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Asmgenproof1.v
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Asmgenproof1.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 proof for ARM code generation: auxiliary results. *)
Require Import Coqlib.
Require Import Errors.
Require Import Maps.
Require Import AST.
Require Import Zbits.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Op.
Require Import Locations.
Require Import Mach.
Require Import Compopts.
Require Import Asm.
Require Import Asmgen.
Require Import Conventions.
Require Import Asmgenproof0.
Local Transparent Archi.ptr64.
(** Useful properties of the R14 registers. *)
Lemma ireg_of_not_R14:
forall m r, ireg_of m = OK r -> IR r <> IR IR14.
Proof.
intros. erewrite <- ireg_of_eq; eauto with asmgen.
Qed.
Global Hint Resolve ireg_of_not_R14: asmgen.
Lemma ireg_of_not_R14':
forall m r, ireg_of m = OK r -> r <> IR14.
Proof.
intros. generalize (ireg_of_not_R14 _ _ H). congruence.
Qed.
Global Hint Resolve ireg_of_not_R14': asmgen.
(** [undef_flags] and [nextinstr_nf] *)
Lemma nextinstr_nf_pc:
forall rs, (nextinstr_nf rs)#PC = Val.offset_ptr rs#PC Ptrofs.one.
Proof.
intros. reflexivity.
Qed.
Definition if_preg (r: preg) : bool :=
match r with
| IR _ => true
| FR _ => true
| CR _ => false
| PC => false
end.
Lemma data_if_preg: forall r, data_preg r = true -> if_preg r = true.
Proof.
intros. destruct r; reflexivity || discriminate.
Qed.
Lemma if_preg_not_PC: forall r, if_preg r = true -> r <> PC.
Proof.
intros; red; intros; subst; discriminate.
Qed.
Global Hint Resolve data_if_preg if_preg_not_PC: asmgen.
Lemma nextinstr_nf_inv:
forall r rs, if_preg r = true -> (nextinstr_nf rs)#r = rs#r.
Proof.
intros. destruct r; reflexivity || discriminate.
Qed.
Lemma nextinstr_nf_inv1:
forall r rs, data_preg r = true -> (nextinstr_nf rs)#r = rs#r.
Proof.
intros. destruct r; reflexivity || discriminate.
Qed.
(** Useful simplification tactic *)
Ltac Simplif :=
((rewrite nextinstr_inv by eauto with asmgen)
|| (rewrite nextinstr_inv1 by eauto with asmgen)
|| (rewrite nextinstr_nf_inv by eauto with asmgen)
|| (rewrite Pregmap.gss)
|| (rewrite nextinstr_pc)
|| (rewrite nextinstr_nf_pc)
|| (rewrite Pregmap.gso by eauto with asmgen)); auto with asmgen.
Ltac Simpl := repeat Simplif.
(** * Correctness of ARM constructor functions *)
Section CONSTRUCTORS.
Variable ge: genv.
Variable fn: function.
(** Decomposition of an integer constant *)
Lemma decompose_int_arm_or:
forall N n p x, List.fold_left Int.or (decompose_int_arm N n p) x = Int.or x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.or_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl (Int.repr 3) p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.or_assoc. decEq. rewrite <- Int.and_or_distrib.
rewrite Int.or_not_self. apply Int.and_mone.
Qed.
Lemma decompose_int_arm_xor:
forall N n p x, List.fold_left Int.xor (decompose_int_arm N n p) x = Int.xor x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.xor_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl (Int.repr 3) p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.xor_assoc. decEq. rewrite <- Int.and_xor_distrib.
rewrite Int.xor_not_self. apply Int.and_mone.
Qed.
Lemma decompose_int_arm_add:
forall N n p x, List.fold_left Int.add (decompose_int_arm N n p) x = Int.add x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.add_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl (Int.repr 3) p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.add_assoc. decEq. rewrite Int.add_and.
rewrite Int.or_not_self. apply Int.and_mone. apply Int.and_not_self.
Qed.
Remark decompose_int_arm_nil:
forall N n p, decompose_int_arm N n p = nil -> n = Int.zero.
Proof.
intros. generalize (decompose_int_arm_or N n p Int.zero). rewrite H. simpl.
rewrite Int.or_commut; rewrite Int.or_zero; auto.
Qed.
Lemma decompose_int_thumb_or:
forall N n p x, List.fold_left Int.or (decompose_int_thumb N n p) x = Int.or x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.or_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl Int.one p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.or_assoc. decEq. rewrite <- Int.and_or_distrib.
rewrite Int.or_not_self. apply Int.and_mone.
Qed.
Lemma decompose_int_thumb_xor:
forall N n p x, List.fold_left Int.xor (decompose_int_thumb N n p) x = Int.xor x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.xor_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl Int.one p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.xor_assoc. decEq. rewrite <- Int.and_xor_distrib.
rewrite Int.xor_not_self. apply Int.and_mone.
Qed.
