From 4af1682d04244bab9f793e00eb24090153a36a0f Mon Sep 17 00:00:00 2001 From: xleroy Date: Thu, 28 Jul 2011 12:51:16 +0000 Subject: Added animation of the CompCert C semantics (ccomp -interp) test/regression: int main() so that interpretation works Revised once more implementation of __builtin_memcpy (to check for PPC & ARM) git-svn-id: https://yquem.inria.fr/compcert/svn/compcert/trunk@1688 fca1b0fc-160b-0410-b1d3-a4f43f01ea2e --- cfrontend/Cexec.v | 1814 +++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 1814 insertions(+) create mode 100644 cfrontend/Cexec.v (limited to 'cfrontend/Cexec.v') diff --git a/cfrontend/Cexec.v b/cfrontend/Cexec.v new file mode 100644 index 0000000..3dd34c1 --- /dev/null +++ b/cfrontend/Cexec.v @@ -0,0 +1,1814 @@ +Require Import Axioms. +Require Import Coqlib. +Require Import Errors. +Require Import Maps. +Require Import Integers. +Require Import Floats. +Require Import Values. +Require Import AST. +Require Import Memory. +Require Import Events. +Require Import Globalenvs. +Require Import Determinism. +Require Import Csyntax. +Require Import Csem. +Require Cstrategy. + +(** Animating the CompCert C semantics *) + +Lemma type_eq: forall (ty1 ty2: type), {ty1=ty2} + {ty1<>ty2} +with typelist_eq: forall (tyl1 tyl2: typelist), {tyl1=tyl2} + {tyl1<>tyl2} +with fieldlist_eq: forall (fld1 fld2: fieldlist), {fld1=fld2} + {fld1<>fld2}. +Proof. + assert (forall (x y: intsize), {x=y} + {x<>y}). decide equality. + assert (forall (x y: signedness), {x=y} + {x<>y}). decide equality. + assert (forall (x y: floatsize), {x=y} + {x<>y}). decide equality. + generalize ident_eq zeq. intros E1 E2. + decide equality. + decide equality. + generalize ident_eq. intros E1. + decide equality. +Defined. + +Opaque type_eq. + +(** Error monad with options or lists *) + +Notation "'do' X <- A ; B" := (match A with Some X => B | None => None end) + (at level 200, X ident, A at level 100, B at level 200) + : option_monad_scope. + +Notation " 'check' A ; B" := (if A then B else None) + (at level 200, A at level 100, B at level 200) + : option_monad_scope. + +Notation "'do' X <- A ; B" := (match A with Some X => B | None => nil end) + (at level 200, X ident, A at level 100, B at level 200) + : list_monad_scope. + +Notation " 'check' A ; B" := (if A then B else nil) + (at level 200, A at level 100, B at level 200) + : list_monad_scope. + +Definition is_val (a: expr) : option (val * type) := + match a with + | Eval v ty => Some(v, ty) + | _ => None + end. + +Lemma is_val_inv: + forall a v ty, is_val a = Some(v, ty) -> a = Eval v ty. +Proof. + intros until ty. destruct a; simpl; congruence. +Qed. + +Definition is_loc (a: expr) : option (block * int * type) := + match a with + | Eloc b ofs ty => Some(b, ofs, ty) + | _ => None + end. + +Lemma is_loc_inv: + forall a b ofs ty, is_loc a = Some(b, ofs, ty) -> a = Eloc b ofs ty. +Proof. + intros until ty. destruct a; simpl; congruence. +Qed. + +Local Open Scope option_monad_scope. + +Fixpoint is_val_list (al: exprlist) : option (list (val * type)) := + match al with + | Enil => Some nil + | Econs a1 al => do vt1 <- is_val a1; do vtl <- is_val_list al; Some(vt1::vtl) + end. + +Definition is_skip (s: statement) : {s = Sskip} + {s <> Sskip}. +Proof. + destruct s; (left; congruence) || (right; congruence). +Qed. + +(** * Reduction of expressions *) + +Section EXEC. + +Variable ge: genv. + +Inductive reduction: Type := + | Lred (l': expr) (m': mem) + | Rred (r': expr) (m': mem) + | Callred (fd: fundef) (args: list val) (tyres: type) (m': mem). + +Section EXPRS. + +Variable e: env. + +Fixpoint sem_cast_arguments (vtl: list (val * type)) (tl: typelist) : option (list val) := + match vtl, tl with + | nil, Tnil => Some nil + | (v1,t1)::vtl, Tcons t1' tl => + do v <- sem_cast v1 t1 t1'; do vl <- sem_cast_arguments vtl tl; Some(v::vl) + | _, _ => None + end. + +(** The result of stepping an expression can be +- [None] denoting that the expression is stuck; +- [Some nil] meaning that the expression is fully reduced + (it's [Eval] for a r-value and [Eloc] for a l-value); +- [Some ll] meaning that the expression can reduce to any of + the elements of [ll]. Each element is a pair of a context + and a reduction inside this context (see type [reduction] above). +*) + +Definition reducts (A: Type): Type := option (list ((expr -> A) * reduction)). + +Definition topred (r: reduction) : reducts expr := + Some (((fun (x: expr) => x), r) :: nil). + +Definition incontext {A B: Type} (ctx: A -> B) (r: reducts A) : reducts B := + match r with + | None => None + | Some l => Some (map (fun z => ((fun (x: expr) => ctx(fst z x)), snd z)) l) + end. + +Definition incontext2 {A1 A2 B: Type} + (ctx1: A1 -> B) (r1: reducts A1) + (ctx2: A2 -> B) (r2: reducts A2) : reducts B := + match r1, r2 with + | None, _ => None + | _, None => None + | Some l1, Some l2 => + Some (map (fun z => ((fun (x: expr) => ctx1(fst z x)), snd z)) l1 + ++ map (fun z => ((fun (x: expr) => ctx2(fst z x)), snd z)) l2) + end. + +Fixpoint step_expr (k: kind) (a: expr) (m: mem): reducts expr := + match k, a with + | LV, Eloc b ofs ty => + Some nil + | LV, Evar x ty => + match e!x with + | Some(b, ty') => + check type_eq ty ty'; + topred (Lred (Eloc b Int.zero ty) m) + | None => + do b <- Genv.find_symbol ge x; + do ty' <- type_of_global ge b; + check type_eq ty ty'; + topred (Lred (Eloc b Int.zero ty) m) + end + | LV, Ederef r ty => + match is_val r with + | Some(Vptr b ofs, ty') => + topred (Lred (Eloc b ofs ty) m) + | Some _ => + None + | None => + incontext (fun x => Ederef x ty) (step_expr RV r m) + end + | LV, Efield l f ty => + match is_loc l with + | Some(b, ofs, ty') => + match ty' with + | Tstruct id fList => + match field_offset f fList with + | Error _ => None + | OK delta => topred (Lred (Eloc b (Int.add ofs (Int.repr delta)) ty) m) + end + | Tunion id fList => + topred (Lred (Eloc b ofs ty) m) + | _ => None + end + | None => + incontext (fun x => Efield x f ty) (step_expr LV l m) + end + | RV, Eval v ty => + Some nil + | RV, Evalof l ty => + match is_loc l with + | Some(b, ofs, ty') => + check type_eq ty ty'; + do v <- load_value_of_type ty m b ofs; + topred (Rred (Eval v ty) m) + | None => + incontext (fun x => Evalof x ty) (step_expr LV l m) + end + | RV, Eaddrof l ty => + match is_loc l with + | Some(b, ofs, ty') => topred (Rred (Eval (Vptr b ofs) ty) m) + | None => incontext (fun x => Eaddrof x ty) (step_expr LV l m) + end + | RV, Eunop op r1 ty => + match is_val r1 with + | Some(v1, ty1) => + do v <- sem_unary_operation op v1 ty1; + topred (Rred (Eval v ty) m) + | None => + incontext (fun x => Eunop op x ty) (step_expr RV r1 m) + end + | RV, Ebinop op r1 r2 ty => + match is_val r1, is_val r2 with + | Some(v1, ty1), Some(v2, ty2) => + do v <- sem_binary_operation op v1 ty1 v2 ty2 m; + topred (Rred (Eval v ty) m) + | _, _ => + incontext2 (fun x => Ebinop op x r2 ty) (step_expr RV r1 m) + (fun x => Ebinop op r1 x ty) (step_expr RV r2 m) + end + | RV, Ecast r1 ty => + match is_val r1 with + | Some(v1, ty1) => + do v <- sem_cast v1 ty1 ty; + topred (Rred (Eval v ty) m) + | None => + incontext (fun x => Ecast x ty) (step_expr RV r1 m) + end + | RV, Econdition r1 r2 r3 ty => + match is_val r1 with + | Some(v1, ty1) => + do b <- bool_val v1 ty1; + topred (Rred (Eparen (if b then r2 else r3) ty) m) + | None => + incontext (fun x => Econdition x r2 r3 ty) (step_expr RV r1 m) + end + | RV, Esizeof ty' ty => + topred (Rred (Eval (Vint (Int.repr (sizeof ty'))) ty) m) + | RV, Eassign l1 r2 ty => + match is_loc l1, is_val r2 with + | Some(b, ofs, ty1), Some(v2, ty2) => + check type_eq ty1 ty; + do v <- sem_cast v2 ty2 ty1; + do m' <- store_value_of_type ty1 m b ofs v; + topred (Rred (Eval v ty) m') + | _, _ => + incontext2 (fun x => Eassign x r2 ty) (step_expr LV l1 m) + (fun x => Eassign l1 x ty) (step_expr RV r2 m) + end + | RV, Eassignop op l1 r2 tyres ty => + match is_loc l1, is_val r2 with + | Some(b, ofs, ty1), Some(v2, ty2) => + check type_eq ty1 ty; + do