Module Smallstep


Tools for small-step operational semantics

This module defines generic operations and theorems over the one-step transition relations that are used to specify operational semantics in small-step style.

Require Import Relations.
Require Import Wellfounded.
Require Import Coqlib.
Require Import Events.
Require Import Globalenvs.
Require Import Integers.

Set Implicit Arguments.

Closures of transitions relations


Section CLOSURES.

Variable genv: Type.
Variable state: Type.

A one-step transition relation has the following signature. It is parameterized by a global environment, which does not change during the transition. It relates the initial state of the transition with its final state. The trace parameter captures the observable events possibly generated during the transition.

Variable step: genv -> state -> trace -> state -> Prop.

No transitions: stuck state

Definition nostep (ge: genv) (s: state) : Prop :=
  forall t s', ~(step ge s t s').

Zero, one or several transitions. Also known as Kleene closure, or reflexive transitive closure.

Inductive star (ge: genv): state -> trace -> state -> Prop :=
  | star_refl: forall s,
      star ge s E0 s
  | star_step: forall s1 t1 s2 t2 s3 t,
      step ge s1 t1 s2 -> star ge s2 t2 s3 -> t = t1 ** t2 ->
      star ge s1 t s3.

Lemma star_one:
  forall ge s1 t s2, step ge s1 t s2 -> star ge s1 t s2.
Proof.
  intros. eapply star_step; eauto. apply star_refl. traceEq.
Qed.

Lemma star_two:
  forall ge s1 t1 s2 t2 s3 t,
  step ge s1 t1 s2 -> step ge s2 t2 s3 -> t = t1 ** t2 ->
  star ge s1 t s3.
Proof.
  intros. eapply star_step; eauto. apply star_one; auto.
Qed.

Lemma star_three:
  forall ge s1 t1 s2 t2 s3 t3 s4 t,
  step ge s1 t1 s2 -> step ge s2 t2 s3 -> step ge s3 t3 s4 -> t = t1 ** t2 ** t3 ->
  star ge s1 t s4.
Proof.
  intros. eapply star_step; eauto. eapply star_two; eauto.
Qed.

Lemma star_four:
  forall ge s1 t1 s2 t2 s3 t3 s4 t4 s5 t,
  step ge s1 t1 s2 -> step ge s2 t2 s3 ->
  step ge s3 t3 s4 -> step ge s4 t4 s5 -> t = t1 ** t2 ** t3 ** t4 ->
  star ge s1 t s5.
Proof.
  intros. eapply star_step; eauto. eapply star_three; eauto.
Qed.

Lemma star_trans:
  forall ge s1 t1 s2, star ge s1 t1 s2 ->
  forall t2 s3 t, star ge s2 t2 s3 -> t = t1 ** t2 -> star ge s1 t s3.
Proof.
  induction 1; intros.
  rewrite H0. simpl. auto.
  eapply star_step; eauto. traceEq.
Qed.

Lemma star_left:
  forall ge s1 t1 s2 t2 s3 t,
  step ge s1 t1 s2 -> star ge s2 t2 s3 -> t = t1 ** t2 ->
  star ge s1 t s3.
Proof star_step.

Lemma star_right:
  forall ge s1 t1 s2 t2 s3 t,
  star ge s1 t1 s2 -> step ge s2 t2 s3 -> t = t1 ** t2 ->
  star ge s1 t s3.
Proof.
  intros. eapply star_trans. eauto. apply star_one. eauto. auto.
Qed.

Lemma star_E0_ind:
  forall ge (P: state -> state -> Prop),
  (forall s, P s s) ->
  (forall s1 s2 s3, step ge s1 E0 s2 -> P s2 s3 -> P s1 s3) ->
  forall s1 s2, star ge s1 E0 s2 -> P s1 s2.
Proof.
  intros ge P BASE REC.
  assert (forall s1 t s2, star ge s1 t s2 -> t = E0 -> P s1 s2).
    induction 1; intros; subst.
    auto.
    destruct (Eapp_E0_inv _ _ H2). subst. eauto.
  eauto.
Qed.

One or several transitions. Also known as the transitive closure.

Inductive plus (ge: genv): state -> trace -> state -> Prop :=
  | plus_left: forall s1 t1 s2 t2 s3 t,
      step ge s1 t1 s2 -> star ge s2 t2 s3 -> t = t1 ** t2 ->
      plus ge s1 t s3.

Lemma plus_one:
  forall ge s1 t s2,
  step ge s1 t s2 -> plus ge s1 t s2.
Proof.
  intros. econstructor; eauto. apply star_refl. traceEq.
Qed.

Lemma plus_two:
  forall ge s1 t1 s2 t2 s3 t,
  step ge s1 t1 s2 -> step ge s2 t2 s3 -> t = t1 ** t2 ->
  plus ge s1 t s3.
Proof.
  intros. eapply plus_left; eauto. apply star_one; auto.
Qed.

Lemma plus_three:
  forall ge s1 t1 s2 t2 s3 t3 s4 t,
  step ge s1 t1 s2 -> step ge s2 t2 s3 -> step ge s3 t3 s4 -> t = t1 ** t2 ** t3 ->
  plus ge s1 t s4.
Proof.
  intros. eapply plus_left; eauto. eapply star_two; eauto.
Qed.

Lemma plus_four:
  forall ge s1 t1 s2 t2 s3 t3 s4 t4 s5 t,
  step ge s1 t1 s2 -> step ge s2 t2 s3 ->
  step ge s3 t3 s4 -> step ge s4 t4 s5 -> t = t1 ** t2 ** t3 ** t4 ->
  plus ge s1 t s5.
Proof.
  intros. eapply plus_left; eauto. eapply star_three; eauto.
Qed.

Lemma plus_star:
  forall ge s1 t s2, plus ge s1 t s2 -> star ge s1 t s2.
Proof.
  intros. inversion H; subst.
  eapply star_step; eauto.
Qed.

Lemma plus_right:
  forall ge s1 t1 s2 t2 s3 t,
  star ge s1 t1 s2 -> step ge s2 t2 s3 -> t = t1 ** t2 ->
  plus ge s1 t s3.
Proof.
  intros. inversion H; subst. simpl. apply plus_one. auto.
  rewrite Eapp_assoc. eapply plus_left; eauto.
  eapply star_right; eauto.
Qed.

Lemma plus_left':
  forall ge s1 t1 s2 t2 s3 t,
  step ge s1 t1 s2 -> plus ge s2 t2 s3 -> t = t1 ** t2 ->
  plus ge s1 t s3.
Proof.
  intros. eapply plus_left; eauto. apply plus_star; auto.
Qed.

Lemma plus_right':
  forall ge s1 t1 s2 t2 s3 t,
  plus ge s1 t1 s2 -> step ge s2 t2 s3 -> t = t1 ** t2 ->
  plus ge s1 t s3.
Proof.
  intros. eapply plus_right; eauto. apply plus_star; auto.
Qed.

Lemma plus_star_trans:
  forall ge s1 t1 s2 t2 s3 t,
  plus ge s1 t1 s2 -> star ge s2 t2 s3 -> t = t1 ** t2 -> plus ge s1 t s3.
Proof.
  intros. inversion H; subst.
  econstructor; eauto. eapply star_trans; eauto.
  traceEq.
Qed.

Lemma star_plus_trans:
  forall ge s1 t1 s2 t2 s3 t,
  star ge s1 t1 s2 -> plus ge s2 t2 s3 -> t = t1 ** t2 -> plus ge s1 t s3.
Proof.
  intros. inversion H; subst.
  simpl; auto.
  rewrite Eapp_assoc.
  econstructor. eauto. eapply star_trans. eauto.
  apply plus_star. eauto. eauto. auto.
Qed.

Lemma plus_trans:
  forall ge s1 t1 s2 t2 s3 t,
  plus ge s1 t1 s2 -> plus ge s2 t2 s3 -> t = t1 ** t2 -> plus ge s1 t s3.
Proof.
  intros. eapply plus_star_trans. eauto. apply plus_star. eauto. auto.
Qed.

Lemma plus_inv:
  forall ge s1 t s2,
  plus ge s1 t s2 ->
  step ge s1 t s2 \/ exists s', exists t1, exists t2, step ge s1 t1 s' /\ plus ge s' t2 s2 /\ t = t1 ** t2.
Proof.
  intros. inversion H; subst. inversion H1; subst.
  left. rewrite E0_right. auto.
  right. exists s3; exists t1; exists (t0 ** t3); split. auto.
  split. econstructor; eauto. auto.
Qed.

Lemma star_inv:
  forall ge s1 t s2,
  star ge s1 t s2 ->
  (s2 = s1 /\ t = E0) \/ plus ge s1 t s2.
Proof.
  intros. inv H. left; auto. right; econstructor; eauto.
Qed.