Lemma decompose_int_thumb_add:
forall N n p x, List.fold_left Int.add (decompose_int_thumb N n p) x = Int.add x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.add_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl Int.one p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.add_assoc. decEq. rewrite Int.add_and.
rewrite Int.or_not_self. apply Int.and_mone. apply Int.and_not_self.
Qed.
Remark decompose_int_thumb_nil:
forall N n p, decompose_int_thumb N n p = nil -> n = Int.zero.
Proof.
intros. generalize (decompose_int_thumb_or N n p Int.zero). rewrite H. simpl.
rewrite Int.or_commut; rewrite Int.or_zero; auto.
Qed.
Lemma decompose_int_general:
forall (f: val -> int -> val) (g: int -> int -> int),
(forall v1 n2 n3, f (f v1 n2) n3 = f v1 (g n2 n3)) ->
(forall n1 n2 n3, g (g n1 n2) n3 = g n1 (g n2 n3)) ->
(forall n, g Int.zero n = n) ->
(forall N n p x, List.fold_left g (decompose_int_arm N n p) x = g x n) ->
(forall N n p x, List.fold_left g (decompose_int_thumb N n p) x = g x n) ->
forall n v,
List.fold_left f (decompose_int n) v = f v n.
Proof.
intros f g DISTR ASSOC ZERO DECOMP1 DECOMP2.
assert (A: forall l x y, g x (fold_left g l y) = fold_left g l (g x y)).
induction l; intros; simpl. auto. rewrite IHl. decEq. rewrite ASSOC; auto.
assert (B: forall l v n, fold_left f l (f v n) = f v (fold_left g l n)).
induction l; intros; simpl.
auto.
rewrite IHl. rewrite DISTR. decEq. decEq. auto.
intros. unfold decompose_int, decompose_int_base.
destruct (thumb tt); [destruct (is_immed_arith_thumb_special n)|].
- reflexivity.
- destruct (decompose_int_thumb 24%nat n Int.zero) eqn:DB.
+ simpl. exploit decompose_int_thumb_nil; eauto. congruence.
+ simpl. rewrite B. decEq.
generalize (DECOMP2 24%nat n Int.zero Int.zero).
rewrite DB; simpl. rewrite ! ZERO. auto.
- destruct (decompose_int_arm 12%nat n Int.zero) eqn:DB.
+ simpl. exploit decompose_int_arm_nil; eauto. congruence.
+ simpl. rewrite B. decEq.
generalize (DECOMP1 12%nat n Int.zero Int.zero).
rewrite DB; simpl. rewrite ! ZERO. auto.
Qed.
Lemma decompose_int_or:
forall n v,
List.fold_left (fun v i => Val.or v (Vint i)) (decompose_int n) v = Val.or v (Vint n).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.or v (Vint n)) (g := Int.or).
intros. rewrite Val.or_assoc. auto.
apply Int.or_assoc.
intros. rewrite Int.or_commut. apply Int.or_zero.
apply decompose_int_arm_or. apply decompose_int_thumb_or.
Qed.
Lemma decompose_int_bic:
forall n v,
List.fold_left (fun v i => Val.and v (Vint (Int.not i))) (decompose_int n) v = Val.and v (Vint (Int.not n)).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.and v (Vint (Int.not n))) (g := Int.or).
intros. rewrite Val.and_assoc. simpl. decEq. decEq. rewrite Int.not_or_and_not. auto.
apply Int.or_assoc.
intros. rewrite Int.or_commut. apply Int.or_zero.
apply decompose_int_arm_or. apply decompose_int_thumb_or.
Qed.
Lemma decompose_int_xor:
forall n v,
List.fold_left (fun v i => Val.xor v (Vint i)) (decompose_int n) v = Val.xor v (Vint n).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.xor v (Vint n)) (g := Int.xor).
intros. rewrite Val.xor_assoc. auto.
apply Int.xor_assoc.
intros. rewrite Int.xor_commut. apply Int.xor_zero.
apply decompose_int_arm_xor. apply decompose_int_thumb_xor.
Qed.
Lemma decompose_int_add:
forall n v,
List.fold_left (fun v i => Val.add v (Vint i)) (decompose_int n) v = Val.add v (Vint n).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.add v (Vint n)) (g := Int.add).
intros. rewrite Val.add_assoc. auto.
apply Int.add_assoc.
intros. rewrite Int.add_commut. apply Int.add_zero.
apply decompose_int_arm_add. apply decompose_int_thumb_add.
Qed.
Lemma decompose_int_sub:
forall n v,
List.fold_left (fun v i => Val.sub v (Vint i)) (decompose_int n) v = Val.sub v (Vint n).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.sub v (Vint n)) (g := Int.add).
intros. repeat rewrite Val.sub_add_opp. rewrite Val.add_assoc. decEq. simpl. decEq.
rewrite Int.neg_add_distr; auto.
apply Int.add_assoc.
intros. rewrite Int.add_commut. apply Int.add_zero.
apply decompose_int_arm_add. apply decompose_int_thumb_add.
Qed.
