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SelectOpproof.v
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SelectOpproof.v
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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. *)
(* *)
(* *********************************************************************)
(** Correctness of instruction selection for operators *)
Require Import Coqlib.
Require Import AST Integers Floats.
Require Import Values Memory Builtins Globalenvs.
Require Import Cminor Op CminorSel.
Require Import Compopts.
Require Import SelectOp.
Local Open Scope cminorsel_scope.
Local Transparent Archi.ptr64.
(** * Useful lemmas and tactics *)
(** The following are trivial lemmas and custom tactics that help
perform backward (inversion) and forward reasoning over the evaluation
of operator applications. *)
Ltac EvalOp := eapply eval_Eop; eauto with evalexpr.
Ltac InvEval1 :=
match goal with
| [ H: (eval_expr _ _ _ _ _ (Eop _ Enil) _) |- _ ] =>
inv H; InvEval1
| [ H: (eval_expr _ _ _ _ _ (Eop _ (_ ::: Enil)) _) |- _ ] =>
inv H; InvEval1
| [ H: (eval_expr _ _ _ _ _ (Eop _ (_ ::: _ ::: Enil)) _) |- _ ] =>
inv H; InvEval1
| [ H: (eval_exprlist _ _ _ _ _ Enil _) |- _ ] =>
inv H; InvEval1
| [ H: (eval_exprlist _ _ _ _ _ (_ ::: _) _) |- _ ] =>
inv H; InvEval1
| _ =>
idtac
end.
Ltac InvEval2 :=
match goal with
| [ H: (eval_operation _ _ _ nil _ = Some _) |- _ ] =>
simpl in H; inv H
| [ H: (eval_operation _ _ _ (_ :: nil) _ = Some _) |- _ ] =>
simpl in H; FuncInv
| [ H: (eval_operation _ _ _ (_ :: _ :: nil) _ = Some _) |- _ ] =>
simpl in H; FuncInv
| [ H: (eval_operation _ _ _ (_ :: _ :: _ :: nil) _ = Some _) |- _ ] =>
simpl in H; FuncInv
| _ =>
idtac
end.
Ltac InvEval := InvEval1; InvEval2; InvEval2.
Ltac TrivialExists :=
match goal with
| [ |- exists v, _ /\ Val.lessdef ?a v ] => exists a; split; [EvalOp | auto]
end.
(** * Correctness of the smart constructors *)
Section CMCONSTR.
Variable ge: genv.
Variable sp: val.
Variable e: env.
Variable m: mem.
(** We now show that the code generated by "smart constructor" functions
such as [SelectOp.notint] behaves as expected. Continuing the
[notint] example, we show that if the expression [e]
evaluates to some value [v], then [SelectOp.notint e]
evaluates to a value [v'] which is either [Val.notint v] or more defined
than [Val.notint v].
All proofs follow a common pattern:
- Reasoning by case over the result of the classification functions
(such as [add_match] for integer addition), gathering additional
information on the shape of the argument expressions in the non-default
cases.
- Inversion of the evaluations of the arguments, exploiting the additional
information thus gathered.
- Equational reasoning over the arithmetic operations performed,
using the lemmas from the [Int], [Float] and [Value] modules.
- Construction of an evaluation derivation for the expression returned
by the smart constructor.
*)
Definition unary_constructor_sound (cstr: expr -> expr) (sem: val -> val) : Prop :=
forall le a x,
eval_expr ge sp e m le a x ->
exists v, eval_expr ge sp e m le (cstr a) v /\ Val.lessdef (sem x) v.
Definition binary_constructor_sound (cstr: expr -> expr -> expr) (sem: val -> val -> val) : Prop :=
forall le a x b y,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
exists v, eval_expr ge sp e m le (cstr a b) v /\ Val.lessdef (sem x y) v.
Theorem eval_addrsymbol:
forall le id ofs,
exists v, eval_expr ge sp e m le (addrsymbol id ofs) v /\ Val.lessdef (Genv.symbol_address ge id ofs) v.
Proof.
intros. unfold addrsymbol. econstructor; split.
EvalOp. simpl; eauto.
auto.
Qed.
Theorem eval_addrstack:
forall le ofs,
exists v, eval_expr ge sp e m le (addrstack ofs) v /\ Val.lessdef (Val.offset_ptr sp ofs) v.
Proof.
intros. unfold addrstack. econstructor; split.
EvalOp. simpl; eauto.
auto.
Qed.
Theorem eval_notint: unary_constructor_sound notint Val.notint.
Proof.
assert (forall v, Val.lessdef (Val.notint (Val.notint v)) v).
destruct v; simpl; auto. rewrite Int.not_involutive; auto.
unfold notint; red; intros until x; case (notint_match a); intros; InvEval.
TrivialExists.
subst. exists v1; split; auto.
subst. TrivialExists.
subst. TrivialExists.
subst. TrivialExists.
subst. exists (Val.and v1 v0); split; auto. EvalOp.
subst. exists (Val.or v1 v0); split; auto. EvalOp.
subst. exists (Val.xor v1 v0); split; auto. EvalOp.
subst. exists (Val.or v0 (Val.notint v1)); split. EvalOp.
destruct v0; destruct v1; simpl; auto. rewrite Int.not_and_or_not. rewrite Int.not_involutive.
rewrite Int.or_commut. auto.
subst. exists (Val.and v0 (Val.notint v1)); split. EvalOp.
destruct v0; destruct v1; simpl; auto. rewrite Int.not_or_and_not. rewrite Int.not_involutive.
rewrite Int.and_commut. auto.
subst x. TrivialExists. simpl. rewrite Val.not_xor. rewrite Val.xor_assoc. auto.