Lemma plus_ind2:
  forall ge (P: state -> trace -> state -> Prop),
  (forall s1 t s2, step ge s1 t s2 -> P s1 t s2) ->
  (forall s1 t1 s2 t2 s3 t,
   step ge s1 t1 s2 -> plus ge s2 t2 s3 -> P s2 t2 s3 -> t = t1 ** t2 ->
   P s1 t s3) ->
  forall s1 t s2, plus ge s1 t s2 -> P s1 t s2.
Proof.
  intros ge P BASE IND.
  assert (forall s1 t s2, star ge s1 t s2 ->
         forall s0 t0, step ge s0 t0 s1 ->
         P s0 (t0 ** t) s2).
  induction 1; intros.
  rewrite E0_right. apply BASE; auto.
  eapply IND. eauto. econstructor; eauto. subst t. eapply IHstar; eauto. auto.

  intros. inv H0. eauto.
Qed.

Lemma plus_E0_ind:
  forall ge (P: state -> state -> Prop),
  (forall s1 s2 s3, step ge s1 E0 s2 -> star ge s2 E0 s3 -> P s1 s3) ->
  forall s1 s2, plus ge s1 E0 s2 -> P s1 s2.
Proof.
  intros. inv H0. exploit Eapp_E0_inv; eauto. intros [A B]; subst. eauto.
Qed.

Counted sequences of transitions

Inductive starN (ge: genv): nat -> state -> trace -> state -> Prop :=
  | starN_refl: forall s,
      starN ge O s E0 s
  | starN_step: forall n s t t1 s' t2 s'',
      step ge s t1 s' -> starN ge n s' t2 s'' -> t = t1 ** t2 ->
      starN ge (S n) s t s''.

Remark starN_star:
  forall ge n s t s', starN ge n s t s' -> star ge s t s'.
Proof.
  induction 1; econstructor; eauto.
Qed.

Remark star_starN:
  forall ge s t s', star ge s t s' -> exists n, starN ge n s t s'.
Proof.
  induction 1.
  exists O; constructor.
  destruct IHstar as [n P]. exists (S n); econstructor; eauto.
Qed.

Infinitely many transitions

CoInductive forever (ge: genv): state -> traceinf -> Prop :=
  | forever_intro: forall s1 t s2 T,
      step ge s1 t s2 -> forever ge s2 T ->
      forever ge s1 (t *** T).

Lemma star_forever:
  forall ge s1 t s2, star ge s1 t s2 ->
  forall T, forever ge s2 T ->
  forever ge s1 (t *** T).
Proof.
  induction 1; intros. simpl. auto.
  subst t. rewrite Eappinf_assoc.
  econstructor; eauto.
Qed.

An alternate, equivalent definition of forever that is useful for coinductive reasoning.

Variable A: Type.
Variable order: A -> A -> Prop.

CoInductive forever_N (ge: genv) : A -> state -> traceinf -> Prop :=
  | forever_N_star: forall s1 t s2 a1 a2 T1 T2,
      star ge s1 t s2 ->
      order a2 a1 ->
      forever_N ge a2 s2 T2 ->
      T1 = t *** T2 ->
      forever_N ge a1 s1 T1
  | forever_N_plus: forall s1 t s2 a1 a2 T1 T2,
      plus ge s1 t s2 ->
      forever_N ge a2 s2 T2 ->
      T1 = t *** T2 ->
      forever_N ge a1 s1 T1.

Hypothesis order_wf: well_founded order.

Lemma forever_N_inv:
  forall ge a s T,
  forever_N ge a s T ->
  exists t, exists s', exists a', exists T',
  step ge s t s' /\ forever_N ge a' s' T' /\ T = t *** T'.
Proof.
  intros ge a0. pattern a0. apply (well_founded_ind order_wf).
  intros. inv H0.
  inv H1.
  change (E0 *** T2) with T2. apply H with a2. auto. auto.
  exists t1; exists s0; exists x; exists (t2 *** T2).
  split. auto. split. eapply forever_N_star; eauto.
  apply Eappinf_assoc.
  inv H1.
  exists t1; exists s0; exists a2; exists (t2 *** T2).
  split. auto.
  split. inv H3. auto.
  eapply forever_N_plus. econstructor; eauto. eauto. auto.
  apply Eappinf_assoc.
Qed.

Lemma forever_N_forever:
  forall ge a s T, forever_N ge a s T -> forever ge s T.
Proof.
  cofix COINDHYP; intros.
  destruct (forever_N_inv H) as [t [s' [a' [T' [P [Q R]]]]]].
  rewrite R. apply forever_intro with s'. auto.
  apply COINDHYP with a'; auto.
Qed.

Yet another alternative definition of forever.

CoInductive forever_plus (ge: genv) : state -> traceinf -> Prop :=
  | forever_plus_intro: forall s1 t s2 T1 T2,
      plus ge s1 t s2 ->
      forever_plus ge s2 T2 ->
      T1 = t *** T2 ->
      forever_plus ge s1 T1.

Lemma forever_plus_inv:
  forall ge s T,
  forever_plus ge s T ->
  exists s', exists t, exists T',
  step ge s t s' /\ forever_plus ge s' T' /\ T = t *** T'.
Proof.
  intros. inv H. inv H0. exists s0; exists t1; exists (t2 *** T2).
  split. auto.
  split. exploit star_inv; eauto. intros [[P Q] | R].
    subst. simpl. auto. econstructor; eauto.
  traceEq.
Qed.

Lemma forever_plus_forever:
  forall ge s T, forever_plus ge s T -> forever ge s T.
Proof.
  cofix COINDHYP; intros.
  destruct (forever_plus_inv H) as [s' [t [T' [P [Q R]]]]].
  subst. econstructor; eauto.
Qed.

Infinitely many silent transitions

CoInductive forever_silent (ge: genv): state -> Prop :=
  | forever_silent_intro: forall s1 s2,
      step ge s1 E0 s2 -> forever_silent ge s2 ->
      forever_silent ge s1.

An alternate definition.

CoInductive forever_silent_N (ge: genv) : A -> state -> Prop :=
  | forever_silent_N_star: forall s1 s2 a1 a2,
      star ge s1 E0 s2 ->
      order a2 a1 ->
      forever_silent_N ge a2 s2 ->
      forever_silent_N ge a1 s1
  | forever_silent_N_plus: forall s1 s2 a1 a2,
      plus ge s1 E0 s2 ->
      forever_silent_N ge a2 s2 ->
      forever_silent_N ge a1 s1.

Lemma forever_silent_N_inv:
  forall ge a s,
  forever_silent_N ge a s ->
  exists s', exists a',
  step ge s E0 s' /\ forever_silent_N ge a' s'.
Proof.
  intros ge a0. pattern a0. apply (well_founded_ind order_wf).
  intros. inv H0.
  inv H1.
  apply H with a2. auto. auto.
  exploit Eapp_E0_inv; eauto. intros [P Q]. subst.
  exists s0; exists x.
  split. auto. eapply forever_silent_N_star; eauto.
  inv H1. exploit Eapp_E0_inv; eauto. intros [P Q]. subst.
  exists s0; exists a2.
  split. auto. inv H3. auto.
  eapply forever_silent_N_plus. econstructor; eauto. eauto.
Qed.

Lemma forever_silent_N_forever:
  forall ge a s, forever_silent_N ge a s -> forever_silent ge s.
Proof.
  cofix COINDHYP; intros.
  destruct (forever_silent_N_inv H) as [s' [a' [P Q]]].
  apply forever_silent_intro with s'. auto.
  apply COINDHYP with a'; auto.
Qed.

Infinitely many non-silent transitions

CoInductive forever_reactive (ge: genv): state -> traceinf -> Prop :=
  | forever_reactive_intro: forall s1 s2 t T,
      star ge s1 t s2 -> t <> E0 -> forever_reactive ge s2 T ->
      forever_reactive ge s1 (t *** T).

Lemma star_forever_reactive:
  forall ge s1 t s2 T,
  star ge s1 t s2 -> forever_reactive ge s2 T ->
  forever_reactive ge s1 (t *** T).
Proof.
  intros. inv H0. rewrite <- Eappinf_assoc. econstructor.
  eapply star_trans; eauto.
  red; intro. exploit Eapp_E0_inv; eauto. intros [P Q]. contradiction.
  auto.
Qed.

End CLOSURES.

Transition semantics


The general form of a transition semantics.

Record semantics : Type := Semantics {
  state: Type;
  funtype: Type;
  vartype: Type;
  step : Genv.t funtype vartype -> state -> trace -> state -> Prop;
  initial_state: state -> Prop;
  final_state: state -> int -> Prop;
  globalenv: Genv.t funtype vartype
}.

Handy notations.

Notation " 'Step' L " := (step L (globalenv L)) (at level 1) : smallstep_scope.
Notation " 'Star' L " := (star (step L) (globalenv L)) (at level 1) : smallstep_scope.
Notation " 'Plus' L " := (plus (step L) (globalenv L)) (at level 1) : smallstep_scope.
Notation " 'Forever_silent' L " := (forever_silent (step L) (globalenv L)) (at level 1) : smallstep_scope.
Notation " 'Forever_reactive' L " := (forever_reactive (step L) (globalenv L)) (at level 1) : smallstep_scope.
Notation " 'Nostep' L " := (nostep (step L) (globalenv L)) (at level 1) : smallstep_scope.

