Section EVAL_EXPR.
Variable sp: val.
Variable e: env.
Variable m: mem.
Inductive eval_expr: letenv -> expr -> val -> Prop :=
| eval_Evar: forall le id v,
PTree.get id e = Some v ->
eval_expr le (Evar id) v
| eval_Eop: forall le op al vl v,
eval_exprlist le al vl ->
eval_operation ge sp op vl m = Some v ->
eval_expr le (Eop op al) v
| eval_Eload: forall le chunk addr al vl vaddr v,
eval_exprlist le al vl ->
eval_addressing ge sp addr vl = Some vaddr ->
Mem.loadv chunk m vaddr = Some v ->
eval_expr le (Eload chunk addr al) v
| eval_Econdition: forall le a b c va v,
eval_condexpr le a va ->
eval_expr le (if va then b else c) v ->
eval_expr le (Econdition a b c) v
| eval_Elet: forall le a b v1 v2,
eval_expr le a v1 ->
eval_expr (v1 :: le) b v2 ->
eval_expr le (Elet a b) v2
| eval_Eletvar: forall le n v,
nth_error le n = Some v ->
eval_expr le (Eletvar n) v
| eval_Ebuiltin: forall le ef al vl v,
eval_exprlist le al vl ->
external_call ef ge vl m E0 v m ->
eval_expr le (Ebuiltin ef al) v
| eval_Eexternal: forall le id sg al b ef vl v,
Genv.find_symbol ge id = Some b ->
Genv.find_funct_ptr ge b = Some (External ef) ->
ef_sig ef = sg ->
eval_exprlist le al vl ->
external_call ef ge vl m E0 v m ->
eval_expr le (Eexternal id sg al) v
with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
| eval_Enil: forall le,
eval_exprlist le Enil nil
| eval_Econs: forall le a1 al v1 vl,
eval_expr le a1 v1 -> eval_exprlist le al vl ->
eval_exprlist le (Econs a1 al) (v1 :: vl)
with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
| eval_CEcond: forall le cond al vl vb,
eval_exprlist le al vl ->
eval_condition cond vl m = Some vb ->
eval_condexpr le (CEcond cond al) vb
| eval_CEcondition: forall le a b c va v,
eval_condexpr le a va ->
eval_condexpr le (if va then b else c) v ->
eval_condexpr le (CEcondition a b c) v
| eval_CElet: forall le a b v1 v2,
eval_expr le a v1 ->
eval_condexpr (v1 :: le) b v2 ->
eval_condexpr le (CElet a b) v2.
Scheme eval_expr_ind3 := Minimality for eval_expr Sort Prop
with eval_exprlist_ind3 := Minimality for eval_exprlist Sort Prop
with eval_condexpr_ind3 := Minimality for eval_condexpr Sort Prop.
Inductive eval_exitexpr: letenv -> exitexpr -> nat -> Prop :=
| eval_XEexit: forall le x,
eval_exitexpr le (XEexit x) x
| eval_XEjumptable: forall le a tbl n x,
eval_expr le a (Vint n) ->
list_nth_z tbl (Int.unsigned n) = Some x ->
eval_exitexpr le (XEjumptable a tbl) x
| eval_XEcondition: forall le a b c va x,
eval_condexpr le a va ->
eval_exitexpr le (if va then b else c) x ->
eval_exitexpr le (XEcondition a b c) x
| eval_XElet: forall le a b v x,
eval_expr le a v ->
eval_exitexpr (v :: le) b x ->
eval_exitexpr le (XElet a b) x.
Inductive eval_expr_or_symbol: letenv -> expr + ident -> val -> Prop :=
| eval_eos_e: forall le e v,
eval_expr le e v ->
eval_expr_or_symbol le (inl _ e) v
| eval_eos_s: forall le id b,
Genv.find_symbol ge id = Some b ->
eval_expr_or_symbol le (inr _ id) (Vptr b Ptrofs.zero).
Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
| eval_BA: forall a v,
eval_expr nil a v ->
eval_builtin_arg (BA a) v
| eval_BA_int: forall n,
eval_builtin_arg (BA_int n) (Vint n)
| eval_BA_long: forall n,
eval_builtin_arg (BA_long n) (Vlong n)
| eval_BA_float: forall n,
eval_builtin_arg (BA_float n) (Vfloat n)
| eval_BA_single: forall n,
eval_builtin_arg (BA_single n) (Vsingle n)
| eval_BA_loadstack: forall chunk ofs v,
Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
eval_builtin_arg (BA_loadstack chunk ofs) v
| eval_BA_addrstack: forall ofs,
eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
| eval_BA_loadglobal: forall chunk id ofs v,
Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
eval_builtin_arg (BA_loadglobal chunk id ofs) v
| eval_BA_addrglobal: forall id ofs,
eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
| eval_BA_splitlong: forall a1 a2 v1 v2,
eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
| eval_BA_addptr: forall a1 v1 a2 v2,
eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
eval_builtin_arg (BA_addptr a1 a2)
(if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).