TrivialExists.
Qed.
Remark shift_symbol_address:
forall id ofs delta,
Genv.symbol_address ge id (Ptrofs.add ofs (Ptrofs.of_int delta)) = Val.add (Genv.symbol_address ge id ofs) (Vint delta).
Proof.
intros. unfold Genv.symbol_address. destruct (Genv.find_symbol ge id); auto.
Qed.
Theorem eval_addimm:
forall n, unary_constructor_sound (addimm n) (fun x => Val.add x (Vint n)).
Proof.
red; unfold addimm; intros until x.
predSpec Int.eq Int.eq_spec n Int.zero.
subst n. intros. exists x; split; auto.
destruct x; simpl; auto. rewrite Int.add_zero. auto. rewrite Ptrofs.add_zero. auto.
case (addimm_match a); intros; InvEval; simpl; TrivialExists; simpl.
rewrite Int.add_commut. auto.
unfold Genv.symbol_address. destruct (Genv.find_symbol ge s); simpl; auto. rewrite Ptrofs.add_commut; auto.
destruct sp; simpl; auto. rewrite Ptrofs.add_assoc. do 3 f_equal. apply Ptrofs.add_commut.
subst. rewrite Val.add_assoc. rewrite Int.add_commut. auto.
subst. rewrite Ptrofs.add_commut. rewrite shift_symbol_address. rewrite ! Val.add_assoc. f_equal. f_equal. apply Val.add_commut.
Qed.
Theorem eval_addsymbol:
forall s ofs, unary_constructor_sound (addsymbol s ofs) (Val.add (Genv.symbol_address ge s ofs)).
Proof.
red; unfold addsymbol; intros until x.
case (addsymbol_match a); intros; InvEval; simpl; TrivialExists; simpl.
rewrite shift_symbol_address. auto.
rewrite shift_symbol_address. subst x. rewrite Val.add_assoc. f_equal. f_equal.
apply Val.add_commut.
Qed.
Theorem eval_add: binary_constructor_sound add Val.add.
Proof.
red; intros until y.
unfold add; case (add_match a b); intros; InvEval.
- rewrite Val.add_commut. apply eval_addimm; auto.
- apply eval_addimm; auto.
- apply eval_addsymbol; auto.
- rewrite Val.add_commut. apply eval_addsymbol; auto.
- subst.
replace (Val.add (Val.add v1 (Vint n1)) (Val.add v0 (Vint n2)))
with (Val.add (Val.add v1 v0) (Val.add (Vint n1) (Vint n2))).
apply eval_addimm. EvalOp.
repeat rewrite Val.add_assoc. decEq. apply Val.add_permut.
- subst.
replace (Val.add (Val.add v1 (Vint n1)) y)
with (Val.add (Val.add v1 y) (Vint n1)).
apply eval_addimm. EvalOp.
repeat rewrite Val.add_assoc. decEq. apply Val.add_commut.
- subst. TrivialExists.
econstructor. EvalOp. simpl. reflexivity. econstructor. eauto. constructor.
simpl. rewrite Val.add_permut, Val.add_commut. do 2 f_equal.
destruct sp; simpl; auto. rewrite Ptrofs.add_assoc; auto.
- replace (Val.add x y) with
(Val.add (Genv.symbol_address ge s (Ptrofs.add ofs (Ptrofs.of_int n))) (Val.add v1 v0)).
apply eval_addsymbol; auto. EvalOp.
subst. rewrite shift_symbol_address. rewrite ! Val.add_assoc. f_equal.
rewrite Val.add_permut. f_equal. apply Val.add_commut.
- subst. rewrite Val.add_assoc. apply eval_addsymbol. EvalOp.
- subst. rewrite <- Val.add_assoc. apply eval_addimm. EvalOp.
- subst. rewrite Val.add_permut. apply eval_addsymbol. EvalOp.
- TrivialExists.
Qed.
Theorem eval_subimm:
forall n, unary_constructor_sound (subimm n) (fun v => Val.sub (Vint n) v).
Proof.
intros; red; intros until x. unfold subimm. destruct (subimm_match a); intros.
InvEval. TrivialExists.
InvEval. subst x. TrivialExists. unfold eval_operation. destruct v1; simpl; auto.
rewrite ! Int.sub_add_opp. rewrite Int.add_assoc. f_equal. f_equal. f_equal.
rewrite Int.neg_add_distr. apply Int.add_commut.
TrivialExists.
Qed.
Theorem eval_sub: binary_constructor_sound sub Val.sub.
Proof.
red; intros until y.
unfold sub; case (sub_match a b); intros; InvEval.
rewrite Val.sub_add_opp. apply eval_addimm; auto.
apply eval_subimm; auto.
subst. rewrite Val.sub_add_l. rewrite Val.sub_add_r.
rewrite Val.add_assoc. simpl. rewrite Int.add_commut. rewrite <- Int.sub_add_opp.
apply eval_addimm; EvalOp.
subst. rewrite Val.sub_add_l. apply eval_addimm; EvalOp.
subst. rewrite Val.sub_add_r. apply eval_addimm; EvalOp.
TrivialExists.
Qed.
Theorem eval_negint: unary_constructor_sound negint (fun v => Val.sub Vzero v).
Proof.
red; intros. unfold negint. apply eval_subimm; auto.
Qed.
Lemma eval_rolm:
forall amount mask,
unary_constructor_sound (fun a => rolm a amount mask)
(fun x => Val.rolm x amount mask).
Proof.
red; intros until x. unfold rolm; case (rolm_match a); intros; InvEval.