Open Scope smallstep_scope.

Forward simulations between two transition semantics.


The general form of a forward simulation.

Record forward_simulation (L1 L2: semantics) : Type :=
  Forward_simulation {
    fsim_index: Type;
    fsim_order: fsim_index -> fsim_index -> Prop;
    fsim_order_wf: well_founded fsim_order;
    fsim_match_states :> fsim_index -> state L1 -> state L2 -> Prop;
    fsim_match_initial_states:
      forall s1, initial_state L1 s1 ->
      exists i, exists s2, initial_state L2 s2 /\ fsim_match_states i s1 s2;
    fsim_match_final_states:
      forall i s1 s2 r,
      fsim_match_states i s1 s2 -> final_state L1 s1 r -> final_state L2 s2 r;
    fsim_simulation:
      forall s1 t s1', Step L1 s1 t s1' ->
      forall i s2, fsim_match_states i s1 s2 ->
      exists i', exists s2',
         (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ fsim_order i' i))
      /\ fsim_match_states i' s1' s2';
    fsim_symbols_preserved:
      forall id, Genv.find_symbol (globalenv L2) id = Genv.find_symbol (globalenv L1) id
  }.

Implicit Arguments forward_simulation [].

An alternate form of the simulation diagram

Lemma fsim_simulation':
  forall L1 L2 (S: forward_simulation L1 L2),
  forall i s1 t s1', Step L1 s1 t s1' ->
  forall s2, S i s1 s2 ->
  (exists i', exists s2', Plus L2 s2 t s2' /\ S i' s1' s2')
  \/ (exists i', fsim_order S i' i /\ t = E0 /\ S i' s1' s2).
Proof.
  intros. exploit fsim_simulation; eauto.
  intros [i' [s2' [A B]]]. intuition.
  left; exists i'; exists s2'; auto.
  inv H2.
  right; exists i'; auto.
  left; exists i'; exists s2'; split; auto. econstructor; eauto.
Qed.

Forward simulation diagrams.


Various simulation diagrams that imply forward simulation

Section FORWARD_SIMU_DIAGRAMS.

Variable L1: semantics.
Variable L2: semantics.

Hypothesis symbols_preserved:
  forall id, Genv.find_symbol (globalenv L2) id = Genv.find_symbol (globalenv L1) id.

Variable match_states: state L1 -> state L2 -> Prop.

Hypothesis match_initial_states:
  forall s1, initial_state L1 s1 ->
  exists s2, initial_state L2 s2 /\ match_states s1 s2.

Hypothesis match_final_states:
  forall s1 s2 r,
  match_states s1 s2 ->
  final_state L1 s1 r ->
  final_state L2 s2 r.

Simulation when one transition in the first program corresponds to zero, one or several transitions in the second program. However, there is no stuttering: infinitely many transitions in the source program must correspond to infinitely many transitions in the second program.

Section SIMULATION_STAR_WF.

order is a well-founded ordering associated with states of the first semantics. Stuttering steps must correspond to states that decrease w.r.t. order.

Variable order: state L1 -> state L1 -> Prop.
Hypothesis order_wf: well_founded order.

Hypothesis simulation:
  forall s1 t s1', Step L1 s1 t s1' ->
  forall s2, match_states s1 s2 ->
  exists s2',
  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ order s1' s1))
  /\ match_states s1' s2'.

Lemma forward_simulation_star_wf: forward_simulation L1 L2.
Proof.
  apply Forward_simulation with
    (fsim_order := order)
    (fsim_match_states := fun idx s1 s2 => idx = s1 /\ match_states s1 s2);
  auto.
  intros. exploit match_initial_states; eauto. intros [s2 [A B]].
    exists s1; exists s2; auto.
  intros. destruct H. eapply match_final_states; eauto.
  intros. destruct H0. subst i. exploit simulation; eauto. intros [s2' [A B]].
  exists s1'; exists s2'; intuition.
Qed.

End SIMULATION_STAR_WF.

Section SIMULATION_STAR.

We now consider the case where we have a nonnegative integer measure associated with states of the first semantics. It must decrease when we take a stuttering step.

Variable measure: state L1 -> nat.

Hypothesis simulation:
  forall s1 t s1', Step L1 s1 t s1' ->
  forall s2, match_states s1 s2 ->
  (exists s2', Plus L2 s2 t s2' /\ match_states s1' s2')
  \/ (measure s1' < measure s1 /\ t = E0 /\ match_states s1' s2)%nat.

Lemma forward_simulation_star: forward_simulation L1 L2.
Proof.
  apply forward_simulation_star_wf with (ltof _ measure).
  apply well_founded_ltof.
  intros. exploit simulation; eauto. intros [[s2' [A B]] | [A [B C]]].
  exists s2'; auto.
  exists s2; split. right; split. rewrite B. apply star_refl. auto. auto.
Qed.

End SIMULATION_STAR.

Simulation when one transition in the first program corresponds to one or several transitions in the second program.

Section SIMULATION_PLUS.

Hypothesis simulation:
  forall s1 t s1', Step L1 s1 t s1' ->
  forall s2, match_states s1 s2 ->
  exists s2', Plus L2 s2 t s2' /\ match_states s1' s2'.

Lemma forward_simulation_plus: forward_simulation L1 L2.
Proof.
  apply forward_simulation_star with (measure := fun _ => O).
  intros. exploit simulation; eauto.
Qed.

End SIMULATION_PLUS.

Lock-step simulation: each transition in the first semantics corresponds to exactly one transition in the second semantics.

Section SIMULATION_STEP.

Hypothesis simulation:
  forall s1 t s1', Step L1 s1 t s1' ->
  forall s2, match_states s1 s2 ->
  exists s2', Step L2 s2 t s2' /\ match_states s1' s2'.

Lemma forward_simulation_step: forward_simulation L1 L2.
Proof.
  apply forward_simulation_plus.
  intros. exploit simulation; eauto. intros [s2' [A B]].
  exists s2'; split; auto. apply plus_one; auto.
Qed.

End SIMULATION_STEP.

Simulation when one transition in the first program corresponds to zero or one transitions in the second program. However, there is no stuttering: infinitely many transitions in the source program must correspond to infinitely many transitions in the second program.

Section SIMULATION_OPT.

Variable measure: state L1 -> nat.

Hypothesis simulation:
  forall s1 t s1', Step L1 s1 t s1' ->
  forall s2, match_states s1 s2 ->
  (exists s2', Step L2 s2 t s2' /\ match_states s1' s2')
  \/ (measure s1' < measure s1 /\ t = E0 /\ match_states s1' s2)%nat.

Lemma forward_simulation_opt: forward_simulation L1 L2.
Proof.
  apply forward_simulation_star with measure.
  intros. exploit simulation; eauto. intros [[s2' [A B]] | [A [B C]]].
  left; exists s2'; split; auto. apply plus_one; auto.
  right; auto.
Qed.

End SIMULATION_OPT.

End FORWARD_SIMU_DIAGRAMS.

Forward simulation of transition sequences


Section SIMULATION_SEQUENCES.

Variable L1: semantics.
Variable L2: semantics.
Variable S: forward_simulation L1 L2.

Lemma simulation_star:
  forall s1 t s1', Star L1 s1 t s1' ->
  forall i s2, S i s1 s2 ->
  exists i', exists s2', Star L2 s2 t s2' /\ S i' s1' s2'.
Proof.
  induction 1; intros.
  exists i; exists s2; split; auto. apply star_refl.
  exploit fsim_simulation; eauto. intros [i' [s2' [A B]]].
  exploit IHstar; eauto. intros [i'' [s2'' [C D]]].
  exists i''; exists s2''; split; auto. eapply star_trans; eauto.
  intuition. apply plus_star; auto.
Qed.

Lemma simulation_plus:
  forall s1 t s1', Plus L1 s1 t s1' ->
  forall i s2, S i s1 s2 ->
  (exists i', exists s2', Plus L2 s2 t s2' /\ S i' s1' s2')
  \/ (exists i', clos_trans _ (fsim_order S) i' i /\ t = E0 /\ S i' s1' s2).
Proof.
  induction 1 using plus_ind2; intros.
  exploit fsim_simulation'; eauto. intros [A | [i' A]].
  left; auto.
  right; exists i'; intuition.
  exploit fsim_simulation'; eauto. intros [[i' [s2' [A B]]] | [i' [A [B C]]]].
  exploit simulation_star. apply plus_star; eauto. eauto.
  intros [i'' [s2'' [P Q]]].
  left; exists i''; exists s2''; split; auto. eapply plus_star_trans; eauto.
  exploit IHplus; eauto. intros [[i'' [s2'' [P Q]]] | [i'' [P [Q R]]]].
  subst. simpl. left; exists i''; exists s2''; auto.
  subst. simpl. right; exists i''; intuition.
  eapply t_trans; eauto. eapply t_step; eauto.
Qed.