Lemma iterate_op_correct:
forall op1 op2 (f: val -> int -> val) (rs: regset) (r: ireg) m v0 n k,
(forall (rs:regset) n,
exec_instr ge fn (op2 (SOimm n)) rs m =
Next (nextinstr_nf (rs#r <- (f (rs#r) n))) m) ->
(forall n,
exec_instr ge fn (op1 (SOimm n)) rs m =
Next (nextinstr_nf (rs#r <- (f v0 n))) m) ->
exists rs',
exec_straight ge fn (iterate_op op1 op2 (decompose_int n) k) rs m k rs' m
/\ rs'#r = List.fold_left f (decompose_int n) v0
/\ forall r': preg, r' <> r -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros until k; intros SEM2 SEM1.
unfold iterate_op.
destruct (decompose_int n) as [ | i tl] eqn:DI.
unfold decompose_int in DI. destruct (decompose_int_base n); congruence.
revert k. pattern tl. apply List.rev_ind.
(* base case *)
intros; simpl. econstructor.
split. apply exec_straight_one. rewrite SEM1. reflexivity. reflexivity.
intuition Simpl.
(* inductive case *)
intros.
rewrite List.map_app. simpl. rewrite app_ass. simpl.
destruct (H (op2 (SOimm x) :: k)) as [rs' [A [B C]]].
econstructor.
split. eapply exec_straight_trans. eexact A. apply exec_straight_one.
rewrite SEM2. reflexivity. reflexivity.
split. rewrite fold_left_app; simpl. Simpl. rewrite B. auto.
intros; Simpl.
Qed.
(** Loading a constant. *)
Lemma loadimm_correct:
forall r n k rs m,
exists rs',
exec_straight ge fn (loadimm r n k) rs m k rs' m
/\ rs'#r = Vint n
/\ forall r': preg, r' <> r -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold loadimm.
set (l1 := length (decompose_int n)).
set (l2 := length (decompose_int (Int.not n))).
destruct (Nat.leb l1 1%nat).
{ (* single mov *)
econstructor; split. apply exec_straight_one. simpl; reflexivity. auto.
split; intros; Simpl. }
destruct (Nat.leb l2 1%nat).
{ (* single movn *)
econstructor; split. apply exec_straight_one.
simpl. rewrite Int.not_involutive. reflexivity. auto.
split; intros; Simpl. }
destruct Archi.thumb2_support.
{ (* movw / movt *)
unfold loadimm_word. destruct (Int.eq (Int.shru n (Int.repr 16)) Int.zero).
econstructor; split.
apply exec_straight_one. simpl; eauto. auto. split; intros; Simpl.
econstructor; split.
eapply exec_straight_two. simpl; reflexivity. simpl; reflexivity. auto. auto.
split; intros; Simpl. simpl. f_equal. rewrite Int.zero_ext_and by lia.
rewrite Int.and_assoc. change 65535 with (two_p 16 - 1). rewrite Int.and_idem.
apply Int.same_bits_eq; intros.
rewrite Int.bits_or, Int.bits_and, Int.bits_shl, Int.testbit_repr by auto.
rewrite Ztestbit_two_p_m1 by lia. change (Int.unsigned (Int.repr 16)) with 16.
destruct (zlt i 16).
rewrite andb_true_r, orb_false_r; auto.
rewrite andb_false_r; simpl. rewrite Int.bits_shru by lia.
change (Int.unsigned (Int.repr 16)) with 16. rewrite zlt_true by lia. f_equal; lia.
}
destruct (Nat.leb l1 l2).
{ (* mov - orr* *)
replace (Vint n) with (List.fold_left (fun v i => Val.or v (Vint i)) (decompose_int n) Vzero).
apply iterate_op_correct.
auto.
intros; simpl. rewrite Int.or_commut; rewrite Int.or_zero; auto.
rewrite decompose_int_or. simpl. rewrite Int.or_commut; rewrite Int.or_zero; auto.
}
{ (* mvn - bic* *)
replace (Vint n) with (List.fold_left (fun v i => Val.and v (Vint (Int.not i))) (decompose_int (Int.not n)) (Vint Int.mone)).
apply iterate_op_correct.
auto.
intros. simpl. rewrite Int.and_commut; rewrite Int.and_mone; auto.
rewrite decompose_int_bic. simpl. rewrite Int.not_involutive. rewrite Int.and_commut. rewrite Int.and_mone; auto.
}
Qed.
(** Add integer immediate. *)
Lemma addimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (addimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.add rs#r2 (Vint n)
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold addimm.
destruct (Int.ltu (Int.repr (-256)) n).
(* sub *)
econstructor; split. apply exec_straight_one; simpl; auto.
split; intros; Simpl. apply Val.sub_opp_add.
destruct (Nat.leb (length (decompose_int n)) (length (decompose_int (Int.neg n)))).
(* add - add* *)
replace (Val.add (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.add v (Vint i)) (decompose_int n) (rs r2)).
apply iterate_op_correct.
auto.
auto.
apply decompose_int_add.
(* sub - sub* *)
replace (Val.add (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.sub v (Vint i)) (decompose_int (Int.neg n)) (rs r2)).
apply iterate_op_correct.
auto.
auto.
rewrite decompose_int_sub. apply Val.sub_opp_add.
