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Diffstat (limited to 'kvx/lib/RTLpathSchedulerproof.v')
-rw-r--r-- | kvx/lib/RTLpathSchedulerproof.v | 303 |
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diff --git a/kvx/lib/RTLpathSchedulerproof.v b/kvx/lib/RTLpathSchedulerproof.v new file mode 100644 index 00000000..db1f6be7 --- /dev/null +++ b/kvx/lib/RTLpathSchedulerproof.v @@ -0,0 +1,303 @@ +Require Import AST Linking Values Maps Globalenvs Smallstep Registers. +Require Import Coqlib Maps Events Errors Op. +Require Import RTL RTLpath RTLpathLivegen RTLpathLivegenproof RTLpathSE_theory. +Require Import RTLpathScheduler. + +Definition match_prog (p tp: program) := + match_program (fun _ f tf => transf_fundef f = OK tf) eq p tp. + +Lemma transf_program_match: + forall prog tprog, transf_program prog = OK tprog -> match_prog prog tprog. +Proof. + intros. eapply match_transform_partial_program_contextual; eauto. +Qed. + +Section PRESERVATION. + +Variable prog: program. +Variable tprog: program. + +Hypothesis TRANSL: match_prog prog tprog. + +Let pge := Genv.globalenv prog. +Let tpge := Genv.globalenv tprog. + +(* Was a Hypothesis *) +Theorem all_fundef_liveness_ok: forall b fd, + Genv.find_funct_ptr pge b = Some fd -> liveness_ok_fundef fd. +Proof. +Admitted. + +Lemma symbols_preserved s: Genv.find_symbol tpge s = Genv.find_symbol pge s. +Proof. + rewrite <- (Genv.find_symbol_match TRANSL). reflexivity. +Qed. + +Lemma senv_preserved: + Senv.equiv pge tpge. +Proof. + eapply (Genv.senv_match TRANSL). +Qed. + +Lemma functions_preserved: + forall (v: val) (f: fundef), + Genv.find_funct pge v = Some f -> + exists tf cunit, transf_fundef f = OK tf /\ Genv.find_funct tpge v = Some tf /\ linkorder cunit prog. +Proof. + intros. exploit (Genv.find_funct_match TRANSL); eauto. + intros (cu & tf & A & B & C). + repeat eexists; intuition eauto. + + unfold incl; auto. + + eapply linkorder_refl. +Qed. + +Lemma function_ptr_preserved: + forall v f, + Genv.find_funct_ptr pge v = Some f -> + exists tf, + Genv.find_funct_ptr tpge v = Some tf /\ transf_fundef f = OK tf. +Proof. + intros. + exploit (Genv.find_funct_ptr_transf_partial TRANSL); eauto. +Qed. + +Lemma function_sig_preserved: + forall f tf, transf_fundef f = OK tf -> funsig tf = funsig f. +Proof. + intros. destruct f. + - simpl in H. monadInv H. simpl. symmetry. + eapply transf_function_preserves. + assumption. + - simpl in H. monadInv H. reflexivity. +Qed. + +Theorem transf_initial_states: + forall s1, initial_state prog s1 -> + exists s2, initial_state tprog s2 /\ match_states s1 s2. +Proof. + intros. inv H. + exploit function_ptr_preserved; eauto. intros (tf & FIND & TRANSF). + exists (Callstate nil tf nil m0). + split. + - econstructor; eauto. + + intros; apply (Genv.init_mem_match TRANSL); assumption. + + replace (prog_main tprog) with (prog_main prog). rewrite symbols_preserved. eauto. + symmetry. eapply match_program_main. eauto. + + destruct f. + * monadInv TRANSF. rewrite <- H3. symmetry; eapply transf_function_preserves. assumption. + * monadInv TRANSF. assumption. + - constructor; eauto. + + constructor. + + apply transf_fundef_correct; auto. + + eapply all_fundef_liveness_ok; eauto. +Qed. + +Theorem transf_final_states s1 s2 r: + final_state s1 r -> match_states s1 s2 -> final_state s2 r. +Proof. + unfold final_state. + intros H; inv H. + intros H; inv H; simpl in * |- *; try congruence. + inv H1. + destruct st; simpl in * |- *; try congruence. + inv STACKS. constructor. +Qed. + + +Let ge := Genv.globalenv (RTLpath.transf_program prog). +Let