End EVAL_EXPR.
(** Update local environment with the result of a builtin. *)
Definition set_builtin_res (res: builtin_res ident) (v: val) (e: env) : env :=
match res with
| BR id => PTree.set id v e
| _ => e
end.
(** Pop continuation until a call or stop *)
Fixpoint call_cont (k: cont) : cont :=
match k with
| Kseq s k => call_cont k
| Kblock k => call_cont k
| _ => k
end.
Definition is_call_cont (k: cont) : Prop :=
match k with
| Kstop => True
| Kcall _ _ _ _ _ => True
| _ => False
end.
(** Find the statement and manufacture the continuation corresponding to a label *)
Fixpoint find_label (lbl: label) (s: stmt) (k: cont)
{struct s}: option (stmt * cont) :=
match s with
| Sseq s1 s2 =>
match find_label lbl s1 (Kseq s2 k) with
| Some sk => Some sk
| None => find_label lbl s2 k
end
| Sifthenelse c s1 s2 =>
match find_label lbl s1 k with
| Some sk => Some sk
| None => find_label lbl s2 k
end
| Sloop s1 =>
find_label lbl s1 (Kseq (Sloop s1) k)
| Sblock s1 =>
find_label lbl s1 (Kblock k)
| Slabel lbl' s' =>
if ident_eq lbl lbl' then Some(s', k) else find_label lbl s' k
| _ => None
end.
(** One step of execution *)
Inductive step: state -> trace -> state -> Prop :=
| step_skip_seq: forall f s k sp e m,
step (State f Sskip (Kseq s k) sp e m)
E0 (State f s k sp e m)
| step_skip_block: forall f k sp e m,
step (State f Sskip (Kblock k) sp e m)
E0 (State f Sskip k sp e m)
| step_skip_call: forall f k sp e m m',
is_call_cont k ->
Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
E0 (Returnstate Vundef k m')
| step_assign: forall f id a k sp e m v,
eval_expr sp e m nil a v ->
step (State f (Sassign id a) k sp e m)
E0 (State f Sskip k sp (PTree.set id v e) m)
| step_store: forall f chunk addr al b k sp e m vl v vaddr m',
eval_exprlist sp e m nil al vl ->
eval_expr sp e m nil b v ->
eval_addressing ge sp addr vl = Some vaddr ->
Mem.storev chunk m vaddr v = Some m' ->
step (State f (Sstore chunk addr al b) k sp e m)
E0 (State f Sskip k sp e m')
| step_call: forall f optid sig a bl k sp e m vf vargs fd,
eval_expr_or_symbol sp e m nil a vf ->
eval_exprlist sp e m nil bl vargs ->
Genv.find_funct ge vf = Some fd ->
funsig fd = sig ->
step (State f (Scall optid sig a bl) k sp e m)
E0 (Callstate fd vargs (Kcall optid f sp e k) m)
| step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
Genv.find_funct ge vf = Some fd ->
funsig fd = sig ->
Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
E0 (Callstate fd vargs (call_cont k) m')
| step_builtin: forall f res ef al k sp e m vl t v m',
list_forall2 (eval_builtin_arg sp e m) al vl ->
external_call ef ge vl m t v m' ->
step (State f (Sbuiltin res ef al) k sp e m)
t (State f Sskip k sp (set_builtin_res res v e) m')
| step_seq: forall f s1 s2 k sp e m,
step (State f (Sseq s1 s2) k sp e m)
E0 (State f s1 (Kseq s2 k) sp e m)
| step_ifthenelse: forall f c s1 s2 k sp e m b,
eval_condexpr sp e m nil c b ->
step (State f (Sifthenelse c s1 s2) k sp e m)
E0 (State f (if b then s1 else s2) k sp e m)
| step_loop: forall f s k sp e m,
step (State f (Sloop s) k sp e m)
E0 (State f s (Kseq (Sloop s) k) sp e m)
| step_block: forall f s k sp e m,
step (State f (Sblock s) k sp e m)
E0 (State f s (Kblock k) sp e m)
| step_exit_seq: forall f n s k sp e m,
step (State f (Sexit n) (Kseq s k) sp e m)
E0 (State f (Sexit n) k sp e m)
| step_exit_block_0: forall f k sp e m,
step (State f (Sexit O) (Kblock k) sp e m)
E0 (State f Sskip k sp e m)
| step_exit_block_S: forall f n k sp e m,
step (State f (Sexit (S n)) (Kblock k) sp e m)
E0 (State f (Sexit n) k sp e m)
| step_switch: forall f a k sp e m n,