Qed.
(* And integer immediate *)
Lemma andimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (andimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.and rs#r2 (Vint n)
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold andimm. destruct (is_immed_arith n).
(* andi *)
exists (nextinstr_nf (rs#r1 <- (Val.and rs#r2 (Vint n)))).
split. apply exec_straight_one; auto. split; intros; Simpl.
(* bic - bic* *)
replace (Val.and (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.and v (Vint (Int.not i))) (decompose_int (Int.not n)) (rs r2)).
apply iterate_op_correct.
auto. auto.
rewrite decompose_int_bic. rewrite Int.not_involutive. auto.
Qed.
(** Reverse sub immediate *)
Lemma rsubimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (rsubimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.sub (Vint n) rs#r2
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold rsubimm.
(* rsb - add* *)
replace (Val.sub (Vint n) (rs r2))
with (List.fold_left (fun v i => Val.add v (Vint i)) (decompose_int n) (Val.neg (rs r2))).
apply iterate_op_correct.
auto.
intros. simpl. destruct (rs r2); auto. simpl. rewrite Int.sub_add_opp.
rewrite Int.add_commut; auto.
rewrite decompose_int_add.
destruct (rs r2); simpl; auto. rewrite Int.sub_add_opp. rewrite Int.add_commut; auto.
Qed.
(** Or immediate *)
Lemma orimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (orimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.or rs#r2 (Vint n)
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold orimm.
(* ori - ori* *)
replace (Val.or (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.or v (Vint i)) (decompose_int n) (rs r2)).
apply iterate_op_correct.
auto.
auto.
apply decompose_int_or.
Qed.
(** Xor immediate *)
Lemma xorimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (xorimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.xor rs#r2 (Vint n)
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold xorimm.
(* xori - xori* *)
replace (Val.xor (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.xor v (Vint i)) (decompose_int n) (rs r2)).
apply iterate_op_correct.
auto.
auto.
apply decompose_int_xor.
Qed.
(** Indexed memory loads. *)
Lemma indexed_memory_access_correct:
forall (P: regset -> Prop) (mk_instr: ireg -> int -> instruction)
(mk_immed: int -> int) (base: ireg) n k (rs: regset) m m',
(forall (r1: ireg) (rs1: regset) n1 k,
Val.add rs1#r1 (Vint n1) = Val.add rs#base (Vint n) ->
(forall (r: preg), if_preg r = true -> r <> IR14 -> rs1 r = rs r) ->
exists rs',
exec_straight ge fn (mk_instr r1 n1 :: k) rs1 m k rs' m' /\ P rs') ->
exists rs',
exec_straight ge fn
(indexed_memory_access mk_instr mk_immed base n k) rs m
k rs' m'
/\ P rs'.
Proof.
intros until m'; intros SEM.
unfold indexed_memory_access.
destruct (Int.eq n (mk_immed n)).
- apply SEM; auto.
- destruct (addimm_correct IR14 base (Int.sub n (mk_immed n)) (mk_instr IR14 (mk_immed n) :: k) rs m)
as (rs1 & A & B & C).
destruct (SEM IR14 rs1 (mk_immed n) k) as (rs2 & D & E).
rewrite B. rewrite Val.add_assoc. f_equal. simpl.
rewrite Int.sub_add_opp. rewrite Int.add_assoc.
rewrite (Int.add_commut (Int.neg (mk_immed n))).
rewrite Int.add_neg_zero. rewrite Int.add_zero. auto.
auto with asmgen.
exists rs2; split; auto. eapply exec_straight_trans; eauto.
Qed.
Lemma loadind_int_correct:
forall (base: ireg) ofs dst (rs: regset) m v k,
Mem.loadv Mint32 m (Val.offset_ptr rs#base ofs) = Some v ->
exists rs',
exec_straight ge fn (loadind_int base ofs dst k) rs m k rs' m
/\ rs'#dst = v
/\ forall r, if_preg r = true -> r <> IR14 -> r <> dst -> rs'#r = rs#r.
Proof.
intros; unfold loadind_int.
assert (Val.offset_ptr (rs base) ofs = Val.add (rs base) (Vint (Ptrofs.to_int ofs))).
{ destruct (rs base); try discriminate. simpl. f_equal; f_equal. symmetry; auto with ptrofs. }
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H1, <- H0, H. eauto. auto.
split; intros; Simpl.
Qed.
Lemma loadind_correct:
forall (base: ireg) ofs ty dst k c (rs: regset) m v,
loadind base ofs ty dst k = OK c ->
Mem.loadv (chunk_of_type ty) m (Val.offset_ptr rs#base ofs) = Some v ->
exists rs',
exec_straight ge fn c rs m k rs' m
/\ rs'#(preg_of dst) = v
/\ forall r, if_preg r = true -> r <> IR14 -> r <> preg_of dst -> rs'#r = rs#r.
