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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. *)
(* *)
(* *********************************************************************)
Require Import Coqlib.
Require Import Compopts.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Op.
Require Import ValueDomain.
Require Import RTL.
(** Value analysis for ARM operators *)
Definition eval_static_shift (s: shift) (v: aval): aval :=
match s with
| Slsl x => shl v (I x)
| Slsr x => shru v (I x)
| Sasr x => shr v (I x)
| Sror x => ror v (I x)
end.
Definition eval_static_condition (cond: condition) (vl: list aval): abool :=
match cond, vl with
| Ccomp c, v1 :: v2 :: nil => cmp_bool c v1 v2
| Ccompu c, v1 :: v2 :: nil => cmpu_bool c v1 v2
| Ccompshift c s, v1 :: v2 :: nil => cmp_bool c v1 (eval_static_shift s v2)
| Ccompushift c s, v1 :: v2 :: nil => cmpu_bool c v1 (eval_static_shift s v2)
| Ccompimm c n, v1 :: nil => cmp_bool c v1 (I n)
| Ccompuimm c n, v1 :: nil => cmpu_bool c v1 (I n)
| Ccompf c, v1 :: v2 :: nil => cmpf_bool c v1 v2
| Cnotcompf c, v1 :: v2 :: nil => cnot (cmpf_bool c v1 v2)
| Ccompfzero c, v1 :: nil => cmpf_bool c v1 (F Float.zero)
| Cnotcompfzero c, v1 :: nil => cnot (cmpf_bool c v1 (F Float.zero))
| Ccompfs c, v1 :: v2 :: nil => cmpfs_bool c v1 v2
| Cnotcompfs c, v1 :: v2 :: nil => cnot (cmpfs_bool c v1 v2)
| Ccompfszero c, v1 :: nil => cmpfs_bool c v1 (FS Float32.zero)
| Cnotcompfszero c, v1 :: nil => cnot (cmpfs_bool c v1 (FS Float32.zero))
| _, _ => Bnone
end.
Definition eval_static_addressing (addr: addressing) (vl: list aval): aval :=
match addr, vl with
| Aindexed n, v1::nil => add v1 (I n)
| Aindexed2, v1::v2::nil => add v1 v2
| Aindexed2shift s, v1::v2::nil => add v1 (eval_static_shift s v2)
| Ainstack ofs, nil => Ptr(Stk ofs)
| _, _ => Vbot
end.
Definition eval_static_operation (op: operation) (vl: list aval): aval :=
match op, vl with
| Omove, v1::nil => v1
| Ointconst n, nil => I n
| Ofloatconst n, nil => if propagate_float_constants tt then F n else ftop
| Osingleconst n, nil => if propagate_float_constants tt then FS n else ftop
| Oaddrsymbol id ofs, nil => Ptr (Gl id ofs)
| Oaddrstack ofs, nil => Ptr (Stk ofs)
| Ocast8signed, v1 :: nil => sign_ext 8 v1
| Ocast16signed, v1 :: nil => sign_ext 16 v1
| Oadd, v1::v2::nil => add v1 v2
| Oaddshift s, v1::v2::nil => add v1 (eval_static_shift s v2)
| Oaddimm n, v1::nil => add v1 (I n)
| Osub, v1::v2::nil => sub v1 v2
| Osubshift s, v1::v2::nil => sub v1 (eval_static_shift s v2)
| Orsubshift s, v1::v2::nil => sub (eval_static_shift s v2) v1
| Orsubimm n, v1::nil => sub (I n) v1
| Omul, v1::v2::nil => mul v1 v2
| Omla, v1::v2::v3::nil => add (mul v1 v2) v3
| Omulhs, v1::v2::nil => mulhs v1 v2
| Omulhu, v1::v2::nil => mulhu v1 v2
| Odiv, v1::v2::nil => divs v1 v2
| Odivu, v1::v2::nil => divu v1 v2
| Oand, v1::v2::nil => and v1 v2
| Oandshift s, v1::v2::nil => and v1 (eval_static_shift s v2)
| Oandimm n, v1::nil => and v1 (I n)
| Oor, v1::v2::nil => or v1 v2
| Oorshift s, v1::v2::nil => or v1 (eval_static_shift s v2)
| Oorimm n, v1::nil => or v1 (I n)
| Oxor, v1::v2::nil => xor v1 v2
| Oxorshift s, v1::v2::nil => xor v1 (eval_static_shift s v2)
| Oxorimm n, v1::nil => xor v1 (I n)
| Obic, v1::v2::nil => and v1 (notint v2)
| Obicshift s, v1::v2::nil => and v1 (notint (eval_static_shift s v2))
| Onot, v1::nil => notint v1
| Onotshift s, v1::nil => notint (eval_static_shift s v1)