TrivialExists.
subst. rewrite Val.rolm_rolm. TrivialExists.
subst. rewrite <- Val.rolm_zero. rewrite Val.rolm_rolm.
rewrite (Int.add_commut Int.zero). rewrite Int.add_zero. TrivialExists.
TrivialExists.
Qed.
Theorem eval_shlimm:
forall n, unary_constructor_sound (fun a => shlimm a n)
(fun x => Val.shl x (Vint n)).
Proof.
red; intros. unfold shlimm.
predSpec Int.eq Int.eq_spec n Int.zero.
subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.shl_zero; auto.
destruct (Int.ltu n Int.iwordsize) eqn:?.
rewrite Val.shl_rolm; auto. apply eval_rolm; auto.
TrivialExists. econstructor. eauto. econstructor. EvalOp. simpl; eauto. constructor. auto.
Qed.
Theorem eval_shruimm:
forall n, unary_constructor_sound (fun a => shruimm a n)
(fun x => Val.shru x (Vint n)).
Proof.
red; intros. unfold shruimm.
predSpec Int.eq Int.eq_spec n Int.zero.
subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.shru_zero; auto.
destruct (Int.ltu n Int.iwordsize) eqn:?.
rewrite Val.shru_rolm; auto. apply eval_rolm; auto.
TrivialExists. econstructor. eauto. econstructor. EvalOp. simpl; eauto. constructor. auto.
Qed.
Theorem eval_shrimm:
forall n, unary_constructor_sound (fun a => shrimm a n)
(fun x => Val.shr x (Vint n)).
Proof.
red; intros until x. unfold shrimm.
predSpec Int.eq Int.eq_spec n Int.zero.
intros. subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.shr_zero; auto.
destruct (Int.ltu n Int.iwordsize) eqn:WS.
case (shrimm_match a); intros.
InvEval. exists (Vint (Int.shr n1 n)); split. EvalOp. simpl; rewrite WS; auto.
simpl; destruct (Int.lt mask1 Int.zero) eqn:?.
TrivialExists.
replace (Val.shr x (Vint n)) with (Val.shru x (Vint n)).
apply eval_shruimm; auto.
destruct x; simpl; auto. rewrite WS.
decEq. symmetry. InvEval. destruct v1; simpl in H0; inv H0.
apply Int.shr_and_is_shru_and; auto.
simpl. TrivialExists.
intros. simpl. TrivialExists.
constructor. eauto. constructor. EvalOp. simpl; eauto. constructor. auto.
Qed.
Lemma eval_mulimm_base:
forall n, unary_constructor_sound (mulimm_base n) (fun x => Val.mul x (Vint n)).
Proof.
intros; red; intros; unfold mulimm_base.
generalize (Int.one_bits_decomp n).
generalize (Int.one_bits_range n).
destruct (Int.one_bits n).
intros. TrivialExists.
destruct l.
intros. rewrite H1. simpl.
rewrite Int.add_zero.
replace (Vint (Int.shl Int.one i)) with (Val.shl Vone (Vint i)). rewrite Val.shl_mul.
apply eval_shlimm. auto. simpl. rewrite H0; auto with coqlib.
destruct l.
intros. destruct (optim_for_size tt). TrivialExists.
rewrite H1. simpl.
exploit (eval_shlimm i (x :: le) (Eletvar 0) x). constructor; auto. intros [v1 [A1 B1]].
exploit (eval_shlimm i0 (x :: le) (Eletvar 0) x). constructor; auto. intros [v2 [A2 B2]].
exists (Val.add v1 v2); split.
econstructor. eauto. EvalOp.
rewrite Int.add_zero.
replace (Vint (Int.add (Int.shl Int.one i) (Int.shl Int.one i0)))
with (Val.add (Val.shl Vone (Vint i)) (Val.shl Vone (Vint i0))).
rewrite Val.mul_add_distr_r.
repeat rewrite Val.shl_mul. apply Val.add_lessdef; auto.
simpl. repeat rewrite H0; auto with coqlib.
intros. TrivialExists.
Qed.
Theorem eval_mulimm:
forall n, unary_constructor_sound (mulimm n) (fun x => Val.mul x (Vint n)).
Proof.
intros; red; intros until x; unfold mulimm.
predSpec Int.eq Int.eq_spec n Int.zero.
intros. exists (Vint Int.zero); split. EvalOp.
destruct x; simpl; auto. subst n. rewrite Int.mul_zero. auto.
predSpec Int.eq Int.eq_spec n Int.one.
intros. exists x; split; auto.
destruct x; simpl; auto. subst n. rewrite Int.mul_one. auto.
case (mulimm_match a); intros; InvEval.
TrivialExists. simpl. rewrite Int.mul_commut; auto.
subst. rewrite Val.mul_add_distr_l.
exploit eval_mulimm_base; eauto. instantiate (1 := n). intros [v' [A1 B1]].
exploit (eval_addimm (Int.mul n n2) le (mulimm_base n t2) v'). auto. intros [v'' [A2 B2]].
exists v''; split; auto. eapply Val.lessdef_trans. eapply Val.add_lessdef; eauto.
rewrite Val.mul_commut; auto.
apply eval_mulimm_base; auto.
Qed.
Theorem eval_mul: binary_constructor_sound mul Val.mul.
Proof.
red; intros until y.
unfold mul; case (mul_match a b); intros; InvEval.
rewrite Val.mul_commut. apply eval_mulimm. auto.
apply eval_mulimm. auto.
TrivialExists.
Qed.
Theorem eval_mulhs: binary_constructor_sound mulhs Val.mulhs.
Proof.
unfold mulhs; red; intros; TrivialExists.
Qed.
