diff options
Diffstat (limited to 'powerpc')
| -rw-r--r-- | powerpc/Asm.v | 18 | ||||
| -rw-r--r-- | powerpc/Asmgen.v | 24 | ||||
| -rw-r--r-- | powerpc/Asmgenproof.v | 24 | ||||
| -rw-r--r-- | powerpc/Asmgenproof1.v | 214 | ||||
| -rw-r--r-- | powerpc/Asmgenretaddr.v | 10 | ||||
| -rw-r--r-- | powerpc/ConstpropOp.v | 856 | ||||
| -rw-r--r-- | powerpc/ConstpropOp.vp | 277 | ||||
| -rw-r--r-- | powerpc/ConstpropOpproof.v | 549 | ||||
| -rw-r--r-- | powerpc/Op.v | 1265 | ||||
| -rw-r--r-- | powerpc/PrintOp.ml | 2 | ||||
| -rw-r--r-- | powerpc/SelectOp.v | 1018 | ||||
| -rw-r--r-- | powerpc/SelectOp.vp | 432 | ||||
| -rw-r--r-- | powerpc/SelectOpproof.v | 1192 |
13 files changed, 2207 insertions, 3674 deletions
diff --git a/powerpc/Asm.v b/powerpc/Asm.v index 321b074..7174f79 100644 --- a/powerpc/Asm.v +++ b/powerpc/Asm.v @@ -496,10 +496,10 @@ Definition compare_sint (rs: regset) (v1 v2: val) := #CR0_2 <- (Val.cmp Ceq v1 v2) #CR0_3 <- Vundef. -Definition compare_uint (rs: regset) (v1 v2: val) := - rs#CR0_0 <- (Val.cmpu Clt v1 v2) - #CR0_1 <- (Val.cmpu Cgt v1 v2) - #CR0_2 <- (Val.cmpu Ceq v1 v2) +Definition compare_uint (rs: regset) (m: mem) (v1 v2: val) := + rs#CR0_0 <- (Val.cmpu (Mem.valid_pointer m) Clt v1 v2) + #CR0_1 <- (Val.cmpu (Mem.valid_pointer m) Cgt v1 v2) + #CR0_2 <- (Val.cmpu (Mem.valid_pointer m) Ceq v1 v2) #CR0_3 <- Vundef. Definition compare_float (rs: regset) (v1 v2: val) := @@ -596,9 +596,9 @@ Definition exec_instr (c: code) (i: instruction) (rs: regset) (m: mem) : outcome | _ => Error end | Pcmplw r1 r2 => - OK (nextinstr (compare_uint rs rs#r1 rs#r2)) m + OK (nextinstr (compare_uint rs m rs#r1 rs#r2)) m | Pcmplwi r1 cst => - OK (nextinstr (compare_uint rs rs#r1 (const_low cst))) m + OK (nextinstr (compare_uint rs m rs#r1 (const_low cst))) m | Pcmpw r1 r2 => OK (nextinstr (compare_sint rs rs#r1 rs#r2)) m | Pcmpwi r1 cst => @@ -606,9 +606,9 @@ Definition exec_instr (c: code) (i: instruction) (rs: regset) (m: mem) : outcome | Pcror bd b1 b2 => OK (nextinstr (rs#(reg_of_crbit bd) <- (Val.or rs#(reg_of_crbit b1) rs#(reg_of_crbit b2)))) m | Pdivw rd r1 r2 => - OK (nextinstr (rs#rd <- (Val.divs rs#r1 rs#r2))) m + OK (nextinstr (rs#rd <- (Val.maketotal (Val.divs rs#r1 rs#r2)))) m | Pdivwu rd r1 r2 => - OK (nextinstr (rs#rd <- (Val.divu rs#r1 rs#r2))) m + OK (nextinstr (rs#rd <- (Val.maketotal (Val.divu rs#r1 rs#r2)))) m | Peqv rd r1 r2 => OK (nextinstr (rs#rd <- (Val.notint (Val.xor rs#r1 rs#r2)))) m | Pextsb rd r1 => @@ -635,7 +635,7 @@ Definition exec_instr (c: code) (i: instruction) (rs: regset) (m: mem) : outcome | Pfcmpu r1 r2 => OK (nextinstr (compare_float rs rs#r1 rs#r2)) m | Pfcti rd r1 => - OK (nextinstr (rs#FPR13 <- Vundef #rd <- (Val.intoffloat rs#r1))) m + OK (nextinstr (rs#FPR13 <- Vundef #rd <- (Val.maketotal (Val.intoffloat rs#r1)))) m | Pfdiv rd r1 r2 => OK (nextinstr (rs#rd <- (Val.divf rs#r1 rs#r2))) m | Pfmadd rd r1 r2 r3 => diff --git a/powerpc/Asmgen.v b/powerpc/Asmgen.v index 790b2b9..6d1a1fc 100644 --- a/powerpc/Asmgen.v +++ b/powerpc/Asmgen.v @@ -69,7 +69,7 @@ Definition addimm (r1 r2: ireg) (n: int) (k: code) := Paddis r1 r2 (Cint (high_s n)) :: Paddi r1 r1 (Cint (low_s n)) :: k. -Definition andimm (r1 r2: ireg) (n: int) (k: code) := +Definition andimm_base (r1 r2: ireg) (n: int) (k: code) := if Int.eq (high_u n) Int.zero then Pandi_ r1 r2 (Cint n) :: k else if Int.eq (low_u n) Int.zero then @@ -77,6 +77,12 @@ Definition andimm (r1 r2: ireg) (n: int) (k: code) := else loadimm GPR0 n (Pand_ r1 r2 GPR0 :: k). +Definition andimm (r1 r2: ireg) (n: int) (k: code) := + if is_rlw_mask n then + Prlwinm r1 r2 Int.zero n :: k + else + andimm_base r1 r2 n k. + Definition orimm (r1 r2: ireg) (n: int) (k: code) := if Int.eq (high_u n) Int.zero then Pori r1 r2 (Cint n) :: k @@ -95,6 +101,12 @@ Definition xorimm (r1 r2: ireg) (n: int) (k: code) := Pxoris r1 r2 (Cint (high_u n)) :: Pxori r1 r1 (Cint (low_u n)) :: k. +Definition rolm (r1 r2: ireg) (amount mask: int) (k: code) := + if is_rlw_mask mask then + Prlwinm r1 r2 amount mask :: k + else + Prlwinm r1 r2 amount Int.mone :: andimm_base r1 r1 mask k. + (** Accessing slots in the stack frame. *) Definition loadind (base: ireg) (ofs: int) (ty: typ) (dst: mreg) (k: code) := @@ -166,9 +178,9 @@ Definition transl_cond | Cnotcompf cmp, a1 :: a2 :: nil => floatcomp cmp (freg_of a1) (freg_of a2) k | Cmaskzero n, a1 :: nil => - andimm GPR0 (ireg_of a1) n k + andimm_base GPR0 (ireg_of a1) n k | Cmasknotzero n, a1 :: nil => - andimm GPR0 (ireg_of a1) n k + andimm_base GPR0 (ireg_of a1) n k | _, _ => k (**r never happens for well-typed code *) end. @@ -302,12 +314,8 @@ Definition transl_op addimm (ireg_of r) GPR1 n k | Ocast8signed, a1 :: nil => Pextsb (ireg_of r) (ireg_of a1) :: k - | Ocast8unsigned, a1 :: nil => - Prlwinm (ireg_of r) (ireg_of a1) Int.zero (Int.repr 255) :: k | Ocast16signed, a1 :: nil => Pextsh (ireg_of r) (ireg_of a1) :: k - | Ocast16unsigned, a1 :: nil => - Prlwinm (ireg_of r) (ireg_of a1) Int.zero (Int.repr 65535) :: k | Oadd, a1 :: a2 :: nil => Padd (ireg_of r) (ireg_of a1) (ireg_of a2) :: k | Oaddimm n, a1 :: nil => @@ -360,7 +368,7 @@ Definition transl_op | Oshru, a1 :: a2 :: nil => Psrw (ireg_of r) (ireg_of a1) (ireg_of a2) :: k | Orolm amount mask, a1 :: nil => - Prlwinm (ireg_of r) (ireg_of a1) amount mask :: k + rolm (ireg_of r) (ireg_of a1) amount mask k | Oroli amount mask, a1 :: a2 :: nil => if mreg_eq a1 r then (**r should always be true *) Prlwimi (ireg_of r) (ireg_of a2) amount mask :: k diff --git a/powerpc/Asmgenproof.v b/powerpc/Asmgenproof.v index 27b2108..e7b7385 100644 --- a/powerpc/Asmgenproof.v +++ b/powerpc/Asmgenproof.v @@ -343,12 +343,21 @@ Proof. Qed. Hint Rewrite addimm_label: labels. +Remark andimm_base_label: + forall r1 r2 n k, find_label lbl (andimm_base r1 r2 n k) = find_label lbl k. +Proof. + intros; unfold andimm_base. + case (Int.eq (high_u n) Int.zero). reflexivity. + case (Int.eq (low_u n) Int.zero). reflexivity. + autorewrite with labels. reflexivity. +Qed. +Hint Rewrite andimm_base_label: labels. + Remark andimm_label: forall r1 r2 n k, find_label lbl (andimm r1 r2 n k) = find_label lbl k. Proof. intros; unfold andimm. - case (Int.eq (high_u n) Int.zero). reflexivity. - case (Int.eq (low_u n) Int.zero). reflexivity. + case (is_rlw_mask n). reflexivity. autorewrite with labels. reflexivity. Qed. Hint Rewrite andimm_label: labels. @@ -371,6 +380,15 @@ Proof. Qed. Hint Rewrite xorimm_label: labels. +Remark rolm_label: + forall r1 r2 amount mask k, find_label lbl (rolm r1 r2 amount mask k) = find_label lbl k. +Proof. + intros; unfold rolm. + case (is_rlw_mask mask). reflexivity. + simpl. autorewrite with labels. auto. +Qed. +Hint Rewrite rolm_label: labels. + Remark loadind_label: forall base ofs ty dst k, find_label lbl (loadind base ofs ty dst k) = find_label lbl k. Proof. @@ -405,7 +423,7 @@ Proof. case (Int.eq (high_u i) Int.zero). reflexivity. autorewrite with labels; reflexivity. apply floatcomp_label. apply floatcomp_label. - apply andimm_label. apply andimm_label. + apply andimm_base_label. apply andimm_base_label. Qed. Hint Rewrite transl_cond_label: labels. diff --git a/powerpc/Asmgenproof1.v b/powerpc/Asmgenproof1.v index 0b7f4d0..77a19af 100644 --- a/powerpc/Asmgenproof1.v +++ b/powerpc/Asmgenproof1.v @@ -595,11 +595,11 @@ Proof. Qed. Lemma compare_uint_spec: - forall rs v1 v2, - let rs1 := nextinstr (compare_uint rs v1 v2) in - rs1#CR0_0 = Val.cmpu Clt v1 v2 - /\ rs1#CR0_1 = Val.cmpu Cgt v1 v2 - /\ rs1#CR0_2 = Val.cmpu Ceq v1 v2 + forall rs m v1 v2, + let rs1 := nextinstr (compare_uint rs m v1 v2) in + rs1#CR0_0 = Val.cmpu (Mem.valid_pointer m) Clt v1 v2 + /\ rs1#CR0_1 = Val.cmpu (Mem.valid_pointer m) Cgt v1 v2 + /\ rs1#CR0_2 = Val.cmpu (Mem.valid_pointer m) Ceq v1 v2 /\ forall r', r' <> CR0_0 -> r' <> CR0_1 -> r' <> CR0_2 -> r' <> CR0_3 -> r' <> PC -> rs1#r' = rs#r'. Proof. intros. unfold rs1. @@ -687,17 +687,17 @@ Qed. (** And integer immediate. *) -Lemma andimm_correct: +Lemma andimm_base_correct: forall r1 r2 n k (rs : regset) m, r2 <> GPR0 -> let v := Val.and rs#r2 (Vint n) in exists rs', - exec_straight (andimm r1 r2 n k) rs m k rs' m + exec_straight (andimm_base r1 r2 n k) rs m k rs' m /\ rs'#r1 = v /\ rs'#CR0_2 = Val.cmp Ceq v Vzero /\ forall r', is_data_reg r' = true -> r' <> r1 -> rs'#r' = rs#r'. Proof. - intros. unfold andimm. + intros. unfold andimm_base. case (Int.eq (high_u n) Int.zero). (* andi *) exists (nextinstr (compare_sint (rs#r1 <- v) v Vzero)). @@ -734,6 +734,25 @@ Proof. intros. rewrite D; auto with ppcgen. SIMP. Qed. +Lemma andimm_correct: + forall r1 r2 n k (rs : regset) m, + r2 <> GPR0 -> + exists rs', + exec_straight (andimm r1 r2 n k) rs m k rs' m + /\ rs'#r1 = Val.and rs#r2 (Vint n) + /\ forall r', is_data_reg r' = true -> r' <> r1 -> rs'#r' = rs#r'. +Proof. + intros. unfold andimm. destruct (is_rlw_mask n). + (* turned into rlw *) + exists (nextinstr (rs#r1 <- (Val.and rs#r2 (Vint n)))). + split. apply exec_straight_one. simpl. rewrite Val.rolm_zero. auto. reflexivity. + split. SIMP. apply Pregmap.gss. + intros. SIMP. apply Pregmap.gso; auto with ppcgen. + (* andimm_base *) + destruct (andimm_base_correct r1 r2 n k rs m) as [rs' [A [B [C D]]]]; auto. + exists rs'; auto. +Qed. + (** Or integer immediate. *) Lemma orimm_correct: @@ -797,6 +816,33 @@ Proof. intros. repeat SIMP. Qed. +(** Rotate and mask. *) + +Lemma rolm_correct: + forall r1 r2 amount mask k (rs : regset) m, + r1 <> GPR0 -> + exists rs', + exec_straight (rolm r1 r2 amount mask k) rs m k rs' m + /\ rs'#r1 = Val.rolm rs#r2 amount mask + /\ forall r', is_data_reg r' = true -> r' <> r1 -> rs'#r' = rs#r'. +Proof. + intros. unfold rolm. destruct (is_rlw_mask mask). + (* rlwinm *) + exists (nextinstr (rs#r1 <- (Val.rolm rs#r2 amount mask))). + split. apply exec_straight_one; auto. + split. SIMP. apply Pregmap.gss. + intros. SIMP. apply Pregmap.gso; auto. + (* rlwinm ; andimm *) + set (rs1 := nextinstr (rs#r1 <- (Val.rolm rs#r2 amount Int.mone))). + destruct (andimm_base_correct r1 r1 mask k rs1 m) as [rs' [A [B [C D]]]]; auto. + exists rs'. + split. eapply exec_straight_step; eauto. auto. auto. + split. rewrite B. unfold rs1. SIMP. rewrite Pregmap.gss. + destruct (rs r2); simpl; auto. unfold Int.rolm. rewrite Int.and_assoc. + decEq; decEq; decEq. rewrite Int.and_commut. apply Int.and_mone. + intros. rewrite D; auto. unfold rs1; SIMP. apply Pregmap.gso; auto. +Qed. + (** Indexed memory loads. *) Lemma loadind_correct: @@ -947,13 +993,14 @@ Lemma transl_cond_correct_1: exec_straight (transl_cond cond args k) rs m k rs' m /\ rs'#(reg_of_crbit (fst (crbit_for_cond cond))) = (if snd (crbit_for_cond cond) - then eval_condition_total cond (map rs (map preg_of args)) - else Val.notbool (eval_condition_total cond (map rs (map preg_of args)))) + then Val.of_optbool (eval_condition cond (map rs (map preg_of args)) m) + else Val.notbool (Val.of_optbool (eval_condition cond (map rs (map preg_of args)) m))) /\ forall r, is_data_reg r = true -> rs'#r = rs#r. Proof. intros. destruct cond; simpl in H; TypeInv; simpl; UseTypeInfo. (* Ccomp *) + fold (Val.cmp c (rs (ireg_of m0)) (rs (ireg_of m1))). destruct (compare_sint_spec rs (rs (ireg_of m0)) (rs (ireg_of m1))) as [A [B [C D]]]. econstructor; split. @@ -962,7 +1009,8 @@ Proof. case c; simpl; auto; rewrite <- Val.negate_cmp; simpl; auto. auto with ppcgen. (* Ccompu *) - destruct (compare_uint_spec rs (rs (ireg_of m0)) (rs (ireg_of m1))) + fold (Val.cmpu (Mem.valid_pointer m) c (rs (ireg_of m0)) (rs (ireg_of m1))). + destruct (compare_uint_spec rs m (rs (ireg_of m0)) (rs (ireg_of m1))) as [A [B [C D]]]. econstructor; split. apply exec_straight_one. simpl; reflexivity. reflexivity. @@ -970,6 +1018,7 @@ Proof. case c; simpl; auto; rewrite <- Val.negate_cmpu; simpl; auto. auto with ppcgen. (* Ccompimm *) + fold (Val.cmp c (rs (ireg_of m0)) (Vint i)). case (Int.eq (high_s i) Int.zero). destruct (compare_sint_spec rs (rs (ireg_of m0)) (Vint i)) as [A [B [C D]]]. @@ -992,8 +1041,9 @@ Proof. case c; simpl; auto; rewrite <- Val.negate_cmp; simpl; auto. intros. rewrite H; rewrite D; auto with ppcgen. (* Ccompuimm *) + fold (Val.cmpu (Mem.valid_pointer m) c (rs (ireg_of m0)) (Vint i)). case (Int.eq (high_u i) Int.zero). - destruct (compare_uint_spec rs (rs (ireg_of m0)) (Vint i)) + destruct (compare_uint_spec rs m (rs (ireg_of m0)) (Vint i)) as [A [B [C D]]]. econstructor; split. apply exec_straight_one. simpl. eauto. reflexivity. @@ -1002,10 +1052,10 @@ Proof. auto with ppcgen. generalize (loadimm_correct GPR0 i (Pcmplw (ireg_of m0) GPR0 :: k) rs m). intros [rs1 [EX1 [RES1 OTH1]]]. - destruct (compare_uint_spec rs1 (rs (ireg_of m0)) (Vint i)) + destruct (compare_uint_spec rs1 m (rs (ireg_of m0)) (Vint i)) as [A [B [C D]]]. assert (rs1 (ireg_of m0) = rs (ireg_of m0)). apply OTH1; auto with ppcgen. - exists (nextinstr (compare_uint rs1 (rs1 (ireg_of m0)) (Vint i))). + exists (nextinstr (compare_uint rs1 m (rs1 (ireg_of m0)) (Vint i))). split. eapply exec_straight_trans. eexact EX1. apply exec_straight_one. simpl. rewrite RES1; rewrite H; auto. reflexivity. @@ -1013,32 +1063,33 @@ Proof. case c; simpl; auto; rewrite <- Val.negate_cmpu; simpl; auto. intros. rewrite H; rewrite D; auto with ppcgen. (* Ccompf *) + fold (Val.cmpf c (rs (freg_of m0)) (rs (freg_of m1))). destruct (floatcomp_correct c (freg_of m0) (freg_of m1) k rs m) as [rs' [EX [RES OTH]]]. exists rs'. split. auto. split. apply RES. auto with ppcgen. (* Cnotcompf *) + rewrite Val.notbool_negb_3. rewrite Val.notbool_idem4. + fold (Val.cmpf c (rs (freg_of m0)) (rs (freg_of m1))). destruct (floatcomp_correct c (freg_of m0) (freg_of m1) k rs m) as [rs' [EX [RES OTH]]]. exists rs'. split. auto. - split. rewrite RES. - assert (forall v1 v2, Val.notbool (Val.notbool (Val.cmpf c v1 v2)) = Val.cmpf c v1 v2). - intros v1 v2; unfold Val.cmpf; destruct v1; destruct v2; auto. - apply Val.notbool_idem2. - rewrite H. case (snd (crbit_for_fcmp c)); simpl; auto. + split. rewrite RES. destruct (snd (crbit_for_fcmp c)); auto. auto with ppcgen. (* Cmaskzero *) - destruct (andimm_correct GPR0 (ireg_of m0) i k rs m) + destruct (andimm_base_correct GPR0 (ireg_of m0) i k rs m) as [rs' [A [B [C D]]]]. auto with ppcgen. exists rs'. split. assumption. - split. rewrite C. auto. + split. rewrite C. destruct (rs (ireg_of m0)); auto. auto with ppcgen. (* Cmasknotzero *) - destruct (andimm_correct GPR0 (ireg_of m0) i k rs m) + destruct (andimm_base_correct GPR0 (ireg_of m0) i k rs m) as [rs' [A [B [C D]]]]. auto with ppcgen. exists rs'. split. assumption. - split. rewrite C. rewrite Val.notbool_idem3. reflexivity. + split. rewrite C. destruct (rs (ireg_of m0)); auto. + fold (option_map negb (Some (Int.eq (Int.and i0 i) Int.zero))). + rewrite Val.notbool_negb_3. rewrite Val.notbool_idem4. auto. auto with ppcgen. Qed. @@ -1055,9 +1106,10 @@ Lemma transl_cond_correct_2: /\ forall r, is_data_reg r = true -> rs'#r = rs#r. Proof. intros. - assert (eval_condition_total cond rs ## (preg_of ## args) = Val.of_bool b). - apply eval_condition_weaken with m. auto. - rewrite <- H1. eapply transl_cond_correct_1; eauto. + replace (Val.of_bool b) + with (Val.of_optbool (eval_condition cond rs ## (preg_of ## args) m)). + eapply transl_cond_correct_1; eauto. + rewrite H0; auto. Qed. Lemma transl_cond_correct: @@ -1128,46 +1180,43 @@ Proof. Qed. Lemma transl_cond_op_correct: - forall cond args r k rs m b, + forall cond args r k rs m, mreg_type r = Tint -> map mreg_type args = type_of_condition cond -> - eval_condition cond (map rs (map preg_of args)) m = Some b -> exists rs', exec_straight (transl_cond_op cond args r k) rs m k rs' m - /\ rs'#(ireg_of r) = Val.of_bool b + /\ rs'#(ireg_of r) = Val.of_optbool (eval_condition cond (map rs (map preg_of args)) m) /\ forall r', is_data_reg r' = true -> r' <> ireg_of r -> rs'#r' = rs#r'. Proof. intros until args. unfold transl_cond_op. destruct (classify_condition cond args); - intros until b; intros TY1 TY2 EV; simpl in TY2. + intros until m; intros TY1 TY2; simpl in TY2. (* eq 0 *) - inv TY2. simpl in EV. unfold preg_of in *; rewrite H0 in *. + inv TY2. simpl. unfold preg_of; rewrite H0. econstructor; split. eapply exec_straight_two; simpl; reflexivity. - split. repeat SIMP. destruct (rs (ireg_of r)); inv EV. simpl. + split. repeat SIMP. destruct (rs (ireg_of r)); simpl; auto. apply add_carry_eq0. intros; repeat SIMP. (* ne 0 *) - inv TY2. simpl in EV. unfold preg_of in *; rewrite H0 in *. + inv TY2. simpl. unfold preg_of; rewrite H0. econstructor; split. eapply exec_straight_two; simpl; reflexivity. split. repeat SIMP. rewrite gpr_or_zero_not_zero; auto with ppcgen. - destruct (rs (ireg_of r)); inv EV. simpl. + destruct (rs (ireg_of r)); simpl; auto. apply add_carry_ne0. intros; repeat SIMP. (* ge 0 *) - inv TY2. simpl in EV. unfold preg_of in *; rewrite H0 in *. + inv TY2. simpl. unfold preg_of; rewrite H0. econstructor; split. eapply exec_straight_two; simpl; reflexivity. - split. repeat SIMP. rewrite Val.rolm_ge_zero. - destruct (rs (ireg_of r)); simpl; congruence. + split. repeat SIMP. rewrite Val.rolm_ge_zero. auto. intros; repeat SIMP. (* lt 0 *) - inv TY2. simpl in EV. unfold preg_of in *; rewrite H0 in *. + inv TY2. simpl. unfold preg_of; rewrite H0. econstructor; split. apply exec_straight_one; simpl; reflexivity. - split. repeat SIMP. rewrite Val.rolm_lt_zero. - destruct (rs (ireg_of r)); simpl; congruence. + split. repeat SIMP. rewrite Val.rolm_lt_zero. auto. intros; repeat SIMP. (* default *) set (bit := fst (crbit_for_cond c)). @@ -1177,7 +1226,7 @@ Proof. (if isset then k else Pxori (ireg_of r) (ireg_of r) (Cint Int.one) :: k)). - generalize (transl_cond_correct_2 c rl k1 rs m b TY2 EV). + generalize (transl_cond_correct_1 c rl k1 rs m TY2). fold bit; fold isset. intros [rs1 [EX1 [RES1 AG1]]]. destruct isset. @@ -1188,7 +1237,8 @@ Proof. (* bit clear *) econstructor; split. eapply exec_straight_trans. eexact EX1. unfold k1. eapply exec_straight_two; simpl; reflexivity. - split. repeat SIMP. rewrite RES1. destruct b; compute; reflexivity. + split. repeat SIMP. rewrite RES1. + destruct (eval_condition c rs ## (preg_of ## rl) m). destruct b; auto. auto. intros; repeat SIMP. Qed. @@ -1210,26 +1260,23 @@ Lemma transl_op_correct_aux: match op with Omove => is_data_reg r = true | _ => is_nontemp_reg r = true end -> r <> preg_of res -> rs'#r = rs#r. Proof. - intros. - exploit eval_operation_weaken; eauto. intro EV. - inv H. + intros until v; intros WT EV. + inv WT. (* Omove *) - simpl in *. + simpl in *. inv EV. exists (nextinstr (rs#(preg_of res) <- (rs#(preg_of r1)))). - split. unfold preg_of. rewrite <- H2. + split. unfold preg_of. rewrite <- H0. destruct (mreg_type r1); apply exec_straight_one; auto. split. repeat SIMP. intros; repeat SIMP. (* Other instructions *) - destruct op; simpl; simpl in H5; injection H5; clear H5; intros; - TypeInv; simpl in *; UseTypeInfo; try (TranslOpSimpl). - (* Omove again *) - congruence. + destruct op; simpl; simpl in H3; injection H3; clear H3; intros; + TypeInv; simpl in *; UseTypeInfo; inv EV; try (TranslOpSimpl). (* Ointconst *) destruct (loadimm_correct (ireg_of res) i k rs m) as [rs' [A [B C]]]. exists rs'. split. auto. split. auto. auto with ppcgen. (* Oaddrsymbol *) - change (find_symbol_offset ge i i0) with (symbol_offset ge i i0) in *. - set (v' := symbol_offset ge i i0) in *. + change (symbol_address ge i i0) with (symbol_offset ge i i0). + set (v' := symbol_offset ge i i0). caseEq (symbol_is_small_data i i0); intro SD. (* small data *) econstructor; split. apply exec_straight_one; simpl; reflexivity. @@ -1249,18 +1296,6 @@ Opaque Val.add. destruct (addimm_correct (ireg_of res) GPR1 i k rs m) as [rs' [EX [RES OTH]]]. auto with ppcgen. congruence. exists rs'; auto with ppcgen. - (* Ocast8unsigned *) - econstructor; split. apply exec_straight_one; simpl; reflexivity. - split. repeat SIMP. - destruct (rs (ireg_of m0)); simpl; auto. - rewrite Int.rolm_zero. rewrite Int.zero_ext_and. auto. compute; auto. - intros; repeat SIMP. - (* Ocast16unsigned *) - econstructor; split. apply exec_straight_one; simpl; reflexivity. - split. repeat SIMP. - destruct (rs (ireg_of m0)); simpl; auto. - rewrite Int.rolm_zero. rewrite Int.zero_ext_and. auto. compute; auto. - intros; repeat SIMP. (* Oaddimm *) destruct (addimm_correct (ireg_of res) (ireg_of m0) i k rs m) as [rs' [A [B C]]]; auto with ppcgen. exists rs'; auto with ppcgen. @@ -1280,6 +1315,14 @@ Opaque Val.add. eapply exec_straight_trans. eexact EX. apply exec_straight_one; simpl; reflexivity. split. repeat SIMP. rewrite RES. rewrite OTH; auto with ppcgen. intros; repeat SIMP. + (* Odivs *) + replace v with (Val.maketotal (Val.divs (rs (ireg_of m0)) (rs (ireg_of m1)))). + TranslOpSimpl. + rewrite H2; auto. + (* Odivu *) + replace v with (Val.maketotal (Val.divu (rs (ireg_of m0)) (rs (ireg_of m1)))). + TranslOpSimpl. + rewrite H2; auto. (* Oand *) set (v' := Val.and (rs (ireg_of m0)) (rs (ireg_of m1))) in *. pose (rs1 := rs#(ireg_of res) <- v'). @@ -1289,7 +1332,7 @@ Opaque Val.add. split. rewrite D; auto with ppcgen. unfold