Lemma simulation_forever_silent:
  forall i s1 s2,
  Forever_silent L1 s1 -> S i s1 s2 ->
  Forever_silent L2 s2.
Proof.
  assert (forall i s1 s2,
          Forever_silent L1 s1 -> S i s1 s2 ->
          forever_silent_N (step L2) (fsim_order S) (globalenv L2) i s2).
    cofix COINDHYP; intros.
    inv H. destruct (fsim_simulation S _ _ _ H1 _ _ H0) as [i' [s2' [A B]]].
    destruct A as [C | [C D]].
    eapply forever_silent_N_plus; eauto.
    eapply forever_silent_N_star; eauto.
  intros. eapply forever_silent_N_forever; eauto. apply fsim_order_wf.
Qed.

Lemma simulation_forever_reactive:
  forall i s1 s2 T,
  Forever_reactive L1 s1 T -> S i s1 s2 ->
  Forever_reactive L2 s2 T.
Proof.
  cofix COINDHYP; intros.
  inv H.
  destruct (simulation_star H1 i _ H0) as [i' [st2' [A B]]].
  econstructor; eauto.
Qed.

End SIMULATION_SEQUENCES.

Composing two forward simulations


Section COMPOSE_SIMULATIONS.

Variable L1: semantics.
Variable L2: semantics.
Variable L3: semantics.
Variable S12: forward_simulation L1 L2.
Variable S23: forward_simulation L2 L3.

Let ff_index : Type := (fsim_index S23 * fsim_index S12)%type.

Let ff_order : ff_index -> ff_index -> Prop :=
  lex_ord (clos_trans _ (fsim_order S23)) (fsim_order S12).

Let ff_match_states (i: ff_index) (s1: state L1) (s3: state L3) : Prop :=
  exists s2, S12 (snd i) s1 s2 /\ S23 (fst i) s2 s3.

Lemma compose_forward_simulation: forward_simulation L1 L3.
Proof.
  apply Forward_simulation with (fsim_order := ff_order) (fsim_match_states := ff_match_states).
  unfold ff_order. apply wf_lex_ord. apply wf_clos_trans. apply fsim_order_wf. apply fsim_order_wf.
  intros. exploit (fsim_match_initial_states S12); eauto. intros [i [s2 [A B]]].
  exploit (fsim_match_initial_states S23); eauto. intros [i' [s3 [C D]]].
  exists (i', i); exists s3; split; auto. exists s2; auto.
  intros. destruct H as [s3 [A B]].
  eapply (fsim_match_final_states S23); eauto.
  eapply (fsim_match_final_states S12); eauto.
  intros. destruct H0 as [s3 [A B]]. destruct i as [i2 i1]; simpl in *.
  exploit (fsim_simulation' S12); eauto. intros [[i1' [s3' [C D]]] | [i1' [C [D E]]]].
  exploit simulation_plus; eauto. intros [[i2' [s2' [P Q]]] | [i2' [P [Q R]]]].
  exists (i2', i1'); exists s2'; split. auto. exists s3'; auto.
  exists (i2', i1'); exists s2; split.
  right; split. subst t; apply star_refl. red. left. auto.
  exists s3'; auto.
  exists (i2, i1'); exists s2; split.
  right; split. subst t; apply star_refl. red. right. auto.
  exists s3; auto.
  intros. transitivity (Genv.find_symbol (globalenv L2) id); apply fsim_symbols_preserved; auto.
Qed.

End COMPOSE_SIMULATIONS.

Receptiveness and determinacy


Definition single_events (L: semantics) : Prop :=
  forall s t s', Step L s t s' -> (length t <= 1)%nat.

Record receptive (L: semantics) : Prop :=
  Receptive {
    sr_receptive: forall s t1 s1 t2,
      Step L s t1 s1 -> match_traces (globalenv L) t1 t2 -> exists s2, Step L s t2 s2;
    sr_traces:
      single_events L
  }.

Record determinate (L: semantics) : Prop :=
  Determinate {
    sd_determ: forall s t1 s1 t2 s2,
      Step L s t1 s1 -> Step L s t2 s2 ->
      match_traces (globalenv L) t1 t2 /\ (t1 = t2 -> s1 = s2);
    sd_traces:
      single_events L;
    sd_initial_determ: forall s1 s2,
      initial_state L s1 -> initial_state L s2 -> s1 = s2;
    sd_final_nostep: forall s r,
      final_state L s r -> Nostep L s;
    sd_final_determ: forall s r1 r2,
      final_state L s r1 -> final_state L s r2 -> r1 = r2
  }.

Section DETERMINACY.

Variable L: semantics.
Hypothesis DET: determinate L.

Lemma sd_determ_1:
  forall s t1 s1 t2 s2,
  Step L s t1 s1 -> Step L s t2 s2 -> match_traces (globalenv L) t1 t2.
Proof.
  intros. eapply sd_determ; eauto.
Qed.

Lemma sd_determ_2:
  forall s t s1 s2,
  Step L s t s1 -> Step L s t s2 -> s1 = s2.
Proof.
  intros. eapply sd_determ; eauto.
Qed.

Lemma star_determinacy:
  forall s t s', Star L s t s' ->
  forall s'', Star L s t s'' -> Star L s' E0 s'' \/ Star L s'' E0 s'.
Proof.
  induction 1; intros.
  auto.
  inv H2.
  right. eapply star_step; eauto.
  exploit sd_determ_1. eexact H. eexact H3. intros MT.
  exploit (sd_traces DET). eexact H. intros L1.
  exploit (sd_traces DET). eexact H3. intros L2.
  assert (t1 = t0 /\ t2 = t3).
    destruct t1. inv MT. auto.
    destruct t1; simpl in L1; try omegaContradiction.
    destruct t0. inv MT. destruct t0; simpl in L2; try omegaContradiction.
    simpl in H5. split. congruence. congruence.
  destruct H1; subst.
  assert (s2 = s4) by (eapply sd_determ_2; eauto). subst s4.
  auto.
Qed.

End DETERMINACY.

Backward simulations between two transition semantics.


Definition safe (L: semantics) (s: state L) : Prop :=
  forall s',
  Star L s E0 s' ->
  (exists r, final_state L s' r)
  \/ (exists t, exists s'', Step L s' t s'').

Lemma star_safe:
  forall (L: semantics) s s',
  Star L s E0 s' -> safe L s -> safe L s'.
Proof.
  intros; red; intros. apply H0. eapply star_trans; eauto.
Qed.

The general form of a backward simulation.

Record backward_simulation (L1 L2: semantics) : Type :=
  Backward_simulation {
    bsim_index: Type;
    bsim_order: bsim_index -> bsim_index -> Prop;
    bsim_order_wf: well_founded bsim_order;
    bsim_match_states :> bsim_index -> state L1 -> state L2 -> Prop;
    bsim_initial_states_exist:
      forall s1, initial_state L1 s1 -> exists s2, initial_state L2 s2;
    bsim_match_initial_states:
      forall s1 s2, initial_state L1 s1 -> initial_state L2 s2 ->
      exists i, exists s1', initial_state L1 s1' /\ bsim_match_states i s1' s2;
    bsim_match_final_states:
      forall i s1 s2 r,
      bsim_match_states i s1 s2 -> safe L1 s1 -> final_state L2 s2 r ->
      exists s1', Star L1 s1 E0 s1' /\ final_state L1 s1' r;
    bsim_progress:
      forall i s1 s2,
      bsim_match_states i s1 s2 -> safe L1 s1 ->
      (exists r, final_state L2 s2 r) \/
      (exists t, exists s2', Step L2 s2 t s2');
    bsim_simulation:
      forall s2 t s2', Step L2 s2 t s2' ->
      forall i s1, bsim_match_states i s1 s2 -> safe L1 s1 ->
      exists i', exists s1',
         (Plus L1 s1 t s1' \/ (Star L1 s1 t s1' /\ bsim_order i' i))
      /\ bsim_match_states i' s1' s2';
    bsim_symbols_preserved:
      forall id, Genv.find_symbol (globalenv L2) id = Genv.find_symbol (globalenv L1) id
  }.

An alternate form of the simulation diagram

Lemma bsim_simulation':
  forall L1 L2 (S: backward_simulation L1 L2),
  forall i s2 t s2', Step L2 s2 t s2' ->
  forall s1, S i s1 s2 -> safe L1 s1 ->
  (exists i', exists s1', Plus L1 s1 t s1' /\ S i' s1' s2')
  \/ (exists i', bsim_order S i' i /\ t = E0 /\ S i' s1 s2').
Proof.
  intros. exploit bsim_simulation; eauto.
  intros [i' [s1' [A B]]]. intuition.
  left; exists i'; exists s1'; auto.
  inv H3.
  right; exists i'; auto.
  left; exists i'; exists s1'; split; auto. econstructor; eauto.
Qed.

