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-rw-r--r--backend/Machtyping.v245
1 files changed, 2 insertions, 243 deletions
diff --git a/backend/Machtyping.v b/backend/Machtyping.v
index 93ac00c..95ceafe 100644
--- a/backend/Machtyping.v
+++ b/backend/Machtyping.v
@@ -87,7 +87,7 @@ Inductive wt_instr : instruction -> Prop :=
| wt_Mjumptable:
forall arg tbl,
mreg_type arg = Tint ->
- list_length_z tbl * 4 <= Int.max_signed ->
+ list_length_z tbl * 4 <= Int.max_unsigned ->
wt_instr (Mjumptable arg tbl)
| wt_Mreturn:
wt_instr Mreturn.
@@ -96,24 +96,7 @@ 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)
+ 0 <= f.(fn_stacksize) <= Int.max_unsigned
}.
Inductive wt_fundef: fundef -> Prop :=
@@ -125,227 +108,3 @@ Inductive wt_fundef: fundef -> Prop :=
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_undef_temps:
- forall rs, wt_regset rs -> wt_regset (undef_temps rs).
-Proof.
- unfold undef_temps.
- generalize (int_temporaries ++ float_temporaries).
- induction l; simpl; intros. auto.
- apply IHl. red; intros. unfold Regmap.set.
- destruct (RegEq.eq r a). constructor. auto.
-Qed.
-
-Lemma wt_undef_op:
- forall op rs, wt_regset rs -> wt_regset (undef_op op rs).
-Proof.
- intros. set (W := wt_undef_temps rs H).
- destruct op; simpl; auto.
-Qed.
-
-Lemma wt_undef_getparam:
- forall rs, wt_regset rs -> wt_regset (rs#IT1 <- Vundef).
-Proof.
- intros; red; intros. unfold Regmap.set.
- destruct (RegEq.eq r IT1). constructor. auto.
-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.
-
-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 WTI;
- eapply wt_state_intro; eauto with coqlib).
-
- apply wt_setreg; auto. inv WTI. eapply wt_get_slot; eauto.
-
- eapply wt_set_slot; eauto. inv WTI; auto.
-
- assert (mreg_type dst = ty).
- inv WTI; auto.
- assert (wt_frame (parent_frame s)).
- destruct s; simpl. apply wt_empty_frame.
- generalize (STK s (in_eq _ _)); intro. inv H1. auto.
- apply wt_setreg; auto.
- rewrite H0. eapply wt_get_slot; eauto.
- apply wt_undef_getparam; auto.
-
-(* op *)
- apply wt_setreg; auto.
- inv WTI.
- (* move *)
- simpl in H. inv H. rewrite <- H1. apply WTRS.
- (* not move *)
- replace (mreg_type res) with (snd (type_of_operation op)).
- apply type_of_operation_sound with fundef unit ge rs##args sp; auto.
- rewrite <- H4; reflexivity.
- apply wt_undef_op; auto.
-
-(* load *)
- apply wt_setreg; auto. inv WTI. rewrite H6. eapply type_of_chunk_correct; eauto.
- apply wt_undef_temps; auto.
-
-(* store *)
- apply wt_undef_temps; auto.
-
-(* call *)
- 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.
-
-(* tailcall *)
- 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.
-
-(* extcall *)
- apply wt_setreg; auto.
- inv WTI. rewrite H4. eapply external_call_well_typed; eauto.
- apply wt_undef_temps; auto.
-
-(* goto *)
- apply is_tail_find_label with lbl; congruence.
-(* cond *)
- apply is_tail_find_label with lbl; congruence. apply wt_undef_temps; auto.
- apply wt_undef_temps; auto.
-(* jumptable *)
- apply is_tail_find_label with lbl; congruence. apply wt_undef_temps; auto.
-
-(* return *)
- econstructor; eauto.
-
-(* internal function *)
- econstructor; eauto with coqlib. inv H5; auto. exact I.
- apply wt_empty_frame.
-
-(* external function *)
- econstructor; eauto. apply wt_setreg; auto.
- generalize (external_call_well_typed _ _ _ _ _ _ _ H).
- unfold proj_sig_res, loc_result.
- destruct (sig_res (ef_sig ef)).
- destruct t0; simpl; auto.
- simpl; auto.
-
-(* returnstate *)
- generalize (H1 _ (in_eq _ _)); intro. inv H.
- econstructor; eauto.
- eauto with coqlib.
-Qed.
-
-End SUBJECT_REDUCTION.
-
generated by cgit v1.2.3 (git 2.39.1) at 2024-06-28 14:31:53 +0000