v1 <- load_value_of_type ty1 m b ofs; + do v <- sem_binary_operation op v1 ty1 v2 ty2 m; + do v' <- sem_cast v tyres ty1; + do m' <- store_value_of_type ty1 m b ofs v'; + topred (Rred (Eval v' ty) m') + | _, _ => + incontext2 (fun x => Eassignop op x r2 tyres ty) (step_expr LV l1 m) + (fun x => Eassignop op l1 x tyres ty) (step_expr RV r2 m) + end + | RV, Epostincr id l ty => + match is_loc l with + | Some(b, ofs, ty1) => + check type_eq ty1 ty; + do v1 <- load_value_of_type ty m b ofs; + do v2 <- sem_incrdecr id v1 ty; + do v3 <- sem_cast v2 (typeconv ty) ty; + do m' <- store_value_of_type ty m b ofs v3; + topred (Rred (Eval v1 ty) m') + | None => + incontext (fun x => Epostincr id x ty) (step_expr LV l m) + end + | RV, Ecomma r1 r2 ty => + match is_val r1 with + | Some _ => + check type_eq (typeof r2) ty; + topred (Rred r2 m) + | None => + incontext (fun x => Ecomma x r2 ty) (step_expr RV r1 m) + end + | RV, Eparen r1 ty => + match is_val r1 with + | Some (v1, ty1) => + do v <- sem_cast v1 ty1 ty; + topred (Rred (Eval v ty) m) + | None => + incontext (fun x => Eparen x ty) (step_expr RV r1 m) + end + | RV, Ecall r1 rargs ty => + match is_val r1, is_val_list rargs with + | Some(vf, tyf), Some vtl => + match classify_fun tyf with + | fun_case_f tyargs tyres => + do fd <- Genv.find_funct ge vf; + do vargs <- sem_cast_arguments vtl tyargs; + check type_eq (type_of_fundef fd) (Tfunction tyargs tyres); + topred (Callred fd vargs ty m) + | _ => None + end + | _, _ => + incontext2 (fun x => Ecall x rargs ty) (step_expr RV r1 m) + (fun x => Ecall r1 x ty) (step_exprlist rargs m) + end + | _, _ => None + end + +with step_exprlist (rl: exprlist) (m: mem): reducts exprlist := + match rl with + | Enil => + Some nil + | Econs r1 rs => + incontext2 (fun x => Econs x rs) (step_expr RV r1 m) + (fun x => Econs r1 x) (step_exprlist rs m) + end. + +(** Soundness: if [step_expr] returns [Some ll], then every element + of [ll] is a reduct. *) + +Lemma context_compose: + forall k2 k3 C2, context k2 k3 C2 -> + forall k1 C1, context k1 k2 C1 -> + context k1 k3 (fun x => C2(C1 x)) +with contextlist_compose: + forall k2 C2, contextlist k2 C2 -> + forall k1 C1, context k1 k2 C1 -> + contextlist k1 (fun x => C2(C1 x)). +Proof. + induction 1; intros; try (constructor; eauto). + replace (fun x => C1 x) with C1. auto. apply extensionality; auto. + induction 1; intros; constructor; eauto. +Qed. + +Hint Constructors context contextlist. +Hint Resolve context_compose contextlist_compose. + +Definition reduction_ok (a: expr) (m: mem) (rd: reduction) : Prop := + match rd with + | Lred l' m' => lred ge e a m l' m' + | Rred r' m' => rred a m r' m' + | Callred fd args tyres m' => callred ge a fd args tyres /\ m' = m + end. + +Definition reduction_kind (rd: reduction): kind := + match rd with + | Lred l' m' => LV + | Rred r' m' => RV + | Callred fd args tyres m' => RV + end. + +Ltac monadInv := + match goal with + | [ H: match ?x with Some _ => _ | None => None end = Some ?y |- _ ] => + destruct x as []_eqn; [monadInv|discriminate] + | [ H: match ?x with left _ => _ | right _ => None end = Some ?y |- _ ] => + destruct x; [monadInv|discriminate] + | _ => idtac + end. + +Lemma sem_cast_arguments_sound: + forall rargs vtl tyargs vargs, + is_val_list rargs = Some vtl -> + sem_cast_arguments vtl tyargs = Some vargs -> + cast_arguments rargs tyargs vargs. +Proof. + induction rargs; simpl; intros. + inv H. destruct tyargs; simpl in H0; inv H0. constructor. + monadInv. inv H. simpl in H0. destruct p as [v1 t1]. destruct tyargs; try congruence. monadInv. + inv H0. rewrite (is_val_inv _ _ _ Heqo). constructor. auto. eauto. +Qed. + +Definition reducts_ok (k: kind) (a: expr) (m: mem) (res: reducts expr) : Prop := + match res with + | None => True + | Some nil => match k with LV => is_loc a <> None | RV => is_val a <> None end + | Some ll => + forall C rd, + In (C, rd) ll -> + context (reduction_kind rd) k C /\ exists a', a = C a' /\ reduction_ok a' m rd + end. + +Definition list_reducts_ok (al: exprlist) (m: mem) (res: reducts exprlist) : Prop := + match res with + | None => True + | Some nil => is_val_list al <> None + | Some ll => + forall C rd, + In (C, rd) ll -> + contextlist (reduction_kind rd) C /\ exists a', al = C a' /\ reduction_ok a' m rd + end. + +Lemma topred_ok: + forall k a m rd, + reduction_ok a m rd -> + k = reduction_kind rd -> + reducts_ok k a m (topred rd). +Proof. + intros. unfold topred; red. simpl; intros. destruct H1; try contradiction. + inv H1. split. auto. exists a; auto. +Qed. + +Lemma incontext_ok: + forall k a m C res k' a', + reducts_ok k' a' m res -> + a = C a' -> + context k' k C -> + match k' with LV => is_loc a' = None | RV => is_val a' = None end -> + reducts_ok k a m (incontext C res). +Proof. + unfold reducts_ok; intros. destruct res; simpl. destruct l. +(* res = Some nil *) + destruct k'; congruence. +(* res = Some nonempty-list *) + simpl map at 1. hnf. intros. + exploit list_in_map_inv; eauto. intros [[C1 rd1] [P Q]]. inv P. + exploit H; eauto. intros [U [a'' [V W]]]. + split. eapply context_compose; eauto. exists a''; split; auto. congruence. +(* res = None *) + auto. +Qed. + +Remark incontext2_inv: + forall {A1 A2 B: Type} (C1: A1 -> B) res1 (C2: A2 -> B) res2, + match incontext2 C1 res1 C2 res2 with + | None => res1 = None \/ res2 = None + | Some nil => res1 = Some nil /\ res2 = Some nil + | Some ll => + exists ll1, exists ll2, + res1 = Some ll1 /\ res2 = Some ll2 /\ + forall C rd, In (C, rd) ll -> + (exists C', C = (fun x => C1(C' x)) /\ In (C', rd) ll1) + \/ (exists C', C = (fun x => C2(C' x)) /\ In (C', rd) ll2) + end. +Proof. + intros. unfold incontext2. destruct res1 as [ll1|]; auto. destruct res2 as [ll2|]; auto. + set (ll := map + (fun z : (expr -> A1) * reduction => + (fun x : expr => C1 (fst z x), snd z)) ll1 ++ + map + (fun z : (expr -> A2) * reduction => + (fun x : expr => C2 (fst z x), snd z)) ll2). + destruct ll as []_eqn. + destruct (app_eq_nil _ _ Heql). + split. destruct ll1; auto || discriminate. destruct ll2; auto || discriminate. + rewrite <- Heql. exists ll1; exists ll2. split. auto. split. auto. + unfold ll; intros. + rewrite in_app in H. destruct H. + exploit list_in_map_inv; eauto. intros [[C' rd'] [P Q]]. inv P. + left; exists C'; auto. + exploit list_in_map_inv; eauto. intros [[C' rd'] [P Q]]. inv P. + right; exists C'; auto. +Qed. + +Lemma incontext2_ok: + forall k a m k1 a1 res1 k2 a2 res2 C1 C2, + reducts_ok k1 a1 m res1 -> + reducts_ok k2 a2 m res2 -> + a = C1 a1 -> a = C2 a2 -> + context k1 k C1 -> context k2 k C2 -> + match k1 with LV => is_loc a1 = None | RV => is_val a1 = None end + \/ match k2 with LV => is_loc a2 = None | RV => is_val a2 = None end -> + reducts_ok k a m (incontext2 C1 res1 C2 res2). +Proof. + unfold reducts_ok; intros. + generalize (incontext2_inv C1 res1 C2 res2). + destruct (incontext2 C1 res1 C2 res2) as [ll|]; auto. + destruct ll. + intros [EQ1 EQ2]. subst. destruct H5. destruct k1; congruence. destruct k2; congruence. + intros [ll1 [ll2 [EQ1 [EQ2 IN]]]]. subst. intros. + exploit IN; eauto. intros [[C' [P Q]] | [C' [P Q]]]; subst. + destruct ll1; try contradiction. exploit H; eauto. + intros [U [a' [V W]]]. split. eauto. exists a'; split. congruence. auto. + destruct ll2; try contradiction. exploit H0; eauto. + intros [U [a' [V W]]]. split. eauto. exists a'; split. congruence. auto. +Qed. + +Lemma incontext2_list_ok: + forall a1 a2 ty m res1 res2, + reducts_ok RV a1 m res1 -> + list_reducts_ok a2 m res2 -> + is_val a1 = None \/ is_val_list a2 = None -> + reducts_ok RV (Ecall a1 a2 ty) m + (incontext2 (fun x => Ecall x a2 ty) res1 + (fun x => Ecall a1 x ty) res2). +Proof. + unfold reducts_ok, list_reducts_ok; intros. + set (C1 := fun x => Ecall x a2 ty). set (C2 := fun x => Ecall a1 x ty). + generalize (incontext2_inv C1 res1 C2 res2). + destruct (incontext2 C1 res1 C2 res2) as [ll|]; auto. + destruct ll. + intros [EQ1 EQ2]. subst. intuition congruence. + intros [ll1 [ll2 [EQ1 [EQ2 