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(* *********************************************************************)
(* *)
(* The Compcert verified compiler *)
(* *)
(* Xavier Leroy, INRIA Paris-Rocquencourt *)
(* *)
(* Copyright Institut National de Recherche en Informatique et en *)
(* Automatique. All rights reserved. This file is distributed *)
(* under the terms of the INRIA Non-Commercial License Agreement. *)
(* *)
(* *********************************************************************)
(** Type system for the Mach intermediate language. *)
Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Mem.
Require Import Integers.
Require Import Values.
Require Import Mem.
Require Import Events.
Require Import Globalenvs.
Require Import Op.
Require Import Locations.
Require Conventions.
Require Import Mach.
(** * Typing rules *)
Inductive wt_instr : instruction -> Prop :=
| wt_Mlabel:
forall lbl,
wt_instr (Mlabel lbl)
| wt_Mgetstack:
forall ofs ty r,
mreg_type r = ty ->
wt_instr (Mgetstack ofs ty r)
| wt_Msetstack:
forall ofs ty r,
mreg_type r = ty ->
wt_instr (Msetstack r ofs ty)
| wt_Mgetparam:
forall ofs ty r,
mreg_type r = ty ->
wt_instr (Mgetparam ofs ty r)
| wt_Mopmove:
forall r1 r,
mreg_type r1 = mreg_type r ->
wt_instr (Mop Omove (r1 :: nil) r)
| wt_Mop:
forall op args res,
op <> Omove ->
(List.map mreg_type args, mreg_type res) = type_of_operation op ->
wt_instr (Mop op args res)
| wt_Mload:
forall chunk addr args dst,
List.map mreg_type args = type_of_addressing addr ->
mreg_type dst = type_of_chunk chunk ->
wt_instr (Mload chunk addr args dst)
| wt_Mstore:
forall chunk addr args src,
List.map mreg_type args = type_of_addressing addr ->
mreg_type src = type_of_chunk chunk ->
wt_instr (Mstore chunk addr args src)
| wt_Mcall:
forall sig ros,
match ros with inl r => mreg_type r = Tint | inr s => True end ->
wt_instr (Mcall sig ros)
| wt_Mtailcall:
forall sig ros,
Conventions.tailcall_possible sig ->
match ros with inl r => mreg_type r = Tint | inr s => True end ->
wt_instr (Mtailcall sig ros)
| wt_Mgoto:
forall lbl,
wt_instr (Mgoto lbl)
| wt_Mcond:
forall cond args lbl,
List.map mreg_type args = type_of_condition cond ->
wt_instr (Mcond cond args lbl)
| wt_Mjumptable:
forall arg tbl,
mreg_type arg = Tint ->
list_length_z tbl * 4 <= Int.max_signed ->
wt_instr (Mjumptable arg tbl)
| wt_Mreturn:
wt_instr Mreturn.
Record wt_function (f: function) : Prop := mk_wt_function {
wt_function_instrs:
forall instr, In instr f.(fn_code) -> wt_instr instr;
wt_function_stacksize:
f.(fn_stacksize) >= 0;
wt_function_framesize:
0 <= f.(fn_framesize) <= -Int.min_signed;
wt_function_framesize_aligned:
(4 | f.(fn_framesize));
wt_function_link:
0 <= Int.signed f.(fn_link_ofs)
/\ Int.signed f.(fn_link_ofs) + 4 <= f.(fn_framesize);
wt_function_link_aligned:
(4 | Int.signed f.(fn_link_ofs));
wt_function_retaddr:
0 <= Int.signed f.(fn_retaddr_ofs)
/\ Int.signed f.(fn_retaddr_ofs) + 4 <= f.(fn_framesize);
wt_function_retaddr_aligned:
(4 | Int.signed f.(fn_retaddr_ofs));
wt_function_link_retaddr:
Int.signed f.(fn_retaddr_ofs) + 4 <= Int.signed f.(fn_link_ofs)
\/ Int.signed f.(fn_link_ofs) + 4 <= Int.signed f.(fn_retaddr_ofs)
}.
Inductive wt_fundef: fundef -> Prop :=
| wt_fundef_external: forall ef,
wt_fundef (External ef)
| wt_function_internal: forall f,
wt_function f ->
wt_fundef (Internal f).
Definition wt_program (p: program) : Prop :=
forall i f, In (i, f) (prog_funct p) -> wt_fundef f.
(** * Type soundness *)
Require Import Machabstr.
(** We show a weak type soundness result for the abstract semantics
of Mach: for a well-typed Mach program, if a transition is taken
from a state where registers hold values of their static types,
registers in the final state hold values of their static types
as well. This is a subject reduction theorem for our type system.
It is used in the proof of implication from the abstract Mach
semantics to the concrete Mach semantics (file [Machabstr2concr]).
*)
Definition wt_regset (rs: regset) : Prop :=
forall r, Val.has_type (rs r) (mreg_type r).
Definition wt_frame (fr: frame) : Prop :=
forall ty ofs, Val.has_type (fr ty ofs) ty.
Lemma wt_setreg:
forall (rs: regset) (r: mreg) (v: val),
Val.has_type v (mreg_type r) ->
wt_regset rs -> wt_regset (rs#r <- v).
Proof.
intros; red; intros. unfold Regmap.set.
case (RegEq.eq r0 r); intro.
subst r0; assumption.
apply H0.
Qed.
