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(** Correctness proof for code duplication *)
Require Import AST Linking Errors Globalenvs Smallstep.
Require Import Coqlib Maps Events Values.
Require Import Op RTL Duplicate.
Local Open Scope positive_scope.
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.
(* est-ce plus simple de prendre is_copy: node -> node, avec un noeud hors CFG à la place de None ? *)
Inductive match_inst (is_copy: node -> option node): instruction -> instruction -> Prop :=
| match_inst_nop: forall n n',
is_copy n' = (Some n) -> match_inst is_copy (Inop n) (Inop n')
| match_inst_op: forall n n' op lr r,
is_copy n' = (Some n) -> match_inst is_copy (Iop op lr r n) (Iop op lr r n')
| match_inst_load: forall n n' m a lr r,
is_copy n' = (Some n) -> match_inst is_copy (Iload m a lr r n) (Iload m a lr r n')
| match_inst_store: forall n n' m a lr r,
is_copy n' = (Some n) -> match_inst is_copy (Istore m a lr r n) (Istore m a lr r n')
| match_inst_call: forall n n' s ri lr r,
is_copy n' = (Some n) -> match_inst is_copy (Icall s ri lr r n) (Icall s ri lr r n')
| match_inst_tailcall: forall n n' s ri lr,
is_copy n' = (Some n) -> match_inst is_copy (Itailcall s ri lr) (Itailcall s ri lr)
| match_inst_builtin: forall n n' ef la br,
is_copy n' = (Some n) -> match_inst is_copy (Ibuiltin ef la br n) (Ibuiltin ef la br n')
| match_inst_cond: forall ifso ifso' ifnot ifnot' c lr,
is_copy ifso' = (Some ifso) -> is_copy ifnot' = (Some ifnot) ->
match_inst is_copy (Icond c lr ifso ifnot) (Icond c lr ifso' ifnot')
| match_inst_jumptable: forall ln ln' r,
list_forall2 (fun n n' => (is_copy n' = (Some n))) ln ln' ->
match_inst is_copy (Ijumptable r ln) (Ijumptable r ln')
| match_inst_return: forall or, match_inst is_copy (Ireturn or) (Ireturn or).
Lemma verify_mapping_mn_rec_step:
forall lb b f xf,
In b lb ->
verify_mapping_mn_rec f xf lb = OK tt ->
verify_mapping_mn f xf b = OK tt.
Proof.
induction lb; intros.
- monadInv H0. inversion H.
- inversion H.
+ subst. monadInv H0. destruct x. assumption.
+ monadInv H0. destruct x0. eapply IHlb; assumption.
Qed.
Inductive rtl_one_successor : node -> node -> instruction -> instruction -> Prop :=
| inop_one_succ : forall n n', rtl_one_successor n n' (Inop n) (Inop n')
| iop_one_succ : forall op lr r n n', rtl_one_successor n n' (Iop op lr r n) (Iop op lr r n')
| iload_one_succ : forall m a lr r n n', rtl_one_successor n n' (Iload m a lr r n) (Iload m a lr r n')
| istore_one_succ : forall m a lr r n n', rtl_one_successor n n' (Istore m a lr r n) (Istore m a lr r n')
| icall_one_succ : forall s ri lr r n n', rtl_one_successor n n' (Icall s ri lr r n) (Icall s ri lr r n')
| ibuiltin_one_succ : forall ef lbr br n n', rtl_one_successor n n' (Ibuiltin ef lbr br n) (Ibuiltin ef lbr br n')
.
Lemma verify_is_copy_correct_one:
forall xf n n' i i',
verify_is_copy (fn_revmap xf) n n' = OK tt ->
rtl_one_successor n n' i i' ->
match_inst (fun nn => (fn_revmap xf) ! nn) i i'.
Proof.
intros. unfold verify_is_copy in H. destruct (_ ! n') eqn:REVM; [|inversion H].
destruct (n ?= p) eqn:NP; try (inversion H; fail). clear H.
eapply Pos.compare_eq in NP. subst.
inv H0.
all: constructor; eauto.
Qed.
Lemma verify_match_inst_correct:
forall xf i i',
verify_match_inst (fn_revmap xf) i i' = OK tt ->
match_inst (fun nn => (fn_revmap xf) ! nn) i i'.
Proof.
intros. unfold verify_match_inst in H.
destruct i; try (inversion H; fail).
(* Inop *)
- destruct i'; try (inversion H; fail). monadInv H. eapply verify_is_copy_correct_one. destruct x. eassumption.
constructor; eauto.