| Oshl, v1::v2::nil => shl v1 v2
| Oshr, v1::v2::nil => shr v1 v2
| Oshru, v1::v2::nil => shru v1 v2
| Oshift s, v1::nil => eval_static_shift s v1
| Oshrximm n, v1::nil => shrx v1 (I n)
| Onegf, v1::nil => negf v1
| Oabsf, v1::nil => absf v1
| Oaddf, v1::v2::nil => addf v1 v2
| Osubf, v1::v2::nil => subf v1 v2
| Omulf, v1::v2::nil => mulf v1 v2
| Odivf, v1::v2::nil => divf v1 v2
| Onegfs, v1::nil => negfs v1
| Oabsfs, v1::nil => absfs v1
| Oaddfs, v1::v2::nil => addfs v1 v2
| Osubfs, v1::v2::nil => subfs v1 v2
| Omulfs, v1::v2::nil => mulfs v1 v2
| Odivfs, v1::v2::nil => divfs v1 v2
| Osingleoffloat, v1::nil => singleoffloat v1
| Ofloatofsingle, v1::nil => floatofsingle v1
| Ointoffloat, v1::nil => intoffloat v1
| Ointuoffloat, v1::nil => intuoffloat v1
| Ofloatofint, v1::nil => floatofint v1
| Ofloatofintu, v1::nil => floatofintu v1
| Ointofsingle, v1::nil => intofsingle v1
| Ointuofsingle, v1::nil => intuofsingle v1
| Osingleofint, v1::nil => singleofint v1
| Osingleofintu, v1::nil => singleofintu v1
| Omakelong, v1::v2::nil => longofwords v1 v2
| Olowlong, v1::nil => loword v1
| Ohighlong, v1::nil => hiword v1
| Ocmp c, _ => of_optbool (eval_static_condition c vl)
| _, _ => Vbot
end.
Section SOUNDNESS.
Variable bc: block_classification.
Variable ge: genv.
Hypothesis GENV: genv_match bc ge.
Variable sp: block.
Hypothesis STACK: bc sp = BCstack.
Lemma eval_static_shift_sound:
forall s v a,
vmatch bc v a ->
vmatch bc (eval_shift s v) (eval_static_shift s a).
Proof.
intros. unfold eval_shift, eval_static_shift. destruct s; eauto with va.
Qed.
Hint Resolve eval_static_shift_sound: va.
Theorem eval_static_condition_sound:
forall cond vargs m aargs,
list_forall2 (vmatch bc) vargs aargs ->
cmatch (eval_condition cond vargs m) (eval_static_condition cond aargs).
Proof.
intros until aargs; intros VM.
inv VM.
destruct cond; auto with va.
inv H0.
destruct cond; simpl; eauto with va.
inv H2.
destruct cond; simpl; eauto with va.
destruct cond; auto with va.
Qed.
Lemma symbol_address_sound:
forall id ofs,
vmatch bc (Genv.symbol_address ge id ofs) (Ptr (Gl id ofs)).
Proof.
intros; apply symbol_address_sound; apply GENV.
Qed.
Hint Resolve symbol_address_sound: va.
Ltac InvHyps :=
match goal with
| [H: None = Some _ |- _ ] => discriminate
| [H: Some _ = Some _ |- _] => inv H
| [H1: match ?vl with nil => _ | _ :: _ => _ end = Some _ ,
H2: list_forall2 _ ?vl _ |- _ ] => inv H2; InvHyps
| _ => idtac
end.
Theorem eval_static_addressing_sound:
forall addr vargs vres aargs,
eval_addressing ge (Vptr sp Int.zero) addr vargs = Some vres ->
list_forall2 (vmatch bc) vargs aargs ->
vmatch bc vres (eval_static_addressing addr aargs).
Proof.
unfold eval_addressing, eval_static_addressing; intros;
destruct addr; InvHyps; eauto with va.
rewrite Int.add_zero_l; auto with va.
Qed.
Theorem eval_static_operation_sound:
forall op vargs m vres aargs,
eval_operation ge (Vptr sp Int.zero) op vargs m = Some vres ->
list_forall2 (vmatch bc) vargs aargs ->
vmatch bc vres (eval_static_operation op aargs).
Proof.
unfold eval_operation, eval_static_operation; intros;
destruct op; InvHyps; eauto with va.
destruct (propagate_float_constants tt); constructor.
destruct (propagate_float_constants tt); constructor.
rewrite Int.add_zero_l; eauto with va.
fold (Val.sub (Vint i) a1). auto with va.
apply of_optbool_sound. eapply eval_static_condition_sound; eauto.
Qed.
End SOUNDNESS.
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