diff options
Diffstat (limited to 'driver')
-rw-r--r-- | driver/ForwardSimulationBlock.v | 236 |
1 files changed, 236 insertions, 0 deletions
diff --git a/driver/ForwardSimulationBlock.v b/driver/ForwardSimulationBlock.v new file mode 100644 index 00000000..43cf58c3 --- /dev/null +++ b/driver/ForwardSimulationBlock.v @@ -0,0 +1,236 @@ +(*** + +Variante de Forward Simulation pour les blocks. + +***) + +Require Import Relations. +Require Import Wellfounded. +Require Import Coqlib. +Require Import Events. +Require Import Globalenvs. +Require Import Smallstep. + + +Local Open Scope nat_scope. + + +Section ForwardSimuBlock. + +Variable L1 L2: semantics. + +Local Hint Resolve starN_refl starN_step. + + +(* TODO: faut-il se baser sur [starN] ou un type inductif équivalent + (qui ferait les step dans l'ordre ci-dessous) ? + + => Voir ce qui est le plus facile pour prouver l'hypothèse [simu_end_block] ci-dessous... +*) + +Lemma starN_last_step n s t1 s': + starN (step L1) (globalenv L1) n s t1 s' -> + forall (t t2:trace) s'', + Step L1 s' t2 s'' -> t = t1 ** t2 -> starN (step L1) (globalenv L1) (S n) s t s''. +Proof. + induction 1; simpl. + + intros t t1 s0; autorewrite with trace_rewrite. + intros; subst; eapply starN_step; eauto. + autorewrite with trace_rewrite; auto. + + intros. eapply starN_step; eauto. + intros; subst; autorewrite with trace_rewrite; auto. +Qed. + +(** Hypothèses de la preuve *) + +Variable dist_end_block: state L1 -> nat. + +Hypothesis simu_mid_block: + forall s1 t s1', Step L1 s1 t s1' -> (dist_end_block s1)<>0 -> t = E0 /\ dist_end_block s1=S (dist_end_block s1'). + +Hypothesis public_preserved: + forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id. + +Variable build_block: state L1 -> state L2. + +Hypothesis match_initial_states: + forall s1, initial_state L1 s1 -> initial_state L2 (build_block s1). + +Hypothesis match_final_states: + forall s1 r, final_state L1 s1 r -> final_state L2 (build_block s1) r. + +Hypothesis final_states_end_block: + forall s1 t s1' r, Step L1 s1 t s1' -> final_state L1 s1' r -> dist_end_block s1 = 0. + +Hypothesis simu_end_block: + forall s1 t s1', starN (step L1) (globalenv L1) (S (dist_end_block s1)) s1 t s1' -> Step L2 (build_block s1) t (build_block s1'). + + +(** Introduction d'une sémantique par bloc sur L1 appelée "memoL1" *) + +Definition star_in_block (head current: state L1): Prop := + dist_end_block head >= dist_end_block current + /\ starN (step L1) (globalenv L1) (minus (dist_end_block head) (dist_end_block current)) head E0 current. + +Lemma star_in_block_step (head previous next: state L1): + forall t, star_in_block head previous -> Step L1 previous t next -> (dist_end_block previous)<>0 -> star_in_block head next. +Proof. + intros t [H1 H2] H3 H4. + destruct (simu_mid_block _ _ _ H3 H4) as [H5 H6]; subst. + constructor 1. + + omega. + + cutrewrite (dist_end_block head - dist_end_block next = S (dist_end_block head - dist_end_block previous)). + - eapply starN_last_step; eauto. + - omega. +Qed. + +Lemma star_in_block_init (head current: state L1): + forall t, Step L1 head t current -> (dist_end_block head)<>0 -> star_in_block head current. +Proof. + intros t H3 H4. + destruct (simu_mid_block _ _ _ H3 H4) as [H5 H6]; subst. + constructor 1. + + omega. + + cutrewrite (dist_end_block head - dist_end_block current = 1). + - eapply starN_last_step; eauto. + - omega. +Qed. + + +Record memostate := { + real: state L1; + memorized: option (state L1); + memo_star: forall head, memorized = Some head -> star_in_block head real; + memo_final: forall r, final_state L1 real r -> memorized = None +}. + +Definition head (s: memostate): state L1 := + match memorized s with + | None => real s + | Some s' => s' + end. + +Lemma head_star (s: memostate): star_in_block (head s) (real s). +Proof. + destruct s as [rs ms Hs]. simpl. + destruct ms as [ms|]; unfold head; simpl; auto. + constructor 