Proof.
unfold loadind; intros.
assert (Val.offset_ptr (rs base) ofs = Val.add (rs base) (Vint (Ptrofs.to_int ofs))).
{ destruct (rs base); try discriminate. simpl. f_equal; f_equal. symmetry; auto with ptrofs. }
destruct ty; destruct (preg_of dst); inv H; simpl in H0.
- (* int *)
apply loadind_int_correct; auto.
- (* float *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H, <- H1, H0. eauto. auto.
split; intros; Simpl.
- (* single *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H, <- H1, H0. eauto. auto.
split; intros; Simpl.
- (* any32 *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H, <- H1, H0. eauto. auto.
split; intros; Simpl.
- (* any64 *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H, <- H1, H0. eauto. auto.
split; intros; Simpl.
Qed.
(** Indexed memory stores. *)
Lemma storeind_correct:
forall (base: ireg) ofs ty src k c (rs: regset) m m',
storeind src base ofs ty k = OK c ->
Mem.storev (chunk_of_type ty) m (Val.offset_ptr rs#base ofs) (rs#(preg_of src)) = Some m' ->
exists rs',
exec_straight ge fn c rs m k rs' m'
/\ forall r, if_preg r = true -> r <> IR14 -> rs'#r = rs#r.
Proof.
unfold storeind; intros.
assert (DATA: data_preg (preg_of src) = true) by eauto with asmgen.
assert (Val.offset_ptr (rs base) ofs = Val.add (rs base) (Vint (Ptrofs.to_int ofs))).
{ destruct (rs base); try discriminate. simpl. f_equal; f_equal. symmetry; auto with ptrofs. }
destruct ty; destruct (preg_of src); inv H; simpl in H0.
- (* int *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
- (* float *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
- (* single *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
- (* any32 *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
- (* any64 *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
Qed.
(** Saving the link register *)
Lemma save_lr_correct:
forall ofs k (rs: regset) m m',
Mem.storev Mint32 m (Val.offset_ptr rs#IR13 ofs) (rs#IR14) = Some m' ->
exists rs',
exec_straight ge fn (save_lr ofs k) rs m k rs' m'
/\ (forall r, if_preg r = true -> r <> IR12 -> rs'#r = rs#r)
/\ (save_lr_preserves_R12 ofs = true -> rs'#IR12 = rs#IR12).
Proof.
intros; unfold save_lr, save_lr_preserves_R12.
set (n := Ptrofs.to_int ofs). set (n1 := mk_immed_mem_word n).
assert (EQ: Val.offset_ptr rs#IR13 ofs = Val.add rs#IR13 (Vint n)).
{ destruct rs#IR13; try discriminate. simpl. f_equal; f_equal. unfold n; symmetry; auto with ptrofs. }
destruct (Int.eq n n1).
- econstructor; split. apply exec_straight_one. simpl; unfold exec_store. rewrite <- EQ, H; reflexivity. auto.
split. intros; Simpl. intros; Simpl.
- destruct (addimm_correct IR12 IR13 (Int.sub n n1) (Pstr IR14 IR12 (SOimm n1) :: k) rs m)
as (rs1 & A & B & C).
econstructor; split.
eapply exec_straight_trans. eexact A.
apply exec_straight_one. simpl; unfold exec_store.
rewrite B. rewrite Val.add_assoc. simpl.
rewrite Int.sub_add_opp. rewrite Int.add_assoc.
rewrite (Int.add_commut (Int.neg n1)).
rewrite Int.add_neg_zero. rewrite Int.add_zero.
rewrite <- EQ. rewrite C by eauto with asmgen. rewrite H. reflexivity.
auto.
split. intros; Simpl. congruence.
Qed.
(** Translation of shift immediates *)
Lemma transl_shift_correct:
forall s (r: ireg) (rs: regset),
eval_shift_op (transl_shift s r) rs = eval_shift s (rs#r).
Proof.
intros. destruct s; simpl; auto.
Qed.
(** Translation of conditions *)
Lemma compare_int_spec:
forall rs v1 v2 m,
let rs1 := nextinstr (compare_int rs v1 v2 m) in
rs1#CN = Val.negative (Val.sub v1 v2)
/\ rs1#CZ = Val.cmpu (Mem.valid_pointer m) Ceq v1 v2
/\ rs1#CC = Val.cmpu (Mem.valid_pointer m) Cge v1 v2
/\ rs1#CV = Val.sub_overflow v1 v2.
Proof.
intros. unfold rs1. intuition.
Qed.
Lemma compare_int_inv:
forall rs v1 v2 m,
let rs1 := nextinstr (compare_int rs v1 v2 m) in
forall r', data_preg r' = true -> rs1#r' = rs#r'.
Proof.
intros. unfold rs1, compare_int.
repeat Simplif.
Qed.
Lemma int_signed_eq:
forall x y, Int.eq x y = zeq (Int.signed x) (Int.signed y).