IN]]]]. subst. intros. + exploit IN; eauto. intros [[C' [P Q]] | [C' [P Q]]]; subst. + destruct ll1; try contradiction. exploit H; eauto. + intros [U [a' [V W]]]. split. unfold C1. auto. exists a'; split. unfold C1; congruence. auto. + destruct ll2; try contradiction. exploit H0; eauto. + intros [U [a' [V W]]]. split. unfold C2. auto. exists a'; split. unfold C2; congruence. auto. +Qed. + +Lemma incontext2_list_ok': + forall a1 a2 m res1 res2, + reducts_ok RV a1 m res1 -> + list_reducts_ok a2 m res2 -> + list_reducts_ok (Econs a1 a2) m + (incontext2 (fun x => Econs x a2) res1 + (fun x => Econs a1 x) res2). +Proof. + unfold reducts_ok, list_reducts_ok; intros. + set (C1 := fun x => Econs x a2). set (C2 := fun x => Econs a1 x). + generalize (incontext2_inv C1 res1 C2 res2). + destruct (incontext2 C1 res1 C2 res2) as [ll|]; auto. + destruct ll. + intros [EQ1 EQ2]. subst. + simpl. destruct (is_val a1); try congruence. destruct (is_val_list a2); congruence. + intros [ll1 [ll2 [EQ1 [EQ2 IN]]]]. subst. intros. + exploit IN; eauto. intros [[C' [P Q]] | [C' [P Q]]]; subst. + destruct ll1; try contradiction. exploit H; eauto. + intros [U [a' [V W]]]. split. unfold C1. auto. exists a'; split. unfold C1; congruence. auto. + destruct ll2; try contradiction. exploit H0; eauto. + intros [U [a' [V W]]]. split. unfold C2. auto. exists a'; split. unfold C2; congruence. auto. +Qed. + +Ltac mysimpl := + match goal with + | [ |- reducts_ok _ _ _ (match ?x with Some _ => _ | None => None end) ] => + destruct x as []_eqn; [mysimpl|exact I] + | [ |- reducts_ok _ _ _ (match ?x with left _ => _ | right _ => None end) ] => + destruct x as []_eqn; [subst;mysimpl|exact I] + | _ => + idtac + end. + +Theorem step_expr_sound: + forall a k m, reducts_ok k a m (step_expr k a m) +with step_exprlist_sound: + forall al m, list_reducts_ok al m (step_exprlist al m). +Proof with try (exact I). + induction a; destruct k; intros; simpl... +(* Eval *) + congruence. +(* Evar *) + destruct (e!x) as [[b ty'] | ]_eqn; mysimpl. + apply topred_ok; auto. apply red_var_local; auto. + apply topred_ok; auto. apply red_var_global; auto. +(* Efield *) + destruct (is_loc a) as [[[b ofs] ty'] | ]_eqn. + destruct ty'... + (* top struct *) + destruct (field_offset f f0) as [delta|]_eqn... + rewrite (is_loc_inv _ _ _ _ Heqo). apply topred_ok; auto. apply red_field_struct; auto. + (* top union *) + rewrite (is_loc_inv _ _ _ _ Heqo). apply topred_ok; auto. apply red_field_union; auto. + (* in depth *) + eapply incontext_ok; eauto. +(* Evalof *) + destruct (is_loc a) as [[[b ofs] ty'] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. rewrite (is_loc_inv _ _ _ _ Heqo). apply red_rvalof; auto. + (* depth *) + eapply incontext_ok; eauto. +(* Ederef *) + destruct (is_val a) as [[v ty'] | ]_eqn. + (* top *) + destruct v... mysimpl. apply topred_ok; auto. rewrite (is_val_inv _ _ _ Heqo). apply red_deref; auto. + (* depth *) + eapply incontext_ok; eauto. +(* Eaddrof *) + destruct (is_loc a) as [[[b ofs] ty'] | ]_eqn. + (* top *) + apply topred_ok; auto. rewrite (is_loc_inv _ _ _ _ Heqo). apply red_addrof; auto. + (* depth *) + eapply incontext_ok; eauto. +(* unop *) + destruct (is_val a) as [[v ty'] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. rewrite (is_val_inv _ _ _ Heqo). apply red_unop; auto. + (* depth *) + eapply incontext_ok; eauto. +(* binop *) + destruct (is_val a1) as [[v1 ty1] | ]_eqn. + destruct (is_val a2) as [[v2 ty2] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. + rewrite (is_val_inv _ _ _ Heqo). rewrite (is_val_inv _ _ _ Heqo0). apply red_binop; auto. + (* depth *) + eapply incontext2_ok; eauto. + eapply incontext2_ok; eauto. +(* cast *) + destruct (is_val a) as [[v ty'] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. + rewrite (is_val_inv _ _ _ Heqo). apply red_cast; auto. + (* depth *) + eapply incontext_ok; eauto. +(* condition *) + destruct (is_val a1) as [[v ty'] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. + rewrite (is_val_inv _ _ _ Heqo). eapply red_condition; eauto. + (* depth *) + eapply incontext_ok; eauto. +(* sizeof *) + apply topred_ok; auto. apply red_sizeof. +(* assign *) + destruct (is_loc a1) as [[[b ofs] ty1] | ]_eqn. + destruct (is_val a2) as [[v2 ty2] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. + rewrite (is_loc_inv _ _ _ _ Heqo). rewrite (is_val_inv _ _ _ Heqo0). apply red_assign; auto. + (* depth *) + eapply incontext2_ok; eauto. + eapply incontext2_ok; eauto. +(* assignop *) + destruct (is_loc a1) as [[[b ofs] ty1] | ]_eqn. + destruct (is_val a2) as [[v2 ty2] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. + rewrite (is_loc_inv _ _ _ _ Heqo). rewrite (is_val_inv _ _ _ Heqo0). eapply red_assignop; eauto. + (* depth *) + eapply incontext2_ok; eauto. + eapply incontext2_ok; eauto. +(* postincr *) + destruct (is_loc a) as [[[b ofs] ty'] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. + rewrite (is_loc_inv _ _ _ _ Heqo). eapply red_postincr; eauto. + (* depth *) + eapply incontext_ok; eauto. +(* comma *) + destruct (is_val a1) as [[v ty'] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. + rewrite (is_val_inv _ _ _ Heqo). apply red_comma; auto. + (* depth *) + eapply incontext_ok; eauto. +(* call *) + destruct (is_val a) as [[vf tyf] | ]_eqn. + destruct (is_val_list rargs) as [vtl | ]_eqn. + (* top *) + destruct (classify_fun tyf) as [tyargs tyres|]_eqn... + mysimpl. apply topred_ok; auto. + rewrite (is_val_inv _ _ _ Heqo). red. split; auto. eapply red_Ecall; eauto. + eapply sem_cast_arguments_sound; eauto. + (* depth *) + eapply incontext2_list_ok; eauto. + eapply incontext2_list_ok; eauto. +(* loc *) + congruence. +(* paren *) + destruct (is_val a) as [[v ty'] | ]_eqn. + (* top *) + mysimpl. apply topred_ok; auto. + rewrite (is_val_inv _ _ _ Heqo). apply red_paren; auto. + (* depth *) + eapply incontext_ok; eauto. + + induction al; simpl; intros. +(* nil *) + congruence. +(* cons *) + eapply incontext2_list_ok'; eauto. +Qed. + + +Lemma step_exprlist_val_list: + forall m al, is_val_list al <> None -> step_exprlist al m = Some nil. +Proof. + induction al; simpl; intros. + auto. + destruct (is_val r1) as [[v1 ty1]|]_eqn; try congruence. + destruct (is_val_list al) as []_eqn; try congruence. + rewrite (is_val_inv _ _ _ Heqo). + rewrite IHal. auto. congruence. +Qed. + +(** Completeness, part 1: if [step_expr] returns [Some ll], + then [ll] contains all possible reducts. *) + +Lemma lred_topred: + forall l1 m1 l2 m2, + lred ge e l1 m1 l2 m2 -> + step_expr LV l1 m1 = topred (Lred l2 m2). +Proof. + induction 1; simpl. +(* var local *) + rewrite H. rewrite dec_eq_true; auto. +(* var global *) + rewrite H; rewrite H0; rewrite H1. rewrite dec_eq_true; auto. +(* deref *) + auto. +(* field struct *) + rewrite H; auto. +(* field union *) + auto. +Qed. + +Lemma rred_topred: + forall r1 m1 r2 m2, + rred r1 m1 r2 m2 -> + step_expr RV r1 m1 = topred (Rred r2 m2). +Proof. + induction 1; simpl. +(* valof *) + rewrite dec_eq_true; auto. rewrite H; auto. +(* addrof *) + auto. +(* unop *) + rewrite H; auto. +(* binop *) + rewrite H; auto. +(* cast *) + rewrite H; auto. +(* condition *) + rewrite H; auto. +(* sizeof *) + auto. +(* assign *) + rewrite dec_eq_true; auto. rewrite H; rewrite H0; auto. +(* assignop *) + rewrite dec_eq_true; auto. rewrite H; rewrite H0; rewrite H1; rewrite H2; auto. +(* postincr *) + rewrite dec_eq_true; auto. rewrite H; rewrite H0; rewrite H1; rewrite H2; auto. +(* comma *) + rewrite H; rewrite dec_eq_true; auto. +(* paren *) + rewrite H; auto. +Qed. + +Lemma sem_cast_arguments_complete: + forall al tyl vl, + cast_arguments al tyl vl -> + exists vtl, is_val_list al = Some vtl /\ sem_cast_arguments vtl tyl = Some vl. +Proof. + induction 1. + exists (@nil (val * type)); auto. + destruct IHcast_arguments as [vtl [A B]]. + exists ((v, ty) :: vtl); simpl. rewrite A; rewrite B; rewrite H. auto. +Qed. + +Lemma callred_topred: + forall a fd args ty m, + callred ge a fd args ty -> + step_expr RV a m = topred (Callred fd args