Backward simulation diagrams.


Various simulation diagrams that imply backward simulation.

Section BACKWARD_SIMU_DIAGRAMS.

Variable L1: semantics.
Variable L2: semantics.

Hypothesis symbols_preserved:
  forall id, Genv.find_symbol (globalenv L2) id = Genv.find_symbol (globalenv L1) id.

Variable match_states: state L1 -> state L2 -> Prop.

Hypothesis initial_states_exist:
  forall s1, initial_state L1 s1 -> exists s2, initial_state L2 s2.

Hypothesis match_initial_states:
  forall s1 s2, initial_state L1 s1 -> initial_state L2 s2 ->
  exists s1', initial_state L1 s1' /\ match_states s1' s2.

Hypothesis match_final_states:
  forall s1 s2 r,
  match_states s1 s2 -> final_state L2 s2 r -> final_state L1 s1 r.

Hypothesis progress:
  forall s1 s2,
  match_states s1 s2 -> safe L1 s1 ->
  (exists r, final_state L2 s2 r) \/
  (exists t, exists s2', Step L2 s2 t s2').

Section BACKWARD_SIMULATION_PLUS.

Hypothesis simulation:
  forall s2 t s2', Step L2 s2 t s2' ->
  forall s1, match_states s1 s2 -> safe L1 s1 ->
  exists s1', Plus L1 s1 t s1' /\ match_states s1' s2'.

Lemma backward_simulation_plus: backward_simulation L1 L2.
Proof.
  apply Backward_simulation with
    (bsim_order := fun (x y: unit) => False)
    (bsim_match_states := fun (i: unit) s1 s2 => match_states s1 s2);
  auto.
  red; intros; constructor; intros. contradiction.
  intros. exists tt; eauto.
  intros. exists s1; split. apply star_refl. eauto.
  intros. exploit simulation; eauto. intros [s1' [A B]].
  exists tt; exists s1'; auto.
Qed.

End BACKWARD_SIMULATION_PLUS.

End BACKWARD_SIMU_DIAGRAMS.

Backward simulation of transition sequences


Section BACKWARD_SIMULATION_SEQUENCES.

Variable L1: semantics.
Variable L2: semantics.
Variable S: backward_simulation L1 L2.

Lemma bsim_E0_star:
  forall s2 s2', Star L2 s2 E0 s2' ->
  forall i s1, S i s1 s2 -> safe L1 s1 ->
  exists i', exists s1', Star L1 s1 E0 s1' /\ S i' s1' s2'.
Proof.
  intros s20 s20' STAR0. pattern s20, s20'. eapply star_E0_ind; eauto.
  intros. exists i; exists s1; split; auto. apply star_refl.
  intros. exploit bsim_simulation; eauto. intros [i' [s1' [A B]]].
  assert (Star L1 s0 E0 s1'). intuition. apply plus_star; auto.
  exploit H0. eauto. eapply star_safe; eauto. intros [i'' [s1'' [C D]]].
  exists i''; exists s1''; split; auto. eapply star_trans; eauto.
Qed.

Lemma bsim_safe:
  forall i s1 s2,
  S i s1 s2 -> safe L1 s1 -> safe L2 s2.
Proof.
  intros; red; intros.
  exploit bsim_E0_star; eauto. intros [i' [s1' [A B]]].
  eapply bsim_progress; eauto. eapply star_safe; eauto.
Qed.

Lemma bsim_E0_plus:
  forall s2 t s2', Plus L2 s2 t s2' -> t = E0 ->
  forall i s1, S i s1 s2 -> safe L1 s1 ->
     (exists i', exists s1', Plus L1 s1 E0 s1' /\ S i' s1' s2')
  \/ (exists i', clos_trans _ (bsim_order S) i' i /\ S i' s1 s2').
Proof.
  induction 1 using plus_ind2; intros; subst t.
  exploit bsim_simulation'; eauto. intros [[i' [s1' [A B]]] | [i' [A [B C]]]].
  left; exists i'; exists s1'; auto.
  right; exists i'; intuition.
  exploit Eapp_E0_inv; eauto. intros [EQ1 EQ2]; subst.
  exploit bsim_simulation'; eauto. intros [[i' [s1' [A B]]] | [i' [A [B C]]]].
  exploit bsim_E0_star. apply plus_star; eauto. eauto. eapply star_safe; eauto. apply plus_star; auto.
  intros [i'' [s1'' [P Q]]].
  left; exists i''; exists s1''; intuition. eapply plus_star_trans; eauto.
  exploit IHplus; eauto. intros [P | [i'' [P Q]]].
  left; auto.
  right; exists i''; intuition. eapply t_trans; eauto. apply t_step; auto.
Qed.

Lemma star_non_E0_split:
  forall s2 t s2', Star L2 s2 t s2' -> (length t = 1)%nat ->
  exists s2x, exists s2y, Star L2 s2 E0 s2x /\ Step L2 s2x t s2y /\ Star L2 s2y E0 s2'.
Proof.
  induction 1; intros.
  simpl in H; discriminate.
  subst t.
  assert (EITHER: t1 = E0 \/ t2 = E0).
    unfold Eapp in H2; rewrite app_length in H2.
    destruct t1; auto. destruct t2; auto. simpl in H2; omegaContradiction.
  destruct EITHER; subst.
  exploit IHstar; eauto. intros [s2x [s2y [A [B C]]]].
  exists s2x; exists s2y; intuition. eapply star_left; eauto.
  rewrite E0_right. exists s1; exists s2; intuition. apply star_refl.
Qed.

End BACKWARD_SIMULATION_SEQUENCES.

Composing two backward simulations


Section COMPOSE_BACKWARD_SIMULATIONS.

Variable L1: semantics.
Variable L2: semantics.
Variable L3: semantics.
Hypothesis L3_single_events: single_events L3.
Variable S12: backward_simulation L1 L2.
Variable S23: backward_simulation L2 L3.

Let bb_index : Type := (bsim_index S12 * bsim_index S23)%type.

Let bb_order : bb_index -> bb_index -> Prop :=
  lex_ord (clos_trans _ (bsim_order S12)) (bsim_order S23).

Inductive bb_match_states: bb_index -> state L1 -> state L3 -> Prop :=
  | bb_match_later: forall i1 i2 s1 s3 s2x s2y,
      S12 i1 s1 s2x -> Star L2 s2x E0 s2y -> S23 i2 s2y s3 ->
      bb_match_states (i1, i2) s1 s3.

Lemma bb_match_at: forall i1 i2 s1 s3 s2,
  S12 i1 s1 s2 -> S23 i2 s2 s3 ->
  bb_match_states (i1, i2) s1 s3.
Proof.
  intros. econstructor; eauto. apply star_refl.
Qed.

Lemma bb_simulation_base:
  forall s3 t s3', Step L3 s3 t s3' ->
  forall i1 s1 i2 s2, S12 i1 s1 s2 -> S23 i2 s2 s3 -> safe L1 s1 ->
  exists i', exists s1',
    (Plus L1 s1 t s1' \/ (Star L1 s1 t s1' /\ bb_order i' (i1, i2)))
    /\ bb_match_states i' s1' s3'.
Proof.
  intros.
  exploit (bsim_simulation' S23); eauto. eapply bsim_safe; eauto.
  intros [ [i2' [s2' [PLUS2 MATCH2]]] | [i2' [ORD2 [EQ MATCH2]]]].
  assert (EITHER: t = E0 \/ (length t = 1)%nat).
    exploit L3_single_events; eauto.
    destruct t; auto. destruct t; auto. simpl. intros. omegaContradiction.
  destruct EITHER.
  subst t. exploit bsim_E0_plus; eauto.
  intros [ [i1' [s1' [PLUS1 MATCH1]]] | [i1' [ORD1 MATCH1]]].
  exists (i1', i2'); exists s1'; split. auto. eapply bb_match_at; eauto.
  exists (i1', i2'); exists s1; split.
  right; split. apply star_refl. left; auto.
  eapply bb_match_at; eauto.
  exploit star_non_E0_split. apply plus_star; eauto. auto.
  intros [s2x [s2y [P [Q R]]]].
  exploit bsim_E0_star. eexact P. eauto. auto. intros [i1' [s1x [X Y]]].
  exploit bsim_simulation'. eexact Q. eauto. eapply star_safe; eauto.
  intros [[i1'' [s1y [U V]]] | [i1'' [U [V W]]]]; try (subst t; discriminate).
  exists (i1'', i2'); exists s1y; split.
  left. eapply star_plus_trans; eauto. eapply bb_match_later; eauto.
  subst. exists (i1, i2'); exists s1; split.
  right; split. apply star_refl. right; auto.
  eapply bb_match_at; eauto.
Qed.