(* Iop *)
- destruct i'; try (inversion H; fail). monadInv H.
destruct (eq_operation _ _); try discriminate.
destruct (list_eq_dec _ _ _); try discriminate.
destruct (Pos.eq_dec _ _); try discriminate. clear EQ0. subst.
eapply verify_is_copy_correct_one. destruct x. eassumption.
constructor.
(* Iload *)
- destruct i'; try (inversion H; fail). monadInv H.
destruct (chunk_eq _ _); try discriminate.
destruct (eq_addressing _ _); try discriminate.
destruct (list_eq_dec _ _ _); try discriminate.
destruct (Pos.eq_dec _ _); try discriminate. clear EQ0. subst.
eapply verify_is_copy_correct_one. destruct x. eassumption.
constructor.
(* Istore *)
- destruct i'; try (inversion H; fail). monadInv H.
destruct (chunk_eq _ _); try discriminate.
destruct (eq_addressing _ _); try discriminate.
destruct (list_eq_dec _ _ _); try discriminate.
destruct (Pos.eq_dec _ _); try discriminate. clear EQ0. subst.
eapply verify_is_copy_correct_one. destruct x. eassumption.
constructor.
(* Icall *)
- destruct i'; try (inversion H; fail). monadInv H.
destruct (signature_eq _ _); try discriminate.
destruct (product_eq _ _ _ _); try discriminate.
destruct (list_eq_dec _ _ _); try discriminate.
destruct (Pos.eq_dec _ _); try discriminate.
eapply verify_is_copy_correct_one. destruct x. eassumption. subst.
constructor.
Qed.
Lemma verify_mapping_mn_correct:
forall mp n n' i f xf tc,
mp ! n' = Some n ->
(fn_code f) ! n = Some i ->
(fn_code (fn_RTL xf)) = tc ->
fn_revmap xf = mp ->
verify_mapping_mn f xf (n', n) = OK tt ->
exists i',
tc ! n' = Some i'
/\ match_inst (fun nn => mp ! nn) i i'.
Proof.
intros. unfold verify_mapping_mn in H3. rewrite H0 in H3. clear H0. rewrite H1 in H3. clear H1.
destruct (tc ! n') eqn:TCN; [| inversion H3].
exists i0. split; auto. rewrite <- H2.
eapply verify_match_inst_correct. assumption.
Qed.
Lemma verify_mapping_mn_rec_correct:
forall mp n n' i f xf tc,
mp ! n' = Some n ->
(fn_code f) ! n = Some i ->
(fn_code (fn_RTL xf)) = tc ->
fn_revmap xf = mp ->
verify_mapping_mn_rec f xf (PTree.elements mp) = OK tt ->
exists i',
tc ! n' = Some i'
/\ match_inst (fun nn => mp ! nn) i i'.
Proof.
intros. exploit PTree.elements_correct. eapply H. intros IN.
eapply verify_mapping_mn_rec_step in H3; eauto.
eapply verify_mapping_mn_correct; eauto.
Qed.
Theorem revmap_correct: forall f xf n n',
transf_function_aux f = OK xf ->
(fn_revmap xf)!n' = Some n ->
(forall i, (fn_code f)!n = Some i -> exists i', (fn_code (fn_RTL xf))!n' = Some i' /\ match_inst (fun n' => (fn_revmap xf)!n') i i').
Proof.
intros until n'. intros TRANSF REVM i FNC.
unfold transf_function_aux in TRANSF. destruct (duplicate_aux f) as (tcte & mp). destruct tcte as (tc & te). monadInv TRANSF.
simpl in *. monadInv EQ. clear EQ0.
unfold verify_mapping_match_nodes in EQ. simpl in EQ. destruct x1.
eapply verify_mapping_mn_rec_correct. 5: eapply EQ. all: eauto.
Qed.
Theorem revmap_entrypoint:
forall f xf, transf_function_aux f = OK xf -> (fn_revmap xf)!(fn_entrypoint (fn_RTL xf)) = Some (fn_entrypoint f).
Proof.
intros. unfold transf_function_aux in H. destruct (duplicate_aux _) as (tcte & mp). destruct tcte as (tc & te).
monadInv H. simpl. monadInv EQ. unfold verify_mapping_entrypoint in EQ0. simpl in EQ0.
destruct (mp ! te) eqn:PT; try discriminate.
destruct (n ?= fn_entrypoint f) eqn:EQQ; try discriminate. inv EQ0.
apply Pos.compare_eq in EQQ. congruence.
Qed.
