Library compcert.backend.CleanupLabelsproof
Correctness proof for clean-up of labels
Require Import Coqlib.
Require Import Ordered.
Require Import FSets.
Require Import AST.
Require Import Integers.
Require Import Values.
Require Import Memory.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Op.
Require Import Locations.
Require Import Linear.
Require Import CleanupLabels.
Module LabelsetFacts := FSetFacts.Facts(Labelset).
CompCertX:test-compcert-param-memory We create section WITHMEM and associated
contexts to parameterize the proof over the memory model. CompCertX:test-compcert-param-extcall Actually, we also need to parameterize
over external functions. To this end, we created a CompilerConfiguration class
(cf. Events) which is designed to be the single class on which the whole CompCert is to be
parameterized. It includes all operations and properties on which CompCert depends:
memory model, semantics of external functions and their preservation through
compilation.
Section WITHCONFIG.
Context `{compiler_config: CompilerConfiguration}.
Section CLEANUP.
Variable prog: program.
Let tprog := transf_program prog.
Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.
Lemma symbols_preserved:
forall (s: ident), Genv.find_symbol tge s = Genv.find_symbol ge s.
Proof.
intros; unfold ge, tge, tprog, transf_program.
apply Genv.find_symbol_transf.
Qed.
Lemma genv_next_preserved:
Genv.genv_next tge = Genv.genv_next ge.
Proof.
apply Genv.genv_next_transf.
Qed.
Lemma varinfo_preserved:
forall b, Genv.find_var_info tge b = Genv.find_var_info ge b.
Proof.
intros; unfold ge, tge, tprog, transf_program.
apply Genv.find_var_info_transf.
Qed.
Lemma functions_translated:
forall (v: val) (f: fundef),
Genv.find_funct ge v = Some f ->
Genv.find_funct tge v = Some (transf_fundef f).
Proof.
intros.
exact (Genv.find_funct_transf transf_fundef _ _ H).
Qed.
Lemma function_ptr_translated:
forall (b: block) (f: fundef),
Genv.find_funct_ptr ge b = Some f ->
Genv.find_funct_ptr tge b = Some (transf_fundef f).
Proof.
intros.
exact (Genv.find_funct_ptr_transf transf_fundef _ _ H).
Qed.
Lemma sig_function_translated:
forall f,
funsig (transf_fundef f) = funsig f.
Proof.
intros. destruct f; reflexivity.
Qed.
Lemma find_function_translated:
forall ros ls f,
find_function ge ros ls = Some f ->
find_function tge ros ls = Some (transf_fundef f).
Proof.
unfold find_function; intros; destruct ros; simpl.
apply functions_translated; auto.
rewrite symbols_preserved. destruct (Genv.find_symbol ge i).
apply function_ptr_translated; auto.
congruence.
Qed.
Context `{compiler_config: CompilerConfiguration}.
Section CLEANUP.
Variable prog: program.
Let tprog := transf_program prog.
Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.
Lemma symbols_preserved:
forall (s: ident), Genv.find_symbol tge s = Genv.find_symbol ge s.
Proof.
intros; unfold ge, tge, tprog, transf_program.
apply Genv.find_symbol_transf.
Qed.
Lemma genv_next_preserved:
Genv.genv_next tge = Genv.genv_next ge.
Proof.
apply Genv.genv_next_transf.
Qed.
Lemma varinfo_preserved:
forall b, Genv.find_var_info tge b = Genv.find_var_info ge b.
Proof.
intros; unfold ge, tge, tprog, transf_program.
apply Genv.find_var_info_transf.
Qed.
Lemma functions_translated:
forall (v: val) (f: fundef),
Genv.find_funct ge v = Some f ->
Genv.find_funct tge v = Some (transf_fundef f).
Proof.
intros.
exact (Genv.find_funct_transf transf_fundef _ _ H).
Qed.
Lemma function_ptr_translated:
forall (b: block) (f: fundef),
Genv.find_funct_ptr ge b = Some f ->
Genv.find_funct_ptr tge b = Some (transf_fundef f).