Lemma bb_simulation:
  forall s3 t s3', Step L3 s3 t s3' ->
  forall i s1, bb_match_states i s1 s3 -> safe L1 s1 ->
  exists i', exists s1',
    (Plus L1 s1 t s1' \/ (Star L1 s1 t s1' /\ bb_order i' i))
    /\ bb_match_states i' s1' s3'.
Proof.
  intros. inv H0.
  exploit star_inv; eauto. intros [[EQ1 EQ2] | PLUS].
  subst. eapply bb_simulation_base; eauto.
  exploit bsim_E0_plus; eauto.
  intros [[i1' [s1' [A B]]] | [i1' [A B]]].
  exploit bb_simulation_base. eauto. auto. eexact B. eauto.
    eapply star_safe; eauto. eapply plus_star; eauto.
  intros [i'' [s1'' [C D]]].
  exists i''; exists s1''; split; auto.
  left. eapply plus_star_trans; eauto.
  destruct C as [P | [P Q]]. apply plus_star; eauto. eauto.
  traceEq.
  exploit bb_simulation_base. eauto. auto. eexact B. eauto. auto.
  intros [i'' [s1'' [C D]]].
  exists i''; exists s1''; split; auto.
  intuition. right; intuition.
  inv H6. left. eapply t_trans; eauto. left; auto.
Qed.

Lemma compose_backward_simulation: backward_simulation L1 L3.
Proof.
  apply Backward_simulation with (bsim_order := bb_order) (bsim_match_states := bb_match_states).
  unfold bb_order. apply wf_lex_ord. apply wf_clos_trans. apply bsim_order_wf. apply bsim_order_wf.
  intros. exploit (bsim_initial_states_exist S12); eauto. intros [s2 A].
  eapply (bsim_initial_states_exist S23); eauto.
  intros s1 s3 INIT1 INIT3.
  exploit (bsim_initial_states_exist S12); eauto. intros [s2 INIT2].
  exploit (bsim_match_initial_states S23); eauto. intros [i2 [s2' [INIT2' M2]]].
  exploit (bsim_match_initial_states S12); eauto. intros [i1 [s1' [INIT1' M1]]].
  exists (i1, i2); exists s1'; intuition. eapply bb_match_at; eauto.
  intros i s1 s3 r MS SAFE FIN. inv MS.
  exploit (bsim_match_final_states S23); eauto.
    eapply star_safe; eauto. eapply bsim_safe; eauto.
  intros [s2' [A B]].
  exploit bsim_E0_star. eapply star_trans. eexact H0. eexact A. auto. eauto. auto.
  intros [i1' [s1' [C D]]].
  exploit (bsim_match_final_states S12); eauto. eapply star_safe; eauto.
  intros [s1'' [P Q]].
  exists s1''; split; auto. eapply star_trans; eauto.
  intros i s1 s3 MS SAFE. inv MS.
  eapply (bsim_progress S23). eauto. eapply star_safe; eauto. eapply bsim_safe; eauto.
  exact bb_simulation.
  intros. transitivity (Genv.find_symbol (globalenv L2) id); apply bsim_symbols_preserved; auto.
Qed.

End COMPOSE_BACKWARD_SIMULATIONS.

Converting a forward simulation to a backward simulation


Section FORWARD_TO_BACKWARD.

Variable L1: semantics.
Variable L2: semantics.
Variable FS: forward_simulation L1 L2.
Hypothesis L1_receptive: receptive L1.
Hypothesis L2_determinate: determinate L2.

Exploiting forward simulation

Inductive f2b_transitions: state L1 -> state L2 -> Prop :=
  | f2b_trans_final: forall s1 s2 s1' r,
      Star L1 s1 E0 s1' ->
      final_state L1 s1' r ->
      final_state L2 s2 r ->
      f2b_transitions s1 s2
  | f2b_trans_step: forall s1 s2 s1' t s1'' s2' i' i'',
      Star L1 s1 E0 s1' ->
      Step L1 s1' t s1'' ->
      Plus L2 s2 t s2' ->
      FS i' s1' s2 ->
      FS i'' s1'' s2' ->
      f2b_transitions s1 s2.

Lemma f2b_progress:
  forall i s1 s2, FS i s1 s2 -> safe L1 s1 -> f2b_transitions s1 s2.
Proof.
  intros i0; pattern i0. apply well_founded_ind with (R := fsim_order FS).
  apply fsim_order_wf.
  intros i REC s1 s2 MATCH SAFE.
  destruct (SAFE s1) as [[r FINAL] | [t [s1' STEP1]]]. apply star_refl.
  eapply f2b_trans_final; eauto.
  apply star_refl.
  eapply fsim_match_final_states; eauto.
  exploit (fsim_simulation FS); eauto. intros [i' [s2' [A MATCH']]].
  assert (B: Plus L2 s2 t s2' \/ (s2' = s2 /\ t = E0 /\ fsim_order FS i' i)).
    intuition.
    destruct (star_inv H0); intuition.
  clear A. destruct B as [PLUS2 | [EQ1 [EQ2 ORDER]]].
  eapply f2b_trans_step; eauto. apply star_refl.
  subst. exploit REC; eauto. eapply star_safe; eauto. apply star_one; auto.
  intros TRANS; inv TRANS.
  eapply f2b_trans_final; eauto. eapply star_left; eauto.
  eapply f2b_trans_step; eauto. eapply star_left; eauto.
Qed.

Lemma fsim_simulation_not_E0:
  forall s1 t s1', Step L1 s1 t s1' -> t <> E0 ->
  forall i s2, FS i s1 s2 ->
  exists i', exists s2', Plus L2 s2 t s2' /\ FS i' s1' s2'.
Proof.
  intros. exploit (fsim_simulation FS); eauto. intros [i' [s2' [A B]]].
  exists i'; exists s2'; split; auto.
  destruct A. auto. destruct H2. exploit star_inv; eauto. intros [[EQ1 EQ2] | P]; auto.
  congruence.
Qed.

Exploiting determinacy

Remark silent_or_not_silent:
  forall t, t = E0 \/ t <> E0.
Proof.
  intros; unfold E0; destruct t; auto; right; congruence.
Qed.

Remark not_silent_length:
  forall t1 t2, (length (t1 ** t2) <= 1)%nat -> t1 = E0 \/ t2 = E0.
Proof.
  unfold Eapp, E0; intros. rewrite app_length in H.
  destruct t1; destruct t2; auto. simpl in H. omegaContradiction.
Qed.

Lemma f2b_determinacy_inv:
  forall s2 t' s2' t'' s2'',
  Step L2 s2 t' s2' -> Step L2 s2 t'' s2'' ->
  (t' = E0 /\ t'' = E0 /\ s2' = s2'')
  \/ (t' <> E0 /\ t'' <> E0 /\ match_traces (globalenv L1) t' t'').
Proof.
  intros.
  assert (match_traces (globalenv L2) t' t'').
    eapply sd_determ_1; eauto.
  destruct (silent_or_not_silent t').
  subst. inv H1.
  left; intuition. eapply sd_determ_2; eauto.
  destruct (silent_or_not_silent t'').
  subst. inv H1. elim H2; auto.
  right; intuition.
  eapply match_traces_preserved with (ge1 := (globalenv L2)); auto.
  intros; symmetry; apply (fsim_symbols_preserved FS).
Qed.

Lemma f2b_determinacy_star:
  forall s s1, Star L2 s E0 s1 ->
  forall t s2 s3,
  Step L2 s1 t s2 -> t <> E0 ->
  Star L2 s t s3 ->
  Star L2 s1 t s3.
Proof.
  intros s0 s01 ST0. pattern s0, s01. eapply star_E0_ind; eauto.
  intros. inv H3. congruence.
  exploit f2b_determinacy_inv. eexact H. eexact H4.
  intros [[EQ1 [EQ2 EQ3]] | [NEQ1 [NEQ2 MT]]].
  subst. simpl in *. eauto.
  congruence.
Qed.

Orders

Inductive f2b_index : Type :=
  | F2BI_before (n: nat)
  | F2BI_after (n: nat).

Inductive f2b_order: f2b_index -> f2b_index -> Prop :=
  | f2b_order_before: forall n n',
      (n' < n)%nat ->
      f2b_order (F2BI_before n') (F2BI_before n)
  | f2b_order_after: forall n n',
      (n' < n)%nat ->
      f2b_order (F2BI_after n') (F2BI_after n)
  | f2b_order_switch: forall n n',
      f2b_order (F2BI_before n') (F2BI_after n).

Lemma wf_f2b_order:
  well_founded f2b_order.
Proof.
  assert (ACC1: forall n, Acc f2b_order (F2BI_before n)).
    intros n0; pattern n0; apply lt_wf_ind; intros.
    constructor; intros. inv H0. auto.
  assert (ACC2: forall n, Acc f2b_order (F2BI_after n)).
    intros n0; pattern n0; apply lt_wf_ind; intros.
    constructor; intros. inv H0. auto. auto.
  red; intros. destruct a; auto.
Qed.