Proof.
intros. unfold Int.eq. unfold proj_sumbool.
destruct (zeq (Int.unsigned x) (Int.unsigned y));
destruct (zeq (Int.signed x) (Int.signed y)); auto.
elim n. unfold Int.signed. rewrite e; auto.
elim n. apply Int.eqm_small_eq; auto with ints.
eapply Int.eqm_trans. apply Int.eqm_sym. apply Int.eqm_signed_unsigned.
rewrite e. apply Int.eqm_signed_unsigned.
Qed.
Lemma int_not_lt:
forall x y, negb (Int.lt y x) = (Int.lt x y || Int.eq x y).
Proof.
intros. unfold Int.lt. rewrite int_signed_eq. unfold proj_sumbool.
destruct (zlt (Int.signed y) (Int.signed x)).
rewrite zlt_false. rewrite zeq_false. auto. lia. lia.
destruct (zeq (Int.signed x) (Int.signed y)).
rewrite zlt_false. auto. lia.
rewrite zlt_true. auto. lia.
Qed.
Lemma int_lt_not:
forall x y, Int.lt y x = negb (Int.lt x y) && negb (Int.eq x y).
Proof.
intros. rewrite <- negb_orb. rewrite <- int_not_lt. rewrite negb_involutive. auto.
Qed.
Lemma int_not_ltu:
forall x y, negb (Int.ltu y x) = (Int.ltu x y || Int.eq x y).
Proof.
intros. unfold Int.ltu, Int.eq.
destruct (zlt (Int.unsigned y) (Int.unsigned x)).
rewrite zlt_false. rewrite zeq_false. auto. lia. lia.
destruct (zeq (Int.unsigned x) (Int.unsigned y)).
rewrite zlt_false. auto. lia.
rewrite zlt_true. auto. lia.
Qed.
Lemma int_ltu_not:
forall x y, Int.ltu y x = negb (Int.ltu x y) && negb (Int.eq x y).
Proof.
intros. rewrite <- negb_orb. rewrite <- int_not_ltu. rewrite negb_involutive. auto.
Qed.
Lemma cond_for_signed_cmp_correct:
forall c v1 v2 rs m b,
Val.cmp_bool c v1 v2 = Some b ->
eval_testcond (cond_for_signed_cmp c)
(nextinstr (compare_int rs v1 v2 m)) = Some b.
Proof.
intros. generalize (compare_int_spec rs v1 v2 m).
set (rs' := nextinstr (compare_int rs v1 v2 m)).
intros [A [B [C D]]].
destruct v1; destruct v2; simpl in H; inv H.
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
simpl. unfold Val.cmp, Val.cmpu.
rewrite Int.lt_sub_overflow.
destruct c; simpl.
destruct (Int.eq i i0); auto.
destruct (Int.eq i i0); auto.
destruct (Int.lt i i0); auto.
rewrite int_not_lt. destruct (Int.lt i i0); simpl; destruct (Int.eq i i0); auto.
rewrite (int_lt_not i i0). destruct (Int.lt i i0); destruct (Int.eq i i0); reflexivity.
destruct (Int.lt i i0); reflexivity.
Qed.
Lemma cond_for_unsigned_cmp_correct:
forall c v1 v2 rs m b,
Val.cmpu_bool (Mem.valid_pointer m) c v1 v2 = Some b ->
eval_testcond (cond_for_unsigned_cmp c)
(nextinstr (compare_int rs v1 v2 m)) = Some b.
Proof.
intros. generalize (compare_int_spec rs v1 v2 m).
set (rs' := nextinstr (compare_int rs v1 v2 m)).
intros [A [B [C D]]].
unfold eval_testcond. rewrite B; rewrite C. unfold Val.cmpu, Val.cmp.
destruct v1; destruct v2; simpl in H; inv H.
(* int int *)
destruct c; simpl; auto.
destruct (Int.eq i i0); reflexivity.
destruct (Int.eq i i0); auto.
destruct (Int.ltu i i0); auto.
rewrite (int_not_ltu i i0). destruct (Int.ltu i i0); destruct (Int.eq i i0); auto.
rewrite (int_ltu_not i i0). destruct (Int.ltu i i0); destruct (Int.eq i i0); reflexivity.
destruct (Int.ltu i i0); reflexivity.
(* int ptr *)
destruct (Int.eq i Int.zero &&
(Mem.valid_pointer m b0 (Ptrofs.unsigned i0) || Mem.valid_pointer m b0 (Ptrofs.unsigned i0 - 1))) eqn:?; try discriminate.
destruct c; simpl in *; inv H1.
rewrite Heqb1; reflexivity.
rewrite Heqb1; reflexivity.
(* ptr int *)
destruct (Int.eq i0 Int.zero &&
(Mem.valid_pointer m b0 (Ptrofs.unsigned i) || Mem.valid_pointer m b0 (Ptrofs.unsigned i - 1))) eqn:?; try discriminate.
destruct c; simpl in *; inv H1.
rewrite Heqb1; reflexivity.
rewrite Heqb1; reflexivity.