Lemma wt_get_slot:
forall f fr ty ofs v,
get_slot f fr ty ofs v ->
wt_frame fr ->
Val.has_type v ty.
Proof.
induction 1; intros.
subst v. apply H1.
Qed.
Lemma wt_set_slot:
forall f fr ty ofs v fr',
set_slot f fr ty ofs v fr' ->
wt_frame fr ->
Val.has_type v ty ->
wt_frame fr'.
Proof.
intros. induction H. subst fr'; red; intros. unfold update.
destruct (zeq (ofs - f.(fn_framesize)) ofs0).
destruct (typ_eq ty ty0). congruence. exact I.
destruct (zle (ofs0 + AST.typesize ty0) (ofs - f.(fn_framesize))).
apply H0.
destruct (zle (ofs - f.(fn_framesize) + AST.typesize ty) ofs0).
apply H0.
exact I.
Qed.
Lemma wt_empty_frame:
wt_frame empty_frame.
Proof.
intros; red; intros; exact I.
Qed.
Lemma is_tail_find_label:
forall lbl c c', find_label lbl c = Some c' -> is_tail c' c.
Proof.
induction c; simpl.
intros; discriminate.
case (is_label lbl a); intros.
injection H; intro; subst c'. constructor. constructor.
constructor; auto.
Qed.
Lemma wt_event_match:
forall ef args t res,
event_match ef args t res ->
Val.has_type res (proj_sig_res ef.(ef_sig)).
Proof.
induction 1. inversion H0; exact I.
Qed.
Section SUBJECT_REDUCTION.
Inductive wt_stackframe: stackframe -> Prop :=
| wt_stackframe_intro: forall f sp c fr,
wt_function f ->
Val.has_type sp Tint ->
is_tail c f.(fn_code) ->
wt_frame fr ->
wt_stackframe (Stackframe f sp c fr).
Inductive wt_state: state -> Prop :=
| wt_state_intro: forall stk f sp c rs fr m
(STK: forall s, In s stk -> wt_stackframe s)
(WTF: wt_function f)
(WTSP: Val.has_type sp Tint)
(TAIL: is_tail c f.(fn_code))
(WTRS: wt_regset rs)
(WTFR: wt_frame fr),
wt_state (State stk f sp c rs fr m)
| wt_state_call: forall stk f rs m,
(forall s, In s stk -> wt_stackframe s) ->
wt_fundef f ->
wt_regset rs ->
wt_state (Callstate stk f rs m)
| wt_state_return: forall stk rs m,
(forall s, In s stk -> wt_stackframe s) ->
wt_regset rs ->
wt_state (Returnstate stk rs m).
Variable p: program.
Hypothesis wt_p: wt_program p.
Let ge := Genv.globalenv p.
Lemma subject_reduction:
forall s1 t s2, step ge s1 t s2 ->
forall (WTS: wt_state s1), wt_state s2.
Proof.
induction 1; intros; inv WTS;
try (generalize (wt_function_instrs _ WTF _ (is_tail_in TAIL)); intro;
eapply wt_state_intro; eauto with coqlib).
apply wt_setreg; auto.
inversion H0. rewrite H2. eapply wt_get_slot; eauto.
inversion H0. eapply wt_set_slot; eauto.
rewrite <- H2. apply WTRS.
assert (wt_frame (parent_frame s)).
destruct s; simpl. apply wt_empty_frame.
generalize (STK s (in_eq _ _)); intro. inv H1. auto.
inversion H0. apply wt_setreg; auto.
rewrite H3. eapply wt_get_slot; eauto.
apply wt_setreg; auto. inv H0.
simpl in H.
rewrite <- H2. replace v with (rs r1). apply WTRS. congruence.
replace (mreg_type res) with (snd (type_of_operation op)).
apply type_of_operation_sound with fundef ge rs##args sp; auto.
rewrite <- H5; reflexivity.
apply wt_setreg; auto. inversion H1. rewrite H7.
eapply type_of_chunk_correct; eauto.
assert (WTFD: wt_fundef f').
destruct ros; simpl in H.
apply (Genv.find_funct_prop wt_fundef wt_p H).
destruct (Genv.find_symbol ge i); try discriminate.
apply (Genv.find_funct_ptr_prop wt_fundef wt_p H).
econstructor; eauto.
intros. elim H0; intro. subst s0. econstructor; eauto with coqlib.
auto.
assert (WTFD: wt_fundef f').
destruct ros; simpl in H.
apply (Genv.find_funct_prop wt_fundef wt_p H).
destruct (Genv.find_symbol ge i); try discriminate.
apply (Genv.find_funct_ptr_prop wt_fundef wt_p H).
econstructor; eauto.
(* apply wt_setreg; auto. exact I. *)
apply is_tail_find_label with lbl; congruence.
apply is_tail_find_label with lbl; congruence.
apply is_tail_find_label with lbl; congruence.
econstructor; eauto.
econstructor; eauto with coqlib. inv H5; auto. exact I.
apply wt_empty_frame.
econstructor; eauto. apply wt_setreg; auto.
generalize (wt_event_match _ _ _ _ H).
unfold proj_sig_res, Conventions.loc_result.
destruct (sig_res (ef_sig ef)).
destruct t0; simpl; auto.
simpl; auto.
generalize (H1 _ (in_eq _ _)); intro. inv H.
econstructor; eauto.
eauto with coqlib.
Qed.
End SUBJECT_REDUCTION.
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