tge := Genv.globalenv (RTLpath.transf_program tprog). + +Lemma senv_sym x y: Senv.equiv x y -> Senv.equiv y x. +Proof. + unfold Senv.equiv. intuition congruence. +Qed. + +Lemma senv_transitivity x y z: Senv.equiv x y -> Senv.equiv y z -> Senv.equiv x z. +Proof. + unfold Senv.equiv. intuition congruence. +Qed. + +Lemma senv_preserved_RTL: + Senv.equiv ge tge. +Proof. + eapply senv_transitivity. { eapply senv_sym; eapply RTLpath.senv_preserved. } + eapply senv_transitivity. { eapply senv_preserved. } + eapply RTLpath.senv_preserved. +Qed. + +Lemma symbols_preserved_RTL s: Genv.find_symbol tge s = Genv.find_symbol ge s. +Proof. + unfold tge, ge. erewrite RTLpath.symbols_preserved; eauto. + rewrite symbols_preserved. + erewrite RTLpath.symbols_preserved; eauto. +Qed. + +Lemma s_find_function_preserved sp svos rs0 m0 fd: + sfind_function pge ge sp svos rs0 m0 = Some fd -> + exists fd', sfind_function tpge tge sp svos rs0 m0 = Some fd' + /\ transf_fundef fd = OK fd' + /\ liveness_ok_fundef fd. +Proof. + Local Hint Resolve symbols_preserved_RTL: core. + unfold sfind_function. destruct svos; simpl. + + rewrite (seval_preserved ge tge); auto. + destruct (seval_sval _ _ _ _); try congruence. + intros; exploit functions_preserved; eauto. + intros (fd' & cunit & X). eexists. intuition eauto. + eapply find_funct_liveness_ok; eauto. + intros. eapply all_fundef_liveness_ok; eauto. + + rewrite symbols_preserved. destruct (Genv.find_symbol _ _); try congruence. + intros; exploit function_ptr_preserved; eauto. + intros (fd' & X). eexists. intuition eauto. + intros. eapply all_fundef_liveness_ok; eauto. +Qed. + +Lemma sistate_simu f dupmap sp st st' rs m is: + ssem ge sp st rs m is -> + liveness_ok_function f -> + sistate_simu f dupmap st st' -> + exists is', + ssem tge sp st' rs m is' /\ istate_simu f dupmap is is'. +Proof. + intros SEM LIVE X; eapply X; eauto. +Qed. + + +Lemma sfmatch_simu dupmap f f' stk stk' sp st st' rs0 m0 sv sv' rs m t s: + match_function dupmap f f' -> + liveness_ok_function f -> + list_forall2 match_stackframes stk stk' -> + (* s_istate_simu f dupmap st st' -> *) + sfval_simu f dupmap st.(si_pc) st'.(si_pc) sv sv' -> + sfmatch pge ge sp st stk f rs0 m0 sv rs m t s -> + exists s', sfmatch tpge tge sp st' stk' f' rs0 m0 sv' rs m t s' /\ match_states s s'. +Proof. + Local Hint Resolve transf_fundef_correct: core. + intros FUN LIVE STK (* SIS *) SFV. destruct SFV; intros SEM; inv SEM; simpl. + - (* Snone *) + exploit initialize_path. { eapply dupmap_path_entry1; eauto. } + intros (path & PATH). + eexists; split; econstructor; eauto. + eapply eqlive_reg_refl. + - (* Scall *) + exploit s_find_function_preserved; eauto. + intros (fd' & FIND & TRANSF & LIVE'). + erewrite <- function_sig_preserved; eauto. + exploit initialize_path. { eapply dupmap_path_entry1; eauto. } + intros (path & PATH). + eexists; split; econstructor; eauto. + + erewrite <- (list_sval_eval_preserved ge tge); auto. + + simpl. repeat (econstructor; eauto). + - (* Sreturn *) + eexists; split; econstructor; eauto. + + erewrite <- preserv_fnstacksize; eauto. + + destruct os; auto. + erewrite <- seval_preserved; eauto. +Qed. + +(* The main theorem on simulation of symbolic states ! *) +Theorem smatch_sstate_simu dupmap f f' stk stk' sp st st' rs m t s: + match_function dupmap f f' -> + liveness_ok_function f -> + list_forall2 match_stackframes stk stk' -> + smatch pge ge sp st stk f rs m t s -> + sstate_simu f dupmap st st' -> + exists s', smatch tpge tge sp st' stk' f' rs m t s' /\ match_states s s'. +Proof. + intros MFUNC LIVE STACKS