Proof.
intros.
exact (Genv.find_funct_ptr_transf transf_fundef _ _ H).
Qed.
Lemma sig_function_translated:
forall f,
funsig (transf_fundef f) = funsig f.
Proof.
intros. destruct f; reflexivity.
Qed.
Lemma find_function_translated:
forall ros ls f,
find_function ge ros ls = Some f ->
find_function tge ros ls = Some (transf_fundef f).
Proof.
unfold find_function; intros; destruct ros; simpl.
apply functions_translated; auto.
rewrite symbols_preserved. destruct (Genv.find_symbol ge i).
apply function_ptr_translated; auto.
congruence.
Qed.
Correctness of labels_branched_to.
Definition instr_branches_to (i: instruction) (lbl: label) : Prop :=
match i with
| Lgoto lbl´ => lbl = lbl´
| Lcond cond args lbl´ => lbl = lbl´
| Ljumptable arg tbl => In lbl tbl
| _ => False
end.
Remark add_label_branched_to_incr:
forall ls i, Labelset.Subset ls (add_label_branched_to ls i).
Proof.
intros; red; intros; destruct i; simpl; auto.
apply Labelset.add_2; auto.
apply Labelset.add_2; auto.
revert H; induction l; simpl. auto. intros; apply Labelset.add_2; auto.
Qed.
Remark add_label_branched_to_contains:
forall ls i lbl,
instr_branches_to i lbl ->
Labelset.In lbl (add_label_branched_to ls i).
Proof.
destruct i; simpl; intros; try contradiction.
apply Labelset.add_1; auto.
apply Labelset.add_1; auto.
revert H. induction l; simpl; intros.
contradiction.
destruct H. apply Labelset.add_1; auto. apply Labelset.add_2; auto.
Qed.
Lemma labels_branched_to_correct:
forall c i lbl,
In i c -> instr_branches_to i lbl -> Labelset.In lbl (labels_branched_to c).
Proof.
intros.
assert (forall c´ bto,
Labelset.Subset bto (fold_left add_label_branched_to c´ bto)).
induction c´; intros; simpl; red; intros.
auto.
apply IHc´. apply add_label_branched_to_incr; auto.
assert (forall c´ bto,
In i c´ -> Labelset.In lbl (fold_left add_label_branched_to c´ bto)).
induction c´; simpl; intros.
contradiction.
destruct H2.
subst a. apply H1. apply add_label_branched_to_contains; auto.
apply IHc´; auto.
unfold labels_branched_to. auto.
Qed.
Commutation with find_label.
Lemma remove_unused_labels_cons:
forall bto i c,
remove_unused_labels bto (i :: c) =
match i with
| Llabel lbl =>
if Labelset.mem lbl bto then i :: remove_unused_labels bto c else remove_unused_labels bto c
| _ =>
i :: remove_unused_labels bto c
end.
Proof.
unfold remove_unused_labels; intros. rewrite list_fold_right_eq. auto.
Qed.
Lemma find_label_commut:
forall lbl bto,
Labelset.In lbl bto ->
forall c c´,
find_label lbl c = Some c´ ->
find_label lbl (remove_unused_labels bto c) = Some (remove_unused_labels bto c´).
Proof.
induction c; simpl; intros.
congruence.
rewrite remove_unused_labels_cons.
unfold is_label in H0. destruct a; simpl; auto.
destruct (peq lbl l). subst l. inv H0.
rewrite Labelset.mem_1; auto.
simpl. rewrite peq_true. auto.
destruct (Labelset.mem l bto); auto. simpl. rewrite peq_false; auto.
Qed.
Corollary find_label_translated:
forall f i c´ lbl c,
incl (i :: c´) (fn_code f) ->
find_label lbl (fn_code f) = Some c ->
instr_branches_to i lbl ->
find_label lbl (fn_code (transf_function f)) =
Some (remove_unused_labels (labels_branched_to (fn_code f)) c).