ty m). +Proof. + induction 1; simpl. + rewrite H2. exploit sem_cast_arguments_complete; eauto. intros [vtl [A B]]. + rewrite A; rewrite H; rewrite B; rewrite H1; rewrite dec_eq_true. auto. +Qed. + +Definition reducts_incl {A B: Type} (C: A -> B) (res1: reducts A) (res2: reducts B) : Prop := + match res1, res2 with + | Some ll1, Some ll2 => + forall C1 rd, In (C1, rd) ll1 -> In ((fun x => C(C1 x)), rd) ll2 + | None, Some ll2 => False + | _, None => True + end. + +Lemma reducts_incl_trans: + forall (A1 A2: Type) (C: A1 -> A2) res1 res2, reducts_incl C res1 res2 -> + forall (A3: Type) (C': A2 -> A3) res3, + reducts_incl C' res2 res3 -> + reducts_incl (fun x => C'(C x)) res1 res3. +Proof. + unfold reducts_incl; intros. destruct res1; destruct res2; destruct res3; auto. contradiction. +Qed. + +Lemma reducts_incl_nil: + forall (A B: Type) (C: A -> B) res, + reducts_incl C (Some nil) res. +Proof. + intros; red. destruct res; auto. intros; contradiction. +Qed. + +Lemma reducts_incl_val: + forall (A: Type) a m v ty (C: expr -> A) res, + is_val a = Some(v, ty) -> reducts_incl C (step_expr RV a m) res. +Proof. + intros. rewrite (is_val_inv _ _ _ H). apply reducts_incl_nil. +Qed. + +Lemma reducts_incl_loc: + forall (A: Type) a m b ofs ty (C: expr -> A) res, + is_loc a = Some(b, ofs, ty) -> reducts_incl C (step_expr LV a m) res. +Proof. + intros. rewrite (is_loc_inv _ _ _ _ H). apply reducts_incl_nil. +Qed. + +Lemma reducts_incl_listval: + forall (A: Type) a m vtl (C: exprlist -> A) res, + is_val_list a = Some vtl -> reducts_incl C (step_exprlist a m) res. +Proof. + intros. rewrite step_exprlist_val_list. apply reducts_incl_nil. congruence. +Qed. + +Lemma reducts_incl_incontext: + forall (A B: Type) (C: A -> B) res, + reducts_incl C res (incontext C res). +Proof. + intros; unfold reducts_incl. destruct res; simpl; auto. + intros. set (f := fun z : (expr -> A) * reduction => (fun x : expr => C (fst z x), snd z)). + change (In (f (C1, rd)) (map f l)). apply in_map. auto. +Qed. + +Lemma reducts_incl_incontext2_left: + forall (A1 A2 B: Type) (C1: A1 -> B) res1 (C2: A2 -> B) res2, + reducts_incl C1 res1 (incontext2 C1 res1 C2 res2). +Proof. + intros. destruct res1; simpl; auto. destruct res2; auto. + intros. rewrite in_app_iff. left. + set (f := fun z : (expr -> A1) * reduction => (fun x : expr => C1 (fst z x), snd z)). + change (In (f (C0, rd)) (map f l)). apply in_map; auto. +Qed. + +Lemma reducts_incl_incontext2_right: + forall (A1 A2 B: Type) (C1: A1 -> B) res1 (C2: A2 -> B) res2, + reducts_incl C2 res2 (incontext2 C1 res1 C2 res2). +Proof. + intros. destruct res1; destruct res2; simpl; auto. + intros. rewrite in_app_iff. right. + set (f := fun z : (expr -> A2) * reduction => (fun x : expr => C2 (fst z x), snd z)). + change (In (f (C0, rd)) (map f l0)). apply in_map; auto. +Qed. + +Hint Resolve reducts_incl_val reducts_incl_loc reducts_incl_listval + reducts_incl_incontext reducts_incl_incontext2_left reducts_incl_incontext2_right. + +Lemma step_expr_context: + forall from to C, context from to C -> + forall a m, reducts_incl C (step_expr from a m) (step_expr to (C a) m) +with step_exprlist_context: + forall from C, contextlist from C -> + forall a m, reducts_incl C (step_expr from a m) (step_exprlist (C a) m). +Proof. + induction 1; simpl; intros. +(* top *) + red. destruct (step_expr k a m); auto. intros. + replace (fun x => C1 x) with C1; auto. apply extensionality; auto. +(* deref *) + eapply reducts_incl_trans with (C' := fun x => Ederef x ty); eauto. + destruct (is_val (C a)) as [[v ty']|]_eqn; eauto. +(* field *) + eapply reducts_incl_trans with (C' := fun x => Efield x f ty); eauto. + destruct (is_loc (C a)) as [[[b ofs] ty']|]_eqn; eauto. +(* valof *) + eapply reducts_incl_trans with (C' := fun x => Evalof x ty); eauto. + destruct (is_loc (C a)) as [[[b ofs] ty']|]_eqn; eauto. +(* addrof *) + eapply reducts_incl_trans with (C' := fun x => Eaddrof x ty); eauto. + destruct (is_loc (C a)) as [[[b ofs] ty']|]_eqn; eauto. +(* unop *) + eapply reducts_incl_trans with (C' := fun x => Eunop op x ty); eauto. + destruct (is_val (C a)) as [[v ty']|]_eqn; eauto. +(* binop left *) + eapply reducts_incl_trans with (C' := fun x => Ebinop op x e2 ty); eauto. + destruct (is_val (C a)) as [[v ty']|]_eqn; eauto. +(* binop right *) + eapply reducts_incl_trans with (C' := fun x => Ebinop op e1 x ty); eauto. + destruct (is_val e1) as [[v1 ty1]|]_eqn; eauto. + destruct (is_val (C a)) as [[v2 ty2]|]_eqn; eauto. +(* cast *) + eapply reducts_incl_trans with (C' := fun x => Ecast x ty); eauto. + destruct (is_val (C a)) as [[v ty']|]_eqn; eauto. +(* condition *) + eapply reducts_incl_trans with (C' := fun x => Econdition x r2 r3 ty); eauto. + destruct (is_val (C a)) as [[v ty']|]_eqn; eauto. +(* assign left *) + eapply reducts_incl_trans with (C' := fun x => Eassign x e2 ty); eauto. + destruct (is_loc (C a)) as [[[b ofs] ty']|]_eqn; eauto. +(* assign right *) + eapply reducts_incl_trans with (C' := fun x => Eassign e1 x ty); eauto. + destruct (is_loc e1) as [[[b ofs] ty1]|]_eqn; eauto. + destruct (is_val (C a)) as [[v2 ty2]|]_eqn; eauto. +(* assignop left *) + eapply reducts_incl_trans with (C' := fun x => Eassignop op x e2 tyres ty); eauto. + destruct (is_loc (C a)) as [[[b ofs] ty']|]_eqn; eauto. +(* assignop right *) + eapply reducts_incl_trans with (C' := fun x => Eassignop op e1 x tyres ty); eauto. + destruct (is_loc e1) as [[[b ofs] ty1]|]_eqn; eauto. + destruct (is_val (C a)) as [[v2 ty2]|]_eqn; eauto. +(* postincr *) + eapply reducts_incl_trans with (C' := fun x => Epostincr id x ty); eauto. + destruct (is_loc (C a)) as [[[b ofs] ty']|]_eqn; eauto. +(* call left *) + eapply reducts_incl_trans with (C' := fun x => Ecall x el ty); eauto. + destruct (is_val (C a)) as [[v ty']|]_eqn; eauto. +(* call right *) + eapply reducts_incl_trans with (C' := fun x => Ecall e1 x ty). apply step_exprlist_context. auto. + destruct (is_val e1) as [[v1 ty1]|]_eqn; eauto. + destruct (is_val_list (C a)) as [vl|]_eqn; eauto. +(* comma *) + eapply reducts_incl_trans with (C' := fun x => Ecomma x e2 ty); eauto. + destruct (is_val (C a)) as [[v ty']|]_eqn; eauto. +(* paren *) + eapply reducts_incl_trans with (C' := fun x => Eparen x ty); eauto. + destruct (is_val (C a)) as [[v ty']|]_eqn; eauto. + + induction 1; simpl; intros. +(* cons left *) + eapply reducts_incl_trans with (C' := fun x => Econs x el). + apply step_expr_context; eauto. eauto. +(* binop right *) + eapply reducts_incl_trans with (C' := fun x => Econs e1 x). + apply step_exprlist_context; eauto. eauto. +Qed. + +(** Completeness, part 2: given a safe expression, [step_expr] does not return [None]. *) + +Lemma topred_none: + forall rd, topred rd <> None. +Proof. + intros; unfold topred; congruence. +Qed. + +Lemma incontext_none: + forall (A B: Type) (C: A -> B) rds, + rds <> None -> incontext C rds <> None. +Proof. + unfold incontext; intros. destruct rds; congruence. +Qed. + +Lemma incontext2_none: + forall (A1 A2 B: Type) (C1: A1 -> B) rds1 (C2: A2 -> B) rds2, + rds1 <> None -> rds2 <> None -> incontext2 C1 rds1 C2 rds2 <> None. +Proof. + unfold incontext2; intros. destruct rds1; destruct rds2; congruence. +Qed. + +Lemma safe_expr_kind: + forall k C a m, + context k RV C -> + not_stuck ge e (C a) m -> + k = Cstrategy.expr_kind a. +Proof. + intros. + symmetry. eapply Cstrategy.not_imm_stuck_kind; eauto. +Qed. + +Lemma safe_inversion: + forall k C a m, + context k RV C -> + not_stuck ge e (C a) m -> + match a with + | Eloc _ _ _ => True + | Eval _ _ => True + | _ => Cstrategy.invert_expr_prop ge e a m + end. +Proof. + intros. eapply Cstrategy.not_imm_stuck_inv; eauto. +Qed. + +Lemma is_val_list_all_values: + forall al vtl, is_val_list al = Some vtl -> Cstrategy.exprlist_all_values al. +Proof. + induction al; simpl; intros. auto. + destruct (is_val r1) as [[v ty]|]_eqn; try discriminate. + destruct (is_val_list al) as [vtl'|]_eqn; try discriminate. + rewrite (is_val_inv _ _ _ Heqo). eauto. +Qed. + +Theorem step_expr_defined: + forall a k m C, + context k RV C -> + not_stuck ge e (C a) m -> + step_expr k a m <> None +with step_exprlist_defined: + forall al m C, + Cstrategy.contextlist' C -> + not_stuck ge e (C al) m -> + step_exprlist al m <> None. +Proof. + induction a; intros k m C CTX NS; + rewrite (safe_expr_kind _ _ _ _ CTX NS); + rewrite (safe_expr_kind _ _ _ _ CTX NS) in CTX; + simpl in *; + generalize (safe_inversion _ _ _ _ CTX NS); intro INV. +(* val *) + congruence. +(* var *) + red in INV. destruct INV as [b [P | [P [Q R]]]]. + rewrite P; rewrite dec_eq_true. apply topred_none. + rewrite P; rewrite Q; rewrite R; rewrite dec_eq_true. apply topred_none. +(* field *) + destruct (is_loc a) as [[[b ofs] ty']|]_eqn. + rewrite (is_loc_inv _ _ _ _ Heqo) in INV. red in INV. + destruct ty'; try contradiction. destruct INV as [delta EQ]. rewrite EQ. apply topred_none. + apply topred_none. + apply incontext_none. apply IHa with (C := fun x => C(Efield x f ty)); eauto. +(* valof *) + destruct (is_loc a) as [[[b ofs] ty']|]_eqn. + rewrite (is_loc_inv _ _ _ _ Heqo) in INV. red in INV. destruct INV as [EQ [v LD]]. subst. + rewrite dec_eq_true; rewrite LD; apply topred_none. + apply incontext_none. apply IHa with (C := fun x => C(Evalof x ty)); eauto. +(* deref *) + destruct (is_val a) as [[v ty']|]_eqn. + rewrite (is_val_inv _ _ _ Heqo) in INV. red in INV. destruct INV as [b [ofs EQ]]. subst. + apply topred_none. + apply incontext_none. apply IHa with (C := fun x => C(Ederef x ty)); eauto. +(* addrof *) + destruct (is_loc a) as [[[b ofs] ty']|]_eqn. + apply topred_none. + apply incontext_none. apply IHa with (C := fun x => C(Eaddrof x ty)); eauto. +(* unop *) + destruct (is_val a) as [[v1 ty1]|]_eqn. + rewrite (is_val_inv _ _ _ Heqo) in INV. red in INV. destruct INV as [v EQ]. + rewrite EQ; apply topred_none. + apply incontext_none. apply IHa with (C := fun x => C(Eunop op x ty)); eauto. +(* binop *) + destruct (is_val a1) as [[v1 ty1]|]_eqn. + destruct (is_val a2) as [[v2 ty2]|]_eqn. + rewrite (is_val_inv _ _ _ Heqo) in INV. + rewrite (is_val_inv _ _ _ Heqo0) in INV. red in INV. destruct INV as [v EQ]. + rewrite EQ; apply topred_none. + apply incontext2_none. apply IHa1 with (C := fun x => C(Ebinop op x a2 ty)); eauto. apply IHa2 with (C := fun x => C(Ebinop op a1 x ty)); eauto. + apply incontext2_none. apply IHa1 with (C := fun x => C(Ebinop op x a2 ty)); eauto. apply IHa2 with (C := fun x => C(Ebinop op a1 x ty)); eauto. +(* cast *) + destruct (is_val a) as [[v1 ty1]|]_eqn. + rewrite (is_val_inv _ _ _ Heqo) in INV. red in INV. destruct INV as [v EQ]. + rewrite EQ; apply topred_none. + apply incontext_none. apply IHa with (C := fun x => C(Ecast x ty)); eauto. +(* condition *) + destruct (is_val a1) as [[v1 ty1]|]_eqn. + rewrite (is_val_inv _ _ _ Heqo) in INV. red in INV. destruct INV as [v EQ]. + rewrite EQ; apply topred_none. + apply incontext_none. apply IHa1 with (C := fun x => C(Econdition x a2 a3 ty)); eauto. +(* sizeof *) + apply topred_none. +(* assign *) + destruct (is_loc a1) as [[[b ofs] ty1]|]_eqn. + destruct (is_val a2) as [[v2 ty2]|]_eqn. + rewrite (is_loc_inv _ _ _ _ Heqo) in INV. + rewrite (is_val_inv _ _ _ Heqo0) in INV. red in INV. + destruct INV as [v [m' [P [Q R]]]]. + subst. rewrite dec_eq_true; rewrite Q; rewrite R; apply topred_none. + apply incontext2_none. apply IHa1 with (C := fun x => C(Eassign x a2 ty)); eauto. apply IHa2 with (C := fun x => C(Eassign a1 x ty)); eauto. + apply incontext2_none. apply IHa1 with (C := fun x => C(Eassign x a2 ty)); eauto. apply IHa2 with (C := fun x => C(Eassign a1 x ty)); eauto. +(* assignop *) + destruct (is_loc a1) as [[[b ofs] ty1]|]_eqn. + destruct (is_val a2) as [[v2 ty2]|]_eqn. + rewrite (is_loc_inv _ _ _ _ Heqo) in INV. + rewrite (is_val_inv _ _ _ Heqo0) in INV. red in INV. + destruct INV as [v1 [v [v' [m' [P [Q [R [S T]]]]]]]]. + subst. rewrite dec_eq_true; rewrite Q; rewrite R; rewrite S; rewrite T; apply topred_none. + apply incontext2_none. apply IHa1 with (C := fun x => C(Eassignop op x a2 tyres ty)); eauto. apply IHa2 with (C := fun x => C(Eassignop op a1 x tyres ty)); eauto. + apply incontext2_none. apply IHa1 with (C := fun x => C(Eassignop op x a2 tyres ty)); eauto. apply IHa2 with (C := fun x => C(Eassignop op a1 x tyres ty)); eauto. +(* postincr *) + destruct (is_loc a) as [[[b ofs] ty1]|]_eqn. + rewrite (is_loc_inv _ _ _ _ Heqo) in INV. red in INV. + destruct INV as [v1 [v2 [v3 [m' [P [Q [R [S T]]]]]]]]. + subst. rewrite dec_eq_true; rewrite Q; rewrite R; rewrite S; rewrite T; apply topred_none. + apply incontext_none. apply IHa with (C := fun x => C(Epostincr id x ty)); eauto. +(* comma *) + destruct (is_val a1) as [[v1 ty1]|]_eqn. + rewrite (is_val_inv _ _ _ Heqo) in INV. red in INV. rewrite INV. + rewrite dec_eq_true; apply topred_none. + apply incontext_none. apply IHa1 with (C := fun x => C(Ecomma x a2 ty)); eauto. +(* call *) + destruct (is_val a) as [[vf tyf]|]_eqn. + destruct (is_val_list rargs) as [vtl|]_eqn. + rewrite (is_val_inv _ _ _ Heqo) in INV. red in INV. + destruct INV as [tyargs [tyres [fd [vl [P [Q [R S]]]]]]]. + eapply is_val_list_all_values; eauto. + rewrite P; rewrite Q. + exploit sem_cast_arguments_complete; eauto. intros [vtl' [U V]]. + assert (vtl' = vtl) by congruence. subst. rewrite V. rewrite S. rewrite dec_eq_true. + apply topred_none. + apply incontext2_none. apply IHa with (C := fun x => C(Ecall x rargs ty)); eauto. + apply step_exprlist_defined with (C := fun x => C(Ecall a x ty)); auto. + apply Cstrategy.contextlist'_intro with (rl0 := Enil). auto. + apply incontext2_none. apply IHa with (C := fun x => C(Ecall x rargs ty)); eauto. + apply step_exprlist_defined with (C := fun x => C(Ecall a x ty)); auto. + apply Cstrategy.contextlist'_intro with (rl0 := Enil). auto. +(* loc *) + congruence. +(* paren *) + destruct (is_val a) as [[v1 ty1]|]_eqn. + rewrite (is_val_inv _ _ _ Heqo) in INV. red in INV. destruct INV as [v EQ]. + rewrite EQ; apply topred_none. + apply incontext_none. apply IHa with (C := fun x => C(Eparen x ty)); eauto. + + induction al; intros; simpl. +(* nil *) + congruence. +(* cons *) + apply incontext2_none. + apply step_expr_defined with (C := fun x => C(Econs x al)); eauto. + apply Cstrategy.contextlist'_head; auto. + apply IHal with (C := fun x => C(Econs r1 x)); auto. + apply Cstrategy.contextlist'_tail; auto. +Qed. + +(** Connections between [not_stuck] and [step_expr]. *) + +Lemma step_expr_not_imm_stuck: + forall k a m, + step_expr k a m <> None -> + not_imm_stuck ge e k a m. +Proof. + intros. generalize (step_expr_sound a k m). unfold reducts_ok. + destruct (step_expr k a m) as [ll|]. destruct ll. +(* value or location *) + destruct k; destruct a; simpl; intros; try congruence. + apply not_stuck_loc. + apply Csem.not_stuck_val. +(* reducible *) + intros R. destruct p as [C1 rd1]. destruct (R C1 rd1) as [P [a' [U V]]]; auto with coqlib. + subst a. red in V. destruct rd1. + eapply not_stuck_lred; eauto. + eapply not_stuck_rred; eauto. + destruct V. subst m'. eapply not_stuck_callred; eauto. +(* stuck *) + congruence. +Qed. + +Lemma step_expr_not_stuck: + forall a m, + step_expr RV a m <> None -> + not_stuck ge e a m. +Proof. + intros; red; intros. subst a. + apply step_expr_not_imm_stuck. + generalize (step_expr_context _ _ C H0 e' m). unfold reducts_incl. + destruct (step_expr k e' m). congruence. + destruct (step_expr RV (C e') m). tauto. congruence. +Qed. + +Lemma not_stuck_step_expr: + forall a m, + not_stuck ge e a m -> + step_expr RV a m <> None. +Proof. + intros. apply step_expr_defined with (C := fun x => x); auto. +Qed. + +End EXPRS. + +(** * External functions and events. *) + +Section EVENTS. + +Variable F V: Type. +Variable genv: Genv.t F V. + +Definition eventval_of_val (v: val) (t: typ) : option eventval := + match v, t with + | Vint i, AST.Tint => Some (EVint i) + | Vfloat f, AST.Tfloat => Some (EVfloat f) + | Vptr b ofs, AST.Tint => do id <- Genv.invert_symbol genv b; Some (EVptr_global id ofs) + | _, _ => None + end. + +Fixpoint