rs1. SIMP. intros. rewrite D; auto with ppcgen. unfold rs1. SIMP. (* Oandimm *) - destruct (andimm_correct (ireg_of res) (ireg_of m0) i k rs m) as [rs' [A [B [C D]]]]; auto with ppcgen. + destruct (andimm_correct (ireg_of res) (ireg_of m0) i k rs m) as [rs' [A [B C]]]; auto with ppcgen. exists rs'; auto with ppcgen. (* Oorimm *) destruct (orimm_correct (ireg_of res) (ireg_of m0) i k rs m) as [rs' [A [B C]]]. @@ -1300,19 +1343,24 @@ Opaque Val.add. (* Oshrximm *) econstructor; split. eapply exec_straight_two; simpl; reflexivity. - split. repeat SIMP. apply Val.shrx_carry. + split. repeat SIMP. apply Val.shrx_carry. auto. intros; repeat SIMP. + (* Orolm *) + destruct (rolm_correct (ireg_of res) (ireg_of m0) i i0 k rs m) as [rs' [A [B C]]]; auto with ppcgen. + exists rs'; auto with ppcgen. (* Oroli *) destruct (mreg_eq m0 res). subst m0. TranslOpSimpl. econstructor; split. eapply exec_straight_three; simpl; reflexivity. split. repeat SIMP. intros; repeat SIMP. + (* Ointoffloat *) + replace v with (Val.maketotal (Val.intoffloat (rs (freg_of m0)))). + TranslOpSimpl. + rewrite H2; auto. (* Ocmp *) - destruct (eval_condition c rs ## (preg_of ## args) m) as [ b | ] _eqn; try discriminate. - destruct (transl_cond_op_correct c args res k rs m b) as [rs' [A [B C]]]; auto. - exists rs'; intuition auto with ppcgen. - rewrite B. destruct b; inv H0; auto. + destruct (transl_cond_op_correct c args res k rs m) as [rs' [A [B C]]]; auto. + exists rs'; auto with ppcgen. Qed. Lemma transl_op_correct: @@ -1340,14 +1388,14 @@ Lemma transl_load_store_correct: forall (mk1: constant -> ireg -> instruction) (mk2: ireg -> ireg -> instruction) addr args (temp: ireg) k ms sp rs m ms' m', (forall cst (r1: ireg) (rs1: regset) k, - eval_addressing_total ge sp addr (map rs (map preg_of args)) = - Val.add (gpr_or_zero rs1 r1) (const_low ge cst) -> + eval_addressing ge sp addr (map rs (map preg_of args)) = + Some(Val.add (gpr_or_zero rs1 r1) (const_low ge cst)) -> (forall (r: preg), r <> PC -> r <> temp -> rs1 r = rs r) -> exists rs', exec_straight (mk1 cst r1 :: k) rs1 m k rs' m' /\ agree ms' sp rs') -> (forall (r1 r2: ireg) k, - eval_addressing_total ge sp addr (map rs (map preg_of args)) = Val.add rs#r1 rs#r2 -> + eval_addressing ge sp addr (map rs (map preg_of args)) = Some(Val.add rs#r1 rs#r2) -> exists rs', exec_straight (mk2 r1 r2 :: k) rs m k rs' m' /\ agree ms' sp rs') -> @@ -1386,7 +1434,7 @@ Transparent Val.add. (* Aglobal from small data *) apply H. rewrite gpr_or_zero_zero. simpl const_low. rewrite small_data_area_addressing; auto. simpl. - unfold find_symbol_offset, symbol_offset. + unfold symbol_address, symbol_offset. destruct (Genv.find_symbol ge i); auto. rewrite Int.add_zero. auto. auto. (* Aglobal general case *) @@ -1396,7 +1444,7 @@ Transparent Val.add. unfold rs1. rewrite nextinstr_inv. rewrite Pregmap.gss. unfold const_high, const_low. set (v := symbol_offset ge i i0). - symmetry. rewrite Val.add_commut. unfold v. apply low_high_half. + symmetry. rewrite Val.add_commut. unfold v. rewrite low_high_half. auto. discriminate. intros; unfold rs1. rewrite nextinstr_inv; auto. apply Pregmap.gso; auto. intros [rs' [EX' AG']]. @@ -1414,8 +1462,8 @@ Transparent Val.add. rewrite Val.add_assoc. unfold const_high, const_low. set (v := symbol_offset ge i i0). - symmetry. rewrite Val.add_commut. decEq. - unfold v. rewrite Val.add_commut. apply low_high_half. + symmetry. rewrite Val.add_commut. decEq. decEq. + unfold v. rewrite Val.add_commut. rewrite low_high_half. auto. UseTypeInfo. auto. discriminate. intros. unfold rs1. rewrite nextinstr_inv; auto. apply Pregmap.gso; auto. intros [rs' [EX' AG']]. @@ -1465,12 +1513,11 @@ Proof. exploit eval_addressing_lessdef. eapply preg_vals; eauto. eauto. intros [a' [A B]]. exploit Mem.loadv_extends; eauto. intros [v' [C D]]. - exploit eval_addressing_weaken. eexact A. intro E. rewrite <- E in C. apply transl_load_store_correct with ms; auto. (* mk1 *) intros. exists (nextinstr (rs1#(preg_of dst) <- v')). split. apply exec_straight_one. rewrite H. - unfold load1. rewrite <- H6. rewrite C. auto. + unfold load1. rewrite A in H6. inv H6. rewrite C. auto. unfold nextinstr. SIMP. decEq. SIMP. apply sym_not_equal; auto with ppcgen. apply agree_set_mreg with rs1. apply agree_undef_temps with rs; auto with ppcgen. @@ -1479,7 +1526,7 @@ Proof. (* mk2 *) intros. exists (nextinstr (rs#(preg_of dst) <- v')). split. apply exec_straight_one. rewrite H0. - unfold load2. rewrite <- H6. rewrite C. auto. + unfold load2. rewrite A in H6. inv H6. rewrite C. auto. unfold nextinstr. SIMP. decEq. SIMP. apply sym_not_equal; auto with ppcgen. apply agree_set_mreg with rs. apply agree_undef_temps with rs; auto with ppcgen. @@ -1521,13 +1568,12 @@ Proof. intros [a' [A B]]. assert (Z: Val.lessdef (ms src) (rs (preg_of src))). eapply preg_val; eauto. exploit Mem.storev_extends; eauto. intros [m1' [C D]]. - exploit eval_addressing_weaken. eexact A. intro E. rewrite <- E in C. exists m1'; split; auto. apply transl_load_store_correct with ms; auto. (* mk1 *) intros. exploit (H cst r1 rs1 (nextinstr rs1) m1'). - unfold store1. rewrite <- H6. + unfold store1. rewrite A in H6. inv H6. replace (rs1 (preg_of src)) with (rs (preg_of src)). rewrite C. auto. symmetry. apply H7. auto with ppcgen. @@ -1541,7 +1587,7 @@ Proof. (* mk2 *) intros. exploit (H0 r1 r2 rs (nextinstr rs) m1'). - unfold store2. rewrite <- H6. rewrite C. auto. + unfold store2. rewrite A in H6. inv H6. rewrite C. auto. intros [rs3 [U V]]. exists rs3; split. apply exec_straight_one. auto. rewrite V; auto with ppcgen. diff --git a/powerpc/Asmgenretaddr.v b/powerpc/Asmgenretaddr.v index adc1529..081336c 100644 --- a/powerpc/Asmgenretaddr.v +++ b/powerpc/Asmgenretaddr.v @@ -112,6 +112,11 @@ Lemma addimm_tail: Proof. unfold addimm; intros; IsTail. Qed. Hint Resolve addimm_tail: ppcretaddr. +Lemma andimm_base_tail: + forall r1 r2 n k, is_tail k (andimm_base r1 r2 n k). +Proof. unfold andimm_base; intros; IsTail. Qed. +Hint Resolve andimm_base_tail: ppcretaddr. + Lemma andimm_tail: forall r1 r2 n k, is_tail k (andimm r1 r2 n k). Proof. unfold andimm; intros; IsTail. Qed. @@ -127,6 +132,11 @@ Lemma xorimm_tail: Proof. unfold xorimm; intros; IsTail. Qed. Hint Resolve xorimm_tail: ppcretaddr. +Lemma rolm_tail: + forall r1 r2 amount mask k, is_tail k (rolm r1 r2 amount mask k). +Proof. unfold rolm; intros; IsTail. Qed. +Hint Resolve rolm_tail: ppcretaddr. + Lemma loadind_tail: forall base ofs ty dst k, is_tail k (loadind base ofs ty dst k). Proof. unfold loadind; intros. destruct ty; IsTail. Qed. diff --git a/powerpc/ConstpropOp.v b/powerpc/ConstpropOp.v deleted file mode 100644 index 07a1872..0000000 --- a/powerpc/ConstpropOp.v +++ /dev/null @@ -1,856 +0,0 @@ -(* *********************************************************************) -(* *) -(* 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. *) -(* *) -(* *********************************************************************) - -(** Static analysis and strength reduction for operators - and conditions. This is the machine-dependent part of [Constprop]. *) - -Require Import Coqlib. -Require Import AST. -Require Import Integers. -Require Import Floats. -Require Import Values. -Require Import Op. -Require Import Registers. - -(** * Static analysis *) - -(** To each pseudo-register at each program point, the static analysis - associates a compile-time approximation taken from the following set. *) - -Inductive approx : Type := - | Novalue: approx (** No value possible, code is unreachable. *) - | Unknown: approx (** All values are possible, - no compile-time information is available. *) - | I: int -> approx (** A known integer value. *) - | F: float -> approx (** A known floating-point value. *) - | S: ident -> int -> approx. - (** The value is the address of the given global - symbol plus the given integer offset. *) - -(** We now define the abstract interpretations of conditions and operators - over this set of approximations. For instance, the abstract interpretation - of the operator [Oaddf] applied to two expressions [a] and [b] is - [F(Float.add f g)] if [a] and [b] have static approximations [Vfloat f] - and [Vfloat g] respectively, and [Unknown] otherwise. - - The static approximations are defined by large pattern-matchings over - the approximations of the results. We write these matchings in the - indirect style described in file [Cmconstr] to avoid excessive - duplication of cases in proofs. *) - -(* -Definition eval_static_condition (cond: condition) (vl: list approx) := - match cond, vl with - | Ccomp c, I n1 :: I n2 :: nil => Some(Int.cmp c n1 n2) - | Ccompu c, I n1 :: I n2 :: nil => Some(Int.cmpu c n1 n2) - | Ccompimm c n, I n1 :: nil => Some(Int.cmp c n1 n) - | Ccompuimm c n, I n1 :: nil => Some(Int.cmpu c n1 n) - | Ccompf c, F n1 :: F n2 :: nil => Some(Float.cmp c n1 n2) - | Cnotcompf c, F n1 :: F n2 :: nil => Some(negb(Float.cmp c n1 n2)) - | Cmaskzero n, I n1 :: nil => Some(Int.eq (Int.and n1 n) Int.zero) - | Cmasknotzero n, n1::nil => Some(negb(Int.eq (Int.and n1 n) Int.zero)) - | _, _ => None - end. -*) - -Inductive eval_static_condition_cases: forall (cond: condition) (vl: list approx), Type := - | eval_static_condition_case1: - forall c n1 n2, - eval_static_condition_cases (Ccomp c) (I n1 :: I n2 :: nil) - | eval_static_condition_case2: - forall c n1 n2, - eval_static_condition_cases (Ccompu c) (I n1 :: I n2 :: nil) - | eval_static_condition_case3: - forall c n n1, - eval_static_condition_cases (Ccompimm c n) (I n1 :: nil) - | eval_static_condition_case4: - forall c n n1, - eval_static_condition_cases (Ccompuimm c n) (I n1 :: nil) - | eval_static_condition_case5: - forall c n1 n2, - eval_static_condition_cases (Ccompf c) (F n1 :: F n2 :: nil) - | eval_static_condition_case6: - forall c n1 n2, - eval_static_condition_cases (Cnotcompf c) (F n1 :: F n2 :: nil) - | eval_static_condition_case7: - forall n n1, - eval_static_condition_cases (Cmaskzero n) (I n1 :: nil) - | eval_static_condition_case8: - forall n n1, - eval_static_condition_cases (Cmasknotzero n) (I n1 :: nil) - | eval_static_condition_default: - forall (cond: condition) (vl: list approx), - eval_static_condition_cases cond vl. - -Definition eval_static_condition_match (cond: condition) (vl: list approx) := - match cond as z1, vl as z2 return eval_static_condition_cases z1 z2 with - | Ccomp c, I n1 :: I n2 :: nil => - eval_static_condition_case1 c n1 n2 - | Ccompu c, I n1 :: I n2 :: nil => - eval_static_condition_case2 c n1 n2 - | Ccompimm c n, I n1 :: nil => - eval_static_condition_case3 c n n1 - | Ccompuimm c n, I n1 :: nil => - eval_static_condition_case4 c n n1 - | Ccompf c, F n1 :: F n2 :: nil => - eval_static_condition_case5 c n1 n2 - | Cnotcompf c, F n1 :: F n2 :: nil => - eval_static_condition_case6 c n1 n2 - | Cmaskzero n, I n1 :: nil => - eval_static_condition_case7 n n1 - | Cmasknotzero n, I n1 :: nil => - eval_static_condition_case8 n n1 - | cond, vl => - eval_static_condition_default cond vl - end. - -Definition eval_static_condition (cond: condition) (vl: list approx) := - match eval_static_condition_match cond vl with - | eval_static_condition_case1 c n1 n2 => - Some(Int.cmp c n1 n2) - | eval_static_condition_case2 c n1 n2 => - Some(Int.cmpu c n1 n2) - | eval_static_condition_case3 c n n1 => - Some(Int.cmp c n1 n) - | eval_static_condition_case4 c n n1 => - Some(Int.cmpu c n1 n) - | eval_static_condition_case5 c n1 n2 => - Some(Float.cmp c n1 n2) - | eval_static_condition_case6 c n1 n2 => - Some(negb(Float.cmp c n1 n2)) - | eval_static_condition_case7 n n1 => - Some(Int.eq (Int.and n1 n) Int.zero) - | eval_static_condition_case8 n n1 => - Some(negb(Int.eq (Int.and n1 n) Int.zero)) - | eval_static_condition_default cond vl => - None - end. - -(* -Definition eval_static_operation (op: operation) (vl: list approx) := - match op, vl with - | Omove, v1::nil => v1 - | Ointconst n, nil => I n - | Ofloatconst n, nil => F n - | Oaddrsymbol s n, nil => S s n - | Ocast8signed, I n1 :: nil => I(Int.sign_ext 8 n) - | Ocast8unsigned, I n1 :: nil => I(Int.zero_ext 8 n) - | Ocast16signed, I n1 :: nil => I(Int.sign_ext 16 n) - | Ocast16unsigned, I n1 :: nil => I(Int.zero_ext 16 n) - | Oadd, I n1 :: I n2 :: nil => I(Int.add n1 n2) - | Oadd, S s1 n1 :: I n2 :: nil => S s1 (Int.add n1 n2) - | Oaddimm n, I n1 :: nil => I (Int.add n1 n) - | Oaddimm n, S s1 n1 :: nil => S s1 (Int.add n1 n) - | Osub, I n1 :: I n2 :: nil => I(Int.sub n1 n2) - | Osub, S s1 n1 :: I n2 :: nil => S s1 (Int.sub n1 n2) - | Osubimm n, I n1 :: nil => I (Int.sub n n1) - | Omul, I n1 :: I n2 :: nil => I(Int.mul n1 n2) - | Omulimm n, I n1 :: nil => I(Int.mul n1 n) - | Odiv, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2) - | Odivu, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2) - | Oand, I n1 :: I n2 :: nil => I(Int.and n1 n2) - | Oandimm n, I n1 :: nil => I(Int.and n1 n) - | Oor, I n1 :: I n2 :: nil => I(Int.or n1 n2) - | Oorimm n, I n1 :: nil => I(Int.or n1 n) - | Oxor, I n1 :: I n2 :: nil => I(Int.xor n1 n2) - | Oxorimm n, I n1 :: nil => I(Int.xor n1 n) - | Onand, I n1 :: I n2 :: nil => I(Int.xor (Int.and n1 n2) Int.mone) - | Onor, I n1 :: I n2 :: nil => I(Int.xor (Int.or n1 n2) Int.mone) - | Onxor, I n1 :: I n2 :: nil => I(Int.xor (Int.xor n1 n2) Int.mone) - | Oshl, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shl n1 n2) else Unknown - | Oshr, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shr n1 n2) else Unknown - | Oshrimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shr n1 n) else Unknown - | Oshrximm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shrx n1 n) else Unknown - | Oshru, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shru n1 n2) else Unknown - | Orolm amount mask, I n1 :: nil => I(Int.rolm n1 amount mask) - | Onegf, F n1 :: nil => F(Float.neg n1) - | Oabsf, F n1 :: nil => F(Float.abs n1) - | Oaddf, F n1 :: F n2 :: nil => F(Float.add n1 n2) - | Osubf, F n1 :: F n2 :: nil => F(Float.sub n1 n2) - | Omulf, F n1 :: F n2 :: nil => F(Float.mul n1 n2) - | Odivf, F n1 :: F n2 :: nil => F(Float.div n1 n2) - | Omuladdf, F n1 :: F n2 :: F n3 :: nil => F(Float.add (Float.mul n1 n2) n3) - | Omulsubf, F n1 :: F n2 :: F n3 :: nil => F(Float.sub (Float.mul n1 n2) n3) - | Osingleoffloat, F n1 :: nil => F(Float.singleoffloat n1) - | Ointoffloat, F n1 :: nil => match Float.intoffloat n1 with Some x => I x | None => Unknown end - | Ofloatofwords, I n1 :: I n2 :: nil => F(Float.from_words n1 n2) - | Ocmp c, vl => - match eval_static_condition c vl with - | None => Unknown - | Some b => I(if b then Int.one else Int.zero) - end - | _, _ => Unknown - end. -*) - -Inductive eval_static_operation_cases: forall (op: operation) (vl: list approx), Type := - | eval_static_operation_case1: - forall v1, - eval_static_operation_cases (Omove) (v1::nil) - | eval_static_operation_case2: - forall n, - eval_static_operation_cases (Ointconst n) (nil) - | eval_static_operation_case3: - forall n, - eval_static_operation_cases (Ofloatconst n) (nil) - | eval_static_operation_case4: - forall s n, - eval_static_operation_cases (Oaddrsymbol s n) (nil) - | eval_static_operation_case6: - forall n1, - eval_static_operation_cases (Ocast8signed) (I n1 :: nil) - | eval_static_operation_case7: - forall n1, - eval_static_operation_cases (Ocast16signed) (I n1 :: nil) - | eval_static_operation_case8: - forall n1 n2, - eval_static_operation_cases (Oadd) (I n1 :: I n2 :: nil) - | eval_static_operation_case9: - forall s1 n1 n2, - eval_static_operation_cases (Oadd) (S s1 n1 :: I n2 :: nil) - | eval_static_operation_case11: - forall n n1, - eval_static_operation_cases (Oaddimm n) (I n1 :: nil) - | eval_static_operation_case12: - forall n s1 n1, - eval_static_operation_cases (Oaddimm n) (S s1 n1 :: nil) - | eval_static_operation_case13: - forall n1 n2, - eval_static_operation_cases (Osub) (I n1 :: I n2 :: nil) - | eval_static_operation_case14: - forall s1 n1 n2, - eval_static_operation_cases (Osub) (S s1 n1 :: I n2 :: nil) - | eval_static_operation_case15: - forall n n1, - eval_static_operation_cases (Osubimm n) (I n1 :: nil) - | eval_static_operation_case16: - forall n1 n2, - eval_static_operation_cases (Omul) (I n1 :: I n2 :: nil) - | eval_static_operation_case17: - forall n n1, - eval_static_operation_cases (Omulimm n) (I n1 :: nil) - | eval_static_operation_case18: - forall n1 n2, - eval_static_operation_cases (Odiv) (I n1 :: I n2 :: nil) - | eval_static_operation_case19: - forall n1 n2, - eval_static_operation_cases (Odivu) (I n1 :: I n2 :: nil) - | eval_static_operation_case20: - forall n1 n2, - eval_static_operation_cases (Oand) (I n1 :: I n2 :: nil) - | eval_static_operation_case21: - forall n n1, - eval_static_operation_cases (Oandimm n) (I n1 :: nil) - | eval_static_operation_case22: - forall n1 n2, - eval_static_operation_cases (Oor) (I n1 :: I n2 :: nil) - | eval_static_operation_case23: - forall n n1, - eval_static_operation_cases (Oorimm n) (I n1 :: nil) - | eval_static_operation_case24: - forall n1 n2, - eval_static_operation_cases (Oxor) (I n1 :: I n2 :: nil) - | eval_static_operation_case25: - forall n n1, - eval_static_operation_cases (Oxorimm n) (I n1 :: nil) - | eval_static_operation_case26: - forall n1 n2, - eval_static_operation_cases (Onand) (I n1 :: I n2 :: nil) - | eval_static_operation_case27: - forall n1 n2, - eval_static_operation_cases (Onor) (I n1 :: I n2 :: nil) - | eval_static_operation_case28: - forall n1 n2, - eval_static_operation_cases (Onxor) (I n1 :: I n2 :: nil) - | eval_static_operation_case29: - forall n1 n2, - eval_static_operation_cases (Oshl) (I n1 :: I n2 :: nil) - | eval_static_operation_case30: - forall n1 n2, - eval_static_operation_cases (Oshr) (I n1 :: I n2 :: nil) - | eval_static_operation_case31: - forall n n1, - eval_static_operation_cases (Oshrimm n) (I n1 :: nil) - | eval_static_operation_case32: - forall n n1, - eval_static_operation_cases (Oshrximm n) (I n1 :: nil) - | eval_static_operation_case33: - forall n1 n2, - eval_static_operation_cases (Oshru) (I n1 :: I n2 :: nil) - | eval_static_operation_case34: - forall amount mask n1, - eval_static_operation_cases (Orolm amount mask) (I n1 :: nil) - | eval_static_operation_case35: - forall n1, - eval_static_operation_cases (Onegf) (F n1 :: nil) - | eval_static_operation_case36: - forall n1, - eval_static_operation_cases (Oabsf) (F n1 :: nil) - | eval_static_operation_case37: - forall n1 n2, - eval_static_operation_cases (Oaddf) (F n1 :: F n2 :: nil) - | eval_static_operation_case38: - forall n1 n2, - eval_static_operation_cases (Osubf) (F n1 :: F n2 :: nil) - | eval_static_operation_case39: - forall n1 n2, - eval_static_operation_cases (Omulf) (F n1 :: F n2 :: nil) - | eval_static_operation_case40: - forall n1 n2, - eval_static_operation_cases (Odivf) (F n1 :: F n2 :: nil) - | eval_static_operation_case41: - forall n1 n2 n3, - eval_static_operation_cases (Omuladdf) (F n1 :: F n2 :: F n3 :: nil) - | eval_static_operation_case42: - forall n1 n2 n3, - eval_static_operation_cases (Omulsubf) (F n1 :: F n2 :: F n3 :: nil) - | eval_static_operation_case43: - forall n1, - eval_static_operation_cases (Osingleoffloat) (F n1 :: nil) - | eval_static_operation_case44: - forall n1, - eval_static_operation_cases (Ointoffloat) (F n1 :: nil) - | eval_static_operation_case45: - forall n1 n2, - eval_static_operation_cases (Ofloatofwords) (I n1 :: I n2 :: nil) - | eval_static_operation_case47: - forall c vl, - eval_static_operation_cases (Ocmp c) (vl) - | eval_static_operation_case48: - forall n1, - eval_static_operation_cases (Ocast8unsigned) (I n1 :: nil) - | eval_static_operation_case49: - forall n1, - eval_static_operation_cases (Ocast16unsigned) (I n1 :: nil) - | eval_static_operation_default: - forall (op: operation) (vl: list approx), - eval_static_operation_cases op vl. - -Definition eval_static_operation_match (op: operation) (vl: list approx) := - match op as z1, vl as z2 return eval_static_operation_cases z1 z2 with - | Omove, v1::nil => - eval_static_operation_case1 v1 - | Ointconst n, nil => - eval_static_operation_case2 n - | Ofloatconst n, nil => - eval_static_operation_case3 n - | Oaddrsymbol s n, nil => - eval_static_operation_case4 s n - | Ocast8signed, I n1 :: nil => - eval_static_operation_case6 n1 - | Ocast16signed, I n1 :: nil => - eval_static_operation_case7 n1 - | Oadd, I n1 :: I n2 :: nil => - eval_static_operation_case8 n1 n2 - | Oadd, S s1 n1 :: I n2 :: nil => - eval_static_operation_case9 s1 n1 n2 - | Oaddimm n, I n1 :: nil => - eval_static_operation_case11 n n1 - | Oaddimm n, S s1 n1 :: nil => - eval_static_operation_case12 n s1 n1 - | Osub, I n1 :: I n2 :: nil => - eval_static_operation_case13 n1 n2 - | Osub, S s1 n1 :: I n2 :: nil => - eval_static_operation_case14 s1 n1 n2 - | Osubimm n, I n1 :: nil => - eval_static_operation_case15 n n1 - | Omul, I n1 :: I n2 :: nil => - eval_static_operation_case16 n1 n2 - | Omulimm n, I n1 :: nil => - eval_static_operation_case17 n n1 - | Odiv, I n1 :: I n2 :: nil => - eval_static_operation_case18 n1 n2 - | Odivu, I n1 :: I n2 :: nil => - eval_static_operation_case19 n1 n2 - | Oand, I n1 :: I n2 :: nil => - eval_static_operation_case20 n1 n2 - | Oandimm n, I n1 :: nil => - eval_static_operation_case21 n n1 - | Oor, I n1 :: I n2 :: nil => - eval_static_operation_case22 n1 n2 - | Oorimm n, I n1 :: nil => - eval_static_operation_case23 n n1 - | Oxor, I n1 :: I n2 :: nil => - eval_static_operation_case24 n1 n2 - | Oxorimm n, I n1 :: nil => - eval_static_operation_case25 n n1 - | Onand, I n1 :: I n2 :: nil => - eval_static_operation_case26 n1 n2 - | Onor, I n1 :: I n2 :: nil => - eval_static_operation_case27 n1 n2 - | Onxor, I n1 :: I n2 :: nil => - eval_static_operation_case28 n1 n2 - | Oshl, I n1 :: I n2 :: nil => - eval_static_operation_case29 n1 n2 - | Oshr, I n1 :: I n2 :: nil => - eval_static_operation_case30 n1 n2 - | Oshrimm n, I n1 :: nil => - eval_static_operation_case31 n n1 - | Oshrximm n, I n1 :: nil => - eval_static_operation_case32 n n1 - | Oshru, I n1 :: I n2 :: nil => - eval_static_operation_case33 n1 n2 - | Orolm amount mask, I n1 :: nil => - eval_static_operation_case34 amount mask n1 - | Onegf, F n1 :: nil => - eval_static_operation_case35 n1 - | Oabsf, F n1 :: nil => - eval_static_operation_case36 n1 - | Oaddf, F n1 :: F n2 :: nil => - eval_static_operation_case37 n1 n2 - | Osubf, F n1 :: F n2 :: nil => - eval_static_operation_case38 n1 n2 - | Omulf, F n1 :: F n2 :: nil => - eval_static_operation_case39 n1 n2 - | Odivf, F n1 :: F n2 :: nil => - eval_static_operation_case40 n1 n2 - | Omuladdf, F n1 :: F n2 :: F n3 :: nil => - eval_static_operation_case41 n1 n2 n3 - | Omulsubf, F n1 :: F n2 :: F n3 :: nil => - eval_static_operation_case42 n1 n2 n3 - | Osingleoffloat, F n1 :: nil => - eval_static_operation_case43 n1 - | Ointoffloat, F n1 :: nil => - eval_static_operation_case44 n1 - | Ofloatofwords, I n1 :: I n2 :: nil => - eval_static_operation_case45 n1 n2 - | Ocmp c, vl => - eval_static_operation_case47 c vl - | Ocast8unsigned, I n1 :: nil => - eval_static_operation_case48 n1 - | Ocast16unsigned, I n1 :: nil => - eval_static_operation_case49 n1 - | op, vl => - eval_static_operation_default op vl - end. - -Definition eval_static_operation (op: operation) (vl: list approx) := - match eval_static_operation_match op vl with - | eval_static_operation_case1 v1 => - v1 - | eval_static_operation_case2 n => - I n - | eval_static_operation_case3 n => - F n - | eval_static_operation_case4 s n => - S s n - | eval_static_operation_case6 n1 => - I(Int.sign_ext 8 n1) - | eval_static_operation_case7 n1 => - I(Int.sign_ext 16 n1) - | eval_static_operation_case8 n1 n2 => - I(Int.add n1 n2) - | eval_static_operation_case9 s1 n1 n2 => - S s1 (Int.add n1 n2) - | eval_static_operation_case11 n n1 => - I (Int.add n1 n) - | eval_static_operation_case12 n s1 n1 => - S s1 (Int.add n1 n) - | eval_static_operation_case13 n1 n2 => - I(Int.sub n1 n2) - | eval_static_operation_case14 s1 n1 n2 => - S s1 (Int.sub n1 n2) - | eval_static_operation_case15 n n1 => - I (Int.sub n n1) - | eval_static_operation_case16 n1 n2 => - I(Int.mul n1 n2) - | eval_static_operation_case17 n n1 => - I(Int.mul n1 n) - | eval_static_operation_case18 n1 n2 => - if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2) - | eval_static_operation_case19 n1 n2 => - if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2) - | eval_static_operation_case20 n1 n2 => - I(Int.and n1 n2) - | eval_static_operation_case21 n n1 => - I(Int.and n1 n) - | eval_static_operation_case22 n1 n2 => - I(Int.or n1 n2) - | eval_static_operation_case23 n n1 => - I(Int.or n1 n) - | eval_static_operation_case24 n1 n2 => - I(Int.xor n1 n2) - | eval_static_operation_case25 n n1 => - I(Int.xor n1 n) - | eval_static_operation_case26 n1 n2 => - I(Int.xor (Int.and n1 n2) Int.mone) - | eval_static_operation_case27 n1 n2 => - I(Int.xor (Int.or n1 n2) Int.mone) - | eval_static_operation_case28 n1 n2 => - I(Int.xor (Int.xor n1 n2) Int.mone) - | eval_static_operation_case29 n1 n2 => - if Int.ltu n2 Int.iwordsize then I(Int.shl n1 n2) else Unknown - | eval_static_operation_case30 n1 n2 => - if Int.ltu n2 Int.iwordsize then I(Int.shr n1 n2) else Unknown - | eval_static_operation_case31 n n1 => - if Int.ltu n Int.iwordsize then I(Int.shr n1 n) else Unknown - | eval_static_operation_case32 n n1 => - if Int.ltu n Int.iwordsize then I(Int.shrx n1 n) else Unknown - | eval_static_operation_case33 n1 n2 => - if Int.ltu n2 Int.iwordsize then I(Int.shru n1 n2) else Unknown - | eval_static_operation_case34 amount mask n1 => - I(Int.rolm n1 amount mask) - | eval_static_operation_case35 n1 => - F(Float.neg n1) - | eval_static_operation_case36 n1 => - F(Float.abs n1) - | eval_static_operation_case37 n1 n2 => - F(Float.add n1 n2) - | eval_static_operation_case38 n1 n2 => - F(Float.sub n1 n2) - | eval_static_operation_case39 n1 n2 => - F(Float.mul n1 n2) - | eval_static_operation_case40 n1 n2 => - F(Float.div n1 n2) - | eval_static_operation_case41 n1 n2 n3 => - F(Float.add (Float.mul n1 n2) n3) - | eval_static_operation_case42 n1 n2 n3 => - F(Float.sub (Float.mul n1 n2) n3) - | eval_static_operation_case43 n1 => - F(Float.singleoffloat n1) - | eval_static_operation_case44 n1 => - match Float.intoffloat n1 with Some x => I x | None => Unknown end - | eval_static_operation_case45 n1 n2 => - F(Float.from_words n1 n2) - | eval_static_operation_case47 c vl => - match eval_static_condition c vl with - | None => Unknown - | Some b => I(if b then Int.one else Int.zero) - end - | eval_static_operation_case48 n1 => - I(Int.zero_ext 8 n1) - | eval_static_operation_case49 n1 => - I(Int.zero_ext 16 n1) - | eval_static_operation_default op vl => - Unknown - end. - -(** * Operator strength reduction *) - -(** We now define auxiliary functions for strength reduction of - operators and addressing modes: replacing an operator with a cheaper - one if some of its arguments are statically known. These are again - large pattern-matchings expressed in indirect style. *) - -Section STRENGTH_REDUCTION. - -Variable app: reg -> approx. - -Definition intval (r: reg) : option int := - match app r with I n => Some n | _ => None end. - -Inductive cond_strength_reduction_cases: condition -> list reg -> Type := - | csr_case1: - forall c r1 r2, - cond_strength_reduction_cases (Ccomp c) (r1 :: r2 :: nil) - | csr_case2: - forall c r1 r2, - cond_strength_reduction_cases (Ccompu c) (r1 :: r2 :: nil) - | csr_default: - forall c rl, - cond_strength_reduction_cases c rl. - -Definition cond_strength_reduction_match (cond: condition) (rl: list reg) := - match cond as x, rl as y return cond_strength_reduction_cases x y with - | Ccomp c, r1 :: r2 :: nil => - csr_case1 c r1 r2 - | Ccompu c, r1 :: r2 :: nil => - csr_case2 c r1 r2 - | cond, rl => - csr_default cond rl - end. - -Definition cond_strength_reduction - (cond: condition) (args: list reg) : condition * list reg := - match cond_strength_reduction_match cond args with - | csr_case1 c r1 r2 => - match intval r1, intval r2 with - | Some n, _ => - (Ccompimm (swap_comparison c) n, r2 :: nil) - | _, Some n => - (Ccompimm c n, r1 :: nil) - | _, _ => - (cond, args) - end - | csr_case2 c r1 r2 => - match intval r1, intval r2 with - | Some n, _ => - (Ccompuimm (swap_comparison c) n, r2 :: nil) - | _, Some n => - (Ccompuimm c n, r1 :: nil) - | _, _ => - (cond, args) - end - | csr_default cond args => - (cond, args) - end. - -Definition make_addimm (n: int) (r: reg) := - if Int.eq n Int.zero - then (Omove, r :: nil) - else (Oaddimm n, r :: nil). - -Definition make_shlimm (n: int) (r: reg) := - if Int.eq n Int.zero - then (Omove, r :: nil) - else (Orolm n (Int.shl Int.mone n), r :: nil). - -Definition make_shrimm (n: int) (r: reg) := - if Int.eq n Int.zero - then (Omove, r :: nil) - else (Oshrimm n, r :: nil). - -Definition make_shruimm (n: int) (r: reg) := - if Int.eq n Int.zero - then (Omove, r :: nil) - else (Orolm (Int.sub Int.iwordsize n) (Int.shru Int.mone n), r :: nil). - -Definition make_mulimm (n: int) (r: reg) := - if Int.eq n Int.zero then - (Ointconst Int.zero, nil) - else if Int.eq n Int.one then - (Omove, r :: nil) - else - match Int.is_power2 n with - | Some l => make_shlimm l r - | None => (Omulimm n, r :: nil) - end. - -Definition make_andimm (n: int) (r: reg) := - if Int.eq n Int.zero - then (Ointconst Int.zero, nil) - else if Int.eq n Int.mone then (Omove, r :: nil) - else (Oandimm n, r :: nil). - -Definition make_orimm (n: int) (r: reg) := - if Int.eq n Int.zero then (Omove, r :: nil) - else if Int.eq n Int.mone then (Ointconst Int.mone, nil) - else (Oorimm n, r :: nil). - -Definition make_xorimm (n: int) (r: reg) := - if Int.eq n Int.zero - then (Omove, r :: nil) - else (Oxorimm n, r :: nil). - -Inductive op_strength_reduction_cases: operation -> list reg -> Type := - | op_strength_reduction_case1: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Oadd (r1 :: r2 :: nil) - | op_strength_reduction_case2: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Osub (r1 :: r2 :: nil) - | op_strength_reduction_case3: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Omul (r1 :: r2 :: nil) - | op_strength_reduction_case4: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Odiv (r1 :: r2 :: nil) - | op_strength_reduction_case5: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Odivu (r1 :: r2 :: nil) - | op_strength_reduction_case6: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Oand (r1 :: r2 :: nil) - | op_strength_reduction_case7: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Oor (r1 :: r2 :: nil) - | op_strength_reduction_case8: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Oxor (r1 :: r2 :: nil) - | op_strength_reduction_case9: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Oshl (r1 :: r2 :: nil) - | op_strength_reduction_case10: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Oshr (r1 :: r2 :: nil) - | op_strength_reduction_case11: - forall (r1: reg) (r2: reg), - op_strength_reduction_cases Oshru (r1 :: r2 :: nil) - | op_strength_reduction_case12: - forall (c: condition) (rl: list reg), - op_strength_reduction_cases (Ocmp c) rl - | op_strength_reduction_default: - forall (op: operation) (args: list reg), - op_strength_reduction_cases op args. - -Definition op_strength_reduction_match (op: operation) (args: list reg) := - match op as z1, args as z2 return op_strength_reduction_cases z1 z2 with - | Oadd, r1 :: r2 :: nil => - op_strength_reduction_case1 r1 r2 - | Osub, r1 :: r2 :: nil => - op_strength_reduction_case2 r1 r2 - | Omul, r1 :: r2 :: nil => - op_strength_reduction_case3 r1 r2 - | Odiv, r1 :: r2 :: nil => - op_strength_reduction_case4 r1 r2 - | Odivu, r1 :: r2 :: nil => - op_strength_reduction_case5 r1 r2 - | Oand, r1 :: r2 :: nil => - op_strength_reduction_case6 r1 r2 - | Oor, r1 :: r2 :: nil => - op_strength_reduction_case7 r1 r2 - | Oxor, r1 :: r2 :: nil => - op_strength_reduction_case8 r1 r2 - | Oshl, r1 :: r2 :: nil => - op_strength_reduction_case9 r1 r2 - | Oshr, r1 :: r2 :: nil => - op_strength_reduction_case10 r1 r2 - | Oshru, r1 :: r2 :: nil => - op_strength_reduction_case11 r1 r2 - | Ocmp c, rl => - op_strength_reduction_case12 c rl - | op, args => - op_strength_reduction_default op args - end. - -Definition op_strength_reduction (op: operation) (args: list reg) := - match op_strength_reduction_match op args with - | op_strength_reduction_case1 r1 r2 => (* Oadd *) - match intval r1, intval r2 with - | Some n, _ => make_addimm n r2 - | _, Some n => make_addimm n r1 - | _, _ => (op, args) - end - | op_strength_reduction_case2 r1 r2 => (* Osub *) - match intval r1, intval r2 with - | Some n, _ => (Osubimm n, r2 :: nil) - | _, Some n => make_addimm (Int.neg n) r1 - | _, _ => (op, args) - end - | op_strength_reduction_case3 r1 r2 => (* Omul *) - match intval r1, intval r2 with - | Some n, _ => make_mulimm n r2 - | _, Some n => make_mulimm n r1 - | _, _ => (op, args) - end - | op_strength_reduction_case4 r1 r2 => (* Odiv *) - match intval r2 with - | Some n => - match Int.is_power2 n with - | Some l => (Oshrximm l, r1 :: nil) - | None => (op, args) - end - | None => - (op, args) - end - | op_strength_reduction_case5 r1 r2 => (* Odivu *) - match intval r2 with - | Some n => - match Int.is_power2 n with - | Some l => make_shruimm l r1 - | None => (op, args) - end - | None => - (op, args) - end - | op_strength_reduction_case6 r1 r2 => (* Oand *) - match intval r1, intval r2 with - | Some n, _ => make_andimm n r2 - | _, Some n => make_andimm n r1 - | _, _ => (op, args) - end - | op_strength_reduction_case7 r1 r2 => (* Oor *) - match intval r1, intval r2 with - | Some n, _ => make_orimm n r2 - | _, Some n => make_orimm n r1 - | _, _ => (op, args) - end - | op_strength_reduction_case8 r1 r2 => (* Oxor *) - match intval r1, intval r2 with - | Some n, _ => make_xorimm n r2 - | _, Some n => make_xorimm n r1 - | _, _ => (op, args) - end - | op_strength_reduction_case9 r1 r2 => (* Oshl *) - match intval r2 with - | Some n => - if Int.ltu n Int.iwordsize - then make_shlimm n r1 - else (op, args) - | _ => (op, args) - end - | op_strength_reduction_case10 r1 r2 => (* Oshr *) - match intval r2 with - | Some n => - if Int.ltu n Int.iwordsize - then make_shrimm n r1 - else (op, args) - | _ => (op, args) - end - | op_strength_reduction_case11 r1 r2 => (* Oshru *) - match intval r2 with - | Some n => - if Int.ltu n Int.iwordsize - then make_shruimm n r1 - else (op, args) - | _ => (op, args) - end - | op_strength_reduction_case12 c args => (* Ocmp *) - let (c', args') := cond_strength_reduction c args in - (Ocmp c', args') - | op_strength_reduction_default op args => (* default *) - (op, args) - end. - -Inductive addr_strength_reduction_cases: forall (addr: addressing) (args: list reg), Type := - | addr_strength_reduction_case1: - forall (r1: reg) (r2: reg), - addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) - | addr_strength_reduction_case2: - forall (symb: ident) (ofs: int) (r1: reg), - addr_strength_reduction_cases (Abased symb ofs) (r1 :: nil) - | addr_strength_reduction_case3: - forall n r1, - addr_strength_reduction_cases (Aindexed n) (r1 :: nil) - | addr_strength_reduction_default: - forall (addr: addressing) (args: list reg), - addr_strength_reduction_cases addr args. - -Definition addr_strength_reduction_match (addr: addressing) (args: list reg) := - match addr as z1, args as z2 return addr_strength_reduction_cases z1 z2 with - | Aindexed2, r1 :: r2 :: nil => - addr_strength_reduction_case1 r1 r2 - | Abased symb ofs, r1 :: nil => - addr_strength_reduction_case2 symb ofs r1 - | Aindexed n, r1 :: nil => - addr_strength_reduction_case3 n r1 - | addr, args => - addr_strength_reduction_default addr args - end. - -Definition addr_strength_reduction (addr: addressing) (args: list reg) := - match addr_strength_reduction_match addr args with - | addr_strength_reduction_case1 r1 r2 => (* Aindexed2 *) - match app r1, app r2 with - | S symb n1, I n2 => (Aglobal symb (Int.add n1 n2), nil) - | S symb n1, _ => (Abased symb n1, r2 :: nil) - | I n1, S symb n2 => (Aglobal symb (Int.add n1 n2), nil) - | I n1, _ => (Aindexed n1, r2 :: nil) - | _, S symb n2 => (Abased symb n2, r1 :: nil) - | _, I n2 => (Aindexed n2, r1 :: nil) - | _, _ => (addr, args) - end - | addr_strength_reduction_case2 symb ofs r1 => (* Abased *) - match intval r1 with - | Some n => (Aglobal symb (Int.add ofs n), nil) - | _ => (addr, args) - end - | addr_strength_reduction_case3 n r1 => (* Aindexed *) - match app r1 with - | S symb ofs => (Aglobal symb (Int.add ofs n), nil) - | _ => (addr, args) - end - | addr_strength_reduction_default addr args => (* default *) - (addr, args) - end. - -End STRENGTH_REDUCTION. diff --git a/powerpc/ConstpropOp.vp b/powerpc/ConstpropOp.vp new file mode 100644 index 0000000..22e89e3 --- /dev/null +++ b/powerpc/ConstpropOp.vp @@ -0,0 +1,277 @@ +(* *********************************************************************) +(* *) +(* 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. *) +(* *) +(* *********************************************************************) + +(** Static analysis and strength reduction for operators + and conditions. This is the machine-dependent part of [Constprop]. *) + +Require Import Coqlib. +Require Import AST. +Require Import Integers. +Require Import Floats. +Require Import Values. +Require Import Op. +Require Import Registers. + +(** * Static analysis *) + +(** To each pseudo-register at each program point, the static analysis + associates a compile-time approximation taken from the following set. *) + +Inductive approx : Type := + | Novalue: approx (** No value possible, code is unreachable. *) + | Unknown: approx (** All values are possible, + no compile-time information is available. *) + | I: int -> approx (** A known integer value. *) + | F: float -> approx (** A known floating-point value. *) + | G: ident -> int -> approx + (** The value is the address of the given global + symbol plus the given integer offset. *) + | S: int -> approx. (** The value is the stack pointer plus the offset. *) + +(** We now define the abstract interpretations of conditions and operators + over this set of approximations. For instance, the abstract interpretation + of the operator [Oaddf] applied to two expressions [a] and [b] is + [F(Float.add f g)] if [a] and [b] have static approximations [Vfloat f] + and [Vfloat g] respectively, and [Unknown] otherwise. + + The static approximations are defined by large pattern-matchings over + the approximations of the results. We write these matchings in the + indirect style described in file [SelectOp] to avoid excessive + duplication of cases in proofs. *) + +Nondetfunction eval_static_condition (cond: condition) (vl: list approx) := + match cond, vl with + | Ccomp c, I n1 :: I n2 :: nil => Some(Int.cmp c n1 n2) + | Ccompu c, I n1 :: I n2 :: nil => Some(Int.cmpu c n1 n2) + | Ccompimm c n, I n1 :: nil => Some(Int.cmp c n1 n) + | Ccompuimm c n, I n1 :: nil => Some(Int.cmpu c n1 n) + | Ccompf c, F n1 :: F n2 :: nil => Some(Float.cmp c n1 n2) + | Cnotcompf c, F n1 :: F n2 :: nil => Some(negb(Float.cmp c n1 n2)) + | Cmaskzero n, I n1 :: nil => Some(Int.eq (Int.and n1 n) Int.zero) + | Cmasknotzero n, I n1::nil => Some(negb(Int.eq (Int.and n1 n) Int.zero)) + | _, _ => None + end. + +Definition eval_static_condition_val (cond: condition) (vl: list approx) := + match eval_static_condition cond vl with + | None => Unknown + | Some b => I(if b then Int.one else Int.zero) + end. + +Definition eval_static_intoffloat (f: float) := + match Float.intoffloat f with Some x => I x | None => Unknown end. + +Nondetfunction eval_static_operation (op: operation) (vl: list approx) := + match op, vl with + | Omove, v1::nil => v1 + | Ointconst n, nil => I n + | Ofloatconst n, nil => F n + | Oaddrsymbol s n, nil => G s n + | Oaddrstack n, nil => S n + | Ocast8signed, I n1 :: nil => I(Int.sign_ext 8 n1) + | Ocast16signed, I n1 :: nil => I(Int.sign_ext 16 n1) + | Oadd, I n1 :: I n2 :: nil => I(Int.add n1 n2) + | Oadd, G s1 n1 :: I n2 :: nil => G s1 (Int.add n1 n2) + | Oadd, I n1 :: G s2 n2 :: nil => G s2 (Int.add n1 n2) + | Oadd, S n1 :: I n2 :: nil => S (Int.add n1 n2) + | Oadd, I n1 :: S n2 :: nil => S (Int.add n1 n2) + | Oaddimm n, I n1 :: nil => I (Int.add n1 n) + | Oaddimm n, G s1 n1 :: nil => G s1 (Int.add n1 n) + | Oaddimm n, S n1 :: nil => S (Int.add n1 n) + | Osub, I n1 :: I n2 :: nil => I(Int.sub n1 n2) + | Osub, G s1 n1 :: I n2 :: nil => G s1 (Int.sub n1 n2) + | Osub, S n1 :: I n2 :: nil => S (Int.sub n1 n2) + | Osubimm n, I n1 :: nil => I (Int.sub n n1) + | Omul, I n1 :: I n2 :: nil => I(Int.mul n1 n2) + | Omulimm n, I n1 :: nil => I(Int.mul n1 n) + | Odiv, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2) + | Odivu, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2) + | Oand, I n1 :: I n2 :: nil => I(Int.and n1 n2) + | Oandimm n, I n1 :: nil => I(Int.and n1 n) + | Oor, I n1 :: I n2 :: nil => I(Int.or n1 n2) + | Oorimm n, I n1 :: nil => I(Int.or n1 n) + | Oxor, I n1 :: I n2 :: nil => I(Int.xor n1 n2) + | Oxorimm n, I n1 :: nil => I(Int.xor n1 n) + | Onand, I n1 :: I n2 :: nil => I(Int.xor (Int.and n1 n2) Int.mone) + | Onor, I n1 :: I n2 :: nil => I(Int.xor (Int.or n1 n2) Int.mone) + | Onxor, I n1 :: I n2 :: nil => I(Int.xor (Int.xor n1 n2) Int.mone) + | Oshl, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shl n1 n2) else Unknown + | Oshr, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shr n1 n2) else Unknown + | Oshrimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shr n1 n) else Unknown + | Oshrximm n, I n1 :: nil => if Int.ltu n (Int.repr 31) then I(Int.shrx n1 n) else Unknown + | Oshru, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shru n1 n2) else Unknown + | Orolm amount mask, I n1 :: nil => I(Int.rolm n1 amount mask) + | Oroli amount mask, I n1 :: I n2 :: nil => I(Int.or (Int.and n1 (Int.not mask)) (Int.rolm n2 amount mask)) + | Onegf, F n1 :: nil => F(Float.neg n1) + | Oabsf, F n1 :: nil => F(Float.abs n1) + | Oaddf, F n1 :: F n2 :: nil => F(Float.add n1 n2) + | Osubf, F n1 :: F n2 :: nil => F(Float.sub n1 n2) + | Omulf, F