Proof.
intros. unfold transf_function; unfold cleanup_labels; simpl.
apply find_label_commut. eapply labels_branched_to_correct; eauto.
apply H; auto with coqlib.
auto.
Qed.
Lemma find_label_incl:
forall lbl c c´, find_label lbl c = Some c´ -> incl c´ c.
Proof.
induction c; simpl; intros.
discriminate.
destruct (is_label lbl a). inv H; auto with coqlib. auto with coqlib.
Qed.
Correctness of clean-up
Inductive match_stackframes: stackframe -> stackframe -> Prop :=
| match_stackframe_intro:
forall f sp ls c,
incl c f.(fn_code) ->
match_stackframes
(Stackframe f sp ls c)
(Stackframe (transf_function f) sp ls
(remove_unused_labels (labels_branched_to f.(fn_code)) c)).
Inductive match_states: state -> state -> Prop :=
| match_states_intro:
forall s f sp c ls m ts
(STACKS: list_forall2 match_stackframes s ts)
(INCL: incl c f.(fn_code)),
match_states (State s f sp c ls m)
(State ts (transf_function f) sp (remove_unused_labels (labels_branched_to f.(fn_code)) c) ls m)
| match_states_call:
forall s f ls m ts,
list_forall2 match_stackframes s ts ->
match_states (Callstate s f ls m)
(Callstate ts (transf_fundef f) ls m)
| match_states_return:
forall s ls m ts,
list_forall2 match_stackframes s ts ->
match_states (Returnstate s ls m)
(Returnstate ts ls m).
Definition measure (st: state) : nat :=
match st with
| State s f sp c ls m => List.length c
| _ => O
end.
CompCertX:test-compcert-protect-stack-arg We also parameterize over the initial location set (which, in whole-program setting, is Vundef, and, in non-whole-program settings, will contain the caller's callee-save registers and the arguments passed to the first callee.)
Section WITHINITLS.
Variable init_ls: locset.
Lemma match_parent_locset:
forall s ts,
list_forall2 match_stackframes s ts ->
parent_locset init_ls ts = parent_locset init_ls s.
Proof.
induction 1; simpl. auto. inv H; auto.
Qed.
Variable init_ls: locset.
Lemma match_parent_locset:
forall s ts,
list_forall2 match_stackframes s ts ->
parent_locset init_ls ts = parent_locset init_ls s.
Proof.
induction 1; simpl. auto. inv H; auto.
Qed.
CompCertX:test-compcert-protect-stack-arg We also parameterize over a way to mark blocks writable.
Section WITHWRITABLEBLOCK.
Context `{Hwritable_block: WritableBlock}.
Theorem transf_step_correct:
forall s1 t s2, step init_ls ge s1 t s2 ->
forall s1´ (MS: match_states s1 s1´),
(exists s2´, step init_ls tge s1´ t s2´ /\ match_states s2 s2´)
\/ (measure s2 < measure s1 /\ t = E0 /\ match_states s2 s1´)%nat.
Proof.
induction 1; intros; inv MS; try rewrite remove_unused_labels_cons.
left; econstructor; split.
econstructor; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; eauto. instantiate (1 := v). rewrite <- H.
apply eval_operation_preserved. exact symbols_preserved.
econstructor; eauto with coqlib.
assert (eval_addressing tge sp addr (LTL.reglist rs args) = Some a).
rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
left; econstructor; split.
econstructor; eauto.
econstructor; eauto with coqlib.
assert (eval_addressing tge sp addr (LTL.reglist rs args) = Some a).
rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
left; econstructor; split.
econstructor; eauto.
eauto using writable_block_genv_next, genv_next_preserved.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor. eapply find_function_translated; eauto.
symmetry; apply sig_function_translated.
econstructor; eauto. constructor; auto. constructor; eauto with coqlib.
left; econstructor; split.
econstructor. erewrite match_parent_locset; eauto. eapply find_function_translated; eauto.
symmetry; apply sig_function_translated.