Section PRESERVATION.
Variable prog: program.
Variable tprog: program.
Hypothesis TRANSL: match_prog prog tprog.
Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.
Lemma symbols_preserved s: Genv.find_symbol tge s = Genv.find_symbol ge s.
Proof.
rewrite <- (Genv.find_symbol_match TRANSL). reflexivity.
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:
Senv.equiv ge tge.
Proof.
eapply (Genv.senv_match TRANSL).
Qed.
Lemma functions_translated:
forall (v: val) (f: fundef),
Genv.find_funct ge v = Some f ->
exists tf cunit, transf_fundef f = OK tf /\ Genv.find_funct tge 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_translated:
forall v f,
Genv.find_funct_ptr ge v = Some f ->
exists tf,
Genv.find_funct_ptr tge v = Some tf /\ transf_fundef f = OK tf.
Proof.
intros.
exploit (Genv.find_funct_ptr_transf_partial TRANSL); eauto.
Qed.
Lemma function_sig_translated:
forall f tf, transf_fundef f = OK tf -> funsig tf = funsig f.
Proof.
intros. destruct f.
- simpl in H. monadInv H. simpl. symmetry. monadInv EQ. apply transf_function_aux_preserves. assumption.
- simpl in H. monadInv H. (* monadInv EQ. *) reflexivity.
Qed.
Lemma sig_preserved:
forall f tf,
transf_fundef f = OK tf ->
funsig tf = funsig f.
Proof.
unfold transf_fundef, transf_partial_fundef; intros.
destruct f. monadInv H. simpl. symmetry. monadInv EQ. apply transf_function_aux_preserves. assumption.
inv H. reflexivity.
Qed.
Lemma list_nth_z_revmap:
forall ln f ln' (pc pc': node) val,
list_nth_z ln val = Some pc ->
list_forall2 (fun n n' => (fn_revmap f)!n' = Some n) ln ln' ->
exists pc',
list_nth_z ln' val = Some pc'
/\ (fn_revmap f)!pc' = Some pc.
Proof.
induction ln; intros until val; intros LNZ LFA.
- inv LNZ.
- inv LNZ. destruct (zeq val 0) eqn:ZEQ.
+ inv H0. destruct ln'; inv LFA.
simpl. exists p. split; auto.
+ inv LFA. simpl. rewrite ZEQ. exploit IHln. 2: eapply H0. all: eauto.
intros (pc'1 & LNZ & REV). exists pc'1. split; auto. congruence.
Qed.
Inductive match_stackframes: stackframe -> stackframe -> Prop :=
| match_stackframe_intro:
forall res f sp pc rs xf pc'
(TRANSF: transf_function_aux f = OK xf)
(DUPLIC: (fn_revmap xf)!pc' = Some pc),
match_stackframes (Stackframe res f sp pc rs) (Stackframe res (fn_RTL xf) sp pc' rs).
Inductive match_states: state -> state -> Prop :=
| match_states_intro:
forall st f sp pc rs m st' xf pc'
(STACKS: list_forall2 match_stackframes st st')
(TRANSF: transf_function_aux f = OK xf)
(DUPLIC: (fn_revmap xf)!pc' = Some pc),
match_states (State st f sp pc rs m) (State st' (fn_RTL xf) sp pc' rs m)
| match_states_call:
forall st st' f f' args m
(STACKS: list_forall2 match_stackframes st st')
(TRANSF: transf_fundef f = OK f'),
match_states (Callstate st f args m) (Callstate st' f' args m)
| match_states_return:
forall st st' v m
(STACKS: list_forall2 match_stackframes st st'),
match_states (Returnstate st v m) (Returnstate st' v m).
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_translated; eauto. intros (tf & FIND & TRANSF).
eexists. split.
- econstructor.
+ eapply (Genv.init_mem_transf_partial TRANSL); eauto.
+ replace (prog_main tprog) with (prog_main prog). rewrite symbols_preserved. eauto.
symmetry. eapply match_program_main. eauto.
+ exploit function_ptr_translated; eauto.
+ destruct f.
* monadInv TRANSF. monadInv EQ. rewrite <- H3. symmetry; eapply transf_function_aux_preserves. assumption.
* monadInv TRANSF. (* monadInv EQ. *) assumption.
- constructor; eauto. constructor.
Qed.
Theorem transf_final_states:
forall 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.
Theorem step_simulation:
forall s1 t s1', step ge s1 t s1' ->
forall s2 (MS: match_states s1 s2),
exists s2',
step tge s2 t s2'
/\ match_states s1' s2'.