eval_exitexpr sp e m nil a n ->
step (State f (Sswitch a) k sp e m)
E0 (State f (Sexit n) k sp e m)
| step_return_0: forall f k sp e m m',
Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
E0 (Returnstate Vundef (call_cont k) m')
| step_return_1: forall f a k sp e m v m',
eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
E0 (Returnstate v (call_cont k) m')
| step_label: forall f lbl s k sp e m,
step (State f (Slabel lbl s) k sp e m)
E0 (State f s k sp e m)
| step_goto: forall f lbl k sp e m s' k',
find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
step (State f (Sgoto lbl) k sp e m)
E0 (State f s' k' sp e m)
| step_internal_function: forall f vargs k m m' sp e,
Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
step (Callstate (Internal f) vargs k m)
E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
| step_external_function: forall ef vargs k m t vres m',
external_call ef ge vargs m t vres m' ->
step (Callstate (External ef) vargs k m)
t (Returnstate vres k m')
| step_return: forall v optid f sp e k m,
step (Returnstate v (Kcall optid f sp e k) m)
E0 (State f Sskip k sp (set_optvar optid v e) m).
End RELSEM.
Inductive initial_state (p: program): state -> Prop :=
| initial_state_intro: forall b f m0,
let ge := Genv.globalenv p in
Genv.init_mem p = Some m0 ->
Genv.find_symbol ge p.(prog_main) = Some b ->
Genv.find_funct_ptr ge b = Some f ->
funsig f = signature_main ->
initial_state p (Callstate f nil Kstop m0).
Inductive final_state: state -> int -> Prop :=
| final_state_intro: forall r m,
final_state (Returnstate (Vint r) Kstop m) r.
Definition semantics (p: program) :=
Semantics step (initial_state p) final_state (Genv.globalenv p).
Hint Constructors eval_expr eval_exprlist eval_condexpr: evalexpr.
(** * Lifting of let-bound variables *)
(** Instruction selection sometimes generate [Elet] constructs to share the evaluation of a subexpression.  Owing to the use of de Bruijn indices for let-bound variables, we need to shift de Bruijn indices when an expression [b] is put in a [Elet a b] context. *)
Fixpoint lift_expr (p: nat) (a: expr) {struct a}: expr :=
match a with
| Evar id => Evar id
| Eop op bl => Eop op (lift_exprlist p bl)
| Eload chunk addr bl => Eload chunk addr (lift_exprlist p bl)
| Econdition a b c =>
Econdition (lift_condexpr p a) (lift_expr p b) (lift_expr p c)
| Elet b c => Elet (lift_expr p b) (lift_expr (S p) c)
| Eletvar n =>
if le_gt_dec p n then Eletvar (S n) else Eletvar n
| Ebuiltin ef bl => Ebuiltin ef (lift_exprlist p bl)
| Eexternal id sg bl => Eexternal id sg (lift_exprlist p bl)
end
with lift_exprlist (p: nat) (a: exprlist) {struct a}: exprlist :=
match a with
| Enil => Enil
| Econs b cl => Econs (lift_expr p b) (lift_exprlist p cl)
end
with lift_condexpr (p: nat) (a: condexpr) {struct a}: condexpr :=
match a with
| CEcond c al => CEcond c (lift_exprlist p al)
| CEcondition a b c => CEcondition (lift_condexpr p a) (lift_condexpr p b) (lift_condexpr p c)
| CElet a b => CElet (lift_expr p a) (lift_condexpr (S p) b)
end.
Definition lift (a: expr): expr := lift_expr O a.
(** We now relate the evaluation of a lifted expression with that of the original expression. *)
Inductive insert_lenv: letenv -> nat -> val -> letenv -> Prop :=
| insert_lenv_0:
forall le v,
insert_lenv le O v (v :: le)
| insert_lenv_S:
forall le p w le' v,
insert_lenv le p w le' ->
insert_lenv (v :: le) (S p) w (v :: le').

Lemma insert_lenv_lookup2:
forall le p w le',
insert_lenv le p w le' ->
forall n v,
nth_error le n = Some v -> (p <= n)%nat ->
nth_error le' (S n) = Some v.