Theorem eval_mulhu: binary_constructor_sound mulhu Val.mulhu.
Proof.
unfold mulhu; red; intros; TrivialExists.
Qed.
Theorem eval_andimm:
forall n, unary_constructor_sound (andimm n) (fun x => Val.and x (Vint n)).
Proof.
intros; red; intros until x. unfold andimm.
predSpec Int.eq Int.eq_spec n Int.zero.
intros. subst. exists (Vint Int.zero); split. EvalOp.
destruct x; simpl; auto. rewrite Int.and_zero; auto.
predSpec Int.eq Int.eq_spec n Int.mone.
intros. subst. exists x; split. auto.
destruct x; simpl; auto. rewrite Int.and_mone; auto.
clear H H0.
case (andimm_match a); intros.
InvEval. TrivialExists. simpl. rewrite Int.and_commut; auto.
set (n' := Int.and n n2).
destruct (Int.eq (Int.shru (Int.shl n' amount) amount) n' &&
Int.ltu amount Int.iwordsize) eqn:?.
InvEval. destruct (andb_prop _ _ Heqb).
generalize (Int.eq_spec (Int.shru (Int.shl n' amount) amount) n'). rewrite H1; intros.
replace (Val.and x (Vint n))
with (Val.rolm v0 (Int.sub Int.iwordsize amount) (Int.and (Int.shru Int.mone amount) n')).
apply eval_rolm; auto.
subst. destruct v0; simpl; auto. rewrite H3. simpl. decEq. rewrite Int.and_assoc.
rewrite (Int.and_commut n2 n).
transitivity (Int.and (Int.shru i amount) (Int.and n n2)).
rewrite (Int.shru_rolm i); auto. unfold Int.rolm. rewrite Int.and_assoc; auto.
symmetry. apply Int.shr_and_shru_and. auto.
set (e2 := Eop (Oshrimm amount) (t2 ::: Enil)) in *.
InvEval. subst. rewrite Val.and_assoc. simpl. rewrite Int.and_commut. TrivialExists.
InvEval. subst. rewrite Val.and_assoc. simpl. rewrite Int.and_commut. TrivialExists.
InvEval. subst. TrivialExists. simpl.
destruct v1; auto. simpl. unfold Int.rolm. rewrite Int.and_assoc.
decEq. decEq. decEq. apply Int.and_commut.
destruct (Int.eq (Int.shru (Int.shl n amount) amount) n &&
Int.ltu amount Int.iwordsize) eqn:?.
InvEval. destruct (andb_prop _ _ Heqb).
generalize (Int.eq_spec (Int.shru (Int.shl n amount) amount) n). rewrite H0; intros.
replace (Val.and x (Vint n))
with (Val.rolm v1 (Int.sub Int.iwordsize amount) (Int.and (Int.shru Int.mone amount) n)).
apply eval_rolm; auto.
subst x. destruct v1; simpl; auto. rewrite H1; simpl. decEq.
transitivity (Int.and (Int.shru i amount) n).
rewrite (Int.shru_rolm i); auto. unfold Int.rolm. rewrite Int.and_assoc; auto.
symmetry. apply Int.shr_and_shru_and. auto.
TrivialExists.
TrivialExists.
Qed.
Theorem eval_and: binary_constructor_sound and Val.and.
Proof.
red; intros until y; unfold and; case (and_match a b); intros; InvEval.
rewrite Val.and_commut. apply eval_andimm; auto.
apply eval_andimm; auto.
subst. rewrite Val.and_commut. TrivialExists.
subst. TrivialExists.
TrivialExists.
Qed.
Theorem eval_orimm:
forall n, unary_constructor_sound (orimm n) (fun x => Val.or x (Vint n)).
Proof.
intros; red; intros until x. unfold orimm.
predSpec Int.eq Int.eq_spec n Int.zero.
intros. subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.or_zero; auto.
predSpec Int.eq Int.eq_spec n Int.mone.
intros. subst. exists (Vint Int.mone); split. EvalOp. destruct x; simpl; auto. rewrite Int.or_mone; auto.
clear H H0. destruct (orimm_match a); intros; InvEval.
TrivialExists. simpl. rewrite Int.or_commut; auto.
subst. rewrite Val.or_assoc. simpl. rewrite Int.or_commut. TrivialExists.
TrivialExists.
Qed.
Remark eval_same_expr:
forall a1 a2 le v1 v2,
same_expr_pure a1 a2 = true ->
eval_expr ge sp e m le a1 v1 ->
eval_expr ge sp e m le a2 v2 ->
a1 = a2 /\ v1 = v2.
Proof.
intros until v2.
destruct a1; simpl; try (intros; discriminate).
destruct a2; simpl; try (intros; discriminate).
case (ident_eq i i0); intros.
subst i0. inversion H0. inversion H1. split. auto. congruence.
discriminate.
Qed.
Theorem eval_or: binary_constructor_sound or Val.or.
Proof.
red; intros until y; unfold or; case (or_match a b); intros.
(* rolm - rolm *)
destruct (Int.eq amount1 amount2 && same_expr_pure t1 t2) eqn:?.
destruct (andb_prop _ _ Heqb0).
generalize (Int.eq_spec amount1 amount2). rewrite H1. intro. subst amount2.
InvEval. exploit eval_same_expr; eauto. intros [EQ1 EQ2]. subst.
rewrite Val.or_rolm. TrivialExists.
TrivialExists.
(* andimm - rolm *)
destruct (Int.eq mask1 (Int.not mask2) && is_rlw_mask mask2) eqn:?.
destruct (andb_prop _ _ Heqb0).
generalize (Int.eq_spec mask1 (Int.not mask2)); rewrite H1; intros.