SEM (SIMU1 & SIMU2). destruct SEM as [is CONT SEM|is t s' CONT SEM1 SEM2]; simpl. + - (* sem_early *) + exploit sistate_simu; eauto. + unfold istate_simu; rewrite CONT. + intros (is' & SEM' & (path & PATH & (CONT' & RS' & M') & PC')). + exists (State stk' f' sp (ipc is') (irs is') (imem is')). + split. + + eapply smatch_early; auto. congruence. + + rewrite M'. econstructor; eauto. + - (* sem_normal *) + exploit sistate_simu; eauto. + unfold istate_simu; rewrite CONT. + intros (is' & SEM' & (CONT' & RS' & M')). + rewrite <- eqlive_reg_triv in RS'. + exploit sfmatch_simu; eauto. + clear SEM2; intros (s0 & SEM2 & MATCH0). + exploit sfmatch_equiv; eauto. + clear SEM2; rewrite M'; rewrite CONT' in CONT; intros (s1 & EQ & SEM2). + exists s1; split. + + eapply smatch_normal; eauto. + + eapply match_states_equiv; eauto. +Qed. + +Lemma exec_path_simulation dupmap path stk stk' f f' sp rs m pc pc' t s: + (fn_path f)!pc = Some path -> + path_step ge pge path.(psize) stk f sp rs m pc t s -> + list_forall2 match_stackframes stk stk' -> + dupmap ! pc' = Some pc -> + match_function dupmap f f' -> + liveness_ok_function f -> + exists path' s', (fn_path f')!pc' = Some path' /\ path_step tge tpge path'.(psize) stk' f' sp rs m pc' t s' /\ match_states s s'. +Proof. + intros PATH STEP STACKS DUPPC MATCHF LIVE. + exploit initialize_path. { eapply dupmap_path_entry2; eauto. } + intros (path' & PATH'). + exists path'. + exploit (sstep_correct f pc pge ge sp path stk rs m t s); eauto. + intros (st & SYMB & SEM); clear STEP. + exploit dupmap_correct; eauto. + clear SYMB; intros (st' & SYMB & SIMU). + exploit smatch_sstate_simu; eauto. + intros (s0 & SEM0 & MATCH). + exploit sstep_exact; eauto. + intros (s' & STEP' & EQ). + exists s'; intuition. + eapply match_states_equiv; eauto. +Qed. + +Lemma step_simulation s1 t s1' s2: + step ge pge s1 t s1' -> + match_states s1 s2 -> + exists s2', + step tge tpge s2 t s2' + /\ match_states s1' s2'. +Proof. + Local Hint Resolve eqlive_stacks_refl transf_fundef_correct: core. + destruct 1 as [path stack f sp rs m pc t s PATH STEP | | | ]; intros MS; inv MS. +(* exec_path *) + - try_simplify_someHyps. intros. + exploit path_step_eqlive; eauto. { intros. eapply all_fundef_liveness_ok; eauto. } + clear STEP EQUIV rs; intros (s2 & STEP & EQLIVE). + exploit exec_path_simulation; eauto. + clear STEP; intros (path' & s' & PATH' & STEP' & MATCH'). + exists s'; split. + + eapply exec_path; eauto. + + eapply eqlive_match_states; eauto. +(* exec_function_internal *) + - inv LIVE. + exploit initialize_path. { eapply (fn_entry_point_wf f). } + destruct 1 as (path & PATH). + inversion TRANSF as [f0 xf tf MATCHF|]; subst. eexists. split. + + eapply exec_function_internal. erewrite <- preserv_fnstacksize; eauto. + + erewrite preserv_fnparams; eauto. + econstructor; eauto. + { apply preserv_entrypoint; auto. } + { apply eqlive_reg_refl. } +(* exec_function_external *) + - inversion TRANSF as [|]; subst. eexists. split. + + econstructor. eapply external_call_symbols_preserved; eauto. apply senv_preserved_RTL. + + constructor. assumption. +(* exec_return *) + - inv STACKS. destruct b1 as [res' f' sp' pc' rs']. eexists. split. + + constructor. + + inv H1. econstructor; eauto. +Qed. + +Theorem transf_program_correct: + forward_simulation (semantics prog) (semantics tprog). +Proof. + eapply forward_simulation_step with match_states. + - eapply senv_preserved. + - eapply transf_initial_states. + - intros; eapply transf_final_states; eauto. + - intros; eapply step_simulation; eauto. +Qed. + +End PRESERVATION. |