Proof.
induction 1; intros; inv MS.
(* Inop *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3).
inv H3.
eexists. split.
+ eapply exec_Inop; eauto.
+ constructor; eauto.
(* Iop *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3). inv H3.
pose symbols_preserved as SYMPRES.
eexists. split.
+ eapply exec_Iop; eauto. erewrite eval_operation_preserved; eauto.
+ constructor; eauto.
(* Iload *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3). inv H3.
pose symbols_preserved as SYMPRES.
eexists. split.
+ eapply exec_Iload; eauto. erewrite eval_addressing_preserved; eauto.
+ constructor; auto.
(* Istore *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3). inv H3.
pose symbols_preserved as SYMPRES.
eexists. split.
+ eapply exec_Istore; eauto. erewrite eval_addressing_preserved; eauto.
+ constructor; auto.
(* Icall *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3). inv H3.
pose symbols_preserved as SYMPRES.
destruct ros.
* simpl in H0. apply functions_translated in H0.
destruct H0 as (tf & cunit & TFUN & GFIND & LO).
eexists. split.
+ eapply exec_Icall. eassumption. simpl. eassumption.
apply function_sig_translated. assumption.
+ repeat (constructor; auto).
* simpl in H0. destruct (Genv.find_symbol _ _) eqn:GFS; try discriminate.
apply function_ptr_translated in H0. destruct H0 as (tf & GFF & TF).
eexists. split.
+ eapply exec_Icall. eassumption. simpl. rewrite symbols_preserved. rewrite GFS.
eassumption. apply function_sig_translated. assumption.
+ repeat (constructor; auto).
(* Itailcall *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H10 & H11). inv H11.
pose symbols_preserved as SYMPRES.
destruct ros.
* simpl in H0. apply functions_translated in H0.
destruct H0 as (tf & cunit & TFUN & GFIND & LO).
eexists. split.
+ eapply exec_Itailcall. eassumption. simpl. eassumption.
apply function_sig_translated. assumption.
erewrite <- transf_function_aux_fnstacksize; eauto.
+ repeat (constructor; auto).
* simpl in H0. destruct (Genv.find_symbol _ _) eqn:GFS; try discriminate.
apply function_ptr_translated in H0. destruct H0 as (tf & GFF & TF).
eexists. split.
+ eapply exec_Itailcall. eassumption. simpl. rewrite symbols_preserved. rewrite GFS.
eassumption. apply function_sig_translated. assumption.
erewrite <- transf_function_aux_fnstacksize; eauto.
+ repeat (constructor; auto).
(* Ibuiltin *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3). inv H3.
pose symbols_preserved as SYMPRES.
eexists. split.
+ eapply exec_Ibuiltin; eauto. eapply eval_builtin_args_preserved; eauto.
eapply external_call_symbols_preserved; eauto. eapply senv_preserved.
+ constructor; auto.
(* Icond *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3). inv H3.
pose symbols_preserved as SYMPRES.
eexists. split.
+ eapply exec_Icond; eauto.
+ constructor; auto. destruct b; auto.
(* Ijumptable *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3). inv H3.
pose symbols_preserved as SYMPRES.
exploit list_nth_z_revmap; eauto. intros (pc'1 & LNZ & REVM).
eexists. split.
+ eapply exec_Ijumptable; eauto.
+ constructor; auto.
(* Ireturn *)
- eapply revmap_correct in DUPLIC; eauto.
destruct DUPLIC as (i' & H2 & H3). inv H3.
pose symbols_preserved as SYMPRES.
eexists. split.
+ eapply exec_Ireturn; eauto. erewrite <- transf_function_aux_fnstacksize; eauto.
+ constructor; auto.
(* exec_function_internal *)
- monadInv TRANSF. monadInv EQ. eexists. split.
+ eapply exec_function_internal. erewrite <- transf_function_aux_fnstacksize; eauto.
+ erewrite transf_function_aux_fnparams; eauto.
econstructor; eauto. apply revmap_entrypoint. assumption.
(* exec_function_external *)
- monadInv TRANSF. (* monadInv EQ. *) eexists. split.
+ econstructor. eapply external_call_symbols_preserved; eauto. apply senv_preserved.
+ constructor. assumption.
(* exec_return *)
- inv STACKS. destruct b1 as [res' f' sp' pc' rs']. eexists. split.
+ constructor.
+ inv H1. constructor; assumption.
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.
- eapply transf_final_states.
- eapply step_simulation.
Qed.
End PRESERVATION.
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