Definition confluent R := diamond (star R).
Definition semi_confluent R :=
forall x y z, R x y -> star R x z -> joinable (star R) y z.
Definition church_rosser R :=
ecl R <=2 joinable (star R).
Goal forall R, diamond R -> semi_confluent R.
Proof.
intros R A x y z B C.
revert x C y B.
refine (star_simpl_ind _ _).
- intros y C. exists y. eauto using star.
- intros x x' C D IH y E.
destruct (A _ _ _ C E) as [v [F G]].
destruct (IH _ F) as [u [H I]].
assert (J:= starC G H).
exists u. eauto using star.
Qed.
Lemma diamond_to_semi_confluent R :
diamond R -> semi_confluent R.
Proof.
intros A x y z B C. revert y B.
induction C as [|x x' z D _ IH]; intros y B.
- exists y. eauto using star.
- destruct (A _ _ _ B D) as [v [E F]].
destruct (IH _ F) as [u [G H]].
exists u. eauto using star.
Qed.
Lemma semi_confluent_confluent R :
semi_confluent R <-> confluent R.
Proof.
split; intros A x y z B C.
- revert y B.
induction C as [|x x' z D _ IH]; intros y B.
+ exists y. eauto using star.
+ destruct (A _ _ _ D B) as [v [E F]].
destruct (IH _ E) as [u [G H]].
exists u. eauto using (@star_trans R).
- apply (A x y z); eauto using star.
Qed.
Lemma diamond_to_confluent R :
diamond R -> confluent R.
Proof.
intros A. apply semi_confluent_confluent, diamond_to_semi_confluent, A.
Qed.
Lemma confluent_CR R :
church_rosser R <-> confluent R.
Proof.
split; intros A.
- intros x y z B C. apply A.
eauto using (@ecl_trans R), star_ecl, (@ecl_sym R).
- intros x y B. apply semi_confluent_confluent in A.
induction B as [x|x x' y C B IH|x x' y C B IH].
+ exists x. eauto using star.
+ destruct IH as [z [D E]]. exists z. eauto using star.
+ destruct IH as [u [D E]].
destruct (A _ _ _ C D) as [z [F G]].
exists z. eauto using (@star_trans R).
Qed.
(* End Semantics Library *)
Uniform confluence and parametrized confluence
Definition uniform_confluent (R : X -> X -> Prop ) := forall s t1 t2, R s t1 -> R s t2 -> t1 = t2 \/ exists u, R t1 u /\ R t2 u.
Lemma pow_add R n m (s t : X) : pow R (n + m) s t <-> rcomp (pow R n) (pow R m) s t.
Proof.
revert m s t; induction n; intros m s t.
- simpl. split; intros. econstructor. split. unfold pow. simpl. reflexivity. eassumption.
destruct H as [u [H1 H2]]. unfold pow in H1. simpl in *. subst s. eassumption.
- simpl in *; split; intros.
+ destruct H as [u [H1 H2]].
change (it (rcomp R) (n + m) eq) with (pow R (n+m)) in H2.
rewrite IHn in H2.
destruct H2 as [u' [A B]]. unfold pow in A.
econstructor.
split. econstructor. repeat split; repeat eassumption. eassumption.
+ destruct H as [u [H1 H2]].
destruct H1 as [u' [A B]].
econstructor. split. eassumption. change (it (rcomp R) (n + m) eq) with (pow R (n + m)).
rewrite IHn. econstructor. split; eassumption.
Qed.
Lemma rcomp_eq (R S R' S' : X -> X -> Prop) (s t : X) : (R =2 R') -> (S =2 S') -> (rcomp R S s t <-> rcomp R' S' s t).
Proof.
intros A B.
split; intros H; destruct H as [u [H1 H2]];
eapply A in H1; eapply B in H2;
econstructor; split; eassumption.
Qed.
Lemma eq_ref : forall (R : X -> X -> Prop), R =2 R.
Proof.
split; intros s t; tauto.
Qed.
Lemma rcomp_1 (R : X -> X -> Prop): R =2 pow R 1.
Proof.
split; intros s t; unfold pow in *; simpl in *; intros H.
- econstructor. split; eauto.
- destruct H as [u [H1 H2]]; subst u; eassumption.
Qed.
Lemma parametrized_semi_confluence (R : X -> X -> Prop) (m : nat) (s t1 t2 : X) :
uniform_confluent R ->
pow R m s t1 ->
R s t2 ->
exists k l u,
k <= 1 /\ l <= m /\ pow R k t1 u /\ pow R l t2 u /\ m + k = S l.
Proof.
intros unifConfR; revert s t1 t2; induction m; intros s t1 t2 s_to_t1 s_to_t2.
- unfold pow in s_to_t1. simpl in *. subst s.
exists 1, 0, t2.
repeat split; try omega.
econstructor. split; try eassumption; econstructor.
- destruct s_to_t1 as [v [s_to_v v_to_t1]].
destruct (unifConfR _ _ _ s_to_v s_to_t2) as [H | [u [v_to_u t2_to_u]]].
+ subst v. eexists 0, m, t1; repeat split; try omega; eassumption.
+ destruct (IHm _ _ _ v_to_t1 v_to_u) as [k [l [u' H]]].
eexists k, (S l), u'; repeat split; try omega; try tauto.
econstructor. split. eassumption. tauto.
Qed.
Lemma rcomp_comm R m (s t : X) : rcomp R (it (rcomp R) m eq) s t <-> rcomp (it (rcomp R) m eq) R s t.
Proof.
split; intros H;
[rewrite (rcomp_eq s t (rcomp_1 R) (eq_ref _)) in H;
rewrite (rcomp_eq s t (eq_ref _) (rcomp_1 R)) |
rewrite (rcomp_eq s t (eq_ref _) (rcomp_1 R)) in H;
rewrite (rcomp_eq s t (rcomp_1 R) (eq_ref _))];
change ((it (rcomp R) m eq)) with (pow R m) in *;
try rewrite <- pow_add in *;
rewrite plus_comm; eassumption.
Qed.
Lemma parametrized_confluence (R : X -> X -> Prop) (m n : nat) (s t1 t2 : X) :
uniform_confluent R ->
pow R m s t1 ->
pow R n s t2 ->
exists k l u,
k <= n /\ l <= m /\ pow R k t1 u /\ pow R l t2 u /\ m + k = n + l.
Proof.
revert n s t1 t2; induction m; intros n s t1 t2 unifConR s_to_t1 s_to_t2.
- unfold pow in s_to_t1. simpl in s_to_t1. subst s.
exists n, 0, t2. repeat split; try now omega. eassumption.
- unfold pow in s_to_t1. simpl in *.
destruct s_to_t1 as [v [s_to_v v_to_t1]].
destruct (parametrized_semi_confluence unifConR s_to_t2 s_to_v) as
[k [l [u [k_lt_1 [l_lt_n [t2_to_u [v_to_u H]]]]]]].
destruct (IHm _ _ _ _ unifConR v_to_t1 v_to_u) as
[l'[k'[u'[l'_lt_l [k'_lt_m [t1_to_u' [u_to_u' H2]]]]]]].
exists l', (k + k'), u'.
repeat split; try omega. eassumption.
rewrite pow_add.
econstructor; split; eassumption.
Qed.
End FixX.