Constructing the backward simulation

Inductive f2b_match_states: f2b_index -> state L1 -> state L2 -> Prop :=
  | f2b_match_at: forall i s1 s2,
      FS i s1 s2 ->
      f2b_match_states (F2BI_after O) s1 s2
  | f2b_match_before: forall s1 t s1' s2b s2 n s2a i,
      Step L1 s1 t s1' -> t <> E0 ->
      Star L2 s2b E0 s2 ->
      starN (step L2) (globalenv L2) n s2 t s2a ->
      FS i s1 s2b ->
      f2b_match_states (F2BI_before n) s1 s2
  | f2b_match_after: forall n s2 s2a s1 i,
      starN (step L2) (globalenv L2) (S n) s2 E0 s2a ->
      FS i s1 s2a ->
      f2b_match_states (F2BI_after (S n)) s1 s2.

Remark f2b_match_after':
  forall n s2 s2a s1 i,
  starN (step L2) (globalenv L2) n s2 E0 s2a ->
  FS i s1 s2a ->
  f2b_match_states (F2BI_after n) s1 s2.
Proof.
  intros. inv H.
  econstructor; eauto.
  econstructor; eauto. econstructor; eauto.
Qed.

Backward simulation of L2 steps

Lemma f2b_simulation_step:
  forall s2 t s2', Step L2 s2 t s2' ->
  forall i s1, f2b_match_states i s1 s2 -> safe L1 s1 ->
  exists i', exists s1',
    (Plus L1 s1 t s1' \/ (Star L1 s1 t s1' /\ f2b_order i' i))
     /\ f2b_match_states i' s1' s2'.
Proof.
  intros s2 t s2' STEP2 i s1 MATCH SAFE.
  inv MATCH.
  exploit f2b_progress; eauto. intros TRANS; inv TRANS.
  exploit (sd_final_nostep L2_determinate); eauto. contradiction.
  inv H2.
  exploit f2b_determinacy_inv. eexact H5. eexact STEP2.
  intros [[EQ1 [EQ2 EQ3]] | [NOT1 [NOT2 MT]]].
  destruct (silent_or_not_silent t2).
  subst. simpl in *. destruct (star_starN H6) as [n STEPS2].
  exists (F2BI_after n); exists s1''; split.
  left. eapply plus_right; eauto.
  eapply f2b_match_after'; eauto.
  subst. simpl in *. destruct (star_starN H6) as [n STEPS2].
  exists (F2BI_before n); exists s1'; split.
  right; split. auto. constructor.
  econstructor. eauto. auto. apply star_one; eauto. eauto. eauto.
  exploit not_silent_length. eapply (sr_traces L1_receptive); eauto. intros [EQ | EQ].
  congruence.
  subst t2. rewrite E0_right in H1.
  exploit (sr_receptive L1_receptive); eauto. intros [s1''' STEP1].
  exploit fsim_simulation_not_E0. eexact STEP1. auto. eauto.
  intros [i''' [s2''' [P Q]]]. inv P.
  exploit not_silent_length. eapply (sr_traces L1_receptive); eauto. intros [EQ | EQ].
  subst t0. simpl in *. exploit sd_determ_1. eauto. eexact STEP2. eexact H2.
  intros. elim NOT2. inv H8. auto.
  subst t2. rewrite E0_right in *.
  assert (s4 = s2'). eapply sd_determ_2; eauto. subst s4.
  destruct (star_starN H7) as [n STEPS2]. exists (F2BI_after n); exists s1'''; split.
  left. eapply plus_right; eauto.
  eapply f2b_match_after'; eauto.

  inv H2. congruence.
  exploit f2b_determinacy_inv. eexact H4. eexact STEP2.
  intros [[EQ1 [EQ2 EQ3]] | [NOT1 [NOT2 MT]]].
  subst. simpl in *. exists (F2BI_before n0); exists s1; split.
  right; split. apply star_refl. constructor. omega.
  econstructor; eauto. eapply star_right; eauto.
  exploit not_silent_length. eapply (sr_traces L1_receptive); eauto. intros [EQ | EQ].
  congruence.
  subst. rewrite E0_right in *.
  exploit (sr_receptive L1_receptive); eauto. intros [s1''' STEP1].
  exploit fsim_simulation_not_E0. eexact STEP1. auto. eauto.
  intros [i''' [s2''' [P Q]]].
  exploit f2b_determinacy_star. eauto. eexact STEP2. auto. apply plus_star; eauto.
  intro R. inv R. congruence.
  exploit not_silent_length. eapply (sr_traces L1_receptive); eauto. intros [EQ | EQ].
  subst. simpl in *. exploit sd_determ_1. eauto. eexact STEP2. eexact H2.
  intros. elim NOT2. inv H7; auto.
  subst. rewrite E0_right in *.
  assert (s3 = s2'). eapply sd_determ_2; eauto. subst s3.
  destruct (star_starN H6) as [n STEPS2]. exists (F2BI_after n); exists s1'''; split.
  left. apply plus_one; auto.
  eapply f2b_match_after'; eauto.

  inv H. exploit Eapp_E0_inv; eauto. intros [EQ1 EQ2]; subst.
  exploit f2b_determinacy_inv. eexact H2. eexact STEP2.
  intros [[EQ1 [EQ2 EQ3]] | [NOT1 [NOT2 MT]]].
  subst. exists (F2BI_after n); exists s1; split.
  right; split. apply star_refl. constructor; omega.
  eapply f2b_match_after'; eauto.
  congruence.
Qed.

The backward simulation

Lemma forward_to_backward_simulation: backward_simulation L1 L2.
Proof.
  apply Backward_simulation with (bsim_order := f2b_order) (bsim_match_states := f2b_match_states).
  apply wf_f2b_order.
  intros. exploit (fsim_match_initial_states FS); eauto. intros [i [s2 [A B]]].
  exists s2; auto.
  intros. exploit (fsim_match_initial_states FS); eauto. intros [i [s2' [A B]]].
  assert (s2 = s2') by (eapply sd_initial_determ; eauto). subst s2'.
  exists (F2BI_after O); exists s1; split; auto. econstructor; eauto.
  intros. inv H.
  exploit f2b_progress; eauto. intros TRANS; inv TRANS.
  assert (r0 = r) by (eapply (sd_final_determ L2_determinate); eauto). subst r0.
  exists s1'; auto.
  inv H4. exploit (sd_final_nostep L2_determinate); eauto. contradiction.
  inv H5. congruence. exploit (sd_final_nostep L2_determinate); eauto. contradiction.
  inv H2. exploit (sd_final_nostep L2_determinate); eauto. contradiction.
  intros. inv H.
  exploit f2b_progress; eauto. intros TRANS; inv TRANS.
  left; exists r; auto.
  inv H3. right; econstructor; econstructor; eauto.
  inv H4. congruence. right; econstructor; econstructor; eauto.
  inv H1. right; econstructor; econstructor; eauto.
  exact f2b_simulation_step.
  exact (fsim_symbols_preserved FS).
Qed.

End FORWARD_TO_BACKWARD.

Transforming a semantics into a single-event, equivalent semantics


Definition well_behaved_traces (L: semantics) : Prop :=
  forall s t s', Step L s t s' ->
  match t with nil => True | ev :: t' => output_trace t' end.

Section ATOMIC.

Variable L: semantics.

Hypothesis Lwb: well_behaved_traces L.

Inductive atomic_step (ge: Genv.t (funtype L) (vartype L)): (trace * state L) -> trace -> (trace * state L) -> Prop :=
  | atomic_step_silent: forall s s',
      Step L s E0 s' ->
      atomic_step ge (E0, s) E0 (E0, s')
  | atomic_step_start: forall s ev t s',
      Step L s (ev :: t) s' ->
      atomic_step ge (E0, s) (ev :: nil) (t, s')
  | atomic_step_continue: forall ev t s,
      output_trace (ev :: t) ->
      atomic_step ge (ev :: t, s) (ev :: nil) (t, s).

Definition atomic : semantics := {|
  state := (trace * state L)%type;
  funtype := funtype L;
  vartype := vartype L;
  step := atomic_step;
  initial_state := fun s => initial_state L (snd s) /\ fst s = E0;
  final_state := fun s r => final_state L (snd s) r /\ fst s = E0;
  globalenv := globalenv L
|}.

End ATOMIC.

A forward simulation from a semantics L1 to a single-event semantics L2 can be "factored" into a forward simulation from atomic L1 to L2.

Section FACTOR_FORWARD_SIMULATION.

Variable L1: semantics.
Variable L2: semantics.
Variable sim: forward_simulation L1 L2.
Hypothesis L2single: single_events L2.

Inductive ffs_match: fsim_index sim -> (trace * state L1) -> state L2 -> Prop :=
  | ffs_match_at: forall i s1 s2,
      sim i s1 s2 ->
      ffs_match i (E0, s1) s2
  | ffs_match_buffer: forall i ev t s1 s2 s2',
      Star L2 s2 (ev :: t) s2' -> sim i s1 s2' ->
      ffs_match i (ev :: t, s1) s2.