InvEval. subst. TrivialExists.
TrivialExists.
(* rolm - andimm *)
destruct (Int.eq mask2 (Int.not mask1) && is_rlw_mask mask1) eqn:?.
destruct (andb_prop _ _ Heqb0).
generalize (Int.eq_spec mask2 (Int.not mask1)); rewrite H1; intros.
InvEval. subst. rewrite Val.or_commut. TrivialExists.
TrivialExists.
(* intconst *)
InvEval. rewrite Val.or_commut. apply eval_orimm; auto.
InvEval. apply eval_orimm; auto.
(* orc *)
InvEval. subst. rewrite Val.or_commut. TrivialExists.
InvEval. subst. TrivialExists.
(* default *)
TrivialExists.
Qed.
Theorem eval_xorimm:
forall n, unary_constructor_sound (xorimm n) (fun x => Val.xor x (Vint n)).
Proof.
intros; red; intros until x. unfold xorimm.
predSpec Int.eq Int.eq_spec n Int.zero.
intros. subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.xor_zero; auto.
predSpec Int.eq Int.eq_spec n Int.mone.
intros. subst. rewrite <- Val.not_xor. apply eval_notint; auto.
clear H H0. destruct (xorimm_match a); intros; InvEval.
TrivialExists. simpl. rewrite Int.xor_commut; auto.
subst. rewrite Val.xor_assoc. simpl. rewrite Int.xor_commut. TrivialExists.
subst x. TrivialExists. simpl. rewrite Val.not_xor. rewrite Val.xor_assoc.
simpl. rewrite Int.xor_commut; auto.
TrivialExists.
Qed.
Theorem eval_xor: binary_constructor_sound xor Val.xor.
Proof.
red; intros until y; unfold xor; case (xor_match a b); intros; InvEval.
rewrite Val.xor_commut. apply eval_xorimm; auto.
apply eval_xorimm; auto.
subst. rewrite Val.xor_commut. rewrite Val.not_xor. rewrite <- Val.xor_assoc.
rewrite <- Val.not_xor. rewrite Val.xor_commut. TrivialExists.
subst. rewrite Val.not_xor. rewrite <- Val.xor_assoc. rewrite <- Val.not_xor. TrivialExists.
TrivialExists.
Qed.
Theorem eval_divs_base:
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.divs x y = Some z ->
exists v, eval_expr ge sp e m le (divs_base a b) v /\ Val.lessdef z v.
Proof.
intros. unfold divs_base. exists z; split. EvalOp. auto.
Qed.
Lemma eval_mod_aux:
forall divop semdivop,
(forall sp x y m, eval_operation ge sp divop (x :: y :: nil) m = semdivop x y) ->
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
semdivop x y = Some z ->
eval_expr ge sp e m le (mod_aux divop a b) (Val.sub x (Val.mul z y)).
Proof.
intros; unfold mod_aux.
eapply eval_Elet. eexact H0. eapply eval_Elet.
apply eval_lift. eexact H1.
eapply eval_Eop. eapply eval_Econs.
eapply eval_Eletvar. simpl; reflexivity.
eapply eval_Econs. eapply eval_Eop.
eapply eval_Econs. eapply eval_Eop.
eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity.
eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity.
apply eval_Enil.
rewrite H. eauto.
eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity.
apply eval_Enil.
simpl; reflexivity. apply eval_Enil.
reflexivity.
Qed.
Theorem eval_mods_base:
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.mods x y = Some z ->
exists v, eval_expr ge sp e m le (mods_base a b) v /\ Val.lessdef z v.
Proof.
intros; unfold mods_base.
exploit Val.mods_divs; eauto. intros [v [A B]].
subst. econstructor; split; eauto.
apply eval_mod_aux with (semdivop := Val.divs); auto.
Qed.
Theorem eval_divu_base:
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.divu x y = Some z ->
exists v, eval_expr ge sp e m le (divu_base a b) v /\ Val.lessdef z v.
Proof.
intros. unfold divu_base. exists z; split. EvalOp. auto.
Qed.
Theorem eval_modu_base:
forall le a b x y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.modu x y = Some z ->
exists v, eval_expr ge sp e m le (modu_base a b) v /\ Val.lessdef z v.
Proof.
intros; unfold modu_base.
exploit Val.modu_divu; eauto. intros [v [A B]].
subst. econstructor; split; eauto.
apply eval_mod_aux with (semdivop := Val.divu); auto.
Qed.
Theorem eval_shrximm:
forall le a n x z,
eval_expr ge sp e m le a x ->
Val.shrx x (Vint n) = Some z ->
exists v, eval_expr ge sp e m le (shrximm a n) v /\ Val.lessdef z v.
Proof.
intros. unfold shrximm.
predSpec Int.eq Int.eq_spec n Int.zero.
subst n. exists x; split; auto.
destruct x; simpl in H0; try discriminate.
destruct (Int.ltu Int.zero (Int.repr 31)); inv H0.
replace (Int.shrx i Int.zero) with i. auto.
unfold Int.shrx, Int.divs. rewrite Int.shl_zero.
change (Int.signed Int.one) with 1. rewrite Z.quot_1_r. rewrite Int.repr_signed; auto.
econstructor; split. EvalOp. auto.
Qed.
Theorem eval_shl: binary_constructor_sound shl Val.shl.
Proof.
red; intros until y; unfold shl; case (shl_match b); intros.
InvEval. apply eval_shlimm; auto.
TrivialExists.
Qed.
Theorem eval_shr: binary_constructor_sound shr Val.shr.
Proof.
red; intros until y; unfold shr; case (shr_match b); intros.