simpl. eauto.
econstructor; eauto.
left; econstructor; split.
econstructor; eauto.
eapply external_call_writable_block_weak´.
eapply external_call_symbols_preserved´; eauto.
exact symbols_preserved. exact varinfo_preserved.
exact genv_next_preserved.
apply writable_block_genv_next, genv_next_preserved.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; eauto.
eapply external_call_writable_block_weak´.
eapply external_call_symbols_preserved´; eauto.
exact symbols_preserved. exact varinfo_preserved.
exact genv_next_preserved.
apply writable_block_genv_next, genv_next_preserved.
econstructor; eauto with coqlib.
case_eq (Labelset.mem lbl (labels_branched_to (fn_code f))); intros.
left; econstructor; split.
constructor.
econstructor; eauto with coqlib.
right. split. simpl. omega. split. auto. econstructor; eauto with coqlib.
left; econstructor; split.
econstructor. eapply find_label_translated; eauto. red; auto.
econstructor; eauto. eapply find_label_incl; eauto.
left; econstructor; split.
econstructor. auto. eauto. eapply find_label_translated; eauto. red; auto.
econstructor; eauto. eapply find_label_incl; eauto.
left; econstructor; split.
eapply exec_Lcond_false; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor. eauto. eauto. eapply find_label_translated; eauto.
red. eapply list_nth_z_in; eauto. eauto.
econstructor; eauto. eapply find_label_incl; eauto.
left; econstructor; split.
econstructor; eauto.
erewrite <- match_parent_locset; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; simpl; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; eauto.
eapply external_call_writable_block_weak´.
eapply external_call_symbols_preserved´; eauto.
exact symbols_preserved. exact varinfo_preserved.
exact genv_next_preserved.
apply writable_block_genv_next, genv_next_preserved.
econstructor; eauto with coqlib.
inv H3. inv H1. left; econstructor; split.
econstructor; eauto.
econstructor; eauto.
Qed.
End WITHWRITABLEBLOCK.
End WITHINITLS.
Lemma transf_initial_states:
forall st1, initial_state prog st1 ->
exists st2, initial_state tprog st2 /\ match_states st1 st2.
Proof.
intros. inv H.
econstructor; split.
Context `{Hwritable_block: WritableBlock}.
Theorem transf_step_correct:
forall s1 t s2, step init_ls ge s1 t s2 ->
forall s1´ (MS: match_states s1 s1´),
(exists s2´, step init_ls tge s1´ t s2´ /\ match_states s2 s2´)
\/ (measure s2 < measure s1 /\ t = E0 /\ match_states s2 s1´)%nat.
Proof.
induction 1; intros; inv MS; try rewrite remove_unused_labels_cons.
left; econstructor; split.
econstructor; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; eauto. instantiate (1 := v). rewrite <- H.
apply eval_operation_preserved. exact symbols_preserved.
econstructor; eauto with coqlib.
assert (eval_addressing tge sp addr (LTL.reglist rs args) = Some a).
rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
left; econstructor; split.
econstructor; eauto.
econstructor; eauto with coqlib.
assert (eval_addressing tge sp addr (LTL.reglist rs args) = Some a).
rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
left; econstructor; split.
econstructor; eauto.
eauto using writable_block_genv_next, genv_next_preserved.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor. eapply find_function_translated; eauto.
symmetry; apply sig_function_translated.
econstructor; eauto. constructor; auto. constructor; eauto with coqlib.
left; econstructor; split.
econstructor. erewrite match_parent_locset; eauto. eapply find_function_translated; eauto.
symmetry; apply sig_function_translated.
simpl. eauto.
econstructor; eauto.
left; econstructor; split.
econstructor; eauto.
eapply external_call_writable_block_weak´.
eapply external_call_symbols_preserved´; eauto.
exact symbols_preserved. exact varinfo_preserved.
exact genv_next_preserved.
apply writable_block_genv_next, genv_next_preserved.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; eauto.