n1 :: F n2 :: nil => F(Float.mul n1 n2) + | Odivf, F n1 :: F n2 :: nil => F(Float.div n1 n2) + | Omuladdf, F n1 :: F n2 :: F n3 :: nil => F(Float.add (Float.mul n1 n2) n3) + | Omulsubf, F n1 :: F n2 :: F n3 :: nil => F(Float.sub (Float.mul n1 n2) n3) + | Osingleoffloat, F n1 :: nil => F(Float.singleoffloat n1) + | Ointoffloat, F n1 :: nil => eval_static_intoffloat n1 + | Ofloatofwords, I n1 :: I n2 :: nil => F(Float.from_words n1 n2) + | Ocmp c, vl => eval_static_condition_val c vl + | _, _ => Unknown + end. + +(** * Operator strength reduction *) + +(** We now define auxiliary functions for strength reduction of + operators and addressing modes: replacing an operator with a cheaper + one if some of its arguments are statically known. These are again + large pattern-matchings expressed in indirect style. *) + +Section STRENGTH_REDUCTION. + +Nondetfunction cond_strength_reduction + (cond: condition) (args: list reg) (vl: list approx) := + match cond, args, vl with + | Ccomp c, r1 :: r2 :: nil, I n1 :: v2 :: nil => + (Ccompimm (swap_comparison c) n1, r2 :: nil) + | Ccomp c, r1 :: r2 :: nil, v1 :: I n2 :: nil => + (Ccompimm c n2, r1 :: nil) + | Ccompu c, r1 :: r2 :: nil, I n1 :: v2 :: nil => + (Ccompuimm (swap_comparison c) n1, r2 :: nil) + | Ccompu c, r1 :: r2 :: nil, v1 :: I n2 :: nil => + (Ccompuimm c n2, r1 :: nil) + | _, _, _ => + (cond, args) + end. + +Definition make_addimm (n: int) (r: reg) := + if Int.eq n Int.zero + then (Omove, r :: nil) + else (Oaddimm n, r :: nil). + +Definition make_shlimm (n: int) (r1 r2: reg) := + if Int.eq n Int.zero then + (Omove, r1 :: nil) + else if Int.ltu n Int.iwordsize then + (Orolm n (Int.shl Int.mone n), r1 :: nil) + else + (Oshl, r1 :: r2 :: nil). + +Definition make_shrimm (n: int) (r1 r2: reg) := + if Int.eq n Int.zero then + (Omove, r1 :: nil) + else if Int.ltu n Int.iwordsize then + (Oshrimm n, r1 :: nil) + else + (Oshr, r1 :: r2 :: nil). + +Definition make_shruimm (n: int) (r1 r2: reg) := + if Int.eq n Int.zero then + (Omove, r1 :: nil) + else if Int.ltu n Int.iwordsize then + (Orolm (Int.sub Int.iwordsize n) (Int.shru Int.mone n), r1 :: nil) + else + (Oshru, r1 :: r2 :: nil). + +Definition make_mulimm (n: int) (r1 r2: reg) := + if Int.eq n Int.zero then + (Ointconst Int.zero, nil) + else if Int.eq n Int.one then + (Omove, r1 :: nil) + else + match Int.is_power2 n with + | Some l => (Orolm l (Int.shl Int.mone l), r1 :: nil) + | None => (Omulimm n, r1 :: nil) + end. + +Definition make_divimm (n: int) (r1 r2: reg) := + match Int.is_power2 n with + | Some l => if Int.ltu l (Int.repr 31) + then (Oshrximm l, r1 :: nil) + else (Odiv, r1 :: r2 :: nil) + | None => (Odiv, r1 :: r2 :: nil) + end. + +Definition make_divuimm (n: int) (r1 r2: reg) := + match Int.is_power2 n with + | Some l => (Orolm (Int.sub Int.iwordsize l) (Int.shru Int.mone l), r1 :: nil) + | None => (Odivu, r1 :: r2 :: nil) + end. + +Definition make_andimm (n: int) (r: reg) := + if Int.eq n Int.zero + then (Ointconst Int.zero, nil) + else if Int.eq n Int.mone then (Omove, r :: nil) + else (Oandimm n, r :: nil). + +Definition make_orimm (n: int) (r: reg) := + if Int.eq n Int.zero then (Omove, r :: nil) + else if Int.eq n Int.mone then (Ointconst Int.mone, nil) + else (Oorimm n, r :: nil). + +Definition make_xorimm (n: int) (r: reg) := + if Int.eq n Int.zero + then (Omove, r :: nil) + else (Oxorimm n, r :: nil). + +Nondetfunction op_strength_reduction + (op: operation) (args: list reg) (vl: list approx) := + match op, args, vl with + | Oadd, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_addimm n1 r2 + | Oadd, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_addimm n2 r1 + | Osub, r1 :: r2 :: nil, I n1 :: v2 :: nil => (Osubimm n1, r2 :: nil) + | Osub, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_addimm (Int.neg n2) r1 + | Omul, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_mulimm n1 r2 r1 + | Omul, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_mulimm n2 r1 r2 + | Odiv, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_divimm n2 r1 r2 + | Odivu, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_divuimm n2 r1 r2 + | Oand, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_andimm n1 r2 + | Oand, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_andimm n2 r1 + | Oor, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_orimm n1 r2 + | Oor, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_orimm n2 r1 + | Oxor, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_xorimm n1 r2 + | Oxor, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_xorimm n2 r1 + | Oshl, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shlimm n2 r1 r2 + | Oshr, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shrimm n2 r1 r2 + | Oshru, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shruimm n2 r1 r2 + | Ocmp c, args, vl => + let (c', args') := cond_strength_reduction c args vl in (Ocmp c', args') + | _, _, _ => (op, args) + end. + +Nondetfunction addr_strength_reduction + (addr: addressing) (args: list reg) (vl: list approx) := + match addr, args, vl with + | Aindexed2, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil => + (Aglobal symb (Int.add n1 n2), nil) + | Aindexed2, r1 :: r2 :: nil, I n1 :: G symb n2 :: nil => + (Aglobal symb (Int.add n1 n2), nil) + | Aindexed2, r1 :: r2 :: nil, S n1 :: I n2 :: nil => + (Ainstack (Int.add n1 n2), nil) + | Aindexed2, r1 :: r2 :: nil, I n1 :: S n2 :: nil => + (Ainstack (Int.add n1 n2), nil) + | Aindexed2, r1 :: r2 :: nil, G symb n1 :: v2 :: nil => + (Abased symb n1, r2 :: nil) + | Aindexed2, r1 :: r2 :: nil, v1 :: G symb n2 :: nil => + (Abased symb n2, r1 :: nil) + | Aindexed2, r1 :: r2 :: nil, I n1 :: v2 :: nil => + (Aindexed n1, r2 :: nil) + | Aindexed2, r1 :: r2 :: nil, v1 :: I n2 :: nil => + (Aindexed n2, r1 :: nil) + | Abased symb ofs, r1 :: nil, I n1 :: nil => + (Aglobal symb (Int.add ofs n1), nil) + | Aindexed n, r1 :: nil, G symb n1 :: nil => + (Aglobal symb (Int.add n1 n), nil) + | Aindexed n, r1 :: nil, S n1 :: nil => + (Ainstack (Int.add n1 n), nil) + | _, _, _ => + (addr, args) + end. + +End STRENGTH_REDUCTION. diff --git a/powerpc/ConstpropOpproof.v b/powerpc/ConstpropOpproof.v index bf065b7..36444b3 100644 --- a/powerpc/ConstpropOpproof.v +++ b/powerpc/ConstpropOpproof.v @@ -30,6 +30,7 @@ Require Import Constprop. Section ANALYSIS. Variable ge: genv. +Variable sp: val. (** We first show that the dataflow analysis is correct with respect to the dynamic semantics: the approximations (sets of values) @@ -43,7 +44,8 @@ Definition val_match_approx (a: approx) (v: val) : Prop := | Unknown => True | I p => v = Vint p | F p => v = Vfloat p - | S symb ofs => exists b, Genv.find_symbol ge symb = Some b /\ v = Vptr b ofs + | G symb ofs => v = symbol_address ge symb ofs + | S ofs => v = Val.add sp (Vint ofs) | _ => False end. @@ -62,12 +64,10 @@ Ltac SimplVMA := simpl in H; (try subst v); SimplVMA | H: (val_match_approx (F _) ?v) |- _ => simpl in H; (try subst v); SimplVMA - | H: (val_match_approx (S _ _) ?v) |- _ => - simpl in H; - (try (elim H; - let b := fresh "b" in let A := fresh in let B := fresh in - (intros b [A B]; subst v; clear H))); - SimplVMA + | H: (val_match_approx (G _ _) ?v) |- _ => + simpl in H; (try subst v); SimplVMA + | H: (val_match_approx (S _) ?v) |- _ => + simpl in H; (try subst v); SimplVMA | _ => idtac end. @@ -75,9 +75,9 @@ Ltac SimplVMA := Ltac InvVLMA := match goal with | H: (val_list_match_approx nil ?vl) |- _ => - inversion H + inv H | H: (val_list_match_approx (?a :: ?al) ?vl) |- _ => - inversion H; SimplVMA; InvVLMA + inv H; SimplVMA; InvVLMA | _ => idtac end. @@ -99,8 +99,15 @@ Proof. InvVLMA; simpl; congruence. Qed. +Remark shift_symbol_address: + forall symb ofs n, + symbol_address ge symb (Int.add ofs n) = Val.add (symbol_address ge symb ofs) (Vint n). +Proof. + unfold symbol_address; intros. destruct (Genv.find_symbol ge symb); auto. +Qed. + Lemma eval_static_operation_correct: - forall op sp al vl m v, + forall op al vl m v, val_list_match_approx al vl -> eval_operation ge sp op vl m = Some v -> val_match_approx (eval_static_operation op al) v. @@ -108,57 +115,44 @@ Proof. intros until v. unfold eval_static_operation. case (eval_static_operation_match op al); intros; - InvVLMA; simpl in *; FuncInv; try congruence. - - destruct (Genv.find_symbol ge s). exists b. intuition congruence. - congruence. - - rewrite <- H3. replace v0 with (Vint n1). reflexivity. congruence. - rewrite <- H3. replace v0 with (Vint n1). reflexivity. congruence. - - exists b. split. auto. congruence. - exists b. split. auto. congruence. - exists b. split. auto. congruence. + InvVLMA; simpl in *; FuncInv; try subst v; auto. - replace n2 with i0. destruct (Int.eq i0 Int.zero). - discriminate. injection H0; intro; subst v. simpl. congruence. congruence. + rewrite shift_symbol_address; auto. - replace n2 with i0. destruct (Int.eq i0 Int.zero). - discriminate. injection H0; intro; subst v. simpl. congruence. congruence. + rewrite Int.add_commut. rewrite shift_symbol_address. rewrite Val.add_commut. auto. - subst v. unfold Int.not. congruence. - subst v. unfold Int.not. congruence. - subst v. unfold Int.not. congruence. + rewrite Int.add_commut; auto. - replace n2 with i0. destruct (Int.ltu i0 Int.iwordsize). - injection H0; intro; subst v. simpl. congruence. discriminate. congruence. + rewrite Val.add_assoc. rewrite Int.add_commut. auto. - replace n2 with i0. destruct (Int.ltu i0 Int.iwordsize). - injection H0; intro; subst v. simpl. congruence. discriminate. congruence. + change (Val.add (Vint n1) (Val.add sp (Vint n2)) = Val.add sp (Vint (Int.add n1 n2))). + rewrite Val.add_permut. auto. - destruct (Int.ltu n Int.iwordsize). - injection H0; intro; subst v. simpl. congruence. discriminate. + rewrite shift_symbol_address; auto. - destruct (Int.ltu n Int.iwordsize). - injection H0; intro; subst v. simpl. congruence. discriminate. + rewrite Val.add_assoc; auto. - replace n2 with i0. destruct (Int.ltu i0 Int.iwordsize). - injection H0; intro; subst v. simpl. congruence. discriminate. congruence. + unfold symbol_address. destruct (Genv.find_symbol ge s1); auto. - rewrite <- H3. replace v0 with (Vfloat n1). reflexivity. congruence. + rewrite Val.sub_add_opp. rewrite Val.add_assoc. simpl. rewrite Int.sub_add_opp. auto. - inv H4. destruct (Float.intoffloat f); simpl in H0; inv H0. red; auto. + destruct (Int.eq n2 Int.zero); inv H0; simpl; auto. + destruct (Int.eq n2 Int.zero); inv H0; simpl; auto. - caseEq (eval_static_condition c vl0). - intros. generalize (eval_static_condition_correct _ _ _ m _ H H1). - intro. rewrite H2 in H0. - destruct b; injection H0; intro; subst v; simpl; auto. - intros; simpl; auto. + destruct (Int.ltu n2 Int.iwordsize); simpl; auto. + destruct (Int.ltu n2 Int.iwordsize); simpl; auto. + destruct (Int.ltu n Int.iwordsize); simpl; auto. + destruct (Int.ltu n (Int.repr 31)); inv H0. simpl; auto. + destruct (Int.ltu n2 Int.iwordsize); simpl; auto. - rewrite <- H3. replace v0 with (Vint n1). reflexivity. congruence. - rewrite <- H3. replace v0 with (Vint n1). reflexivity. congruence. + unfold eval_static_intoffloat. destruct (Float.intoffloat n1); simpl in H0; inv H0. + simpl; auto. - auto. + unfold eval_static_condition_val, Val.of_optbool. + destruct (eval_static_condition c vl0) as []_eqn. + rewrite (eval_static_condition_correct _ _ _ m _ H Heqo). + destruct b; simpl; auto. + simpl; auto. Qed. (** * Correctness of strength reduction *) @@ -171,352 +165,243 @@ Qed. Section STRENGTH_REDUCTION. -Variable app: reg -> approx. -Variable sp: val. +Variable app: D.t. Variable rs: regset. Variable m: mem. -Hypothesis MATCH: forall r, val_match_approx (app r) rs#r. +Hypothesis MATCH: forall r, val_match_approx (approx_reg app r) rs#r. -Lemma intval_correct: - forall r n, - intval app r = Some n -> rs#r = Vint n. -Proof. - intros until n. - unfold intval. caseEq (app r); intros; try discriminate. - generalize (MATCH r). unfold val_match_approx. rewrite H. - congruence. -Qed. +Ltac InvApproxRegs := + match goal with + | [ H: _ :: _ = _ :: _ |- _ ] => + injection H; clear H; intros; InvApproxRegs + | [ H: ?v = approx_reg app ?r |- _ ] => + generalize (MATCH r); rewrite <- H; clear H; intro; InvApproxRegs + | _ => idtac + end. Lemma cond_strength_reduction_correct: - forall cond args, - let (cond', args') := cond_strength_reduction app cond args in + forall cond args vl, + vl = approx_regs app args -> + let (cond', args') := cond_strength_reduction cond args vl in eval_condition cond' rs##args' m = eval_condition cond rs##args m. Proof. - intros. unfold cond_strength_reduction. - case (cond_strength_reduction_match cond args); intros. - caseEq (intval app r1); intros. - simpl. rewrite (intval_correct _ _ H). - destruct (rs#r2); auto. rewrite Int.swap_cmp. auto. - caseEq (intval app r2); intros. - simpl. rewrite (intval_correct _ _ H0). auto. - auto. - caseEq (intval app r1); intros. - simpl. rewrite (intval_correct _ _ H). - destruct (rs#r2); auto. rewrite Int.swap_cmpu. auto. - destruct c; reflexivity. - caseEq (intval app r2); intros. - simpl. rewrite (intval_correct _ _ H0). auto. - auto. + intros until vl. unfold cond_strength_reduction. + case (cond_strength_reduction_match cond args vl); simpl; intros; InvApproxRegs; SimplVMA. + rewrite H0. apply Val.swap_cmp_bool. + rewrite H. auto. + rewrite H0. apply Val.swap_cmpu_bool. + rewrite H. auto. auto. Qed. Lemma make_addimm_correct: - forall n r v, + forall n r, let (op, args) := make_addimm n r in - eval_operation ge sp Oadd (rs#r :: Vint n :: nil) m = Some v -> - eval_operation ge sp op rs##args m = Some v. + exists v, eval_operation ge sp op rs##args m = Some v /\ Val.lessdef (Val.add rs#r (Vint n)) v. Proof. - intros; unfold make_addimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intros. - subst n. simpl in *. FuncInv. rewrite Int.add_zero in H. congruence. - rewrite Int.add_zero in H. congruence. - exact H0. + intros. unfold make_addimm. + predSpec Int.eq Int.eq_spec n Int.zero; intros. + subst. exists (rs#r); split; auto. destruct (rs#r); simpl; auto; rewrite Int.add_zero; auto. + exists (Val.add rs#r (Vint n)); auto. Qed. Lemma make_shlimm_correct: - forall n r v, - let (op, args) := make_shlimm n r in - eval_operation ge sp Oshl (rs#r :: Vint n :: nil) m = Some v -> - eval_operation ge sp op rs##args m = Some v. + forall n r1 r2, + rs#r2 = Vint n -> + let (op, args) := make_shlimm n r1 r2 in + exists v, eval_operation ge sp op rs##args m = Some v /\ Val.lessdef (Val.shl rs#r1 (Vint n)) v. Proof. intros; unfold make_shlimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intros. - subst n. simpl in *. FuncInv. rewrite Int.shl_zero in H. congruence. - simpl in *. FuncInv. caseEq (Int.ltu n Int.iwordsize); intros. - rewrite H1 in H0. rewrite Int.shl_rolm in H0. auto. exact H1. - rewrite H1 in H0. discriminate. + predSpec Int.eq Int.eq_spec n Int.zero; intros. subst. + exists (rs#r1); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.shl_zero. auto. + destruct (Int.ltu n Int.iwordsize) as []_eqn; intros. + rewrite Val.shl_rolm; auto. econstructor; split; eauto. auto. + econstructor; split; eauto. simpl. congruence. Qed. Lemma make_shrimm_correct: - forall n r v, - let (op, args) := make_shrimm n r in - eval_operation ge sp Oshr (rs#r :: Vint n :: nil) m = Some v -> - eval_operation ge sp op rs##args m = Some v. + forall n r1 r2, + rs#r2 = Vint n -> + let (op, args) := make_shrimm n r1 r2 in + exists v, eval_operation ge sp op rs##args m = Some v /\ Val.lessdef (Val.shr rs#r1 (Vint n)) v. Proof. intros; unfold make_shrimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intros. - subst n. simpl in *. FuncInv. rewrite Int.shr_zero in H. congruence. - assumption. + predSpec Int.eq Int.eq_spec n Int.zero; intros. subst. + exists (rs#r1); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.shr_zero. auto. + destruct (Int.ltu n Int.iwordsize) as []_eqn. + econstructor; split; eauto. simpl. auto. + econstructor; split; eauto. simpl. congruence. Qed. Lemma make_shruimm_correct: - forall n r v, - let (op, args) := make_shruimm n r in - eval_operation ge sp Oshru (rs#r :: Vint n :: nil) m = Some v -> - eval_operation ge sp op rs##args m = Some v. + forall n r1 r2, + rs#r2 = Vint n -> + let (op, args) := make_shruimm n r1 r2 in + exists v, eval_operation ge sp op rs##args m = Some v /\ Val.lessdef (Val.shru rs#r1 (Vint n)) v. Proof. intros; unfold make_shruimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intros. - subst n. simpl in *. FuncInv. rewrite Int.shru_zero in H. congruence. - simpl in *. FuncInv. caseEq (Int.ltu n Int.iwordsize); intros. - rewrite H1 in H0. rewrite Int.shru_rolm in H0. auto. exact H1. - rewrite H1 in H0. discriminate. + predSpec Int.eq Int.eq_spec n Int.zero; intros. subst. + exists (rs#r1); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.shru_zero. auto. + destruct (Int.ltu n Int.iwordsize) as []_eqn; intros. + rewrite Val.shru_rolm; auto. econstructor; split; eauto. auto. + econstructor; split; eauto. simpl. congruence. Qed. Lemma make_mulimm_correct: - forall n r v, - let (op, args) := make_mulimm n r in - eval_operation ge sp Omul (rs#r :: Vint n :: nil) m = Some v -> - eval_operation ge sp op rs##args m = Some v. + forall n r1 r2, + rs#r2 = Vint n -> + let (op, args) := make_mulimm n r1 r2 in + exists v, eval_operation ge sp op rs##args m = Some v /\ Val.lessdef (Val.mul rs#r1 (Vint n)) v. Proof. intros; unfold make_mulimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intros. - subst n. simpl in H0. FuncInv. rewrite Int.mul_zero in H. simpl. congruence. - generalize (Int.eq_spec n Int.one); case (Int.eq n Int.one); intros. - subst n. simpl in H1. simpl. FuncInv. rewrite Int.mul_one in H0. congruence. - caseEq (Int.is_power2 n); intros. - replace (eval_operation ge sp Omul (rs # r :: Vint n :: nil) m) - with (eval_operation ge sp Oshl (rs # r :: Vint i :: nil) m). - apply make_shlimm_correct. - simpl. generalize (Int.is_power2_range _ _ H1). - change (Z_of_nat Int.wordsize) with 32. intro. rewrite H2. - destruct rs#r; auto. rewrite (Int.mul_pow2 i0 _ _ H1). auto. - exact H2. + predSpec Int.eq Int.eq_spec n Int.zero; intros. subst. + exists (Vint Int.zero); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.mul_zero; auto. + predSpec Int.eq Int.eq_spec n Int.one; intros. subst. + exists (rs#r1); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.mul_one; auto. + destruct (Int.is_power2 n) as []_eqn; intros. + rewrite (Val.mul_pow2 rs#r1 _ _ Heqo). rewrite Val.shl_rolm. + econstructor; split; eauto. auto. + eapply Int.is_power2_range; eauto. + econstructor; split; eauto. auto. +Qed. + +Lemma make_divimm_correct: + forall n r1 r2 v, + Val.divs rs#r1 rs#r2 = Some v -> + rs#r2 = Vint n -> + let (op, args) := make_divimm n r1 r2 in + exists w, eval_operation ge sp op rs##args m = Some w /\ Val.lessdef v w. +Proof. + intros; unfold make_divimm. + destruct (Int.is_power2 n) as []_eqn. + destruct (Int.ltu i (Int.repr 31)) as []_eqn. + exists v; split; auto. simpl. eapply Val.divs_pow2; eauto. congruence. + exists v; auto. + exists v; auto. +Qed. + +Lemma make_divuimm_correct: + forall n r1 r2 v, + Val.divu rs#r1 rs#r2 = Some v -> + rs#r2 = Vint n -> + let (op, args) := make_divuimm n r1 r2 in + exists w, eval_operation ge sp op rs##args m = Some w /\ Val.lessdef v w. +Proof. + intros; unfold make_divuimm. + destruct (Int.is_power2 n) as []_eqn. + econstructor; split. simpl; eauto. + exploit Int.is_power2_range; eauto. intros RANGE. + rewrite <- Val.shru_rolm; auto. rewrite H0 in H. + destruct (rs#r1); simpl in *; inv H. + destruct (Int.eq n Int.zero); inv H2. + rewrite RANGE. rewrite (Int.divu_pow2 i0 _ _ Heqo). auto. + exists v; auto. Qed. Lemma make_andimm_correct: - forall n r v, + forall n r, let (op, args) := make_andimm n r in - eval_operation ge sp Oand (rs#r :: Vint n :: nil) m = Some v -> - eval_operation ge sp op rs##args m = Some v. + exists v, eval_operation ge sp op rs##args m = Some v /\ Val.lessdef (Val.and rs#r (Vint n)) v. Proof. intros; unfold make_andimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intros. - subst n. simpl in *. FuncInv. rewrite Int.and_zero in H. congruence. - generalize (Int.eq_spec n Int.mone); case (Int.eq n Int.mone); intros. - subst n. simpl in *. FuncInv. rewrite Int.and_mone in H0. congruence. - exact H1. + predSpec Int.eq Int.eq_spec n Int.zero; intros. + subst n. exists (Vint Int.zero); split; auto. destruct (rs#r); simpl; auto. rewrite Int.and_zero; auto. + predSpec Int.eq Int.eq_spec n Int.mone; intros. + subst n. exists (rs#r); split; auto. destruct (rs#r); simpl; auto. rewrite Int.and_mone; auto. + econstructor; split; eauto. auto. Qed. Lemma make_orimm_correct: - forall n r v, + forall n r, let (op, args) := make_orimm n r in - eval_operation ge sp Oor (rs#r :: Vint n :: nil) m = Some v -> - eval_operation ge sp op rs##args m = Some v. + exists v, eval_operation ge sp op rs##args m = Some v /\ Val.lessdef (Val.or rs#r (Vint n)) v. Proof. intros; unfold