list_eventval_of_val (vl: list val) (tl: list typ) : option (list eventval) := + match vl, tl with + | nil, nil => Some nil + | v1::vl, t1::tl => + do ev1 <- eventval_of_val v1 t1; + do evl <- list_eventval_of_val vl tl; + Some (ev1 :: evl) + | _, _ => None + end. + +Definition val_of_eventval (ev: eventval) (t: typ) : option val := + match ev, t with + | EVint i, AST.Tint => Some (Vint i) + | EVfloat f, AST.Tfloat => Some (Vfloat f) + | EVptr_global id ofs, AST.Tint => do b <- Genv.find_symbol genv id; Some (Vptr b ofs) + | _, _ => None + end. + +Lemma eventval_of_val_sound: + forall v t ev, eventval_of_val v t = Some ev -> eventval_match genv ev t v. +Proof. + intros. destruct v; destruct t; simpl in H; inv H. + constructor. + constructor. + destruct (Genv.invert_symbol genv b) as [id|]_eqn; inv H1. + constructor. apply Genv.invert_find_symbol; auto. +Qed. + +Lemma eventval_of_val_complete: + forall ev t v, eventval_match genv ev t v -> eventval_of_val v t = Some ev. +Proof. + induction 1; simpl; auto. + rewrite (Genv.find_invert_symbol _ _ H). auto. +Qed. + +Lemma list_eventval_of_val_sound: + forall vl tl evl, list_eventval_of_val vl tl = Some evl -> eventval_list_match genv evl tl vl. +Proof with try discriminate. + induction vl; destruct tl; simpl; intros; inv H. + constructor. + destruct (eventval_of_val a t) as [ev1|]_eqn... + destruct (list_eventval_of_val vl tl) as [evl'|]_eqn... + inv H1. constructor. apply eventval_of_val_sound; auto. eauto. +Qed. + +Lemma list_eventval_of_val_complete: + forall evl tl vl, eventval_list_match genv evl tl vl -> list_eventval_of_val vl tl = Some evl. +Proof. + induction 1; simpl. auto. + rewrite (eventval_of_val_complete _ _ _ H). rewrite IHeventval_list_match. auto. +Qed. + +Lemma val_of_eventval_sound: + forall ev t v, val_of_eventval ev t = Some v -> eventval_match genv ev t v. +Proof. + intros. destruct ev; destruct t; simpl in H; inv H. + constructor. + constructor. + destruct (Genv.find_symbol genv i) as [b|]_eqn; inv H1. + constructor. auto. +Qed. + +Lemma val_of_eventval_complete: + forall ev t v, eventval_match genv ev t v -> val_of_eventval ev t = Some v. +Proof. + induction 1; simpl; auto. rewrite H; auto. +Qed. + +(** System calls and library functions *) + +Definition do_ef_external (name: ident) (sg: signature) + (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) := + do args <- list_eventval_of_val vargs (sig_args sg); + match nextworld_io w name args with + | None => None + | Some(res, w') => + do vres <- val_of_eventval res (proj_sig_res sg); + Some(w', Event_syscall name args res :: E0, vres, m) + end. + +Definition do_ef_volatile_load (chunk: memory_chunk) + (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) := + match vargs with + | Vptr b ofs :: nil => + if block_is_volatile genv b then + do id <- Genv.invert_symbol genv b; + match nextworld_vload w chunk id ofs with + | None => None + | Some(res, w') => + do vres <- val_of_eventval res (type_of_chunk chunk); + Some(w', Event_vload chunk id ofs res :: nil, Val.load_result chunk vres, m) + end + else + do v <- Mem.load chunk m b (Int.unsigned ofs); + Some(w, E0, v, m) + | _ => None + end. + +Definition do_ef_volatile_store (chunk: memory_chunk) + (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) := + match vargs with + | Vptr b ofs :: v :: nil => + if block_is_volatile genv b then + do id <- Genv.invert_symbol genv b; + do ev <- eventval_of_val v (type_of_chunk chunk); + do w' <- nextworld_vstore w chunk id ofs ev; + Some(w', Event_vstore chunk id ofs ev :: nil, Vundef, m) + else + do m' <- Mem.store chunk m b (Int.unsigned ofs) v; + Some(w, E0, Vundef, m') + | _ => None + end. + +Definition do_ef_malloc + (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) := + match vargs with + | Vint n :: nil => + let (m', b) := Mem.alloc m (-4) (Int.unsigned n) in + do m'' <- Mem.store Mint32 m' b (-4) (Vint n); + Some(w, E0, Vptr b Int.zero, m'') + | _ => None + end. + +Definition do_ef_free + (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) := + match vargs with + | Vptr b lo :: nil => + do vsz <- Mem.load Mint32 m b (Int.unsigned lo - 4); + match vsz with + | Vint sz => + check (zlt 0 (Int.unsigned sz)); + do m' <- Mem.free m b (Int.unsigned lo - 4) (Int.unsigned lo + Int.unsigned sz); + Some(w, E0, Vundef, m') + | _ => None + end + | _ => None + end. + +Definition memcpy_args_ok + (sz al: Z) (bdst: block) (odst: Z) (bsrc: block) (osrc: Z) : Prop := + (al = 1 \/ al = 2 \/ al = 4) + /\ sz > 0 + /\ (al | sz) /\ (al | osrc) /\ (al | odst) + /\ (bsrc <> bdst \/ osrc = odst \/ osrc + sz <= odst \/ odst + sz <= osrc). + +Remark memcpy_check_args: + forall sz al bdst odst bsrc osrc, + {memcpy_args_ok sz al bdst odst bsrc osrc} + {~memcpy_args_ok sz al bdst odst bsrc osrc}. +Proof with try (right; intuition omega). + intros. + assert (X: {al = 1 \/ al = 2 \/ al = 4} + {~(al = 1 \/ al = 2 \/ al = 4)}). + destruct (zeq al 1); auto. destruct (zeq al 2); auto. destruct (zeq al 4); auto... + unfold memcpy_args_ok. destruct X... + assert (al > 0) by (intuition omega). + destruct (zlt 0 sz)... + destruct (Zdivide_dec al sz); auto... + destruct (Zdivide_dec al osrc); auto... + destruct (Zdivide_dec al odst); auto... + assert (Y: {bsrc <> bdst \/ osrc = odst \/ osrc + sz <= odst \/ odst + sz <= osrc} + +{~(bsrc <> bdst \/ osrc = odst \/ osrc + sz <= odst \/ odst + sz <= osrc)}). + destruct (eq_block bsrc bdst); auto. + destruct (zeq osrc odst); auto. + destruct (zle (osrc + sz) odst); auto. + destruct (zle (odst + sz) osrc); auto. + right; intuition omega. + destruct Y... left; intuition omega. +Qed. + +Definition do_ef_memcpy (sz al: Z) + (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) := + match vargs with + | Vptr bdst odst :: Vptr bsrc osrc :: nil => + if memcpy_check_args sz al bdst (Int.unsigned odst) bsrc (Int.unsigned osrc) then + do bytes <- Mem.loadbytes m bsrc (Int.unsigned osrc) sz; + do m' <- Mem.storebytes m bdst (Int.unsigned odst) bytes; + Some(w, E0, Vundef, m') + else None + | _ => None + end. + +Definition do_ef_annot (text: ident) (targs: list typ) + (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) := + do args <- list_eventval_of_val vargs targs; + Some(w, Event_annot text args :: E0, Vundef, m). + +Definition do_ef_annot_val (text: ident) (targ: typ) + (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) := + match vargs with + | varg :: nil => + do arg <- eventval_of_val varg targ; + Some(w, Event_annot text (arg :: nil) :: E0, varg, m) + | _ => None + end. + +Definition do_external (ef: external_function): + world -> list val -> mem -> option (world * trace * val * mem) := + match ef with + | EF_external name sg => do_ef_external name sg + | EF_builtin name sg => do_ef_external name sg + | EF_vload chunk => do_ef_volatile_load chunk + | EF_vstore chunk => do_ef_volatile_store chunk + | EF_malloc => do_ef_malloc + | EF_free => do_ef_free + | EF_memcpy sz al => do_ef_memcpy sz al + | EF_annot text targs => do_ef_annot text targs + | EF_annot_val text targ => do_ef_annot_val text targ + end. + +Ltac mydestr := + match goal with + | [ |- None = Some _ -> _ ] => intro X; discriminate + | [ |- Some _ = Some _ -> _ ] => intro X; inv X + | [ |- match ?x with Some _ => _ | None => _ end = Some _ -> _ ] => destruct x as []_eqn; mydestr + | [ |- match ?x with true => _ | false => _ end = Some _ -> _ ] => destruct x as []_eqn; mydestr + | [ |- match ?x with left _ => _ | right _ => _ end = Some _ -> _ ] => destruct x; mydestr + | _ => idtac + end. + +Lemma do_ef_external_sound: + forall ef w vargs m w' t vres m', + do_external ef w vargs m = Some(w', t, vres, m') -> + external_call ef genv vargs m t vres m' /\ possible_trace w t w'. +Proof with try congruence. + intros until m'. + assert (IO: forall name sg, + do_ef_external name sg w vargs m = Some(w', t, vres, m') -> + extcall_io_sem name sg genv vargs m t vres m' /\ possible_trace w t w'). + intros until sg. unfold do_ef_external. mydestr. destruct