eapply external_call_writable_block_weak´.
eapply external_call_symbols_preserved´; eauto.
exact symbols_preserved. exact varinfo_preserved.
exact genv_next_preserved.
apply writable_block_genv_next, genv_next_preserved.
econstructor; eauto with coqlib.
case_eq (Labelset.mem lbl (labels_branched_to (fn_code f))); intros.
left; econstructor; split.
constructor.
econstructor; eauto with coqlib.
right. split. simpl. omega. split. auto. econstructor; eauto with coqlib.
left; econstructor; split.
econstructor. eapply find_label_translated; eauto. red; auto.
econstructor; eauto. eapply find_label_incl; eauto.
left; econstructor; split.
econstructor. auto. eauto. eapply find_label_translated; eauto. red; auto.
econstructor; eauto. eapply find_label_incl; eauto.
left; econstructor; split.
eapply exec_Lcond_false; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor. eauto. eauto. eapply find_label_translated; eauto.
red. eapply list_nth_z_in; eauto. eauto.
econstructor; eauto. eapply find_label_incl; eauto.
left; econstructor; split.
econstructor; eauto.
erewrite <- match_parent_locset; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; simpl; eauto.
econstructor; eauto with coqlib.
left; econstructor; split.
econstructor; eauto.
eapply external_call_writable_block_weak´.
eapply external_call_symbols_preserved´; eauto.
exact symbols_preserved. exact varinfo_preserved.
exact genv_next_preserved.
apply writable_block_genv_next, genv_next_preserved.
econstructor; eauto with coqlib.
inv H3. inv H1. left; econstructor; split.
econstructor; eauto.
econstructor; eauto.
Qed.
End WITHWRITABLEBLOCK.
End WITHINITLS.
Lemma transf_initial_states:
forall st1, initial_state prog st1 ->
exists st2, initial_state tprog st2 /\ match_states st1 st2.
Proof.
intros. inv H.
econstructor; split.
CompCertX:test-compcert-param-memory Some implicit argument names change under the hood,
in spite of Print Implicit. Probably a bug of Coq 8.4pl1?
eapply initial_state_intro with (f0 := transf_fundef f).
eapply Genv.init_mem_transf; eauto.
rewrite symbols_preserved; eauto.
apply function_ptr_translated; auto.
rewrite sig_function_translated. auto.
constructor; auto. constructor.
Qed.
Lemma transf_final_states:
forall st1 st2 r,
match_states st1 st2 -> final_state st1 r -> final_state st2 r.
Proof.
intros. inv H0. inv H. inv H6. econstructor; eauto.
Qed.
eapply Genv.init_mem_transf; eauto.
rewrite symbols_preserved; eauto.
apply function_ptr_translated; auto.
rewrite sig_function_translated. auto.
constructor; auto. constructor.
Qed.
Lemma transf_final_states:
forall st1 st2 r,
match_states st1 st2 -> final_state st1 r -> final_state st2 r.
Proof.
intros. inv H0. inv H. inv H6. econstructor; eauto.
Qed.
CompCertX:test-compcert-protect-stack-arg For whole programs, all blocks are writable.
Local Existing Instance writable_block_always_ops.
Local Existing Instance writable_block_always.
Theorem transf_program_correct:
forward_simulation (Linear.semantics prog) (Linear.semantics tprog).
Proof.
eapply forward_simulation_opt.
eexact symbols_preserved.
eexact transf_initial_states.
eexact transf_final_states.
eexact (transf_step_correct (Locmap.init Vundef)).
Qed.
End CLEANUP.
End WITHCONFIG.
Local Existing Instance writable_block_always.
Theorem transf_program_correct:
forward_simulation (Linear.semantics prog) (Linear.semantics tprog).
Proof.
eapply forward_simulation_opt.
eexact symbols_preserved.
eexact transf_initial_states.
eexact transf_final_states.
eexact (transf_step_correct (Locmap.init Vundef)).
Qed.
End CLEANUP.
End WITHCONFIG.