(* ptr ptr *)
simpl.
fold (Mem.weak_valid_pointer m b0 (Ptrofs.unsigned i)) in *.
fold (Mem.weak_valid_pointer m b1 (Ptrofs.unsigned i0)) in *.
destruct (eq_block b0 b1).
destruct (Mem.weak_valid_pointer m b0 (Ptrofs.unsigned i) &&
Mem.weak_valid_pointer m b1 (Ptrofs.unsigned i0)); inversion H1.
destruct c; simpl; auto.
destruct (Ptrofs.eq i i0); reflexivity.
destruct (Ptrofs.eq i i0); auto.
destruct (Ptrofs.ltu i i0); auto.
rewrite (Ptrofs.not_ltu i i0). destruct (Ptrofs.ltu i i0); simpl; destruct (Ptrofs.eq i i0); auto.
rewrite (Ptrofs.ltu_not i i0). destruct (Ptrofs.ltu i i0); destruct (Ptrofs.eq i i0); reflexivity.
destruct (Ptrofs.ltu i i0); reflexivity.
destruct (Mem.valid_pointer m b0 (Ptrofs.unsigned i) &&
Mem.valid_pointer m b1 (Ptrofs.unsigned i0)); try discriminate.
destruct c; simpl in *; inv H1; reflexivity.
Qed.
Lemma compare_float_spec:
forall rs f1 f2,
let rs1 := nextinstr (compare_float rs (Vfloat f1) (Vfloat f2)) in
rs1#CN = Val.of_bool (Float.cmp Clt f1 f2)
/\ rs1#CZ = Val.of_bool (Float.cmp Ceq f1 f2)
/\ rs1#CC = Val.of_bool (negb (Float.cmp Clt f1 f2))
/\ rs1#CV = Val.of_bool (negb (Float.cmp Ceq f1 f2 || Float.cmp Clt f1 f2 || Float.cmp Cgt f1 f2)).
Proof.
intros. intuition.
Qed.
Lemma compare_float_inv:
forall rs v1 v2,
let rs1 := nextinstr (compare_float rs v1 v2) in
forall r', data_preg r' = true -> rs1#r' = rs#r'.
Proof.
intros. unfold rs1, compare_float.
assert (nextinstr (rs#CN <- Vundef #CZ <- Vundef #CC <- Vundef #CV <- Vundef) r' = rs r').
{ repeat Simplif. }
destruct v1; destruct v2; auto.
repeat Simplif.
Qed.
Lemma compare_float_nextpc:
forall rs v1 v2,
nextinstr (compare_float rs v1 v2) PC = Val.offset_ptr (rs PC) Ptrofs.one.
Proof.
intros. unfold compare_float. destruct v1; destruct v2; reflexivity.
Qed.
Lemma cond_for_float_cmp_correct:
forall c n1 n2 rs,
eval_testcond (cond_for_float_cmp c)
(nextinstr (compare_float rs (Vfloat n1) (Vfloat n2))) =
Some(Float.cmp c n1 n2).
Proof.
intros.
generalize (compare_float_spec rs n1 n2).
set (rs' := nextinstr (compare_float rs (Vfloat n1) (Vfloat n2))).
intros [A [B [C D]]].
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
destruct c; simpl.
(* eq *)
destruct (Float.cmp Ceq n1 n2); auto.
(* ne *)
rewrite Float.cmp_ne_eq. destruct (Float.cmp Ceq n1 n2); auto.
(* lt *)
destruct (Float.cmp Clt n1 n2); auto.
(* le *)
rewrite Float.cmp_le_lt_eq.
destruct (Float.cmp Clt n1 n2); destruct (Float.cmp Ceq n1 n2); auto.
(* gt *)
destruct (Float.cmp Ceq n1 n2) eqn:EQ;
destruct (Float.cmp Clt n1 n2) eqn:LT;
destruct (Float.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
exfalso; eapply Float.cmp_gt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
(* ge *)
rewrite Float.cmp_ge_gt_eq.
destruct (Float.cmp Ceq n1 n2) eqn:EQ;
destruct (Float.cmp Clt n1 n2) eqn:LT;
destruct (Float.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float.cmp_lt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
Qed.
Lemma cond_for_float_not_cmp_correct:
forall c n1 n2 rs,
eval_testcond (cond_for_float_not_cmp c)
(nextinstr (compare_float rs (Vfloat n1) (Vfloat n2)))=
Some(negb(Float.cmp c n1 n2)).
Proof.
intros.
generalize (compare_float_spec rs n1 n2).
set (rs' := nextinstr (compare_float rs (Vfloat n1) (Vfloat n2))).
intros [A [B [C D]]].
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
destruct c; simpl.
(* eq *)
destruct (Float.cmp Ceq n1 n2); auto.
(* ne *)
rewrite Float.cmp_ne_eq. destruct (Float.cmp Ceq n1 n2); auto.
(* lt *)
destruct (Float.cmp Clt n1 n2); auto.
(* le *)
rewrite Float.cmp_le_lt_eq.
destruct (Float.cmp Clt n1 n2) eqn:LT; destruct (Float.cmp Ceq n1 n2) eqn:EQ; auto.