1. + omega. + cutrewrite ((dist_end_block rs - dist_end_block rs)%nat=O). + + apply starN_refl; auto. + + omega. +Qed. + +Inductive is_well_memorized (s s': memostate): Prop := + | StartBloc: + dist_end_block (real s) <> O -> + memorized s = None -> + memorized s' = Some (real s) -> + is_well_memorized s s' + | MidBloc: + dist_end_block (real s) <> O -> + memorized s <> None -> + memorized s' = memorized s -> + is_well_memorized s s' + | ExitBloc: + dist_end_block (real s) = O -> + memorized s' = None -> + is_well_memorized s s'. + +Local Hint Resolve StartBloc MidBloc ExitBloc. + +Definition memoL1 := {| + state := memostate; + genvtype := genvtype L1; + step := fun ge s t s' => + step L1 ge (real s) t (real s') + /\ is_well_memorized s s' ; + initial_state := fun s => initial_state L1 (real s) /\ memorized s = None; + final_state := fun s r => final_state L1 (real s) r; + globalenv:= globalenv L1; + symbolenv:= symbolenv L1 +|}. + + +(** Preuve des 2 forward simulations: L1 -> memoL1 et memoL1 -> L2 *) + +Lemma discr_dist_end s: + {dist_end_block s = O} + {dist_end_block s <> O}. +Proof. + destruct (dist_end_block s); simpl; intuition. +Qed. + +Lemma memo_simulation_step: + forall s1 t s1', Step L1 s1 t s1' -> + forall s2, s1 = (real s2) -> exists s2', Step memoL1 s2 t s2' /\ s1' = (real s2'). +Proof. + intros s1 t s1' H1 [rs2 ms2 Hmoi] H2. simpl in H2; subst. + destruct (discr_dist_end rs2) as [H3 | H3]. + + refine (ex_intro _ {|real:=s1'; memorized:=None |} _); simpl. + intuition. + + destruct ms2 as [s|]. + - refine (ex_intro _ {|real:=s1'; memorized:=Some s |} _); simpl. + intuition. + - refine (ex_intro _ {|real:=s1'; memorized:=Some rs2 |} _); simpl. + intuition. + Unshelve. + * intros; discriminate. + * intros; auto. + * intros head X; injection X; clear X; intros; subst. + eapply star_in_block_step; eauto. + * intros r X; erewrite final_states_end_block in H3; intuition eauto. + * intros head X; injection X; clear X; intros; subst. + eapply star_in_block_init; eauto. + * intros r X; erewrite final_states_end_block in H3; intuition eauto. +Qed. + +Lemma forward_memo_simulation_1: forward_simulation L1 memoL1. +Proof. + apply forward_simulation_step with (match_states:=fun s1 s2 => s1 = (real s2)); auto. + + intros s1 H; eapply ex_intro with (x:={|real:=s1; memorized:=None |}); simpl. + intuition. + + intros; subst; auto. + + intros; exploit memo_simulation_step; eauto. + Unshelve. + * intros; discriminate. + * auto. +Qed. + +Lemma forward_memo_simulation_2: forward_simulation memoL1 L2. +Proof. + apply forward_simulation_opt with (measure:=fun s => dist_end_block (real s)) (match_states:=fun s1 s2 => s2 = (build_block (head s1))); auto. + + unfold memoL1; simpl. intros s1 [H0 H1]; eapply ex_intro with (x:=(build_block (real s1))). + unfold head. rewrite H1. intuition. + + intros s1 s2 r X H0. subst. unfold head. + erewrite memo_final; eauto. + eapply H0. + + unfold memoL1; simpl. + intros s1 t s1' [H1 H2] s H; subst. + destruct H2. + - (* StartBloc *) + constructor 2. destruct (simu_mid_block (real s1) t (real s1')) as [H3 H4]; auto. + unfold head. rewrite H0. rewrite H2. rewrite H4. intuition. + - (* MidBloc *) + constructor 2. destruct (simu_mid_block (real s1) t (real s1')) as [H3 H4]; auto. + unfold head. rewrite H2. rewrite H4. intuition. + destruct (memorized s1); simpl; auto. destruct H0; auto. + - (* EndBloc *) + constructor 1. + eapply ex_intro; intuition eauto. + apply simu_end_block. + destruct (head_star s1) as [H2 H3]. + cutrewrite (dist_end_block (head s1) - dist_end_block (real s1) = dist_end_block (head s1)) in H3. + unfold head; rewrite H0; simpl. + eapply starN_last_step; eauto. + omega. +Qed. + +Lemma forward_simulation_block: forward_simulation L1 L2. +Proof. + eapply compose_forward_simulations. + eapply forward_memo_simulation_1. + apply forward_memo_simulation_2. +Qed. + + +End ForwardSimuBlock.
\ No newline at end of file |