Lemma star_non_E0_split':
  forall s2 t s2', Star L2 s2 t s2' ->
  match t with
  | nil => True
  | ev :: t' => exists s2x, Plus L2 s2 (ev :: nil) s2x /\ Star L2 s2x t' s2'
  end.
Proof.
  induction 1. simpl. auto.
  exploit L2single; eauto. intros LEN.
  destruct t1. simpl in *. subst. destruct t2. auto.
  destruct IHstar as [s2x [A B]]. exists s2x; split; auto.
  eapply plus_left. eauto. apply plus_star; eauto. auto.
  destruct t1. simpl in *. subst t. exists s2; split; auto. apply plus_one; auto.
  simpl in LEN. omegaContradiction.
Qed.

Lemma ffs_simulation:
  forall s1 t s1', Step (atomic L1) s1 t s1' ->
  forall i s2, ffs_match i s1 s2 ->
  exists i', exists s2',
     (Plus L2 s2 t s2' \/ (Star L2 s2 t s2') /\ fsim_order sim i' i)
  /\ ffs_match i' s1' s2'.
Proof.
  induction 1; intros.
  inv H0.
  exploit (fsim_simulation sim); eauto.
  intros [i' [s2' [A B]]].
  exists i'; exists s2'; split. auto. constructor; auto.
  inv H0.
  exploit (fsim_simulation sim); eauto.
  intros [i' [s2' [A B]]].
  destruct t as [ | ev' t].
  exists i'; exists s2'; split. auto. constructor; auto.
  assert (C: Star L2 s2 (ev :: ev' :: t) s2'). intuition. apply plus_star; auto.
  exploit star_non_E0_split'. eauto. simpl. intros [s2x [P Q]].
  exists i'; exists s2x; split. auto. econstructor; eauto.
  inv H0.
  exploit star_non_E0_split'. eauto. simpl. intros [s2x [P Q]].
  destruct t.
  exists i; exists s2'; split. left. eapply plus_star_trans; eauto. constructor; auto.
  exists i; exists s2x; split. auto. econstructor; eauto.
Qed.

Theorem factor_forward_simulation:
  forward_simulation (atomic L1) L2.
Proof.
  apply Forward_simulation with (fsim_match_states := ffs_match) (fsim_order := fsim_order sim).
  apply fsim_order_wf.
  intros. destruct s1 as [t1 s1]. simpl in H. destruct H. subst.
  exploit (fsim_match_initial_states sim); eauto. intros [i [s2 [A B]]].
  exists i; exists s2; split; auto. constructor; auto.
  intros. destruct s1 as [t1 s1]. simpl in H0; destruct H0; subst. inv H.
  eapply (fsim_match_final_states sim); eauto.
  exact ffs_simulation.
  simpl. exact (fsim_symbols_preserved sim).
Qed.

End FACTOR_FORWARD_SIMULATION.

Likewise, a backward simulation from a single-event semantics L1 to a semantics L2 can be "factored" as a backward simulation from L1 to atomic L2.

Section FACTOR_BACKWARD_SIMULATION.

Variable L1: semantics.
Variable L2: semantics.
Variable sim: backward_simulation L1 L2.
Hypothesis L1single: single_events L1.
Hypothesis L2wb: well_behaved_traces L2.

Inductive fbs_match: bsim_index sim -> state L1 -> (trace * state L2) -> Prop :=
  | fbs_match_intro: forall i s1 t s2 s1',
      Star L1 s1 t s1' -> sim i s1' s2 ->
      t = E0 \/ output_trace t ->
      fbs_match i s1 (t, s2).

Lemma fbs_simulation:
  forall s2 t s2', Step (atomic L2) s2 t s2' ->
  forall i s1, fbs_match i s1 s2 -> safe L1 s1 ->
  exists i', exists s1',
     (Plus L1 s1 t s1' \/ (Star L1 s1 t s1' /\ bsim_order sim i' i))
     /\ fbs_match i' s1' s2'.
Proof.
  induction 1; intros.
  inv H0.
  exploit (bsim_simulation sim); eauto. eapply star_safe; eauto.
  intros [i' [s1'' [A B]]].
  exists i'; exists s1''; split.
  destruct A as [P | [P Q]]. left. eapply star_plus_trans; eauto. right; split; auto. eapply star_trans; eauto.
  econstructor. apply star_refl. auto. auto.
  inv H0.
  exploit (bsim_simulation sim); eauto. eapply star_safe; eauto.
  intros [i' [s1'' [A B]]].
  assert (C: Star L1 s1 (ev :: t) s1'').
    eapply star_trans. eauto. destruct A as [P | [P Q]]. apply plus_star; eauto. eauto. auto.
  exploit star_non_E0_split'; eauto. simpl. intros [s1x [P Q]].
  exists i'; exists s1x; split.
  left; auto.
  econstructor; eauto.
  exploit L2wb; eauto.
  inv H0. unfold E0 in H8; destruct H8; try congruence.
  exploit star_non_E0_split'; eauto. simpl. intros [s1x [P Q]].
  exists i; exists s1x; split. left; auto. econstructor; eauto. simpl in H0; tauto.
Qed.

Lemma fbs_progress:
  forall i s1 s2,
  fbs_match i s1 s2 -> safe L1 s1 ->
  (exists r, final_state (atomic L2) s2 r) \/
  (exists t, exists s2', Step (atomic L2) s2 t s2').
Proof.
  intros. inv H. destruct t.
  exploit (bsim_progress sim); eauto. eapply star_safe; eauto.
  intros [[r A] | [t [s2' A]]].
  left; exists r; simpl; auto.
  destruct t.
  right; exists E0; exists (nil, s2'). constructor. auto.
  right; exists (e :: nil); exists (t, s2'). constructor. auto.
  unfold E0 in H3; destruct H3. congruence.
  right; exists (e :: nil); exists (t, s3). constructor. auto.
Qed.

Theorem factor_backward_simulation:
  backward_simulation L1 (atomic L2).
Proof.
  apply Backward_simulation with (bsim_match_states := fbs_match) (bsim_order := bsim_order sim).
  apply bsim_order_wf.
  intros. exploit (bsim_initial_states_exist sim); eauto. intros [s2 A].
  exists (E0, s2). simpl; auto.
  intros. destruct s2 as [t s2]; simpl in H0; destruct H0; subst.
  exploit (bsim_match_initial_states sim); eauto. intros [i [s1' [A B]]].
  exists i; exists s1'; split. auto. econstructor. apply star_refl. auto. auto.
  intros. destruct s2 as [t s2]; simpl in H1; destruct H1; subst.
  inv H. exploit (bsim_match_final_states sim); eauto. eapply star_safe; eauto.
  intros [s1'' [A B]]. exists s1''; split; auto. eapply star_trans; eauto.
  exact fbs_progress.
  exact fbs_simulation.
  simpl. exact (bsim_symbols_preserved sim).
Qed.

End FACTOR_BACKWARD_SIMULATION.

Receptiveness of atomic L.

Record strongly_receptive (L: semantics) : Prop :=
  Strongly_receptive {
    ssr_receptive: forall s ev1 t1 s1 ev2,
      Step L s (ev1 :: t1) s1 ->
      match_traces (globalenv L) (ev1 :: nil) (ev2 :: nil) ->
      exists s2, exists t2, Step L s (ev2 :: t2) s2;
    ssr_well_behaved:
      well_behaved_traces L
  }.

Theorem atomic_receptive:
  forall L, strongly_receptive L -> receptive (atomic L).
Proof.
  intros. constructor; intros.
  inv H0.
  inv H1. exists (E0, s'). constructor; auto.
  assert (exists ev2, t2 = ev2 :: nil). inv H1; econstructor; eauto.
  destruct H0 as [ev2 EQ]; subst t2.
  exploit ssr_receptive; eauto. intros [s2 [t2 P]].
  exploit ssr_well_behaved. eauto. eexact P. simpl; intros Q.
  exists (t2, s2). constructor; auto.
  simpl in H2; destruct H2.
  assert (t2 = ev :: nil). inv H1; simpl in H0; tauto.
  subst t2. exists (t, s0). constructor; auto. simpl; auto.
  red. intros. inv H0; simpl; omega.
Qed.

Connections with big-step semantics


The general form of a big-step semantics

Record bigstep_semantics : Type :=
  Bigstep_semantics {
    bigstep_terminates: trace -> int -> Prop;
    bigstep_diverges: traceinf -> Prop
  }.

Soundness with respect to a small-step semantics

Record bigstep_sound (B: bigstep_semantics) (L: semantics) : Prop :=
  Bigstep_sound {
    bigstep_terminates_sound:
      forall t r,
      bigstep_terminates B t r ->
      exists s1, exists s2, initial_state L s1 /\ Star L s1 t s2 /\ final_state L s2 r;
    bigstep_diverges_sound:
      forall T,
      bigstep_diverges B T ->
      exists s1, initial_state L s1 /\ forever (step L) (globalenv L) s1 T
}.

Generated by coq2html