Require compcert.backend.Deadcodeproof.
Require DeadcodeX.
Require ValueAnalysisX.
Require SmallstepX.
Import Coqlib.
Import Errors.
Import AST.
Import Values.
Import Memory.
Import Events.
Import SmallstepX.
Import Globalenvs.
Import DeadcodeX.
Import ValueDomainX.
Import ValueAnalysisX.
Export Deadcodeproof.
Section WITHCONFIG.
Context `{
external_calls_prf:
ExternalCalls}.
Local Existing Instance romem_for_empty_instance.
Variable prog:
RTL.program.
Variable tprog:
RTL.program.
Variable fn_stack_requirements:
ident ->
Z.
Hypothesis TRANSF:
transf_program prog =
OK tprog.
Let MATCH_PROG:
match_prog prog tprog.
Proof.
apply transf_program_match; auto. Qed.
Let ge :
RTL.genv :=
Genv.globalenv prog.
Let tge:
RTL.genv :=
Genv.globalenv tprog.
Lemma genv_next_preserved:
Genv.genv_next tge =
Genv.genv_next ge.
Proof.
unfold transf_program, transf_fundef in TRANSF.
eapply Deadcodeproof.genv_next_preserved; eauto.
Qed.
Lemma transf_initial_states:
forall i init_m sg args,
forall S,
RTLX.initial_state fn_stack_requirements prog i init_m sg args S ->
exists R,
RTLX.initial_state fn_stack_requirements tprog i init_m sg args R /\
match_states prog S R.
Proof.
unfold transf_program, transf_fundef in TRANSF.
intros. inv H.
exploit function_ptr_translated; eauto. intros (? & f' & ? & ? & ?).
eexists (RTL.Callstate nil f' args _ (fn_stack_requirements i)); split.
econstructor; eauto.
erewrite symbols_preserved; eauto.
symmetry. eapply sig_function_translated; eauto.
econstructor; eauto.
econstructor.
exact (val_lessdef_refl _).
apply Mem.extends_refl.
Qed.
Lemma transf_final_states:
forall sg,
forall st1 st2 r,
match_states prog st1 st2 ->
RTLX.final_state sg st1 r ->
final_state_with_extends (
RTLX.final_state sg)
st2 r.
Proof.
intros. inv H0. inv H.
inv STACKS.
edestruct Mem.unrecord_stack_block_extends as (m2' & USB' & EXT'); eauto.
econstructor; eauto.
constructor; auto.
Qed.
To prove that the initial per-function state is sound with respect
to value analysis, we need the following hypotheses, which
actually hold thanks to the properties on the caller in assembly
code (see AsmX.asm_invariant for the invariant on the assembly
state, and Asm.exec_step_external for the conditions local to
the function call site.)
Variable init_m:
mem.
Variable args:
list val.
Hypotheses
(
INJECT_NEUTRAL:
Mem.inject_neutral (
Mem.nextblock init_m)
init_m)
(
GENV_NEXT:
Ple (
Genv.genv_next ge) (
Mem.nextblock init_m))
(
ARGS_INJECT_NEUTRAL:
Val.inject_list (
Mem.flat_inj (
Mem.nextblock init_m))
args args).
Theorem transf_program_correct:
forall i sg,
forward_simulation
(
RTLX.semantics fn_stack_requirements prog i init_m sg args)
(
semantics_with_extends (
RTLX.semantics fn_stack_requirements tprog i init_m sg args)).
Proof.
unfold transf_program, transf_fundef in TRANSF.
intros.
apply forward_simulation_step with
(match_states := fun s1 s2 => sound_state prog s1 /\ match_states prog s1 s2 /\ RTL.stack_inv s2 /\ RTL.stack_equiv_inv s1 s2).
- intros; eapply senv_preserved; eauto.
- simpl; intros. exploit transf_initial_states; eauto. intros [st2 [A B]].
exists st2; intuition. eapply sound_initial; eauto.
eapply RTLX.initial_stack_inv; eauto.
inv H; inv A. red. simpl. apply stack_equiv_refl; auto.
- simpl; intros. destruct H as (? & MS & ?). eapply transf_final_states; eauto.
- simpl; intros. destruct H0 as (SS & MS & SI & SEI).
assert (sound_state prog s1') by (eapply sound_step; eauto).
fold ge; fold tge. exploit step_simulation; eauto. intros [st2' [A B]].
exploit RTL.stack_inv_inv. apply A. eauto. intro SI2.
exploit stack_equiv_inv_step. 4: exact SEI. eauto. auto. eauto. intros.
exists st2'; auto.
Qed.
End WITHCONFIG.