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" := ( 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" := ( 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 => 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 := x, R x x.
  Definition symmetric R := x y, R x y R y x.
  Definition transitive R := x y z, R x y R y z R x z.
  Definition functional R := 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
    ( x x', R x x' star R x' y p x' p x)
     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 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 :=
     z, R x z R y z.

  Definition diamond R :=
     x y z, R x y R x z joinable R y z.

  Definition confluent R := diamond (star R).

  Definition semi_confluent R :=
     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 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 ) := s , R s R s = u, R u R 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 [ ]]. unfold pow in . simpl in *. subst s. eassumption.
    - simpl in *; split; intros.
      + destruct H as [u [ ]].
        change (it (rcomp R) (n + m) eq) with (pow R (n+m)) in .
        rewrite IHn in .
        destruct as [u' [A B]]. unfold pow in A.
        econstructor.
        split. econstructor. repeat split; repeat eassumption. eassumption.
      + destruct H as [u [ ]].
        destruct 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 [ ]];
    eapply A in ; eapply B in ;
    econstructor; split; eassumption.
  Qed.

  Lemma eq_ref : (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 [ ]]; subst u; eassumption.
  Qed.

  Lemma parametrized_semi_confluence (R : X X Prop) (m : ) (s : X) :
    uniform_confluent R
    pow R m s
    R s
     k l u,
      k 1 l m pow R k u pow R l u m + k = S l.
  Proof.
    intros unifConfR; revert s ; induction m; intros s s_to_t1 s_to_t2.
    - unfold pow in s_to_t1. simpl in *. subst s.
      exists 1, 0, .
      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, ; 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 : ) (s : X) :
    uniform_confluent R
    pow R m s
    pow R n s
     k l u,
      k n l m pow R k u pow R l u m + k = n + l.
  Proof.
    revert n s ; induction m; intros n s unifConR s_to_t1 s_to_t2.
    - unfold pow in s_to_t1. simpl in s_to_t1. subst s.
      exists n, 0, . 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'[ [ [t1_to_u' [u_to_u' ]]]]]]].
      exists l', (k + k'), u'.
      repeat split; try omega. eassumption.
      rewrite pow_add.
      econstructor; split; eassumption.
  Qed.

End FixX.