InvEval. apply eval_shrimm; auto.
TrivialExists.
Qed.
Theorem eval_shru: binary_constructor_sound shru Val.shru.
Proof.
red; intros until y; unfold shru; case (shru_match b); intros.
InvEval. apply eval_shruimm; auto.
TrivialExists.
Qed.
Theorem eval_negf: unary_constructor_sound negf Val.negf.
Proof.
red; intros. TrivialExists.
Qed.
Theorem eval_absf: unary_constructor_sound absf Val.absf.
Proof.
red; intros. TrivialExists.
Qed.
Theorem eval_addf: binary_constructor_sound addf Val.addf.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_subf: binary_constructor_sound subf Val.subf.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_mulf: binary_constructor_sound mulf Val.mulf.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_negfs: unary_constructor_sound negfs Val.negfs.
Proof.
red; intros. TrivialExists.
Qed.
Theorem eval_absfs: unary_constructor_sound absfs Val.absfs.
Proof.
red; intros. TrivialExists.
Qed.
Theorem eval_addfs: binary_constructor_sound addfs Val.addfs.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_subfs: binary_constructor_sound subfs Val.subfs.
Proof.
red; intros; TrivialExists.
Qed.
Theorem eval_mulfs: binary_constructor_sound mulfs Val.mulfs.
Proof.
red; intros; TrivialExists.
Qed.
Section COMP_IMM.
Variable default: comparison -> int -> condition.
Variable intsem: comparison -> int -> int -> bool.
Variable sem: comparison -> val -> val -> val.
Hypothesis sem_int: forall c x y, sem c (Vint x) (Vint y) = Val.of_bool (intsem c x y).
Hypothesis sem_undef: forall c v, sem c Vundef v = Vundef.
Hypothesis sem_eq: forall x y, sem Ceq (Vint x) (Vint y) = Val.of_bool (Int.eq x y).
Hypothesis sem_ne: forall x y, sem Cne (Vint x) (Vint y) = Val.of_bool (negb (Int.eq x y)).
Hypothesis sem_default: forall c v n, sem c v (Vint n) = Val.of_optbool (eval_condition (default c n) (v :: nil) m).
Lemma eval_compimm:
forall le c a n2 x,
eval_expr ge sp e m le a x ->
exists v, eval_expr ge sp e m le (compimm default intsem c a n2) v
/\ Val.lessdef (sem c x (Vint n2)) v.
Proof.
intros until x.
unfold compimm; case (compimm_match c a); intros.
(* constant *)
InvEval. rewrite sem_int. TrivialExists. simpl. destruct (intsem c0 n1 n2); auto.
(* eq cmp *)
InvEval. inv H. simpl in H5. inv H5.
destruct (Int.eq_dec n2 Int.zero). subst n2. TrivialExists.
simpl. rewrite eval_negate_condition.
destruct (eval_condition c0 vl m); simpl.
unfold Vtrue, Vfalse. destruct b; simpl; rewrite sem_eq; auto.
rewrite sem_undef; auto.
destruct (Int.eq_dec n2 Int.one). subst n2. TrivialExists.
simpl. destruct (eval_condition c0 vl m); simpl.
unfold Vtrue, Vfalse. destruct b; simpl; rewrite sem_eq; auto.
rewrite sem_undef; auto.
exists (Vint Int.zero); split. EvalOp.
destruct (eval_condition c0 vl m); simpl.
unfold Vtrue, Vfalse. destruct b; rewrite sem_eq; rewrite Int.eq_false; auto.
rewrite sem_undef; auto.
(* ne cmp *)
InvEval. inv H. simpl in H5. inv H5.
destruct (Int.eq_dec n2 Int.zero). subst n2. TrivialExists.
simpl. destruct (eval_condition c0 vl m); simpl.
unfold Vtrue, Vfalse. destruct b; simpl; rewrite sem_ne; auto.
rewrite sem_undef; auto.
destruct (Int.eq_dec n2 Int.one). subst n2. TrivialExists.
simpl. rewrite eval_negate_condition. destruct (eval_condition c0 vl m); simpl.
unfold Vtrue, Vfalse. destruct b; simpl; rewrite sem_ne; auto.
rewrite sem_undef; auto.
exists (Vint Int.one); split. EvalOp.
destruct (eval_condition c0 vl m); simpl.
unfold Vtrue, Vfalse. destruct b; rewrite sem_ne; rewrite Int.eq_false; auto.
rewrite sem_undef; auto.
(* eq andimm *)
destruct (Int.eq_dec n2 Int.zero). InvEval; subst.
econstructor; split. EvalOp. simpl; eauto.
destruct v1; simpl; try (rewrite sem_undef; auto). rewrite sem_eq.
destruct (Int.eq (Int.and i n1) Int.zero); auto.
TrivialExists. simpl. rewrite sem_default. auto.
(* ne andimm *)
destruct (Int.eq_dec n2 Int.zero). InvEval; subst.
econstructor; split. EvalOp. simpl; eauto.
destruct v1; simpl; try (rewrite sem_undef; auto). rewrite sem_ne.
destruct (Int.eq (Int.and i n1) Int.zero); auto.
TrivialExists. simpl. rewrite sem_default. auto.
(* default *)
TrivialExists. simpl. rewrite sem_default. auto.
Qed.
Hypothesis sem_swap:
forall c x y, sem (swap_comparison c) x y = sem c y x.
Lemma eval_compimm_swap:
forall le c a n2 x,
eval_expr ge sp e m le a x ->
exists v, eval_expr ge sp e m le (compimm default intsem (swap_comparison c) a n2) v
/\ Val.lessdef (sem c (Vint n2) x) v.