make_orimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intros. - subst n. simpl in *. FuncInv. rewrite Int.or_zero in H. congruence. - generalize (Int.eq_spec n Int.mone); case (Int.eq n Int.mone); intros. - subst n. simpl in *. FuncInv. rewrite Int.or_mone in H0. congruence. - exact H1. + predSpec Int.eq Int.eq_spec n Int.zero; intros. + subst n. exists (rs#r); split; auto. destruct (rs#r); simpl; auto. rewrite Int.or_zero; auto. + predSpec Int.eq Int.eq_spec n Int.mone; intros. + subst n. exists (Vint Int.mone); split; auto. destruct (rs#r); simpl; auto. rewrite Int.or_mone; auto. + econstructor; split; eauto. auto. Qed. Lemma make_xorimm_correct: - forall n r v, + forall n r, let (op, args) := make_xorimm n r in - eval_operation ge sp Oxor (rs#r :: Vint n :: nil) m = Some v -> - eval_operation ge sp op rs##args m = Some v. + exists v, eval_operation ge sp op rs##args m = Some v /\ Val.lessdef (Val.xor rs#r (Vint n)) v. Proof. intros; unfold make_xorimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intros. - subst n. simpl in *. FuncInv. rewrite Int.xor_zero in H. congruence. - exact H0. + predSpec Int.eq Int.eq_spec n Int.zero; intros. + subst n. exists (rs#r); split; auto. destruct (rs#r); simpl; auto. rewrite Int.xor_zero; auto. + econstructor; split; eauto. auto. Qed. Lemma op_strength_reduction_correct: - forall op args v, - let (op', args') := op_strength_reduction app op args in + forall op args vl v, + vl = approx_regs app args -> eval_operation ge sp op rs##args m = Some v -> - eval_operation ge sp op' rs##args' m = Some v. + let (op', args') := op_strength_reduction op args vl in + exists w, eval_operation ge sp op' rs##args' m = Some w /\ Val.lessdef v w. Proof. - intros; unfold op_strength_reduction; - case (op_strength_reduction_match op args); intros; simpl List.map. - (* Oadd *) - caseEq (intval app r1); intros. - rewrite (intval_correct _ _ H). - replace (eval_operation ge sp Oadd (Vint i :: rs # r2 :: nil) m) - with (eval_operation ge sp Oadd (rs # r2 :: Vint i :: nil) m). - apply make_addimm_correct. - simpl. destruct rs#r2; auto. rewrite Int.add_commut; auto. - caseEq (intval app r2); intros. - rewrite (intval_correct _ _ H0). apply make_addimm_correct. - assumption. - (* Osub *) - caseEq (intval app r1); intros. - rewrite (intval_correct _ _ H) in H0. assumption. - caseEq (intval app r2); intros. - rewrite (intval_correct _ _ H0). - replace (eval_operation ge sp Osub (rs # r1 :: Vint i :: nil) m) - with (eval_operation ge sp Oadd (rs # r1 :: Vint (Int.neg i) :: nil) m). - apply make_addimm_correct. - simpl. destruct rs#r1; auto; rewrite Int.sub_add_opp; auto. - assumption. - (* Omul *) - caseEq (intval app r1); intros. - rewrite (intval_correct _ _ H). - replace (eval_operation ge sp Omul (Vint i :: rs # r2 :: nil) m) - with (eval_operation ge sp Omul (rs # r2 :: Vint i :: nil) m). - apply make_mulimm_correct. - simpl. destruct rs#r2; auto. rewrite Int.mul_commut; auto. - caseEq (intval app r2); intros. - rewrite (intval_correct _ _ H0). apply make_mulimm_correct. - assumption. - (* Odiv *) - caseEq (intval app r2); intros. - caseEq (Int.is_power2 i); intros. - rewrite (intval_correct _ _ H) in H1. - simpl in *; FuncInv. destruct (Int.eq i Int.zero). congruence. - change 32 with (Z_of_nat Int.wordsize). - rewrite (Int.is_power2_range _ _ H0). - rewrite (Int.divs_pow2 i1 _ _ H0) in H1. auto. - assumption. - assumption. - (* Odivu *) - caseEq (intval app r2); intros. - caseEq (Int.is_power2 i); intros. - rewrite (intval_correct _ _ H). - replace (eval_operation ge sp Odivu (rs # r1 :: Vint i :: nil) m) - with (eval_operation ge sp Oshru (rs # r1 :: Vint i0 :: nil) m). - apply make_shruimm_correct. - simpl. destruct rs#r1; auto. - change 32 with (Z_of_nat Int.wordsize). - rewrite (Int.is_power2_range _ _ H0). - generalize (Int.eq_spec i Int.zero); case (Int.eq i Int.zero); intros. - subst i. discriminate. - rewrite (Int.divu_pow2 i1 _ _ H0). auto. - assumption. - assumption. - (* Oand *) - caseEq (intval app r1); intros. - rewrite (intval_correct _ _ H). - replace (eval_operation ge sp Oand (Vint i :: rs # r2 :: nil) m) - with (eval_operation ge sp Oand (rs # r2 :: Vint i :: nil) m). - apply make_andimm_correct. - simpl. destruct rs#r2; auto. rewrite Int.and_commut; auto. - caseEq (intval app r2); intros. - rewrite (intval_correct _ _ H0). apply make_andimm_correct. - assumption. - (* Oor *) - caseEq (intval app r1); intros. - rewrite (intval_correct _ _ H). - replace (eval_operation ge sp Oor (Vint i :: rs # r2 :: nil) m) - with (eval_operation ge sp Oor (rs # r2 :: Vint i :: nil) m). - apply make_orimm_correct. - simpl. destruct rs#r2; auto. rewrite Int.or_commut; auto. - caseEq (intval app r2); intros. - rewrite (intval_correct _ _ H0). apply make_orimm_correct. - assumption. - (* Oxor *) - caseEq (intval app r1); intros. - rewrite (intval_correct _ _ H). - replace (eval_operation ge sp Oxor (Vint i :: rs # r2 :: nil) m) - with (eval_operation ge sp Oxor (rs # r2 :: Vint i :: nil) m). - apply make_xorimm_correct. - simpl. destruct rs#r2; auto. rewrite Int.xor_commut; auto. - caseEq (intval app r2); intros. - rewrite (intval_correct _ _ H0). apply make_xorimm_correct. - assumption. - (* Oshl *) - caseEq (intval app r2); intros. - caseEq (Int.ltu i Int.iwordsize); intros. - rewrite (intval_correct _ _ H). apply make_shlimm_correct. - assumption. - assumption. - (* Oshr *) - caseEq (intval app r2); intros. - caseEq (Int.ltu i Int.iwordsize); intros. - rewrite (intval_correct _ _ H). apply make_shrimm_correct. - assumption. - assumption. - (* Oshru *) - caseEq (intval app r2); intros. - caseEq (Int.ltu i Int.iwordsize); intros. - rewrite (intval_correct _ _ H). apply make_shruimm_correct. - assumption. - assumption. - (* Ocmp *) - generalize (cond_strength_reduction_correct c rl). - destruct (cond_strength_reduction app c rl). - simpl. intro. rewrite H. auto. - (* default *) - assumption. + intros until v; unfold op_strength_reduction; + case (op_strength_reduction_match op args vl); simpl; intros. +(* add *) + InvApproxRegs. SimplVMA. inv H0. rewrite H1. rewrite Val.add_commut. apply make_addimm_correct. + InvApproxRegs. SimplVMA. inv H0. rewrite H. apply make_addimm_correct. +(* sub *) + InvApproxRegs; SimplVMA. inv H0. rewrite H1. econstructor; split; eauto. + InvApproxRegs; SimplVMA. inv H0. rewrite H. rewrite Val.sub_add_opp. apply make_addimm_correct. +(* mul *) + InvApproxRegs; SimplVMA. inv H0. rewrite H1. rewrite Val.mul_commut. apply make_mulimm_correct; auto. + InvApproxRegs; SimplVMA. inv H0. rewrite H. apply make_mulimm_correct; auto. +(* divs *) + assert (rs#r2 = Vint n2). clear H0. InvApproxRegs; SimplVMA; auto. + apply make_divimm_correct; auto. +(* divu *) + assert (rs#r2 = Vint n2). clear H0. InvApproxRegs; SimplVMA; auto. + apply make_divuimm_correct; auto. +(* and *) + InvApproxRegs. SimplVMA. inv H0. rewrite H1. rewrite Val.and_commut. apply make_andimm_correct. + InvApproxRegs. SimplVMA. inv H0. rewrite H. apply make_andimm_correct. +(* or *) + InvApproxRegs. SimplVMA. inv H0. rewrite H1. rewrite Val.or_commut. apply make_orimm_correct. + InvApproxRegs. SimplVMA. inv H0. rewrite H. apply make_orimm_correct. +(* xor *) + InvApproxRegs. SimplVMA. inv H0. rewrite H1. rewrite Val.xor_commut. apply make_xorimm_correct. + InvApproxRegs. SimplVMA. inv H0. rewrite H. apply make_xorimm_correct. +(* shl *) + InvApproxRegs. SimplVMA. inv H0. rewrite H. apply make_shlimm_correct; auto. +(* shr *) + InvApproxRegs. SimplVMA. inv H0. rewrite H. apply make_shrimm_correct; auto. +(* shru *) + InvApproxRegs. SimplVMA. inv H0. rewrite H. apply make_shruimm_correct; auto. +(* cmp *) + generalize (cond_strength_reduction_correct c args0 vl0). + destruct (cond_strength_reduction c args0 vl0) as [c' args']; intros. + rewrite <- H1 in H0; auto. econstructor; split; eauto. +(* default *) + exists v; auto. Qed. - -Ltac KnownApprox := - match goal with - | H: ?approx ?r = ?a |- _ => - generalize (MATCH r); rewrite H; intro; clear H; KnownApprox - | _ => idtac - end. Lemma addr_strength_reduction_correct: - forall addr args, - let (addr', args') := addr_strength_reduction app addr args in + forall addr args vl, + vl = approx_regs app args -> + let (addr', args') := addr_strength_reduction addr args vl in eval_addressing ge sp addr' rs##args' = eval_addressing ge sp addr rs##args. Proof. - intros. - - (* Useful lemmas *) - assert (A0: forall r1 r2, - eval_addressing ge sp Aindexed2 (rs ## (r1 :: r2 :: nil)) = - eval_addressing ge sp Aindexed2 (rs ## (r2 :: r1 :: nil))). - intros. simpl. destruct (rs#r1); destruct (rs#r2); auto; - rewrite Int.add_commut; auto. - - assert (A1: forall r1 r2 n, - val_match_approx (I n) rs#r2 -> - eval_addressing ge sp (Aindexed n) (rs ## (r1 :: nil)) = - eval_addressing ge sp Aindexed2 (rs ## (r1 :: r2 :: nil))). - intros; simpl in *. rewrite H. auto. - - assert (A2: forall r1 r2 n, - val_match_approx (I n) rs#r1 -> - eval_addressing ge sp (Aindexed n) (rs ## (r2 :: nil)) = - eval_addressing ge sp Aindexed2 (rs ## (r1 :: r2 :: nil))). - intros. rewrite A0. apply A1. auto. - - assert (A3: forall r1 r2 id ofs, - val_match_approx (S id ofs) rs#r1 -> - eval_addressing ge sp (Abased id ofs) (rs ## (r2 :: nil)) = - eval_addressing ge sp Aindexed2 (rs ## (r1 :: r2 :: nil))). - intros. elim H. intros b [A B]. simpl. rewrite A; rewrite B. auto. - - assert (A4: forall r1 r2 id ofs, - val_match_approx (S id ofs) rs#r2 -> - eval_addressing ge sp (Abased id ofs) (rs ## (r1 :: nil)) = - eval_addressing ge sp Aindexed2 (rs ## (r1 :: r2 :: nil))). - intros. rewrite A0. apply A3. auto. - - assert (A5: forall r1 r2 id ofs n, - val_match_approx (S id ofs) rs#r1 -> - val_match_approx (I n) rs#r2 -> - eval_addressing ge sp (Aglobal id (Int.add ofs n)) nil = - eval_addressing ge sp Aindexed2 (rs ## (r1 :: r2 :: nil))). - intros. elim H. intros b [A B]. simpl. rewrite A; rewrite B. - simpl in H0. rewrite H0. auto. - - unfold addr_strength_reduction; - case (addr_strength_reduction_match addr args); intros. - - (* Aindexed2 *) - caseEq (app r1); intros; - caseEq (app r2); intros; - try reflexivity; KnownApprox; auto. - rewrite A0. rewrite Int.add_commut. apply A5; auto. - - (* Abased *) - caseEq (intval app r1); intros. - simpl; rewrite (intval_correct _ _ H). auto. + intros until vl. unfold addr_strength_reduction. + destruct (addr_strength_reduction_match addr args vl); simpl; intros; InvApproxRegs; SimplVMA. + rewrite H; rewrite H0. rewrite shift_symbol_address. auto. + rewrite H; rewrite H0. rewrite Int.add_commut. rewrite shift_symbol_address. rewrite Val.add_commut; auto. + rewrite H; rewrite H0. rewrite Val.add_assoc; auto. + rewrite H; rewrite H0. rewrite Val.add_permut; auto. + rewrite H0. auto. + rewrite H. rewrite Val.add_commut. auto. + rewrite H0. rewrite Val.add_commut; auto. + rewrite H; auto. + rewrite H. rewrite shift_symbol_address. auto. + rewrite H. rewrite shift_symbol_address. auto. + rewrite H. rewrite Val.add_assoc. auto. auto. - - (* Aindexed *) - caseEq (app r1); intros; auto. - simpl; KnownApprox. - elim H0. intros b [A B]. rewrite A; rewrite B. auto. - - (* default *) - reflexivity. Qed. End STRENGTH_REDUCTION. diff --git a/powerpc/Op.v b/powerpc/Op.v index 7bd4247..68b349e 100644 --- a/powerpc/Op.v +++ b/powerpc/Op.v @@ -59,9 +59,7 @@ Inductive operation : Type := | Oaddrstack: int -> operation (**r [rd] is set to the stack pointer plus the given offset *) (*c Integer arithmetic: *) | Ocast8signed: operation (**r [rd] is 8-bit sign extension of [r1] *) - | Ocast8unsigned: operation (**r [rd] is 8-bit zero extension of [r1] *) | Ocast16signed: operation (**r [rd] is 16-bit sign extension of [r1] *) - | Ocast16unsigned: operation (**r [rd] is 16-bit zero extension of [r1] *) | Oadd: operation (**r [rd = r1 + r2] *) | Oaddimm: int -> operation (**r [rd = r1 + n] *) | Osub: operation (**r [rd = r1 - r2] *) @@ -131,138 +129,80 @@ Proof. decide equality. Qed. -(** Evaluation of conditions, operators and addressing modes applied - to lists of values. Return [None] when the computation is undefined: - wrong number of arguments, arguments of the wrong types, undefined - operations such as division by zero. [eval_condition] returns a boolean, - [eval_operation] and [eval_addressing] return a value. *) +(** * Evaluation functions *) -Definition eval_compare_mismatch (c: comparison) : option bool := - match c with Ceq => Some false | Cne => Some true | _ => None end. +Definition symbol_address (F V: Type) (genv: Genv.t F V) (id: ident) (ofs: int) : val := + match Genv.find_symbol genv id with + | Some b => Vptr b ofs + | None => Vundef + end. -Definition eval_compare_null (c: comparison) (n: int) : option bool := - if Int.eq n Int.zero then eval_compare_mismatch c else None. +(** Evaluation of conditions, operators and addressing modes applied + to lists of values. Return [None] when the computation can trigger an + error, e.g. integer division by zero. [eval_condition] returns a boolean, + [eval_operation] and [eval_addressing] return a value. *) -Definition eval_condition (cond: condition) (vl: list val) (m: mem): - option bool := +Definition eval_condition (cond: condition) (vl: list val) (m: mem): option bool := match cond, vl with - | Ccomp c, Vint n1 :: Vint n2 :: nil => - Some (Int.cmp c n1 n2) - | Ccompu c, Vint n1 :: Vint n2 :: nil => - Some (Int.cmpu c n1 n2) - | Ccompu c, Vptr b1 n1 :: Vptr b2 n2 :: nil => - if Mem.valid_pointer m b1 (Int.unsigned n1) - && Mem.valid_pointer m b2 (Int.unsigned n2) then - if eq_block b1 b2 - then Some (Int.cmpu c n1 n2) - else eval_compare_mismatch c - else None - | Ccompu c, Vptr b1 n1 :: Vint n2 :: nil => - eval_compare_null c n2 - | Ccompu c, Vint n1 :: Vptr b2 n2 :: nil => - eval_compare_null c n1 - | Ccompimm c n, Vint n1 :: nil => - Some (Int.cmp c n1 n) - | Ccompuimm c n, Vint n1 :: nil => - Some (Int.cmpu c n1 n) - | Ccompuimm c n, Vptr b1 n1 :: nil => - eval_compare_null c n - | Ccompf c, Vfloat f1 :: Vfloat f2 :: nil => - Some (Float.cmp c f1 f2) - | Cnotcompf c, Vfloat f1 :: Vfloat f2 :: nil => - Some (negb (Float.cmp c f1 f2)) - | Cmaskzero n, Vint n1 :: nil => - Some (Int.eq (Int.and n1 n) Int.zero) - | Cmasknotzero n, Vint n1 :: nil => - Some (negb (Int.eq (Int.and n1 n) Int.zero)) - | _, _ => - None - end. - -Definition offset_sp (sp: val) (delta: int) : option val := - match sp with - | Vptr b n => Some (Vptr b (Int.add n delta)) - | _ => None + | Ccomp c, v1 :: v2 :: nil => Val.cmp_bool c v1 v2 + | Ccompu c, v1 :: v2 :: nil => Val.cmpu_bool (Mem.valid_pointer m) c v1 v2 + | Ccompimm c n, v1 :: nil => Val.cmp_bool c v1 (Vint n) + | Ccompuimm c n, v1 :: nil => Val.cmpu_bool (Mem.valid_pointer m) c v1 (Vint n) + | Ccompf c, v1 :: v2 :: nil => Val.cmpf_bool c v1 v2 + | Cnotcompf c, v1 :: v2 :: nil => option_map negb (Val.cmpf_bool c v1 v2) + | Cmaskzero n, Vint n1 :: nil => Some (Int.eq (Int.and n1 n) Int.zero) + | Cmasknotzero n, Vint n1 :: nil => Some (negb (Int.eq (Int.and n1 n) Int.zero)) + | _, _ => None end. Definition eval_operation - (F V: Type) (genv: Genv.t F V) (sp: val) - (op: operation) (vl: list val) (m: mem): option val := + (F V: Type) (genv: Genv.t F V) (sp: val) + (op: operation) (vl: list val) (m: mem): option val := match op, vl with | Omove, v1::nil => Some v1 | Ointconst n, nil => Some (Vint n) | Ofloatconst n, nil => Some (Vfloat n) - | Oaddrsymbol s ofs, nil => - match Genv.find_symbol genv s with - | None => None - | Some b => Some (Vptr b ofs) - end - | Oaddrstack ofs, nil => offset_sp sp ofs - | Ocast8signed, v1 :: nil => Some (Val.sign_ext 8 v1) - | Ocast8unsigned, v1 :: nil => Some (Val.zero_ext 8 v1) - | Ocast16signed, v1 :: nil => Some (Val.sign_ext 16 v1) - | Ocast16unsigned, v1 :: nil => Some (Val.zero_ext 16 v1) - | Oadd, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.add n1 n2)) - | Oadd, Vint n1 :: Vptr b2 n2 :: nil => Some (Vptr b2 (Int.add n2 n1)) - | Oadd, Vptr b1 n1 :: Vint n2 :: nil => Some (Vptr b1 (Int.add n1 n2)) - | Oaddimm n, Vint n1 :: nil => Some (Vint (Int.add n1 n)) - | Oaddimm n, Vptr b1 n1 :: nil => Some (Vptr b1 (Int.add n1 n)) - | Osub, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.sub n1 n2)) - | Osub, Vptr b1 n1 :: Vint n2 :: nil => Some (Vptr b1 (Int.sub n1 n2)) - | Osub, Vptr b1 n1 :: Vptr b2 n2 :: nil => - if eq_block b1 b2 then Some (Vint (Int.sub n1 n2)) else None - | Osubimm n, Vint n1 :: nil => Some (Vint (Int.sub n n1)) - | Omul, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.mul n1 n2)) - | Omulimm n, Vint n1 :: nil => Some (Vint (Int.mul n1 n)) - | Odiv, Vint n1 :: Vint n2 :: nil => - if Int.eq n2 Int.zero then None else Some (Vint (Int.divs n1 n2)) - | Odivu, Vint n1 :: Vint n2 :: nil => - if Int.eq n2 Int.zero then None else Some (Vint (Int.divu n1 n2)) - | Oand, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.and n1 n2)) - | Oandimm n, Vint n1 :: nil => Some (Vint (Int.and n1 n)) - | Oor, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.or n1 n2)) - | Oorimm n, Vint n1 :: nil => Some (Vint (Int.or n1 n)) - | Oxor, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.xor n1 n2)) - | Oxorimm n, Vint n1 :: nil => Some (Vint (Int.xor n1 n)) - | Onand, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.not (Int.and n1 n2))) - | Onor, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.not (Int.or n1 n2))) - | Onxor, Vint n1 :: Vint n2 :: nil => Some (Vint (Int.not (Int.xor n1 n2))) - | Oshl, Vint n1 :: Vint n2 :: nil => - if Int.ltu n2 Int.iwordsize then Some (Vint (Int.shl n1 n2)) else None - | Oshr, Vint n1 :: Vint n2 :: nil => - if Int.ltu n2 Int.iwordsize then Some (Vint (Int.shr n1 n2)) else None - | Oshrimm n, Vint n1 :: nil => - if Int.ltu n Int.iwordsize then Some (Vint (Int.shr n1 n)) else None - | Oshrximm n, Vint n1 :: nil => - if Int.ltu n Int.iwordsize then Some (Vint (Int.shrx n1 n)) else None - | Oshru, Vint n1 :: Vint n2 :: nil => - if Int.ltu n2 Int.iwordsize then Some (Vint (Int.shru n1 n2)) else None - | Orolm amount mask, Vint n1 :: nil => - Some (Vint (Int.rolm n1 amount mask)) - | Oroli amount mask, Vint n1 :: Vint n2 :: nil => - Some (Vint (Int.or (Int.and n1 (Int.not mask)) (Int.rolm n2 amount mask))) - | Onegf, Vfloat f1 :: nil => Some (Vfloat (Float.neg f1)) - | Oabsf, Vfloat f1 :: nil => Some (Vfloat (Float.abs f1)) - | Oaddf, Vfloat f1 :: Vfloat f2 :: nil => Some (Vfloat (Float.add f1 f2)) - | Osubf, Vfloat f1 :: Vfloat f2 :: nil => Some (Vfloat (Float.sub f1 f2)) - | Omulf, Vfloat f1 :: Vfloat f2 :: nil => Some (Vfloat (Float.mul f1 f2)) - | Odivf, Vfloat f1 :: Vfloat f2 :: nil => Some (Vfloat (Float.div f1 f2)) - | Omuladdf, Vfloat f1 :: Vfloat f2 :: Vfloat f3 :: nil => - Some (Vfloat (Float.add (Float.mul f1 f2) f3)) - | Omulsubf, Vfloat f1 :: Vfloat f2 :: Vfloat f3 :: nil => - Some (Vfloat (Float.sub (Float.mul f1 f2) f3)) - | Osingleoffloat, v1 :: nil => - Some (Val.singleoffloat v1) - | Ointoffloat, Vfloat f1 :: nil => - option_map Vint (Float.intoffloat f1) - | Ofloatofwords, Vint i1 :: Vint i2 :: nil => - Some (Vfloat (Float.from_words i1 i2)) - | Ocmp c, _ => - match eval_condition c vl m with - | None => None - | Some false => Some Vfalse - | Some true => Some Vtrue - end + | Oaddrsymbol s ofs, nil => Some (symbol_address genv s ofs) + | Oaddrstack ofs, nil => Some (Val.add sp (Vint ofs)) + | Ocast8signed, v1::nil => Some (Val.sign_ext 8 v1) + | Ocast16signed, v1::nil => Some (Val.sign_ext 16 v1) + | Oadd, v1::v2::nil => Some (Val.add v1 v2) + | Oaddimm n, v1::nil => Some (Val.add v1 (Vint n)) + | Osub, v1::v2::nil => Some (Val.sub v1 v2) + | Osubimm n, v1::nil => Some (Val.sub (Vint n) v1) + | Omul, v1::v2::nil => Some (Val.mul v1 v2) + | Omulimm n, v1::nil => Some (Val.mul v1 (Vint n)) + | Odiv, v1::v2::nil => Val.divs v1 v2 + | Odivu, v1::v2::nil => Val.divu v1 v2 + | Oand, v1::v2::nil => Some(Val.and v1 v2) + | Oandimm n, v1::nil => Some (Val.and v1 (Vint n)) + | Oor, v1::v2::nil => Some(Val.or v1 v2) + | Oorimm n, v1::nil => Some (Val.or v1 (Vint n)) + | Oxor, v1::v2::nil => Some(Val.xor v1 v2) + | Oxorimm n, v1::nil => Some (Val.xor v1 (Vint n)) + | Onand, v1::v2::nil => Some (Val.notint (Val.and v1 v2)) + | Onor, v1::v2::nil => Some (Val.notint (Val.or v1 v2)) + | Onxor, v1::v2::nil => Some (Val.notint (Val.xor v1 v2)) + | Oshl, v1::v2::nil => Some (Val.shl v1 v2) + | Oshr, v1::v2::nil => Some (Val.shr v1 v2) + | Oshrimm n, v1::nil => Some (Val.shr v1 (Vint n)) + | Oshrximm n, v1::nil => Val.shrx v1 (Vint n) + | Oshru, v1::v2::nil => Some (Val.shru v1 v2) + | Orolm amount mask, v1::nil => Some (Val.rolm v1 amount mask) + | Oroli amount mask, v1::v2::nil => + Some(Val.or (Val.and v1 (Vint (Int.not mask))) (Val.rolm v2 amount mask)) + | Onegf, v1::nil => Some(Val.negf v1) + | Oabsf, v1::nil => Some(Val.absf v1) + | Oaddf, v1::v2::nil => Some(Val.addf v1 v2) + | Osubf, v1::v2::nil => Some(Val.subf v1 v2) + | Omulf, v1::v2::nil => Some(Val.mulf v1 v2) + | Odivf, v1::v2::nil => Some(Val.divf v1 v2) + | Omuladdf, v1::v2::v3::nil => Some(Val.addf (Val.mulf v1 v2) v3) + | Omulsubf, v1::v2::v3::nil => Some(Val.subf (Val.mulf v1 v2) v3) + | Osingleoffloat, v1::nil => Some(Val.singleoffloat v1) + | Ointoffloat, v1::nil => Val.intoffloat v1 + | Ofloatofwords, v1::v2::nil => Some(Val.floatofwords v1 v2) + | Ocmp c, _ => Some(Val.of_optbool (eval_condition c vl m)) | _, _ => None end. @@ -270,39 +210,14 @@ Definition eval_addressing (F V: Type) (genv: Genv.t F V) (sp: val) (addr: addressing) (vl: list val) : option val := match addr, vl with - | Aindexed n, Vptr b1 n1 :: nil => - Some (Vptr b1 (Int.add n1 n)) - | Aindexed2, Vptr b1 n1 :: Vint n2 :: nil => - Some (Vptr b1 (Int.add n1 n2)) - | Aindexed2, Vint n1 :: Vptr b2 n2 :: nil => - Some (Vptr b2 (Int.add n2 n1)) - | Aglobal s ofs, nil => - match Genv.find_symbol genv s with - | None => None - | Some b => Some (Vptr b ofs) - end - | Abased s ofs, Vint n1 :: nil => - match Genv.find_symbol genv s with - | None => None - | Some b => Some (Vptr b (Int.add ofs n1)) - end - | Ainstack ofs, nil => - offset_sp sp ofs + | Aindexed n, v1::nil => Some (Val.add v1 (Vint n)) + | Aindexed2, v1::v2::nil => Some (Val.add v1 v2) + | Aglobal s ofs, nil => Some (symbol_address genv s ofs) + | Abased s ofs, v1::nil => Some (Val.add (symbol_address genv s ofs) v1) + | Ainstack ofs, nil => Some(Val.add sp (Vint ofs)) | _, _ => None end. -Definition negate_condition (cond: condition): condition := - match cond with - | Ccomp c => Ccomp(negate_comparison c) - | Ccompu c => Ccompu(negate_comparison c) - | Ccompimm c n => Ccompimm (negate_comparison c) n - | Ccompuimm c n => Ccompuimm (negate_comparison c) n - | Ccompf c => Cnotcompf c - | Cnotcompf c => Ccompf c - | Cmaskzero n => Cmasknotzero n - | Cmasknotzero n => Cmaskzero n - end. - Ltac FuncInv := match goal with | H: (match ?x with nil => _ | _ :: _ => _ end = Some _) |- _ => @@ -315,103 +230,7 @@ Ltac FuncInv := idtac end. -Remark eval_negate_compare_mismatch: - forall c b, - eval_compare_mismatch c = Some b -> - eval_compare_mismatch (negate_comparison c) = Some (negb b). -Proof. - intros until b. unfold eval_compare_mismatch. - destruct c; intro EQ; inv EQ; auto. -Qed. - -Remark eval_negate_compare_null: - forall c i b, - eval_compare_null c i = Some b -> - eval_compare_null (negate_comparison c) i = Some (negb b). -Proof. - unfold eval_compare_null; intros. - destruct (Int.eq i Int.zero). apply eval_negate_compare_mismatch; auto. congruence. -Qed. - -Lemma eval_negate_condition: - forall cond vl m b, - eval_condition cond vl m = Some b -> - eval_condition (negate_condition cond) vl m = Some (negb b). -Proof. - intros. - destruct cond; simpl in H; FuncInv; try subst b; simpl. - rewrite Int.negate_cmp. auto. - rewrite Int.negate_cmpu. auto. - apply eval_negate_compare_null; auto. - apply eval_negate_compare_null; auto. - destruct (Mem.valid_pointer m b0 (Int.unsigned i) && - Mem.valid_pointer m b1 (Int.unsigned i0)); try congruence. - destruct (eq_block b0 b1). rewrite Int.negate_cmpu. congruence. - apply eval_negate_compare_mismatch; auto. - rewrite Int.negate_cmp. auto. - rewrite Int.negate_cmpu. auto. - apply eval_negate_compare_null; auto. - auto. - rewrite negb_elim. auto. - auto. - rewrite negb_elim. auto. -Qed. - -(** [eval_operation] and [eval_addressing] depend on a global environment - for resolving references to global symbols. We show that they give - the same results if a global environment is replaced by another that - assigns the same addresses to the same symbols. *) - -Section GENV_TRANSF. - -Variable F1 F2 V1 V2: Type. -Variable ge1: Genv.t F1 V1. -Variable ge2: Genv.t F2 V2. -Hypothesis agree_on_symbols: - forall (s: ident), Genv.find_symbol ge2 s = Genv.find_symbol ge1 s. - -Lemma eval_operation_preserved: - forall sp op vl m, - eval_operation ge2 sp op vl m = eval_operation ge1 sp op vl m. -Proof. - intros. - unfold eval_operation; destruct op; try rewrite agree_on_symbols; - reflexivity. -Qed. - -Lemma eval_addressing_preserved: - forall sp addr vl, - eval_addressing ge2 sp addr vl = eval_addressing ge1 sp addr vl. -Proof. - intros. - unfold eval_addressing; destruct addr; try rewrite agree_on_symbols; - reflexivity. -Qed. - -End GENV_TRANSF. - -(** Recognition of move operations. *) - -Definition is_move_operation - (A: Type) (op: operation) (args: list A) : option A := - match op, args with - | Omove, arg :: nil => Some arg - | _, _ => None - end. - -Lemma is_move_operation_correct: - forall (A: Type) (op: operation) (args: list A) (a: A), - is_move_operation op args = Some a -> - op = Omove /\ args = a :: nil. -Proof. - intros until a. unfold is_move_operation; destruct op; - try (intros; discriminate). - destruct args. intros; discriminate. - destruct args. intros. intuition congruence. - intros; discriminate. -Qed. - -(** Static typing of conditions, operators and addressing modes. *) +(** * Static typing of conditions, operators and addressing modes. *) Definition type_of_condition (c: condition) : list typ := match c with @@ -433,9 +252,7 @@ Definition type_of_operation (op: operation) : list typ * typ := | Oaddrsymbol _ _ => (nil, Tint) | Oaddrstack _ => (nil, Tint) | Ocast8signed => (Tint :: nil, Tint) - | Ocast8unsigned => (Tint :: nil, Tint) | Ocast16signed => (Tint :: nil, Tint) - | Ocast16unsigned => (Tint :: nil, Tint) | Oadd => (Tint :: Tint :: nil, Tint) | Oaddimm _ => (Tint :: nil, Tint) | Osub => (Tint :: Tint :: nil, Tint) @@ -497,38 +314,52 @@ Lemma type_of_operation_sound: op <> Omove -> eval_operation genv sp op vl m = Some v -> Val.has_type v (snd (type_of_operation op)). -Proof. +Proof with (try exact I). intros. - destruct op; simpl in H0; FuncInv; try subst v; try exact I. + destruct op; simpl in H0; FuncInv; subst; simpl. congruence. - destruct (Genv.find_symbol genv i); simplify_eq H0; intro; subst v; exact I. - simpl. unfold offset_sp in H0. destruct sp; try discriminate. - inversion H0. exact I. - destruct v0; exact I. - destruct v0; exact I. - destruct v0; exact I. - destruct v0; exact I. - destruct (eq_block b b0). injection H0; intro; subst v; exact I. - discriminate. - destruct (Int.eq i0 Int.zero). discriminate. - injection H0; intro; subst v; exact I. - destruct (Int.eq i0 Int.zero). discriminate. - injection H0; intro; subst v; exact I. - destruct (Int.ltu i0 Int.iwordsize). - injection H0; intro; subst v; exact I. discriminate. - destruct (Int.ltu i0 Int.iwordsize). - injection H0; intro; subst v; exact I. discriminate. - destruct (Int.ltu i Int.iwordsize). - injection H0; intro; subst v; exact I. discriminate. - destruct (Int.ltu i Int.iwordsize). - injection H0; intro; subst v; exact I. discriminate. - destruct (Int.ltu i0 Int.iwordsize). - injection H0; intro; subst v; exact I. discriminate. - destruct v0; exact I. - destruct (Float.intoffloat f); simpl in H0; inv H0. exact I. - destruct (eval_condition c vl). - destruct b; injection H0; intro; subst v; exact I. - discriminate. + exact I. + exact I. + unfold symbol_address. destruct (Genv.find_symbol genv i)... + destruct sp... + destruct v0... + destruct v0... + destruct v0; destruct v1... + destruct v0... + destruct v0; destruct v1... simpl. destruct (zeq b b0)... + destruct v0... + destruct v0; destruct v1... + destruct v0... + destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero); inv H2... + destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero); inv H2... + destruct v0; destruct v1... + destruct v0... + destruct v0; destruct v1... + destruct v0... + destruct v0; destruct v1... + destruct v0... + destruct v0; destruct v1... + destruct v0; destruct v1... + destruct v0; destruct v1... + destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)... + destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)... + destruct v0; simpl... destruct (Int.ltu i Int.iwordsize)... + destruct v0; simpl in *; inv H0. destruct (Int.ltu i (Int.repr 31)); inv H2... + destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)... + destruct v0... + destruct v0; destruct v1... + destruct v0... + destruct v0... + destruct v0; destruct v1... + destruct v0; destruct v1... + destruct v0; destruct v1... + destruct v0; destruct v1... + destruct v0; destruct v1; destruct v2... + destruct v0; destruct v1; destruct v2... + destruct v0... + destruct v0; simpl in H0; inv H0. destruct (Float.intoffloat f); inv H2... + destruct v0; destruct v1... + destruct (eval_condition c vl m); simpl... destruct b... Qed. Lemma type_of_chunk_correct: @@ -546,243 +377,436 @@ Qed. End SOUNDNESS. -(** Alternate definition of [eval_condition], [eval_op], [eval_addressing] - as total functions that return [Vundef] when not applicable - (instead of [None]). Used in the proof of [PPCgen]. *) - -Section EVAL_OP_TOTAL. +(** * Manipulating and transforming operations *) -Variable F V: Type. -Variable genv: Genv.t F V. +(** Recognition of move operations. *) -Definition find_symbol_offset (id: ident) (ofs: int) : val := - match Genv.find_symbol genv id with - | Some b => Vptr b ofs - | None => Vundef +Definition is_move_operation + (A: Type) (op: operation) (args: list A) : option A := + match op, args with + | Omove, arg :: nil => Some arg + | _, _ => None end. -Definition eval_condition_total (cond: condition) (vl: list val) : val := - match cond, vl with - | Ccomp c, v1::v2::nil => Val.cmp c v1 v2 - | Ccompu c, v1::v2::nil => Val.cmpu c v1 v2 - | Ccompimm c n, v1::nil => Val.cmp c v1 (Vint n) - | Ccompuimm c n, v1::nil => Val.cmpu c v1 (Vint n) - | Ccompf c, v1::v2::nil => Val.cmpf c v1 v2 - | Cnotcompf c, v1::v2::nil => Val.notbool(Val.cmpf c v1 v2) - | Cmaskzero n, v1::nil => Val.notbool (Val.and v1 (Vint n)) - | Cmasknotzero n, v1::nil => Val.notbool(Val.notbool (Val.and v1 (Vint n))) - | _, _ => Vundef +Lemma is_move_operation_correct: + forall (A: Type) (op: operation) (args: list A) (a: A), + is_move_operation op args = Some a -> + op = Omove /\ args = a :: nil. +Proof. + intros until a. unfold is_move_operation; destruct op; + try (intros; discriminate). + destruct args. intros; discriminate. + destruct args. intros. intuition congruence. + intros; discriminate. +Qed. + +(** [negate_condition cond] returns a condition that is logically + equivalent to the negation of [cond]. *) + +Definition negate_condition (cond: condition): condition := + match cond with + | Ccomp c => Ccomp(negate_comparison c) + | Ccompu c => Ccompu(negate_comparison c) + | Ccompimm c n => Ccompimm (negate_comparison c) n + | Ccompuimm c n => Ccompuimm (negate_comparison c) n + | Ccompf c => Cnotcompf c + | Cnotcompf c => Ccompf c + | Cmaskzero n => Cmasknotzero n + | Cmasknotzero n => Cmaskzero n end. -Definition eval_operation_total (sp: val) (op: operation) (vl: list val) : val := - match op, vl with - | Omove, v1::nil => v1 - | Ointconst n, nil => Vint n - | Ofloatconst n, nil => Vfloat n - | Oaddrsymbol s ofs, nil => find_symbol_offset s ofs - | Oaddrstack ofs, nil => Val.add sp (Vint ofs) - | Ocast8signed, v1::nil => Val.sign_ext 8 v1 - | Ocast8unsigned, v1::nil => Val.zero_ext 8 v1 - | Ocast16signed, v1::nil => Val.sign_ext 16 v1 - | Ocast16unsigned, v1::nil => Val.zero_ext 16 v1 - | Oadd, v1::v2::nil => Val.add v1 v2 - | Oaddimm n, v1::nil => Val.add v1 (Vint n) - | Osub, v1::v2::nil => Val.sub v1 v2 - | Osubimm n, v1::nil => Val.sub (Vint n) v1 - | Omul, v1::v2::nil => Val.mul v1 v2 - | Omulimm n, v1::nil => Val.mul v1 (Vint n) - | Odiv, v1::v2::nil => Val.divs v1 v2 - | Odivu, v1::v2::nil => Val.divu v1 v2 - | Oand, v1::v2::nil => Val.and v1 v2 - | Oandimm n, v1::nil => Val.and v1 (Vint n) - | Oor, v1::v2::nil => Val.or v1 v2 - | Oorimm n, v1::nil => Val.or v1 (Vint n) - | Oxor, v1::v2::nil => Val.xor v1 v2 - | Oxorimm n, v1::nil => Val.xor v1 (Vint n) - | Onand, v1::v2::nil => Val.notint(Val.and v1 v2) - | Onor, v1::v2::nil => Val.notint(Val.or v1 v2) - | Onxor, v1::v2::nil => Val.notint(Val.xor v1 v2) - | Oshl, v1::v2::nil => Val.shl v1 v2 - | Oshr, v1::v2::nil => Val.shr v1 v2 - | Oshrimm n, v1::nil => Val.shr v1 (Vint n) - | Oshrximm n, v1::nil => Val.shrx v1 (Vint n) - | Oshru, v1::v2::nil => Val.shru v1 v2 - | Orolm amount mask, v1::nil => Val.rolm v1 amount mask - | Oroli amount mask, v1::v2::nil => - Val.or (Val.and v1 (Vint (Int.not mask))) (Val.rolm v2 amount mask) - | Onegf, v1::nil => Val.negf v1 - | Oabsf, v1::nil => Val.absf v1 - | Oaddf, v1::v2::nil => Val.addf v1 v2 - | Osubf, v1::v2::nil => Val.subf v1 v2 - | Omulf, v1::v2::nil => Val.mulf v1 v2 - | Odivf, v1::v2::nil => Val.divf v1 v2 - | Omuladdf, v1::v2::v3::nil => Val.addf (Val.mulf v1 v2) v3 - | Omulsubf, v1::v2::v3::nil => Val.subf (Val.mulf v1 v2) v3 - | Osingleoffloat, v1::nil => Val.singleoffloat v1 - | Ointoffloat, v1::nil => Val.intoffloat v1 - | Ofloatofwords, v1::v2::nil => Val.floatofwords v1 v2 - | Ocmp c, _ => eval_condition_total c vl - | _, _ => Vundef +Lemma eval_negate_condition: + forall cond vl m b, + eval_condition cond vl m = Some b -> + eval_condition (negate_condition cond) vl m = Some (negb b). +Proof. + intros. + destruct cond; simpl in H; FuncInv; simpl. + rewrite Val.negate_cmp_bool; rewrite H; auto. + rewrite Val.negate_cmpu_bool; rewrite H; auto. + rewrite Val.negate_cmp_bool; rewrite H; auto. + rewrite Val.negate_cmpu_bool; rewrite H; auto. + rewrite H; auto. + destruct (Val.cmpf_bool c v v0); simpl in H; inv H. rewrite negb_elim; auto. + rewrite H0; auto. + rewrite <- H0. rewrite negb_elim; auto. +Qed. + +(** Shifting stack-relative references. This is used in [Stacking]. *) + +Definition shift_stack_addressing (delta: int) (addr: addressing) := + match addr with + | Ainstack ofs => Ainstack (Int.add delta ofs) + | _ => addr end. -Definition eval_addressing_total - (sp: val) (addr: addressing) (vl: list val) : val := - match addr, vl with - | Aindexed n, v1::nil => Val.add v1 (Vint n) - | Aindexed2, v1::v2::nil => Val.add v1 v2 - | Aglobal s ofs, nil => find_symbol_offset s ofs - | Abased s ofs, v1::nil => Val.add (find_symbol_offset s ofs) v1 - | Ainstack ofs, nil => Val.add sp (Vint ofs) - | _, _ => Vundef +Definition shift_stack_operation (delta: int) (op: operation) := + match op with + | Oaddrstack ofs => Oaddrstack (Int.add delta ofs) + | _ => op end. -Lemma eval_compare_mismatch_weaken: - forall c b, - eval_compare_mismatch c = Some b -> - Val.cmp_mismatch c = Val.of_bool b. +Lemma type_shift_stack_addressing: + forall delta addr, type_of_addressing (shift_stack_addressing delta addr) = type_of_addressing addr. Proof. - unfold eval_compare_mismatch. intros. destruct c; inv H; auto. + intros. destruct addr; auto. Qed. -Lemma eval_compare_null_weaken: - forall n c b, - eval_compare_null c n = Some b -> - (if Int.eq n Int.zero then Val.cmp_mismatch c else Vundef) = Val.of_bool b. +Lemma type_shift_stack_operation: + forall delta op, type_of_operation (shift_stack_operation delta op) = type_of_operation op. Proof. - unfold eval_compare_null. - intros. destruct (Int.eq n Int.zero). apply eval_compare_mismatch_weaken. auto. - discriminate. + intros. destruct op; auto. Qed. -Lemma eval_condition_weaken: - forall c vl b m, - eval_condition c vl m = Some b -> - eval_condition_total c vl = Val.of_bool b. +Lemma eval_shift_stack_addressing: + forall F V (ge: Genv.t F V) sp addr vl delta, + eval_addressing ge sp (shift_stack_addressing delta addr) vl = + eval_addressing ge (Val.add sp (Vint delta)) addr vl. Proof. - intros. - unfold eval_condition in H; destruct c; FuncInv; - try subst b; try reflexivity; simpl; - try (apply eval_compare_null_weaken; auto). - destruct (Mem.valid_pointer m b0 (Int.unsigned i) && - Mem.valid_pointer m b1 (Int.unsigned i0)); try congruence. - unfold eq_block in H. destruct (zeq b0 b1). - congruence. - apply eval_compare_mismatch_weaken; auto. - symmetry. apply Val.notbool_negb_1. - symmetry. apply Val.notbool_negb_1. + intros. destruct addr; simpl; auto. + rewrite Val.add_assoc. simpl. auto. Qed. -Lemma eval_operation_weaken: - forall sp op vl v m, - eval_operation genv sp op vl m = Some v -> - eval_operation_total sp op vl = v. +Lemma eval_shift_stack_operation: + forall F V (ge: Genv.t F V) sp op vl m delta, + eval_operation ge sp (shift_stack_operation delta op) vl m = + eval_operation ge (Val.add sp (Vint delta)) op vl m. Proof. - intros. - unfold eval_operation in H; destruct op; FuncInv; - try subst v; try reflexivity; simpl. - unfold find_symbol_offset. - destruct (Genv.find_symbol genv i); try discriminate. - congruence. - unfold offset_sp in H. - destruct sp; try discriminate. simpl. congruence. - unfold eq_block in H. destruct (zeq b b0); congruence. - destruct (Int.eq i0 Int.zero); congruence. - destruct (Int.eq i0 Int.zero); congruence. - destruct (Int.ltu i0 Int.iwordsize); congruence. - destruct (Int.ltu i0 Int.iwordsize); congruence. - destruct (Int.ltu i Int.iwordsize); congruence. - destruct (Int.ltu i Int.iwordsize); congruence. - destruct (Int.ltu i0 Int.iwordsize); congruence. - destruct (Float.intoffloat f); inv H. auto. - caseEq (eval_condition c vl m); intros; rewrite H0 in H. - replace v with (Val.of_bool b). - eapply eval_condition_weaken; eauto. - destruct b; simpl; congruence. - discriminate. + intros. destruct op; simpl; auto. + rewrite Val.add_assoc. simpl. auto. Qed. -Lemma eval_addressing_weaken: - forall sp addr vl v, - eval_addressing genv sp addr vl = Some v -> - eval_addressing_total sp addr vl = v. +(** Transformation of addressing modes with two operands or more + into an equivalent arithmetic operation. This is used in the [Reload] + pass when a store instruction cannot be reloaded directly because + it runs out of temporary registers. *) + +(** For the PowerPC, there is only one binary addressing mode: [Aindexed2]. + The corresponding operation is [Oadd]. *) + +Definition op_for_binary_addressing (addr: addressing) : operation := Oadd. + +Lemma eval_op_for_binary_addressing: + forall (F V: Type) (ge: Genv.t F V) sp addr args v m, + (length args >= 2)%nat -> + eval_addressing ge sp addr args = Some v -> + eval_operation ge sp (op_for_binary_addressing addr) args m = Some v. Proof. intros. - unfold eval_addressing in H; destruct addr; FuncInv; - try subst v; simpl; try reflexivity. - unfold find_symbol_offset. - destruct (Genv.find_symbol genv i); congruence. - unfold find_symbol_offset. - destruct (Genv.find_symbol genv i); try congruence. - inversion H. reflexivity. - unfold offset_sp in H. destruct sp; simpl; congruence. + destruct addr; simpl in H0; FuncInv; simpl in H; try omegaContradiction. + simpl; congruence. Qed. -Lemma eval_condition_total_is_bool: - forall cond vl, Val.is_bool (eval_condition_total cond vl). +Lemma type_op_for_binary_addressing: + forall addr, + (length (type_of_addressing addr) >= 2)%nat -> + type_of_operation (op_for_binary_addressing addr) = (type_of_addressing addr, Tint). Proof. - intros; destruct cond; - destruct vl; try apply Val.undef_is_bool; - destruct vl; try apply Val.undef_is_bool; - try (destruct vl; try apply Val.undef_is_bool); simpl. - apply Val.cmp_is_bool. - apply Val.cmpu_is_bool. - apply Val.cmp_is_bool. - apply Val.cmpu_is_bool. - apply Val.cmpf_is_bool. - apply Val.notbool_is_bool. - apply Val.notbool_is_bool. - apply Val.notbool_is_bool. + intros. destruct addr; simpl in H; reflexivity || omegaContradiction. Qed. -End EVAL_OP_TOTAL. +(** Two-address operations. There is only one: rotate-mask-insert. *) -(** Compatibility of the evaluation functions with the - ``is less defined'' relation over values. *) +Definition two_address_op (op: operation) : bool := + match op with + | Oroli _ _ => true + | _ => false + end. -Section EVAL_LESSDEF. +(** Operations that are so cheap to recompute that CSE should not factor them out. *) + +Definition is_trivial_op (op: operation) : bool := + match op with + | Omove => true + | Ointconst _ => true + | Oaddrsymbol _ _ => true + | Oaddrstack _ => true + | _ => false + end. + +(** Operations that depend on the memory state. *) + +Definition op_depends_on_memory (op: operation) : bool := + match op with + | Ocmp (Ccompu _) => true + | _ => false + end. + +Lemma op_depends_on_memory_correct: + forall (F V: Type) (ge: Genv.t F V) sp op args m1 m2, + op_depends_on_memory op = false -> + eval_operation ge sp op args m1 = eval_operation ge sp op args m2. +Proof. + intros until m2. destruct op; simpl; try congruence. + destruct c; simpl; auto; discriminate. +Qed. + +(** * Invariance and compatibility properties. *) + +(** [eval_operation] and [eval_addressing] depend on a global environment + for resolving references to global symbols. We show that they give + the same results if a global environment is replaced by another that + assigns the same addresses to the same symbols. *) + +Section GENV_TRANSF. + +Variable F1 F2 V1 V2: Type. +Variable ge1: Genv.t F1 V1. +Variable ge2: Genv.t F2 V2. +Hypothesis agree_on_symbols: + forall (s: ident), Genv.find_symbol ge2 s = Genv.find_symbol ge1 s. + +Lemma eval_operation_preserved: + forall sp op vl m, + eval_operation ge2 sp op vl m = eval_operation ge1 sp op vl m. +Proof. + intros. destruct op; simpl; auto. + destruct vl; auto. decEq. unfold symbol_address. rewrite