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Require Import AST Linking Values Maps Globalenvs Smallstep Registers.
Require Import Coqlib Maps Events Errors Op OptionMonad.
Require Import RTL BTL BTL_SEtheory.
Require Import BTLmatchRTL BTL_Scheduler.
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.
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.
(* TODO gourdinl move this to BTL_Scheduler.v? *)
Inductive match_fundef: fundef -> fundef -> Prop :=
| match_Internal f f': match_function f f' -> match_fundef (Internal f) (Internal f')
| match_External ef: match_fundef (External ef) (External ef).
Inductive match_stackframes: stackframe -> stackframe -> Prop :=
| match_stackframe_intro
res f sp pc rs f'
(TRANSF: match_function f f')
: match_stackframes (BTL.Stackframe res f sp pc rs) (BTL.Stackframe res f' sp pc rs).
Inductive match_states: state -> state -> Prop :=
| match_states_intro
st f sp pc rs rs' m st' f'
(EQREGS: forall r, rs # r = rs' # r)
(STACKS: list_forall2 match_stackframes st st')
(TRANSF: match_function f f')
: match_states (State st f sp pc rs m) (State st' f' sp pc rs' m)
| match_states_call
st st' f f' args m
(STACKS: list_forall2 match_stackframes st st')
(TRANSF: match_fundef f f')
: match_states (Callstate st f args m) (Callstate st' f' args m)
| match_states_return
st st' v m
(STACKS: list_forall2 match_stackframes st st')
: match_states (Returnstate st v m) (Returnstate st' v m)
.
Lemma transf_function_correct f f':
transf_function f = OK f' -> match_function f f'.
Proof.
unfold transf_function. intros H. monadInv H.
econstructor; eauto. eapply check_symbolic_simu_correct; eauto.
Qed.
Lemma transf_fundef_correct f f':
transf_fundef f = OK f' -> match_fundef f f'.
Proof.
Admitted.
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_translated f tf: transf_fundef f = OK tf -> funsig tf = funsig f.
Proof.
intros H; apply transf_fundef_correct in H; destruct H; simpl; eauto.
symmetry; erewrite preserv_fnsig; eauto.
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).
eexists. split.
- econstructor; eauto.
+ intros; apply (Genv.init_mem_match TRANSL); eauto.
+ replace (prog_main tprog) with (prog_main prog). rewrite symbols_preserved. eauto.
symmetry. eapply match_program_main. eauto.
+ erewrite function_sig_translated; eauto.
- constructor; eauto.
+ constructor.
+ apply transf_fundef_correct; auto.
Qed.
Lemma transf_final_states s1 s2 r:
match_states s1 s2 -> final_state s1 r -> final_state s2 r.
Proof.
intros. inv H0. inv H. inv STACKS. constructor.
Qed.
(* TODO gourdinl
generalize with a None version of the lemma *)
Lemma iblock_istep_simulation stk1 stk2 f f' sp pc fin rs1 m rs1' m' rs2 ib1 t s
(ISTEP: iblock_istep pge sp rs1 m ib1 rs1' m' (Some fin))
(FSTEP : final_step tr_inputs pge stk1 f sp rs1' m' fin t s)
(EQREGS: forall r : positive, rs1 # r = rs2 # r)
(STACKS : list_forall2 match_stackframes stk1 stk2)
(TRANSF : match_function f f')
:exists ib2 rs2' s',
(fn_code f') ! pc = Some ib2
/\ iblock_istep tpge sp rs2 m ib2.(entry) rs2' m' (Some fin)
/\ final_step tr_inputs tpge stk2 f' sp rs2' m' fin t s'
/\ match_states s s'.
Proof.
induction ib1; inv ISTEP; eauto.
- (* BF *)
admit.
- (* Bseq continue *)
eauto.
Admitted.
Lemma iblock_step_simulation stk1 stk2 f f' sp rs rs' m ibf t s1 pc
(STEP: iblock_step tr_inputs pge stk1 f sp rs m ibf.(entry) t s1)
(PC: (fn_code f) ! pc = Some ibf)
(EQREGS: forall r : positive, rs # r = rs' # r)
(STACKS: list_forall2 match_stackframes stk1 stk2)
(TRANSF: match_function f f')
:exists ibf' s2,
(fn_code f') ! pc = Some ibf' /\
iblock_step tr_inputs tpge stk2 f' sp rs' m ibf'.(entry) t s2 /\
match_states s1 s2.
Proof.
destruct STEP as (rs1 & m1 & fin & ISTEP & FSTEP).
exploit iblock_istep_simulation; eauto.
intros (ibf2 & rs2' & s' & PC' & ISTEP' & FSTEP' & MS').
repeat eexists; eauto.
Qed.
Local Hint Constructors step: core.
Theorem step_simulation s1 s1' t s2
(STEP: step tr_inputs pge s1 t s1')
(MS: match_states s1 s2)
:exists s2',
step tr_inputs tpge s2 t s2'
/\ match_states s1' s2'.
Proof.
destruct STEP as [stack ibf f sp n rs m t s PC STEP | | | ]; inv MS.
- (* iblock *)
simplify_someHyps. intros PC.
exploit iblock_step_simulation; eauto.
intros (ibf' & s2 & PC2 & STEP2 & MS2).
eexists; split; eauto.
- (* function internal *)
inversion TRANSF as [xf tf MF |]; subst.
eexists; split.
+ econstructor. erewrite <- preserv_fnstacksize; eauto.
+ erewrite preserv_fnparams; eauto.
erewrite preserv_entrypoint; eauto.
econstructor; eauto.
- (* function external *)
inv TRANSF. eexists; split; econstructor; eauto.
eapply external_call_symbols_preserved; eauto. apply senv_preserved.
- (* return *)
inv STACKS. destruct b1 as [res' f' sp' pc' rs'].
eexists; split.
+ econstructor.
+ inv H1. econstructor; eauto.
Qed.
Theorem transf_program_correct:
forward_simulation (fsem prog) (fsem tprog).
Proof.
eapply forward_simulation_step with match_states; simpl; eauto. (* lock-step with respect to match_states *)
- eapply senv_preserved.
- eapply transf_initial_states.
- eapply transf_final_states.
- intros; eapply step_simulation; eauto.
Qed.
End PRESERVATION.
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