Proof.
intros. rewrite <- sem_swap. eapply eval_compimm; eauto.
Qed.
End COMP_IMM.
Theorem eval_comp:
forall c, binary_constructor_sound (comp c) (Val.cmp c).
Proof.
intros; red; intros until y. unfold comp; case (comp_match a b); intros; InvEval.
eapply eval_compimm_swap; eauto.
intros. unfold Val.cmp. rewrite Val.swap_cmp_bool; auto.
eapply eval_compimm; eauto.
TrivialExists.
Qed.
Theorem eval_compu:
forall c, binary_constructor_sound (compu c) (Val.cmpu (Mem.valid_pointer m) c).
Proof.
intros; red; intros until y. unfold compu; case (compu_match a b); intros; InvEval.
eapply eval_compimm_swap; eauto.
intros. unfold Val.cmpu. rewrite Val.swap_cmpu_bool; auto.
eapply eval_compimm; eauto.
TrivialExists.
Qed.
Theorem eval_compf:
forall c, binary_constructor_sound (compf c) (Val.cmpf c).
Proof.
intros; red; intros. unfold compf. TrivialExists.
Qed.
Theorem eval_compfs:
forall c, binary_constructor_sound (compfs c) (Val.cmpfs c).
Proof.
intros; red; intros. unfold compfs.
replace (Val.cmpfs c x y) with
(Val.cmpf c (Val.floatofsingle x) (Val.floatofsingle y)).
TrivialExists. constructor. EvalOp. simpl; reflexivity.
constructor. EvalOp. simpl; reflexivity. constructor.
auto.
destruct x; auto. destruct y; auto. unfold Val.cmpf, Val.cmpfs; simpl.
rewrite Float32.cmp_double. auto.
Qed.
Theorem eval_cast8signed: unary_constructor_sound cast8signed (Val.sign_ext 8).
Proof.
red; intros until x. unfold cast8signed. destruct (cast8signed_match a); intros.
InvEval; TrivialExists.
TrivialExists.
Qed.
Theorem eval_cast8unsigned: unary_constructor_sound cast8unsigned (Val.zero_ext 8).
Proof.
red; intros. unfold cast8unsigned.
rewrite Val.zero_ext_and. apply eval_andimm; auto. lia.
Qed.
Theorem eval_cast16signed: unary_constructor_sound cast16signed (Val.sign_ext 16).
Proof.
red; intros until x. unfold cast16signed. destruct (cast16signed_match a); intros.
InvEval; TrivialExists.
TrivialExists.
Qed.
Theorem eval_cast16unsigned: unary_constructor_sound cast16unsigned (Val.zero_ext 16).
Proof.
red; intros. unfold cast16unsigned.
rewrite Val.zero_ext_and. apply eval_andimm; auto. lia.
Qed.
Theorem eval_singleoffloat: unary_constructor_sound singleoffloat Val.singleoffloat.
Proof.
red; intros. unfold singleoffloat. TrivialExists.
Qed.
Theorem eval_floatofsingle: unary_constructor_sound floatofsingle Val.floatofsingle.
Proof.
red; intros. unfold floatofsingle. TrivialExists.
Qed.
Theorem eval_intoffloat:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.intoffloat x = Some y ->
exists v, eval_expr ge sp e m le (intoffloat a) v /\ Val.lessdef y v.
Proof.
intros; unfold intoffloat. TrivialExists.
Qed.
Theorem eval_intuoffloat:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.intuoffloat x = Some y ->
exists v, eval_expr ge sp e m le (intuoffloat a) v /\ Val.lessdef y v.
Proof.
intros. destruct x; simpl in H0; try discriminate.
destruct (Float.to_intu f) as [n|] eqn:?; simpl in H0; inv H0.
exists (Vint n); split; auto. unfold intuoffloat.
destruct Archi.ppc64.
- apply Float.to_intu_to_long in Heqo.
econstructor. constructor. econstructor. econstructor; eauto. constructor.
simpl; rewrite Heqo; simpl; eauto. constructor.
simpl. unfold Int64.loword. rewrite Int64.unsigned_repr, Int.repr_unsigned. auto.
assert (Int.modulus < Int64.max_unsigned) by (compute; auto).
generalize (Int.unsigned_range n). lia.
- set (im := Int.repr Int.half_modulus).
set (fm := Float.of_intu im).
assert (eval_expr ge sp e m (Vfloat fm :: Vfloat f :: le) (Eletvar (S O)) (Vfloat f)).
constructor. auto.
assert (eval_expr ge sp e m (Vfloat fm :: Vfloat f :: le) (Eletvar O) (Vfloat fm)).
constructor. auto.
econstructor. eauto.
econstructor. instantiate (1 := Vfloat fm). EvalOp.
eapply eval_Econdition with (va := Float.cmp Clt f fm).
eauto with evalexpr.
destruct (Float.cmp Clt f fm) eqn:?.
exploit Float.to_intu_to_int_1; eauto. intro EQ.
EvalOp. simpl. rewrite EQ; auto.
exploit Float.to_intu_to_int_2; eauto.
change Float.ox8000_0000 with im. fold fm. intro EQ.
set (t2 := subf (Eletvar (S O)) (Eletvar O)).
set (t3 := intoffloat t2).
exploit (eval_subf (Vfloat fm :: Vfloat f :: le) (Eletvar (S O)) (Vfloat f) (Eletvar O)); eauto.
fold t2. intros [v2 [A2 B2]]. simpl in B2. inv B2.
exploit (eval_addimm Float.ox8000_0000 (Vfloat fm :: Vfloat f :: le) t3).
unfold t3. unfold intoffloat. EvalOp. simpl. rewrite EQ. simpl. eauto.
intros [v4 [A4 B4]]. simpl in B4. inv B4.
rewrite Int.sub_add_opp in A4. rewrite Int.add_assoc in A4.
rewrite (Int.add_commut (Int.neg im)) in A4.
rewrite Int.add_neg_zero in A4.
rewrite Int.add_zero in A4.
auto.