p as [res w'']; mydestr. + split. econstructor. apply list_eventval_of_val_sound; auto. + apply val_of_eventval_sound; auto. + econstructor. constructor; eauto. constructor. + + destruct ef; simpl. +(* EF_external *) + auto. +(* EF_builtin *) + auto. +(* EF_vload *) + unfold do_ef_volatile_load. destruct vargs... destruct v... destruct vargs... + mydestr. destruct p as [res w'']; mydestr. + split. constructor. apply Genv.invert_find_symbol; auto. auto. + apply val_of_eventval_sound; auto. + econstructor. constructor; eauto. constructor. + split. constructor; auto. constructor. +(* EF_vstore *) + unfold do_ef_volatile_store. destruct vargs... destruct v... destruct vargs... destruct vargs... + mydestr. + split. constructor. apply Genv.invert_find_symbol; auto. auto. + apply eventval_of_val_sound; auto. + econstructor. constructor; eauto. constructor. + split. constructor; auto. constructor. +(* EF_malloc *) + unfold do_ef_malloc. destruct vargs... destruct v... destruct vargs... + destruct (Mem.alloc m (-4) (Int.unsigned i)) as [m1 b]_eqn. mydestr. + split. econstructor; eauto. constructor. +(* EF_free *) + unfold do_ef_free. destruct vargs... destruct v... destruct vargs... + mydestr. destruct v... mydestr. + split. econstructor; eauto. omega. constructor. +(* EF_memcpy *) + unfold do_ef_memcpy. destruct vargs... destruct v... destruct vargs... + destruct v... destruct vargs... mydestr. red in m0. + split. econstructor; eauto; tauto. constructor. +(* EF_annot *) + unfold do_ef_annot. mydestr. + split. constructor. apply list_eventval_of_val_sound; auto. + econstructor. constructor; eauto. constructor. +(* EF_annot_val *) + unfold do_ef_annot_val. destruct vargs... destruct vargs... mydestr. + split. constructor. apply eventval_of_val_sound; auto. + econstructor. constructor; eauto. constructor. +Qed. + +Lemma do_ef_external_complete: + forall ef w vargs m w' t vres m', + external_call ef genv vargs m t vres m' -> possible_trace w t w' -> + do_external ef w vargs m = Some(w', t, vres, m'). +Proof. + intros. + assert (IO: forall name sg, + extcall_io_sem name sg genv vargs m t vres m' -> + do_ef_external name sg w vargs m = Some (w', t, vres, m')). + intros. inv H1. inv H0. inv H8. inv H6. + unfold do_ef_external. rewrite (list_eventval_of_val_complete _ _ _ H2). rewrite H8. + rewrite (val_of_eventval_complete _ _ _ H3). auto. + + destruct ef; simpl in *. +(* EF_external *) + auto. +(* EF_builtin *) + auto. +(* EF_vload *) + inv H; unfold do_ef_volatile_load. + inv H0. inv H8. inv H6. + rewrite H2. rewrite (Genv.find_invert_symbol _ _ H1). rewrite H9. + rewrite (val_of_eventval_complete _ _ _ H3). auto. + inv H0. rewrite H1. rewrite H2. auto. +(* EF_vstore *) + inv H; unfold do_ef_volatile_store. + inv H0. inv H8. inv H6. + rewrite H2. rewrite (Genv.find_invert_symbol _ _ H1). + rewrite (eventval_of_val_complete _ _ _ H3). rewrite H9. auto. + inv H0. rewrite H1. rewrite H2. auto. +(* EF_malloc *) + inv H; unfold do_ef_malloc. + inv H0. rewrite H1. rewrite H2. auto. +(* EF_free *) + inv H; unfold do_ef_free. + inv H0. rewrite H1. rewrite zlt_true. rewrite H3. auto. omega. +(* EF_memcpy *) + inv H; unfold do_ef_memcpy. + inv H0. rewrite pred_dec_true. rewrite H7; rewrite H8; auto. + red. tauto. +(* EF_annot *) + inv H; unfold do_ef_annot. inv H0. inv H6. inv H4. + rewrite (list_eventval_of_val_complete _ _ _ H1). auto. +(* EF_annot_val *) + inv H; unfold do_ef_annot_val. inv H0. inv H6. inv H4. + rewrite (eventval_of_val_complete _ _ _ H1). auto. +Qed. + +End EVENTS. + +(** * Transitions over states. *) + +Fixpoint do_alloc_variables (e: env) (m: mem) (l: list (ident * type)) {struct l} : env * mem := + match l with + | nil => (e,m) + | (id, ty) :: l' => + let (m1,b1) := Mem.alloc m 0 (sizeof ty) in + do_alloc_variables (PTree.set id (b1, ty) e) m1 l' +end. + +Lemma do_alloc_variables_sound: + forall l e m, alloc_variables e m l (fst (do_alloc_variables e m l)) (snd (do_alloc_variables e m l)). +Proof. + induction l; intros; simpl. + constructor. + destruct a as [id ty]. destruct (Mem.alloc m 0 (sizeof ty)) as [m1 b1]_eqn; simpl. + econstructor; eauto. +Qed. + +Lemma do_alloc_variables_complete: + forall e1 m1 l e2 m2, alloc_variables e1 m1 l e2 m2 -> + do_alloc_variables e1 m1 l = (e2, m2). +Proof. + induction 1; simpl. + auto. + rewrite H; rewrite IHalloc_variables; auto. +Qed. + +Function sem_bind_parameters (e: env) (m: mem) (l: list (ident * type)) (lv: list val) + {struct l} : option mem := + match l, lv with + | nil, nil => Some m + | (id, ty) :: params, v1::lv => + match PTree.get id e with + | Some (b, ty') => + check (type_eq ty ty'); + do m1 <- store_value_of_type ty m b Int.zero v1; + sem_bind_parameters e m1 params lv + | None => None + end + | _, _ => None +end. + +Lemma sem_bind_parameters_sound : forall e m l lv m', + sem_bind_parameters e m l lv = Some m' -> + bind_parameters e m l lv m'. +Proof. + intros; functional induction (sem_bind_parameters e m l lv); try discriminate. + inversion H; constructor; auto. + econstructor; eauto. +Qed. + +Lemma sem_bind_parameters_complete : forall e m l lv m', + bind_parameters e m l lv m' -> + sem_bind_parameters e m l lv = Some m'. +Proof. + induction 1; simpl; auto. + rewrite H. rewrite dec_eq_true. + destruct (store_value_of_type ty m b Int.zero v1); try discriminate. + inv H0; auto. +Qed. + +Definition expr_final_state (f: function) (k: cont) (e: env) (C_rd: (expr -> expr) * reduction) := + match snd C_rd with + | Lred a m => (E0, ExprState f (fst C_rd a) k e m) + | Rred a m => (E0, ExprState f (fst C_rd a) k e m) + | Callred fd vargs ty m => (E0, Callstate fd vargs (Kcall f e (fst C_rd) ty k) m) + end. + +Local Open Scope list_monad_scope. + +Definition ret (S: state) : list (trace * state) := (E0, S) :: nil. + +Definition do_step (w: world) (s: state) : list (trace * state) := + match s with + + | ExprState f a k e m => + match is_val a with + | Some(v, ty) => + match k with + | Kdo k => ret (State f Sskip k e m ) + | Kifthenelse s1 s2 k => + do b <- bool_val v ty; ret (State f (if b then s1 else s2) k e m) + | Kwhile1 x s k => + do b <- bool_val v ty; + if b then ret (State f s (Kwhile2 x s k) e m) else ret (State f Sskip k e m) + | Kdowhile2 x s k => + do b <- bool_val v ty; + if b then ret (State f (Sdowhile x s) k e m) else ret (State f Sskip k e m) + | Kfor2 a2 a3 s k => + do b <- bool_val v ty; + if b then ret (State f s (Kfor3 a2 a3 s k) e m) else ret (State f Sskip k e m) + | Kreturn k => + do v' <- sem_cast v ty f.(fn_return); + do m' <- Mem.free_list m (blocks_of_env e); + ret (Returnstate v' (call_cont k) m') + | Kswitch1 sl k => + match v with + | Vint n => ret (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch2 k) e m) + | _ => nil + end + | _ => nil + end + + | None => + match step_expr e RV a m with + | None => nil + | Some ll => map (expr_final_state f k e) ll + end + end + + | State f (Sdo x) k e m => ret(ExprState f x (Kdo k) e m) + + | State f (Ssequence s1 s2) k e m => ret(State f s1 (Kseq s2 k) e m) + | State f Sskip (Kseq s k) e m => ret (State f s k e m) + | State f Scontinue (Kseq s k) e m => ret (State f Scontinue k e m) + | State f Sbreak (Kseq s k) e m => ret (State f Sbreak k e m) + + | State f (Sifthenelse a s1 s2) k e m => ret (ExprState f a (Kifthenelse s1 s2 k) e m) + + | State f (Swhile x s) k e m => ret (ExprState f x (Kwhile1 x s k) e m) + | State f (Sskip|Scontinue) (Kwhile2 x s k) e m => ret (State f (Swhile x s) k e m) + | State f Sbreak (Kwhile2 x s k) e m => ret (State f Sskip k e m) + + | State f (Sdowhile a s) k e m => ret (State f s (Kdowhile1 a s k) e m) + | State f (Sskip|Scontinue) (Kdowhile1 x s k) e m => ret (ExprState f x (Kdowhile2 x s k) e m) + | State f Sbreak (Kdowhile1 x s k) e m => ret (State f Sskip k e m) + + | State f (Sfor a1 a2 a3 s) k e m => + if is_skip a1 + then ret (ExprState f a2 (Kfor2 a2 a3 s k) e m) + else ret (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m) + | State f Sskip (Kfor3 a2 a3 s k) e m => ret (State f a3 (Kfor4 a2 a3 s k) e m) + | State f Scontinue (Kfor3 a2 a3 s k) e m => ret (State f a3 (Kfor4 a2 a3 s k) e m) + | State f Sbreak (Kfor3 a2 a3 s k) e m => ret (State f Sskip k e m) + | State f Sskip (Kfor4 a2 a3 s k) e m => ret (State f (Sfor Sskip a2 a3 s) k e m) + + | State f (Sreturn None) k e m => + do m' <- Mem.free_list m (blocks_of_env e); + ret (Returnstate Vundef (call_cont k) m') + | State f (Sreturn (Some x)) k e m => ret (ExprState f x (Kreturn k) e m) + | State f Sskip ((Kstop | Kcall _ _ _ _ _) as k) e m => + check (type_eq f.