(* gt *)
destruct (Float.cmp Ceq n1 n2) eqn:EQ;
destruct (Float.cmp Clt n1 n2) eqn:LT;
destruct (Float.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
exfalso; eapply Float.cmp_gt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
(* ge *)
rewrite Float.cmp_ge_gt_eq.
destruct (Float.cmp Ceq n1 n2) eqn:EQ;
destruct (Float.cmp Clt n1 n2) eqn:LT;
destruct (Float.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float.cmp_lt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
Qed.
Lemma compare_float32_spec:
forall rs f1 f2,
let rs1 := nextinstr (compare_float32 rs (Vsingle f1) (Vsingle f2)) in
rs1#CN = Val.of_bool (Float32.cmp Clt f1 f2)
/\ rs1#CZ = Val.of_bool (Float32.cmp Ceq f1 f2)
/\ rs1#CC = Val.of_bool (negb (Float32.cmp Clt f1 f2))
/\ rs1#CV = Val.of_bool (negb (Float32.cmp Ceq f1 f2 || Float32.cmp Clt f1 f2 || Float32.cmp Cgt f1 f2)).
Proof.
intros. intuition.
Qed.
Lemma compare_float32_inv:
forall rs v1 v2,
let rs1 := nextinstr (compare_float32 rs v1 v2) in
forall r', data_preg r' = true -> rs1#r' = rs#r'.
Proof.
intros. unfold rs1, compare_float32.
assert (nextinstr (rs#CN <- Vundef #CZ <- Vundef #CC <- Vundef #CV <- Vundef) r' = rs r').
{ repeat Simplif. }
destruct v1; destruct v2; auto.
repeat Simplif.
Qed.
Lemma compare_float32_nextpc:
forall rs v1 v2,
nextinstr (compare_float32 rs v1 v2) PC = Val.offset_ptr (rs PC) Ptrofs.one.
Proof.
intros. unfold compare_float32. destruct v1; destruct v2; reflexivity.
Qed.
Lemma cond_for_float32_cmp_correct:
forall c n1 n2 rs,
eval_testcond (cond_for_float_cmp c)
(nextinstr (compare_float32 rs (Vsingle n1) (Vsingle n2))) =
Some(Float32.cmp c n1 n2).
Proof.
intros.
generalize (compare_float32_spec rs n1 n2).
set (rs' := nextinstr (compare_float32 rs (Vsingle n1) (Vsingle n2))).
intros [A [B [C D]]].
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
destruct c; simpl.
(* eq *)
destruct (Float32.cmp Ceq n1 n2); auto.
(* ne *)
rewrite Float32.cmp_ne_eq. destruct (Float32.cmp Ceq n1 n2); auto.
(* lt *)
destruct (Float32.cmp Clt n1 n2); auto.
(* le *)
rewrite Float32.cmp_le_lt_eq.
destruct (Float32.cmp Clt n1 n2); destruct (Float32.cmp Ceq n1 n2); auto.
(* gt *)
destruct (Float32.cmp Ceq n1 n2) eqn:EQ;
destruct (Float32.cmp Clt n1 n2) eqn:LT;
destruct (Float32.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float32.cmp_lt_gt_false; eauto.
exfalso; eapply Float32.cmp_gt_eq_false; eauto.
exfalso; eapply Float32.cmp_lt_gt_false; eauto.
(* ge *)
rewrite Float32.cmp_ge_gt_eq.
destruct (Float32.cmp Ceq n1 n2) eqn:EQ;
destruct (Float32.cmp Clt n1 n2) eqn:LT;
destruct (Float32.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float32.cmp_lt_eq_false; eauto.
exfalso; eapply Float32.cmp_lt_eq_false; eauto.
exfalso; eapply Float32.cmp_lt_gt_false; eauto.
Qed.
Lemma cond_for_float32_not_cmp_correct:
forall c n1 n2 rs,
eval_testcond (cond_for_float_not_cmp c)
(nextinstr (compare_float32 rs (Vsingle n1) (Vsingle n2)))=
Some(negb(Float32.cmp c n1 n2)).
Proof.
intros.
generalize (compare_float32_spec rs n1 n2).
set (rs' := nextinstr (compare_float32 rs (Vsingle n1) (Vsingle n2))).
intros [A [B [C D]]].
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
destruct c; simpl.
(* eq *)
destruct (Float32.cmp Ceq n1 n2); auto.
(* ne *)
rewrite Float32.cmp_ne_eq. destruct (Float32.cmp Ceq n1 n2); auto.
(* lt *)
destruct (Float32.cmp Clt n1 n2); auto.
(* le *)
rewrite Float32.cmp_le_lt_eq.
destruct (Float32.cmp Clt n1 n2) eqn:LT; destruct (Float32.cmp Ceq n1 n2) eqn:EQ; auto.
(* gt *)
destruct (Float32.cmp Ceq n1 n2) eqn:EQ;
destruct (Float32.cmp Clt n1 n2) eqn:LT;
destruct (Float32.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float32.cmp_lt_gt_false; eauto.
exfalso; eapply Float32.cmp_gt_eq_false; eauto.