Qed.
Theorem eval_floatofint:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.floatofint x = Some y ->
exists v, eval_expr ge sp e m le (floatofint a) v /\ Val.lessdef y v.
Proof.
intros until y. unfold floatofint. destruct (floatofint_match a); intros.
InvEval. TrivialExists.
rename e0 into a. destruct x; simpl in H0; inv H0.
exists (Vfloat (Float.of_int i)); split; auto.
destruct Archi.ppc64.
- rewrite Float.of_int_of_long.
EvalOp. constructor. EvalOp. simpl; eauto. constructor. auto.
- set (t1 := addimm Float.ox8000_0000 a).
set (t2 := Eop Ofloatofwords (Eop (Ointconst Float.ox4330_0000) Enil ::: t1 ::: Enil)).
set (t3 := Eop (Ofloatconst (Float.from_words Float.ox4330_0000 Float.ox8000_0000)) Enil).
exploit (eval_addimm Float.ox8000_0000 le a). eauto. fold t1.
intros [v1 [A1 B1]]. simpl in B1. inv B1.
exploit (eval_subf le t2).
unfold t2. EvalOp. constructor. EvalOp. simpl; eauto. constructor. eauto. constructor.
unfold eval_operation. eauto.
instantiate (2 := t3). unfold t3. EvalOp. simpl; eauto.
intros [v2 [A2 B2]]. simpl in B2. inv B2. rewrite Float.of_int_from_words. auto.
Qed.
Theorem eval_floatofintu:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.floatofintu x = Some y ->
exists v, eval_expr ge sp e m le (floatofintu a) v /\ Val.lessdef y v.
Proof.
intros until y. unfold floatofintu. destruct (floatofintu_match a); intros.
InvEval. TrivialExists.
rename e0 into a. destruct x; simpl in H0; inv H0.
exists (Vfloat (Float.of_intu i)); split; auto.
destruct Archi.ppc64.
- rewrite Float.of_intu_of_long.
EvalOp. constructor. EvalOp. simpl; eauto. constructor. auto.
- set (t2 := Eop Ofloatofwords (Eop (Ointconst Float.ox4330_0000) Enil ::: a ::: Enil)).
set (t3 := Eop (Ofloatconst (Float.from_words Float.ox4330_0000 Int.zero)) Enil).
exploit (eval_subf le t2).
unfold t2. EvalOp. constructor. EvalOp. simpl; eauto. constructor. eauto. constructor.
unfold eval_operation. eauto.
instantiate (2 := t3). unfold t3. EvalOp. simpl; eauto.
intros [v2 [A2 B2]]. simpl in B2. inv B2. rewrite Float.of_intu_from_words. auto.
Qed.
Theorem eval_intofsingle:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.intofsingle x = Some y ->
exists v, eval_expr ge sp e m le (intofsingle a) v /\ Val.lessdef y v.
Proof.
intros; unfold intofsingle.
assert (Val.intoffloat (Val.floatofsingle x) = Some y).
{ destruct x; simpl in H0; try discriminate.
destruct (Float32.to_int f) eqn:F; inv H0.
apply Float32.to_int_double in F.
simpl. unfold Float32.to_double in F; rewrite F; auto.
}
apply eval_intoffloat with (Val.floatofsingle x); auto. EvalOp.
Qed.
Theorem eval_singleofint:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.singleofint x = Some y ->
exists v, eval_expr ge sp e m le (singleofint a) v /\ Val.lessdef y v.
Proof.
intros. unfold singleofint.
assert (exists z, Val.floatofint x = Some z /\ y = Val.singleoffloat z).
{
destruct x; inv H0. simpl. exists (Vfloat (Float.of_int i)); simpl; split; auto.
f_equal. apply Float32.of_int_double.
}
destruct H1 as (z & A & B). subst y.
exploit eval_floatofint; eauto. intros (v & C & D).
exists (Val.singleoffloat v); split. EvalOp. inv D; auto.
Qed.
Theorem eval_intuofsingle:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.intuofsingle x = Some y ->
exists v, eval_expr ge sp e m le (intuofsingle a) v /\ Val.lessdef y v.
Proof.
intros; unfold intuofsingle.
assert (Val.intuoffloat (Val.floatofsingle x) = Some y).
{ destruct x; simpl in H0; try discriminate.
destruct (Float32.to_intu f) eqn:F; inv H0.
apply Float32.to_intu_double in F.
simpl. unfold Float32.to_double in F; rewrite F; auto.
}
apply eval_intuoffloat with (Val.floatofsingle x); auto. EvalOp.
Qed.
Theorem eval_singleofintu:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.singleofintu x = Some y ->
exists v, eval_expr ge sp e m le (singleofintu a) v /\ Val.lessdef y v.
Proof.
intros. unfold singleofintu.
assert (exists z, Val.floatofintu x = Some z /\ y = Val.singleoffloat z).
{
destruct x; inv H0. simpl. exists (Vfloat (Float.of_intu i)); simpl; split; auto.
f_equal. apply Float32.of_intu_double.
}
destruct H1 as (z & A & B). subst y.
exploit eval_floatofintu; eauto. intros (v & C & D).
exists (Val.singleoffloat v); split. EvalOp. inv D; auto.
Qed.