(fn_return) Tvoid); + do m' <- Mem.free_list m (blocks_of_env e); + ret (Returnstate Vundef k m') + + | State f (Sswitch x sl) k e m => ret (ExprState f x (Kswitch1 sl k) e m) + | State f (Sskip|Sbreak) (Kswitch2 k) e m => ret (State f Sskip k e m) + | State f Scontinue (Kswitch2 k) e m => ret (State f Scontinue k e m) + + | State f (Slabel lbl s) k e m => ret (State f s k e m) + | State f (Sgoto lbl) k e m => + match find_label lbl f.(fn_body) (call_cont k) with + | Some(s', k') => ret (State f s' k' e m) + | None => nil + end + + | Callstate (Internal f) vargs k m => + check (list_norepet_dec ident_eq (var_names (fn_params f) ++ var_names (fn_vars f))); + let (e,m1) := do_alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) in + do m2 <- sem_bind_parameters e m1 f.(fn_params) vargs; + ret (State f f.(fn_body) k e m2) + | Callstate (External ef _ _) vargs k m => + match do_external _ _ ge ef w vargs m with + | None => nil + | Some(w',t,v,m') => (t, Returnstate v k m') :: nil + end + + | Returnstate v (Kcall f e C ty k) m => ret (ExprState f (C (Eval v ty)) k e m) + + | _ => nil + end. + +(* +Definition at_external (S: state): option (external_function * list val * mem) := + match S with + | Callstate (External ef _ _) vargs k m => Some (ef, vargs, m) + | _ => None + end. + +Definition after_external (S: state) (v: val) (m: mem): option state := + match S with + | Callstate _ _ k _ => Some (Returnstate v k m) + | _ => None + end. +*) + +Ltac myinv := + match goal with + | [ |- In _ nil -> _ ] => intro X; elim X + | [ |- In _ (ret _) -> _ ] => + intro X; elim X; clear X; + [intro EQ; unfold ret in EQ; inv EQ; myinv | myinv] + | [ |- In _ (_ :: nil) -> _ ] => + intro X; elim X; clear X; [intro EQ; inv EQ; myinv | myinv] + | [ |- In _ (match ?x with Some _ => _ | None => _ end) -> _ ] => destruct x as []_eqn; myinv + | [ |- In _ (match ?x with false => _ | true => _ end) -> _ ] => destruct x as []_eqn; myinv + | [ |- In _ (match ?x with left _ => _ | right _ => _ end) -> _ ] => destruct x; myinv + | _ => idtac + end. + +Hint Extern 3 => exact I. + +Lemma do_step_sound: + forall w S t S', In (t, S') (do_step w S) -> Csem.step ge S t S'. +Proof with try (right; econstructor; eauto; fail). + intros until S'. destruct S; simpl. +(* State *) + destruct s; myinv... + (* skip *) + destruct k; myinv... + (* break *) + destruct k; myinv... + (* continue *) + destruct k; myinv... + (* goto *) + destruct p as [s' k']. myinv... +(* ExprState *) + destruct (is_val r) as [[v ty]|]_eqn. + (* expression is a value *) + rewrite (is_val_inv _ _ _ Heqo). + destruct k; myinv... + destruct v; myinv... + (* expression reduces *) + destruct (step_expr e RV r m) as [ll|]_eqn; try contradiction. intros. + exploit list_in_map_inv; eauto. intros [[C rd] [A B]]. + generalize (step_expr_sound e r RV m). unfold reducts_ok. rewrite Heqr0. + destruct ll; try contradiction. intros SOUND. + exploit SOUND; eauto. intros [CTX [a [EQ RD]]]. subst r. + unfold expr_final_state in A. simpl in A. left. + destruct rd; inv A; simpl in RD. + apply step_lred. auto. apply step_expr_not_stuck; congruence. auto. + apply step_rred. auto. apply step_expr_not_stuck; congruence. auto. + destruct RD; subst m'. apply Csem.step_call. auto. apply step_expr_not_stuck; congruence. auto. +(* callstate *) + destruct fd; myinv. + (* internal *) + destruct (do_alloc_variables empty_env m (fn_params f ++ fn_vars f)) as [e m1]_eqn. + myinv. right; apply step_internal_function with m1. auto. + change e with (fst (e,m1)). change m1 with (snd (e,m1)) at 2. rewrite <- Heqp. + apply do_alloc_variables_sound. apply sem_bind_parameters_sound; auto. + (* external *) + destruct p as [[[w' tr] v] m']. myinv. right; constructor. + eapply do_ef_external_sound; eauto. +(* returnstate *) + destruct k; myinv... +Qed. + +Remark estep_not_val: + forall f a k e m t S, estep ge (ExprState f a k e m) t S -> is_val a = None. +Proof. + intros. + assert (forall b from to C, context from to C -> (C = fun x => x) \/ is_val (C b) = None). + induction 1; simpl; auto. + inv H. + destruct (H0 a0 _ _ _ H10). subst. inv H8; auto. auto. + destruct (H0 a0 _ _ _ H10). subst. inv H8; auto. auto. + destruct (H0 a0 _ _ _ H10). subst. inv H8; auto. auto. +Qed. + +Lemma do_step_complete: + forall w S t S' w', possible_trace w t w' -> Csem.step ge S t S' -> In (t, S') (do_step w S). +Proof with (unfold ret; auto with coqlib). + intros until w'; intro PT; intros. + destruct H. + (* Expression step *) + inversion H; subst; exploit estep_not_val; eauto; intro NOTVAL. +(* lred *) + unfold do_step; rewrite NOTVAL. + destruct (step_expr e RV (C a) m) as [ll|]_eqn. + change (E0, ExprState f (C a') k e m') with (expr_final_state f k e (C, Lred a' m')). + apply in_map. + generalize (step_expr_context e _ _ _ H2 a m). unfold reducts_incl. + rewrite Heqr. destruct (step_expr e LV a m) as [ll'|]_eqn; try tauto. + intro. replace C with (fun x => C x). apply H3. + rewrite (lred_topred _ _ _ _ _ H0) in Heqr0. inv Heqr0. auto with coqlib. + apply extensionality; auto. + exploit not_stuck_step_expr; eauto. +(* rred *) + unfold do_step; rewrite NOTVAL. + destruct (step_expr e RV (C a) m) as [ll|]_eqn. + change (E0, ExprState f (C a') k e m') with (expr_final_state f k e (C, Rred a' m')). + apply in_map. + generalize (step_expr_context e _ _ _ H2 a m). unfold reducts_incl. + rewrite Heqr. destruct (step_expr e RV a m) as [ll'|]_eqn; try tauto. + intro. replace C with (fun x => C x). apply H3. + rewrite (rred_topred _ _ _ _ _ H0) in Heqr0. inv Heqr0. auto with coqlib. + apply extensionality; auto. + exploit not_stuck_step_expr; eauto. +(* callred *) + unfold do_step; rewrite NOTVAL. + destruct (step_expr e RV (C a) m) as [ll|]_eqn. + change (E0, Callstate fd vargs (Kcall f e C ty k) m) with (expr_final_state f k e (C, Callred fd vargs ty m)). + apply in_map. + generalize (step_expr_context e _ _ _ H2 a m). unfold reducts_incl. + rewrite Heqr. destruct (step_expr e RV a m) as [ll'|]_eqn; try tauto. + intro. replace C with (fun x => C x). apply H3. + rewrite (callred_topred _ _ _ _ _ _ H0) in Heqr0. inv Heqr0. auto with coqlib. + apply extensionality; auto. + exploit not_stuck_step_expr; eauto. + + (* Statement step *) + inv H; simpl... + rewrite H0... + rewrite H0... + rewrite H0... + destruct H0; subst s0... + destruct H0; subst s0... + rewrite H0... + rewrite H0... + rewrite pred_dec_false... + rewrite pred_dec_true... + rewrite H0... + rewrite H0... + destruct H0; subst x... + rewrite H0... + rewrite H0; rewrite H1... + rewrite pred_dec_true; auto. rewrite H2. red in H0. destruct k; try contradiction... + destruct H0; subst x... + rewrite H0... + + (* Call step *) + rewrite pred_dec_true; auto. rewrite (do_alloc_variables_complete _ _ _ _ _ H1). + rewrite (sem_bind_parameters_complete _ _ _ _ _ H2)... + exploit do_ef_external_complete; eauto. intro EQ; rewrite EQ. auto with coqlib. +Qed. + +End EXEC. + +Local Open Scope option_monad_scope. + +Definition do_initial_state (p: program): option (genv * state) := + let ge := Genv.globalenv p in + do m0 <- Genv.init_mem p; + do b <- Genv.find_symbol ge p.(prog_main); + do f <- Genv.find_funct_ptr ge b; + check (type_eq (type_of_fundef f) (Tfunction Tnil (Tint I32 Signed))); + Some (ge, Callstate f nil Kstop m0). + +Definition at_final_state (S: state): option int := + match S with + | Returnstate (Vint r) Kstop m => Some r + | _ => None + end. -- cgit v1.2.3