agree_on_symbols. auto. +Qed. + +Lemma eval_addressing_preserved: + forall sp addr vl, + eval_addressing ge2 sp addr vl = eval_addressing ge1 sp addr vl. +Proof. + intros. destruct addr; simpl; auto; unfold symbol_address; rewrite agree_on_symbols; auto. +Qed. + +End GENV_TRANSF. + +(** Compatibility of the evaluation functions with value injections. *) + +Section EVAL_COMPAT. Variable F V: Type. Variable genv: Genv.t F V. +Variable f: meminj. -Ltac InvLessdef := +Hypothesis symbol_address_inj: + forall id ofs, + val_inject f (symbol_address genv id ofs) (symbol_address genv id ofs). + +Variable m1: mem. +Variable m2: mem. + +Hypothesis valid_pointer_inj: + forall b1 ofs b2 delta, + f b1 = Some(b2, delta) -> + Mem.valid_pointer m1 b1 (Int.unsigned ofs) = true -> + Mem.valid_pointer m2 b2 (Int.unsigned (Int.add ofs (Int.repr delta))) = true. + +Hypothesis valid_pointer_no_overflow: + forall b1 ofs b2 delta, + f b1 = Some(b2, delta) -> + Mem.valid_pointer m1 b1 (Int.unsigned ofs) = true -> + 0 <= Int.unsigned ofs + Int.unsigned (Int.repr delta) <= Int.max_unsigned. + +Hypothesis valid_different_pointers_inj: + forall b1 ofs1 b2 ofs2 b1' delta1 b2' delta2, + b1 <> b2 -> + Mem.valid_pointer m1 b1 (Int.unsigned ofs1) = true -> + Mem.valid_pointer m1 b2 (Int.unsigned ofs2) = true -> + f b1 = Some (b1', delta1) -> + f b2 = Some (b2', delta2) -> + b1' <> b2' \/ + Int.unsigned (Int.add ofs1 (Int.repr delta1)) <> Int.unsigned (Int.add ofs2 (Int.repr delta2)). + +Ltac InvInject := match goal with - | [ H: Val.lessdef (Vint _) _ |- _ ] => - inv H; InvLessdef - | [ H: Val.lessdef (Vfloat _) _ |- _ ] => - inv H; InvLessdef - | [ H: Val.lessdef (Vptr _ _) _ |- _ ] => - inv H; InvLessdef - | [ H: Val.lessdef_list nil _ |- _ ] => - inv H; InvLessdef - | [ H: Val.lessdef_list (_ :: _) _ |- _ ] => - inv H; InvLessdef + | [ H: val_inject _ (Vint _) _ |- _ ] => + inv H; InvInject + | [ H: val_inject _ (Vfloat _) _ |- _ ] => + inv H; InvInject + | [ H: val_inject _ (Vptr _ _) _ |- _ ] => + inv H; InvInject + | [ H: val_list_inject _ nil _ |- _ ] => + inv H; InvInject + | [ H: val_list_inject _ (_ :: _) _ |- _ ] => + inv H; InvInject | _ => idtac end. -Lemma eval_condition_lessdef: - forall cond vl1 vl2 b m1 m2, - Val.lessdef_list vl1 vl2 -> - Mem.extends m1 m2 -> +Remark val_add_inj: + forall v1 v1' v2 v2', + val_inject f v1 v1' -> val_inject f v2 v2' -> val_inject f (Val.add v1 v2) (Val.add v1' v2'). +Proof. + intros. inv H; inv H0; simpl; econstructor; eauto. + repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. + repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. +Qed. + +Lemma eval_condition_inj: + forall cond vl1 vl2 b, + val_list_inject f vl1 vl2 -> eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b. Proof. - intros. destruct cond; simpl in *; FuncInv; InvLessdef; auto. - destruct (Mem.valid_pointer m1 b0 (Int.unsigned i) && - Mem.valid_pointer m1 b1 (Int.unsigned i0)) as [] _eqn; try discriminate. - destruct (andb_prop _ _ Heqb2) as [A B]. - assert (forall b ofs, Mem.valid_pointer m1 b ofs = true -> Mem.valid_pointer m2 b ofs = true). - intros until ofs. repeat rewrite Mem.valid_pointer_nonempty_perm. - apply Mem.perm_extends; auto. - rewrite (H _ _ A). rewrite (H _ _ B). auto. +Opaque Int.add. + assert (CMPU: + forall c v1 v2 v1' v2' b, + val_inject f v1 v1' -> + val_inject f v2 v2' -> + Val.cmpu_bool (Mem.valid_pointer m1) c v1 v2 = Some b -> + Val.cmpu_bool (Mem.valid_pointer m2) c v1' v2' = Some b). + intros. inv H; simpl in H1; try discriminate; inv H0; simpl in H1; try discriminate; simpl; auto. + destruct (Mem.valid_pointer m1 b1 (Int.unsigned ofs1)) as []_eqn; try discriminate. + destruct (Mem.valid_pointer m1 b0 (Int.unsigned ofs0)) as []_eqn; try discriminate. + rewrite (valid_pointer_inj _ H2 Heqb4). + rewrite (valid_pointer_inj _ H Heqb0). simpl. + destruct (zeq b1 b0); simpl in H1. + inv H1. rewrite H in H2; inv H2. rewrite zeq_true. + decEq. apply Int.translate_cmpu. + eapply valid_pointer_no_overflow; eauto. + eapply valid_pointer_no_overflow; eauto. + exploit valid_different_pointers_inj; eauto. intros P. + destruct (zeq b2 b3); auto. + destruct P. congruence. + destruct c; simpl in H1; inv H1. + simpl; decEq. rewrite Int.eq_false; auto. congruence. + simpl; decEq. rewrite Int.eq_false; auto. congruence. + + intros. destruct cond; simpl in H0; FuncInv; InvInject; simpl; auto. + inv H3; inv H2; simpl in H0; inv H0; auto. + eauto. + inv H3; simpl in H0; inv H0; auto. + eauto. + inv H3; inv H2; simpl in H0; inv H0; auto. + inv H3; inv H2; simpl in H0; inv H0; auto. Qed. Ltac TrivialExists := match goal with - | [ |- exists v2, Some ?v1 = Some v2 /\ Val.lessdef ?v1 v2 ] => - exists v1; split; [auto | constructor] + | [ |- exists v2, Some ?v1 = Some v2 /\ val_inject _ _ v2 ] => + exists v1; split; auto | _ => idtac end. +Lemma eval_operation_inj: + forall op sp1 vl1 sp2 vl2 v1, + val_inject f sp1 sp2 -> + val_list_inject f vl1 vl2 -> + eval_operation genv sp1 op vl1 m1 = Some v1 -> + exists v2, eval_operation genv sp2 op vl2 m2 = Some v2 /\ val_inject f v1 v2. +Proof. + intros. destruct op; simpl in H1; simpl; FuncInv; InvInject; TrivialExists. + inv H; simpl; econstructor; eauto. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. + inv H4; simpl; auto. + inv H4; simpl; auto. + apply val_add_inj; auto. + apply val_add_inj; auto. + inv H4; inv H2; simpl; auto. econstructor; eauto. + rewrite Int.sub_add_l. auto. + destruct (zeq b1 b0); auto. subst. rewrite H1 in H0. inv H0. rewrite zeq_true. + rewrite Int.sub_shifted. auto. + inv H4; auto. + inv H4; inv H2; simpl; auto. + inv H4; simpl; auto. + inv H4; inv H3; simpl in H1; inv H1. simpl. + destruct (Int.eq i0 Int.zero); inv H2. TrivialExists. + inv H4; inv H3; simpl in H1; inv H1. simpl. + destruct (Int.eq i0 Int.zero); inv H2. TrivialExists. + inv H4; inv H2; simpl; auto. + inv H4; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto. + inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto. + inv H4; simpl; auto. destruct (Int.ltu i Int.iwordsize); auto. + inv H4; simpl in *; inv H1. destruct (Int.ltu i (Int.repr 31)); inv H2. econstructor; eauto. + inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto. + inv H4; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; simpl; auto. + inv H4; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; inv H2; simpl; auto. + inv H4; simpl; auto; inv H2; simpl; auto; inv H3; simpl; auto. + inv H4; simpl; auto; inv H2; simpl; auto; inv H3; simpl; auto. + inv H4; simpl; auto. + inv H4; simpl in H1; inv H1. simpl. destruct (Float.intoffloat f0); simpl in H2; inv H2. + exists (Vint i); auto. + inv H4; inv H2; simpl; auto. + subst v1. destruct (eval_condition c vl1 m1) as []_eqn. + exploit eval_condition_inj; eauto. intros EQ; rewrite EQ. + destruct b; simpl; constructor. + simpl; constructor. +Qed. + +Lemma eval_addressing_inj: + forall addr sp1 vl1 sp2 vl2 v1, + val_inject f sp1 sp2 -> + val_list_inject f vl1 vl2 -> + eval_addressing genv sp1 addr vl1 = Some v1 -> + exists v2, eval_addressing genv sp2 addr vl2 = Some v2 /\ val_inject f v1 v2. +Proof. + intros. destruct addr; simpl in H1; simpl; FuncInv; InvInject; TrivialExists. + apply val_add_inj; auto. + apply val_add_inj; auto. + apply val_add_inj; auto. + apply val_add_inj; auto. +Qed. + +End EVAL_COMPAT. + +(** Compatibility of the evaluation functions with the ``is less defined'' relation over values. *) + +Section EVAL_LESSDEF. + +Variable F V: Type. +Variable genv: Genv.t F V. + +Remark valid_pointer_extends: + forall m1 m2, Mem.extends m1 m2 -> + forall b1 ofs b2 delta, + Some(b1, 0) = Some(b2, delta) -> + Mem.valid_pointer m1 b1 (Int.unsigned ofs) = true -> + Mem.valid_pointer m2 b2 (Int.unsigned (Int.add ofs (Int.repr delta))) = true. +Proof. + intros. inv H0. rewrite Int.add_zero. eapply Mem.valid_pointer_extends; eauto. +Qed. + +Remark valid_pointer_no_overflow_extends: + forall m1 b1 ofs b2 delta, + Some(b1, 0) = Some(b2, delta) -> + Mem.valid_pointer m1 b1 (Int.unsigned ofs) = true -> + 0 <= Int.unsigned ofs + Int.unsigned (Int.repr delta) <= Int.max_unsigned. +Proof. + intros. inv H. rewrite Zplus_0_r. apply Int.unsigned_range_2. +Qed. + +Remark valid_different_pointers_extends: + forall m1 b1 ofs1 b2 ofs2 b1' delta1 b2' delta2, + b1 <> b2 -> + Mem.valid_pointer m1 b1 (Int.unsigned ofs1) = true -> + Mem.valid_pointer m1 b2 (Int.unsigned ofs2) = true -> + Some(b1, 0) = Some (b1', delta1) -> + Some(b2, 0) = Some (b2', delta2) -> + b1' <> b2' \/ + Int.unsigned(Int.add ofs1 (Int.repr delta1)) <> Int.unsigned(Int.add ofs2 (Int.repr delta2)). +Proof. + intros. inv H2; inv H3. auto. +Qed. + +Lemma eval_condition_lessdef: + forall cond vl1 vl2 b m1 m2, + Val.lessdef_list vl1 vl2 -> + Mem.extends m1 m2 -> + eval_condition cond vl1 m1 = Some b -> + eval_condition cond vl2 m2 = Some b. +Proof. + intros. eapply eval_condition_inj with (f := fun b => Some(b, 0)) (m1 := m1). + apply valid_pointer_extends; auto. + apply valid_pointer_no_overflow_extends; auto. + apply valid_different_pointers_extends; auto. + rewrite <- val_list_inject_lessdef. eauto. auto. +Qed. + Lemma eval_operation_lessdef: forall sp op vl1 vl2 v1 m1 m2, Val.lessdef_list vl1 vl2 -> @@ -790,28 +814,18 @@ Lemma eval_operation_lessdef: eval_operation genv sp op vl1 m1 = Some v1 -> exists v2, eval_operation genv sp op vl2 m2 = Some v2 /\ Val.lessdef v1 v2. Proof. - intros. destruct op; simpl in *; FuncInv; InvLessdef; TrivialExists. - exists v2; auto. - destruct (Genv.find_symbol genv i); inv H1. TrivialExists. - exists v1; auto. - exists (Val.sign_ext 8 v2); split. auto. apply Val.sign_ext_lessdef; auto. - exists (Val.zero_ext 8 v2); split. auto. apply Val.zero_ext_lessdef; auto. - exists (Val.sign_ext 16 v2); split. auto. apply Val.sign_ext_lessdef; auto. - exists (Val.zero_ext 16 v2); split. auto. apply Val.zero_ext_lessdef; auto. - destruct (eq_block b b0); inv H1. TrivialExists. - destruct (Int.eq i0 Int.zero); inv H1; TrivialExists. - destruct (Int.eq i0 Int.zero); inv H1; TrivialExists. - destruct (Int.ltu i0 Int.iwordsize); inv H1; TrivialExists. - destruct (Int.ltu i0 Int.iwordsize); inv H1; TrivialExists. - destruct (Int.ltu i Int.iwordsize); inv H1; TrivialExists. - destruct (Int.ltu i Int.iwordsize); inv H1; TrivialExists. - destruct (Int.ltu i0 Int.iwordsize); inv H1; TrivialExists. - exists (Val.singleoffloat v2); split. auto. apply Val.singleoffloat_lessdef; auto. - destruct (Float.intoffloat f); simpl in *; inv H1. TrivialExists. - caseEq (eval_condition c vl1 m1); intros. rewrite H2 in H1. - rewrite (eval_condition_lessdef c H H0 H2). - destruct b; inv H1; TrivialExists. - rewrite H2 in H1. discriminate. + intros. rewrite val_list_inject_lessdef in H. + assert (exists v2 : val, + eval_operation genv sp op vl2 m2 = Some v2 + /\ val_inject (fun b => Some(b, 0)) v1 v2). + eapply eval_operation_inj with (m1 := m1) (sp1 := sp). + intros. rewrite <- val_inject_lessdef; auto. + apply valid_pointer_extends; auto. + apply valid_pointer_no_overflow_extends; auto. + apply valid_different_pointers_extends; auto. + rewrite <- val_inject_lessdef; auto. + eauto. auto. + destruct H2 as [v2 [A B]]. exists v2; split; auto. rewrite val_inject_lessdef; auto. Qed. Lemma eval_addressing_lessdef: @@ -820,40 +834,19 @@ Lemma eval_addressing_lessdef: eval_addressing genv sp addr vl1 = Some v1 -> exists v2, eval_addressing genv sp addr vl2 = Some v2 /\ Val.lessdef v1 v2. Proof. - intros. destruct addr; simpl in *; FuncInv; InvLessdef; TrivialExists. - destruct (Genv.find_symbol genv i); inv H0. TrivialExists. - destruct (Genv.find_symbol genv i); inv H0. TrivialExists. - exists v1; auto. + intros. rewrite val_list_inject_lessdef in H. + assert (exists v2 : val, + eval_addressing genv sp addr vl2 = Some v2 + /\ val_inject (fun b => Some(b, 0)) v1 v2). + eapply eval_addressing_inj with (sp1 := sp). + intros. rewrite <- val_inject_lessdef; auto. + rewrite <- val_inject_lessdef; auto. + eauto. auto. + destruct H1 as [v2 [A B]]. exists v2; split; auto. rewrite val_inject_lessdef; auto. Qed. End EVAL_LESSDEF. -(** Shifting stack-relative references. This is used in [Stacking]. *) - -Definition shift_stack_addressing (delta: int) (addr: addressing) := - match addr with - | Ainstack ofs => Ainstack (Int.add delta ofs) - | _ => addr - end. - -Definition shift_stack_operation (delta: int) (op: operation) := - match op with - | Oaddrstack ofs => Oaddrstack (Int.add delta ofs) - | _ => op - end. - -Lemma type_shift_stack_addressing: - forall delta addr, type_of_addressing (shift_stack_addressing delta addr) = type_of_addressing addr. -Proof. - intros. destruct addr; auto. -Qed. - -Lemma type_shift_stack_operation: - forall delta op, type_of_operation (shift_stack_operation delta op) = type_of_operation op. -Proof. - intros. destruct op; auto. -Qed. - (** Compatibility of the evaluation functions with memory injections. *) Section EVAL_INJECT. @@ -867,20 +860,13 @@ Variable sp2: block. Variable delta: Z. Hypothesis sp_inj: f sp1 = Some(sp2, delta). -Ltac InvInject := - match goal with - | [ H: val_inject _ (Vint _) _ |- _ ] => - inv H; InvInject - | [ H: val_inject _ (Vfloat _) _ |- _ ] => - inv H; InvInject - | [ H: val_inject _ (Vptr _ _) _ |- _ ] => - inv H; InvInject - | [ H: val_list_inject _ nil _ |- _ ] => - inv H; InvInject - | [ H: val_list_inject _ (_ :: _) _ |- _ ] => - inv H; InvInject - | _ => idtac - end. +Remark symbol_address_inject: + forall id ofs, val_inject f (symbol_address genv id ofs) (symbol_address genv id ofs). +Proof. + intros. unfold symbol_address. destruct (Genv.find_symbol genv id) as []_eqn; auto. + exploit (proj1 globals); eauto. intros. + econstructor; eauto. rewrite Int.add_zero; auto. +Qed. Lemma eval_condition_inject: forall cond vl1 vl2 b m1 m2, @@ -889,35 +875,12 @@ Lemma eval_condition_inject: eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b. Proof. - intros. destruct cond; simpl in *; FuncInv; InvInject; auto. - destruct (Mem.valid_pointer m1 b0 (Int.unsigned i)) as [] _eqn; try discriminate. - destruct (Mem.valid_pointer m1 b1 (Int.unsigned i0)) as [] _eqn; try discriminate. - simpl in H1. - exploit Mem.valid_pointer_inject_val. eauto. eexact Heqb0. econstructor; eauto. - intros V1. rewrite V1. - exploit Mem.valid_pointer_inject_val. eauto. eexact Heqb2. econstructor; eauto. - intros V2. rewrite V2. - simpl. - destruct (eq_block b0 b1); inv H1. - rewrite H3 in H5; inv H5. rewrite dec_eq_true. - decEq. apply Int.translate_cmpu. - eapply Mem.valid_pointer_inject_no_overflow; eauto. - eapply Mem.valid_pointer_inject_no_overflow; eauto. - exploit Mem.different_pointers_inject; eauto. intros P. - destruct (eq_block b3 b4); auto. - destruct P. contradiction. - destruct c; unfold eval_compare_mismatch in *; inv H2. - unfold Int.cmpu. rewrite Int.eq_false; auto. congruence. - unfold Int.cmpu. rewrite Int.eq_false; auto. congruence. + intros. eapply eval_condition_inj with (f := f) (m1 := m1); eauto. + intros; eapply Mem.valid_pointer_inject_val; eauto. + intros; eapply Mem.valid_pointer_inject_no_overflow; eauto. + intros; eapply Mem.different_pointers_inject; eauto. Qed. -Ltac TrivialExists2 := - match goal with - | [ |- exists v2, Some ?v1 = Some v2 /\ val_inject _ _ v2 ] => - exists v1; split; [auto | econstructor; eauto] - | _ => idtac - end. - Lemma eval_addressing_inject: forall addr vl1 vl2 v1, val_list_inject f vl1 vl2 -> @@ -926,15 +889,10 @@ Lemma eval_addressing_inject: eval_addressing genv (Vptr sp2 Int.zero) (shift_stack_addressing (Int.repr delta) addr) vl2 = Some v2 /\ val_inject f v1 v2. Proof. - intros. destruct addr; simpl in *; FuncInv; InvInject; TrivialExists2. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - destruct (Genv.find_symbol genv i) as [] _eqn; inv H0. - TrivialExists2. eapply (proj1 globals); eauto. rewrite Int.add_zero; auto. - destruct (Genv.find_symbol genv i) as [] _eqn; inv H0. - TrivialExists2. eapply (proj1 globals); eauto. rewrite Int.add_zero; auto. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. + intros. + rewrite eval_shift_stack_addressing. simpl. + eapply eval_addressing_inj with (sp1 := Vptr sp1 Int.zero); eauto. + exact symbol_address_inject. Qed. Lemma eval_operation_inject: @@ -946,102 +904,89 @@ Lemma eval_operation_inject: eval_operation genv (Vptr sp2 Int.zero) (shift_stack_operation (Int.repr delta) op) vl2 m2 = Some v2 /\ val_inject f v1 v2. Proof. - intros. destruct op; simpl in *; FuncInv; InvInject; TrivialExists2. - exists v'; auto. - destruct (Genv.find_symbol genv i) as [] _eqn; inv H1. - TrivialExists2. eapply (proj1 globals); eauto. rewrite Int.add_zero; auto. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - exists (Val.sign_ext 8 v'); split; auto. inv H4; simpl; auto. - exists (Val.zero_ext 8 v'); split; auto. inv H4; simpl; auto. - exists (Val.sign_ext 16 v'); split; auto. inv H4; simpl; auto. - exists (Val.zero_ext 16 v'); split; auto. inv H4; simpl; auto. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - rewrite Int.sub_add_l. auto. - destruct (eq_block b b0); inv H1. rewrite H3 in H5; inv H5. rewrite dec_eq_true. - rewrite Int.sub_shifted. TrivialExists2. - destruct (Int.eq i0 Int.zero); inv H1. TrivialExists2. - destruct (Int.eq i0 Int.zero); inv H1. TrivialExists2. - destruct (Int.eq i0 Int.zero); inv H1. TrivialExists2. - destruct (Int.ltu i0 Int.iwordsize); inv H2. TrivialExists2. - destruct (Int.ltu i0 Int.iwordsize); inv H2. TrivialExists2. - destruct (Int.ltu i0 Int.iwordsize); inv H1. TrivialExists2. - destruct (Int.ltu i Int.iwordsize); inv H1. TrivialExists2. - destruct (Int.ltu i (Int.repr 31)); inv H1. TrivialExists2. - destruct (Int.ltu i Int.iwordsize); inv H2. TrivialExists2. - destruct (Int.ltu i Int.iwordsize); inv H2. TrivialExists2. - destruct (Int.ltu i0 Int.iwordsize); inv H1. TrivialExists2. - exists (Val.singleoffloat v'); split; auto. inv H4; simpl; auto. - destruct (Float.intoffloat f0); simpl in *; inv H1. TrivialExists2. - destruct (eval_condition c vl1 m1) as [] _eqn; try discriminate. - exploit eval_condition_inject; eauto. intros EQ; rewrite EQ. - destruct b; inv H1; TrivialExists2. + intros. + rewrite eval_shift_stack_operation. simpl. + eapply eval_operation_inj with (sp1 := Vptr sp1 Int.zero) (m1 := m1); eauto. + exact symbol_address_inject. + intros; eapply Mem.valid_pointer_inject_val; eauto. + intros; eapply Mem.valid_pointer_inject_no_overflow; eauto. + intros; eapply Mem.different_pointers_inject; eauto. Qed. End EVAL_INJECT. -(** Transformation of addressing modes with two operands or more - into an equivalent arithmetic operation. This is used in the [Reload] - pass when a store instruction cannot be reloaded directly because - it runs out of temporary registers. *) - -(** For the PowerPC, there is only one binary addressing mode: [Aindexed2]. - The corresponding operation is [Oadd]. *) - -Definition op_for_binary_addressing (addr: addressing) : operation := Oadd. - -Lemma eval_op_for_binary_addressing: - forall (F V: Type) (ge: Genv.t F V) sp addr args v m, - (length args >= 2)%nat -> - eval_addressing ge sp addr args = Some v -> - eval_operation ge sp (op_for_binary_addressing addr) args m = Some v. -Proof. - intros. - unfold eval_addressing in H0; destruct addr; FuncInv; simpl in H; try omegaContradiction; - simpl; congruence. -Qed. - -Lemma type_op_for_binary_addressing: - forall addr, - (length (type_of_addressing addr) >= 2)%nat -> - type_of_operation (op_for_binary_addressing addr) = (type_of_addressing addr, Tint). -Proof. - intros. destruct addr; simpl in H; reflexivity || omegaContradiction. -Qed. - -(** Two-address operations. There is only one: rotate-mask-insert. *) +(** * Masks for rotate and mask instructions *) + +(** Recognition of integers that are acceptable as immediate operands + to the [rlwim] PowerPC instruction. These integers are of the form + [000011110000] or [111100001111], that is, a run of one bits + surrounded by zero bits, or conversely. We recognize these integers by + running the following automaton on the bits. The accepting states are + 2, 3, 4, 5, and 6. +<< + 0 1 0 + / \ / \ / \ + \ / \ / \ / + -0--> [1] --1--> [2] --0--> [3] + / + [0] + \ + -1--> [4] --0--> [5] --1--> [6] + / \ / \ / \ + \ / \ / \ / + 1 0 1 +>> +*) -Definition two_address_op (op: operation) : bool := - match op with - | Oroli _ _ => true - | _ => false +Inductive rlw_state: Type := + | RLW_S0 : rlw_state + | RLW_S1 : rlw_state + | RLW_S2 : rlw_state + | RLW_S3 : rlw_state + | RLW_S4 : rlw_state + | RLW_S5 : rlw_state + | RLW_S6 : rlw_state + | RLW_Sbad : rlw_state. + +Definition rlw_transition (s: rlw_state) (b: bool) : rlw_state := + match s, b with + | RLW_S0, false => RLW_S1 + | RLW_S0, true => RLW_S4 + | RLW_S1, false => RLW_S1 + | RLW_S1, true => RLW_S2 + | RLW_S2, false => RLW_S3 + | RLW_S2, true => RLW_S2 + | RLW_S3, false => RLW_S3 + | RLW_S3, true => RLW_Sbad + | RLW_S4, false => RLW_S5 + | RLW_S4, true => RLW_S4 + | RLW_S5, false => RLW_S5 + | RLW_S5, true => RLW_S6 + | RLW_S6, false => RLW_Sbad + | RLW_S6, true => RLW_S6 + | RLW_Sbad, _ => RLW_Sbad end. -(** Operations that are so cheap to recompute that CSE should not factor them out. *) - -Definition is_trivial_op (op: operation) : bool := - match op with - | Omove => true - | Ointconst _ => true - | Oaddrsymbol _ _ => true - | Oaddrstack _ => true - | _ => false +Definition rlw_accepting (s: rlw_state) : bool := + match s with + | RLW_S0 => false + | RLW_S1 => false + | RLW_S2 => true + | RLW_S3 => true + | RLW_S4 => true + | RLW_S5 => true + | RLW_S6 => true + | RLW_Sbad => false end. -(** Operations that depend on the memory state. *) - -Definition op_depends_on_memory (op: operation) : bool := - match op with - | Ocmp (Ccompu _) => true - | _ => false +Fixpoint is_rlw_mask_rec (n: nat) (s: rlw_state) (x: Z) {struct n} : bool := + match n with + | O => + rlw_accepting s + | S m => + let (b, y) := Int.Z_bin_decomp x in + is_rlw_mask_rec m (rlw_transition s b) y end. -Lemma op_depends_on_memory_correct: - forall (F V: Type) (ge: Genv.t F V) sp op args m1 m2, - op_depends_on_memory op = false -> - eval_operation ge sp op args m1 = eval_operation ge sp op args m2. -Proof. - intros until m2. destruct op; simpl; try congruence. - destruct c; simpl; congruence. -Qed. +Definition is_rlw_mask (x: int) : bool := + is_rlw_mask_rec Int.wordsize RLW_S0 (Int.unsigned x). diff --git a/powerpc/PrintOp.ml b/powerpc/PrintOp.ml index bfac9a9..3b5e98d 100644 --- a/powerpc/PrintOp.ml +++ b/powerpc/PrintOp.ml @@ -54,9 +54,7 @@ let print_operation reg pp = function | Oaddrstack ofs, [] -> fprintf pp "stack(%ld)" (camlint_of_coqint ofs) | Ocast8signed, [r1] -> fprintf pp "int8signed(%a)" reg r1 - | Ocast8unsigned, [r1] -> fprintf pp "int8unsigned(%a)" reg r1 | Ocast16signed, [r1] -> fprintf pp "int16signed(%a)" reg r1 - | Ocast16unsigned, [r1] -> fprintf pp "int16unsigned(%a)" reg r1 | Oadd, [r1;r2] -> fprintf pp "%a + %a" reg r1 reg r2 | Oaddimm n, [r1] -> fprintf pp "%a + %ld" reg r1 (camlint_of_coqint n) | Osub, [r1;r2] -> fprintf pp "%a - %a" reg r1 reg r2 diff --git a/powerpc/SelectOp.v b/powerpc/SelectOp.v deleted file mode 100644 index b188993..0000000 --- a/powerpc/SelectOp.v +++ /dev/null @@ -1,1018 +0,0 @@ -(* *********************************************************************) -(* *) -(* 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. *) -(* *) -(* *********************************************************************) - -(** Instruction selection for operators *) - -(** The instruction selection pass recognizes opportunities for using - combined arithmetic and logical operations and addressing modes - offered by the target processor. For instance, the expression [x + 1] - can take advantage of the "immediate add" instruction of the processor, - and on the PowerPC, the expression [(x >> 6) & 0xFF] can be turned - into a "rotate and mask" instruction. - - This file defines functions for building CminorSel expressions and - statements, especially expressions consisting of operator - applications. These functions examine their arguments to choose - cheaper forms of operators whenever possible. - - For instance, [add e1 e2] will return a CminorSel expression semantically - equivalent to [Eop Oadd (e1 ::: e2 ::: Enil)], but will use a - [Oaddimm] operator if one of the arguments is an integer constant, - or suppress the addition altogether if one of the arguments is the - null integer. In passing, we perform operator reassociation - ([(e + c1) * c2] becomes [(e * c2) + (c1 * c2)]) and a small amount - of constant propagation. - - On top of the "smart constructor" functions defined below, - module [Selection] implements the actual instruction selection pass. -*) - -Require Import Coqlib. -Require Import Maps. -Require Import AST. -Require Import Integers. -Require Import Floats. -Require Import Values. -Require Import Memory. -Require Import Globalenvs. -Require Cminor. -Require Import Op. -Require Import CminorSel. - -Open Local Scope cminorsel_scope. - -(** ** Constants **) - -Definition addrsymbol (id: ident) (ofs: int) := - Eop (Oaddrsymbol id ofs) Enil. - -Definition addrstack (ofs: int) := - Eop (Oaddrstack ofs) Enil. - -(** ** Integer logical negation *) - -(** The natural way to write smart constructors is by pattern-matching - on their arguments, recognizing cases where cheaper operators - or combined operators are applicable. For instance, integer logical - negation has three special cases (not-and, not-or and not-xor), - along with a default case that uses not-or over its arguments and itself. - This is written naively as follows: -<< -Definition notint (e: expr) := - match e with - | Eop (Ointconst n) Enil => Eop (Ointconst (Int.not n)) Enil - | Eop Oand (t1:::t2:::Enil) => Eop Onand (t1:::t2:::Enil) - | Eop Oor (t1:::t2:::Enil) => Eop Onor (t1:::t2:::Enil) - | Eop Oxor (t1:::t2:::Enil) => Eop Onxor (t1:::t2:::Enil) - | _ => Elet e (Eop Onor (Eletvar O ::: Eletvar O ::: Enil)) - end. ->> - However, Coq expands complex pattern-matchings like the above into - elementary matchings over all constructors of an inductive type, - resulting in much duplication of the final catch-all case. - Such duplications generate huge executable code and duplicate - cases in the correctness proofs. - - To limit this duplication, we use the following trick due to - Yves Bertot. We first define a dependent inductive type that - characterizes the expressions that match each of the 4 cases of interest. -*) - -Inductive notint_cases: forall (e: expr), Type := - | notint_case1: - forall n, - notint_cases (Eop (Ointconst n) Enil) - | notint_case2: - forall t1 t2, - notint_cases (Eop Oand (t1:::t2:::Enil)) - | notint_case3: - forall t1 t2, - notint_cases (Eop Oor (t1:::t2:::Enil)) - | notint_case4: - forall t1 t2, - notint_cases (Eop Oxor (t1:::t2:::Enil)) - | notint_default: - forall (e: expr), - notint_cases e. - -(** We then define a classification function that takes an expression - and return the case in which it falls. Note that the catch-all case - [notint_default] does not state that it is mutually exclusive with - the first three, more specific cases. The classification function - nonetheless chooses the specific cases in preference to the catch-all - case. *) - -Definition notint_match (e: expr) := - match e as z1 return notint_cases z1 with - | Eop (Ointconst n) Enil => - notint_case1 n - | Eop Oand (t1:::t2:::Enil) => - notint_case2 t1 t2 - | Eop Oor (t1:::t2:::Enil) => - notint_case3 t1 t2 - | Eop Oxor (t1:::t2:::Enil) => - notint_case4 t1 t2 - | e => - notint_default e - end. - -(** Finally, the [notint] function we need is defined by a 4-case match - over the result of the classification function. Thus, no duplication - of the right-hand sides of this match occur, and the proof has only - 4 cases to consider (it proceeds by case over [notint_match e]). - Since the default case is not obviously exclusive with the three - specific cases, it is important that its right-hand side is - semantically correct for all possible values of [e], which is the - case here and for all other smart constructors. *) - -Definition notint (e: expr) := - match notint_match e with - | notint_case1 n => - Eop (Ointconst (Int.not n)) Enil - | notint_case2 t1 t2 => - Eop Onand (t1:::t2:::Enil) - | notint_case3 t1 t2 => - Eop Onor (t1:::t2:::Enil) - | notint_case4 t1 t2 => - Eop Onxor (t1:::t2:::Enil) - | notint_default e => - Elet e (Eop Onor (Eletvar O ::: Eletvar O ::: Enil)) - end. - -(** This programming pattern will be applied systematically for the - other smart constructors in this file. *) - -(** ** Boolean negation *) - -Definition notbool_base (e: expr) := - Eop (Ocmp (Ccompuimm Ceq Int.zero)) (e ::: Enil). - -Fixpoint notbool (e: expr) {struct e} : expr := - match e with - | Eop (Ointconst n) Enil => - Eop (Ointconst (if Int.eq n Int.zero then Int.one else Int.zero)) Enil - | Eop (Ocmp cond) args => - Eop (Ocmp (negate_condition cond)) args - | Econdition e1 e2 e3 => - Econdition e1 (notbool e2) (notbool e3) - | _ => - notbool_base e - end. - -(** ** Integer addition and pointer addition *) - -(* -Definition addimm (n: int) (e: expr) := - if Int.eq n Int.zero then e else - match e with - | Eop (Ointconst m) Enil => Eop (Ointconst(Int.add n m)) Enil - | Eop (Oaddrsymbol s m) Enil => Eop (Oaddrsymbol s (Int.add n m)) Enil - | Eop (Oaddrstack m) Enil => Eop (Oaddrstack (Int.add n m)) Enil - | Eop (Oaddimm m) (t ::: Enil) => Eop (Oaddimm(Int.add n m)) (t ::: Enil) - | _ => Eop (Oaddimm n) (e ::: Enil) - end. -*) - -(** Addition of an integer constant. *) - -Inductive addimm_cases: forall (e: expr), Type := - | addimm_case1: - forall (m: int), - addimm_cases (Eop (Ointconst m) Enil) - | addimm_case2: - forall (s: ident) (m: int), - addimm_cases (Eop (Oaddrsymbol s m) Enil) - | addimm_case3: - forall (m: int), - addimm_cases (Eop (Oaddrstack m) Enil) - | addimm_case4: - forall (m: int) (t: expr), - addimm_cases (Eop (Oaddimm m) (t ::: Enil)) - | addimm_default: - forall (e: expr), - addimm_cases e. - -Definition addimm_match (e: expr) := - match e as z1 return addimm_cases z1 with - | Eop (Ointconst m) Enil => - addimm_case1 m - | Eop (Oaddrsymbol s m) Enil => - addimm_case2 s m - | Eop (Oaddrstack m) Enil => - addimm_case3 m - | Eop (Oaddimm m) (t ::: Enil) => - addimm_case4 m t - | e => - addimm_default e - end. - -Definition addimm (n: int) (e: expr) := - if Int.eq n Int.zero then e else - match addimm_match e with - | addimm_case1 m => - Eop (Ointconst(Int.add n m)) Enil - | addimm_case2 s m => - Eop (Oaddrsymbol s (Int.add n m)) Enil - | addimm_case3 m => - Eop (Oaddrstack (Int.add n m)) Enil - | addimm_case4 m t => - Eop (Oaddimm(Int.add n m)) (t ::: Enil) - | addimm_default e => - Eop (Oaddimm n) (e ::: Enil) - end. - -(** Addition of two integer or pointer expressions. *) - -(* -Definition add (e1: expr) (e2: expr) := - match e1, e2 with - | Eop (Ointconst n1) Enil, t2 => addimm n1 t2 - | Eop (Oaddimm n1) (t1:::Enil), Eop (Oaddimm n2) (t2:::Enil) => addimm (Int.add n1 n2) (Eop Oadd (t1:::t2:::Enil)) - | Eop(Oaddimm n1) (t1:::Enil)), t2 => addimm n1 (Eop Oadd (t1:::t2:::Enil)) - | t1, Eop (Ointconst n2) Enil => addimm n2 t1 - | t1, Eop (Oaddimm n2) (t2:::Enil) => addimm n2 (Eop Oadd (t1:::t2:::Enil)) - | Eop (Oaddrsymbol s n1) Enil, Eop (Oaddimm n2) (t2:::Enil) => Eop Oadd (Eop (Oaddrsymbol s (Int.add n1 n2)) Enil ::: t2 ::: Enil) - | Eop (Oaddrstack n1) Enil, Eop (Oaddimm n2) (t2:::Enil) => Eop Oadd (Eop (Oaddstack (Int.add n1 n2)) Enil ::: t2 ::: Enil) - | _, _ => Eop Oadd (e1:::e2:::Enil) - end. -*) - -Inductive add_cases: forall (e1: expr) (e2: expr), Type := - | add_case1: - forall (n1: int) (t2: expr), - add_cases (Eop (Ointconst n1) Enil) (t2) - | add_case2: - forall (n1: int) (t1: expr) (n2: int) (t2: expr), - add_cases (Eop (Oaddimm n1) (t1:::Enil)) (Eop (Oaddimm n2) (t2:::Enil)) - | add_case3: - forall (n1: int) (t1: expr) (t2: expr), - add_cases (Eop(Oaddimm n1) (t1:::Enil)) (t2) - | add_case4: - forall (t1: expr) (n2: int), - add_cases (t1) (Eop (Ointconst n2) Enil) - | add_case5: - forall (t1: expr) (n2: int) (t2: expr), - add_cases (t1) (Eop (Oaddimm n2) (t2:::Enil)) - | add_case6: - forall s n1 n2 t2, - add_cases (Eop (Oaddrsymbol s n1) Enil) (Eop (Oaddimm n2) (t2:::Enil)) - | add_case7: - forall n1 n2 t2, - add_cases (Eop (Oaddrstack n1) Enil) (Eop (Oaddimm n2) (t2:::Enil)) - | add_default: - forall (e1: expr) (e2: expr), - add_cases e1 e2. - -Definition add_match_aux (e1: expr) (e2: expr) := - match e2 as z2 return add_cases e1 z2 with - | Eop (Ointconst n2) Enil => - add_case4 e1 n2 - | Eop (Oaddimm n2) (t2:::Enil) => - add_case5 e1 n2 t2 - | e2 => - add_default e1 e2 - end. - -Definition add_match (e1: expr) (e2: expr) := - match e1 as z1, e2 as z2 return add_cases z1 z2 with - | Eop (Ointconst n1) Enil, t2 => - add_case1 n1 t2 - | Eop (Oaddimm n1) (t1:::Enil), Eop (Oaddimm n2) (t2:::Enil) => - add_case2 n1 t1 n2 t2 - | Eop(Oaddimm n1) (t1:::Enil), t2 => - add_case3 n1 t1 t2 - | Eop (Oaddrsymbol s n1) Enil, Eop (Oaddimm n2) (t2:::Enil) => - add_case6 s n1 n2 t2 - | Eop (Oaddrstack n1) Enil, Eop (Oaddimm n2) (t2:::Enil) => - add_case7 n1 n2 t2 - | e1, e2 => - add_match_aux e1 e2 - end. - -Definition add (e1: expr) (e2: expr) := - match add_match e1 e2 with - | add_case1 n1 t2 => - addimm n1 t2 - | add_case2 n1 t1 n2 t2 => - addimm (Int.add n1 n2) (Eop Oadd (t1:::t2:::Enil)) - | add_case3 n1 t1 t2 => - addimm n1 (Eop Oadd (t1:::t2:::Enil)) - | add_case4 t1 n2 => - addimm n2 t1 - | add_case5 t1 n2 t2 => - addimm n2 (Eop Oadd (t1:::t2:::Enil)) - | add_case6 s n1 n2 t2 => - Eop Oadd (Eop (Oaddrsymbol s (Int.add n1 n2)) Enil ::: t2 ::: Enil) - | add_case7 n1 n2 t2 => - Eop Oadd (Eop (Oaddrstack (Int.add n1 n2)) Enil ::: t2 ::: Enil) - | add_default e1 e2 => - Eop Oadd (e1:::e2:::Enil) - end. - -(** ** Integer and pointer subtraction *) - -(* -Definition sub (e1: expr) (e2: expr) := - match e1, e2 with - | t1, Eop (Ointconst n2) Enil => addimm (Int.neg n2) t1 - | Eop (Oaddimm n1) (t1:::Enil), Eop (Oaddimm n2) (t2:::Enil) => addimm -(intsub n1 n2) (Eop Osub (t1:::t2:::Enil)) - | Eop (Oaddimm n1) (t1:::Enil), t2 => addimm n1 (Eop Osub (t1:::t2:::Rni -l)) - | t1, Eop (Oaddimm n2) (t2:::Enil) => addimm (Int.neg n2) (Eop Osub (t1::: -:t2:::Enil)) - | _, _ => Eop Osub (e1:::e2:::Enil) - end. -*) - -Inductive sub_cases: forall (e1: expr) (e2: expr), Type := - | sub_case1: - forall (t1: expr) (n2: int), - sub_cases (t1) (Eop (Ointconst n2) Enil) - | sub_case2: - forall (n1: int) (t1: expr) (n2: int) (t2: expr), - sub_cases (Eop (Oaddimm n1) (t1:::Enil)) (Eop (Oaddimm n2) (t2:::Enil)) - | sub_case3: - forall (n1: int) (t1: expr) (t2: expr), - sub_cases (Eop (Oaddimm n1) (t1:::Enil)) (t2) - | sub_case4: - forall (t1: expr) (n2: int) (t2: expr), - sub_cases (t1) (Eop (Oaddimm n2) (t2:::Enil)) - | sub_default: - forall (e1: expr) (e2: expr), - sub_cases e1 e2. - -Definition sub_match_aux (e1: expr) (e2: expr) := - match e1 as z1 return sub_cases z1 e2 with - | Eop (Oaddimm n1) (t1:::Enil) => - sub_case3 n1 t1 e2 - | e1 => - sub_default e1 e2 - end. - -Definition sub_match (e1: expr) (e2: expr) := - match e2 as z2, e1 as z1 return sub_cases z1 z2 with - | Eop (Ointconst n2) Enil, t1 => - sub_case1 t1 n2 - | Eop (Oaddimm n2) (t2:::Enil), Eop (Oaddimm n1) (t1:::Enil) => - sub_case2 n1 t1 n2 t2 - | Eop (Oaddimm n2) (t2:::Enil), t1 => - sub_case4 t1 n2 t2 - | e2, e1 => - sub_match_aux e1 e2 - end. - -Definition sub (e1: expr) (e2: expr) := - match sub_match e1 e2 with - | sub_case1 t1 n2 => - addimm (Int.neg n2) t1 - | sub_case2 n1 t1 n2 t2 => - addimm (Int.sub n1 n2) (Eop Osub (t1:::t2:::Enil)) - | sub_case3 n1 t1 t2 => - addimm n1 (Eop Osub (t1:::t2:::Enil)) - | sub_case4 t1 n2 t2 => - addimm (Int.neg n2) (Eop Osub (t1:::t2:::Enil)) - | sub_default e1 e2 => - Eop Osub (e1:::e2:::Enil) - end. - -(** ** Rotates and immediate shifts *) - -(** Recognition of integers that are acceptable as immediate operands - to the [rlwim] PowerPC instruction. These integers are of the form - [000011110000] or [111100001111], that is, a run of one bits - surrounded by zero bits, or conversely. We recognize these integers by - running the following automaton on the bits. The accepting states are - 2, 3, 4, 5, and 6. -<< - 0 1 0 - / \ / \ / \ - \ / \ / \ / - -0--> [1] --1--> [2] --0--> [3] - / - [0] - \ - -1--> [4] --0--> [5] --1--> [6] - / \ / \ / \ - \ / \ / \ / - 1 0 1 ->> -*) - -Inductive rlw_state: Type := - | RLW_S0 : rlw_state - | RLW_S1 : rlw_state - | RLW_S2 : rlw_state - | RLW_S3 : rlw_state - | RLW_S4 : rlw_state - | RLW_S5 : rlw_state - | RLW_S6 : rlw_state - | RLW_Sbad : rlw_state. - -Definition rlw_transition (s: rlw_state) (b: bool) : rlw_state := - match s, b with - | RLW_S0, false => RLW_S1 - | RLW_S0, true => RLW_S4 - | RLW_S1, false => RLW_S1 - | RLW_S1, true => RLW_S2 - | RLW_S2, false => RLW_S3 - | RLW_S2, true => RLW_S2 - | RLW_S3, false => RLW_S3 - | RLW_S3, true => RLW_Sbad - | RLW_S4, false => RLW_S5 - | RLW_S4, true => RLW_S4 - | RLW_S5, false => RLW_S5 - | RLW_S5, true => RLW_S6 - | RLW_S6, false => RLW_Sbad - | RLW_S6, true => RLW_S6 - | RLW_Sbad, _ => RLW_Sbad - end. - -Definition rlw_accepting (s: rlw_state) : bool := - match s with - | RLW_S0 => false - | RLW_S1 => false - | RLW_S2 => true - | RLW_S3 => true - | RLW_S4 => true - | RLW_S5 => true - | RLW_S6 => true - | RLW_Sbad => false - end. - -Fixpoint is_rlw_mask_rec (n: nat) (s: rlw_state) (x: Z) {struct n} : bool := - match n with - | O => - rlw_accepting s - | S m => - let (b, y) := Int.Z_bin_decomp x in - is_rlw_mask_rec m (rlw_transition s b) y - end. - -Definition is_rlw_mask (x: int) : bool := - is_rlw_mask_rec Int.wordsize RLW_S0 (Int.unsigned x). - -(* -Definition rolm (e1: expr) := - match e1 with - | Eop (Ointconst n1) Enil => - Eop (Ointconst(Int.and (Int.rol n1 amount2) mask2)) Enil - | Eop (Orolm amount1 mask1) (t1:::Enil) => - let amount := Int.and (Int.add amount1 amount2) Ox1Fl in - let mask := Int.and (Int.rol mask1 amount2) mask2 in - if Int.is_rlw_mask mask - then Eop (Orolm amount mask) (t1:::Enil) - else Eop (Orolm amount2 mask2) (e1:::Enil) - | _ => Eop (Orolm amount2 mask2) (e1:::Enil) - end -*) - -Inductive rolm_cases: forall (e1: expr), Type := - | rolm_case1: - forall (n1: int), - rolm_cases (Eop (Ointconst n1) Enil) - | rolm_case2: - forall (amount1: int) (mask1: int) (t1: expr), - rolm_cases (Eop (Orolm amount1 mask1) (t1:::Enil)) - | rolm_default: - forall (e1: expr), - rolm_cases e1. - -Definition rolm_match (e1: expr) := - match e1 as z1 return rolm_cases z1 with - | Eop (Ointconst n1) Enil => - rolm_case1 n1 - | Eop (Orolm amount1 mask1) (t1:::Enil) => - rolm_case2 amount1 mask1 t1 - | e1 => - rolm_default e1 - end. - -Definition rolm (e1: expr) (amount2 mask2: int) := - match rolm_match e1 with - | rolm_case1 n1 => - Eop (Ointconst(Int.and (Int.rol n1 amount2) mask2)) Enil - | rolm_case2 amount1 mask1 t1 => - let amount := Int.modu (Int.add amount1 amount2) Int.iwordsize in - let mask := Int.and (Int.rol mask1 amount2) mask2 in - if is_rlw_mask mask - then Eop (Orolm amount mask) (t1:::Enil) - else Eop (Orolm amount2 mask2) (e1:::Enil) - | rolm_default e1 => - Eop (Orolm amount2 mask2) (e1:::Enil) - end. - -Definition shlimm (e1: expr) (n2: int) := - if Int.eq n2 Int.zero then - e1 - else if Int.ltu n2 Int.iwordsize then - rolm e1 n2 (Int.shl Int.mone n2) - else - Eop Oshl (e1:::Eop (Ointconst n2) Enil:::Enil). - -Definition shruimm (e1: expr) (n2: int) := - if Int.eq n2 Int.zero then - e1 - else if Int.ltu n2 Int.iwordsize then - rolm e1 (Int.sub Int.iwordsize n2) (Int.shru Int.mone n2) - else - Eop Oshru (e1:::Eop (Ointconst n2) Enil:::Enil). - -(** ** Integer multiply *) - -Definition mulimm_base (n1: int) (e2: expr) := - match Int.one_bits n1 with - | i :: nil => - shlimm e2 i - | i :: j :: nil => - Elet e2 - (Eop Oadd (shlimm (Eletvar 0) i ::: - shlimm (Eletvar 0) j ::: Enil)) - | _ => - Eop (Omulimm n1) (e2:::Enil) - end. - -(* -Definition mulimm (n1: int) (e2: expr) := - if Int.eq n1 Int.zero then - Elet e2 (Eop (Ointconst Int.zero) Enil) - else if Int.eq n1 Int.one then - e2 - else match e2 with - | Eop (Ointconst n2) Enil => Eop (Ointconst(intmul n1 n2)) Enil - | Eop (Oaddimm n2) (t2:::Enil) => addimm (intmul n1 n2) (mulimm_base n1 t2) - | _ => mulimm_base n1 e2 - end. -*) - -Inductive mulimm_cases: forall (e2: expr), Type := - | mulimm_case1: - forall (n2: int), - mulimm_cases (Eop (Ointconst n2) Enil) - | mulimm_case2: - forall (n2: int) (t2: expr), - mulimm_cases (Eop (Oaddimm n2) (t2:::Enil)) - | mulimm_default: - forall (e2: expr), - mulimm_cases e2. - -Definition mulimm_match (e2: expr) := - match e2 as z1 return mulimm_cases z1 with - | Eop (Ointconst n2) Enil => - mulimm_case1 n2 - | Eop (Oaddimm n2) (t2:::Enil) => - mulimm_case2 n2 t2 - | e2 => - mulimm_default e2 - end. - -Definition mulimm (n1: int) (e2: expr) := - if Int.eq n1 Int.zero then - Elet e2 (Eop (Ointconst Int.zero) Enil) - else if Int.eq n1 Int.one then - e2 - else match mulimm_match e2 with - | mulimm_case1 n2 => - Eop (Ointconst(Int.mul n1 n2)) Enil - | mulimm_case2 n2 t2 => - addimm (Int.mul n1 n2) (mulimm_base n1 t2) - | mulimm_default e2 => - mulimm_base n1 e2 - end. - -(* -Definition mul (e1: expr) (e2: expr) := - match e1, e2 with - | Eop (Ointconst n1) Enil, t2 => mulimm n1 t2 - | t1, Eop (Ointconst n2) Enil => mulimm n2 t1 - | _, _ => Eop Omul (e1:::e2:::Enil) - end. -*) - -Inductive mul_cases: forall (e1: expr) (e2: expr), Type := - | mul_case1: - forall (n1: int) (t2: expr), - mul_cases (Eop (Ointconst n1) Enil) (t2) - | mul_case2: - forall (t1: expr) (n2: int), - mul_cases (t1) (Eop (Ointconst n2) Enil) - | mul_default: - forall (e1: expr) (e2: expr), - mul_cases e1 e2. - -Definition mul_match_aux (e1: expr) (e2: expr) := - match e2 as z2 return mul_cases e1 z2 with - | Eop (Ointconst n2) Enil => - mul_case2 e1 n2 - | e2 => - mul_default e1 e2 - end. - -Definition mul_match (e1: expr) (e2: expr) := - match e1 as z1 return mul_cases z1 e2 with - | Eop (Ointconst n1) Enil => - mul_case1 n1 e2 - | e1 => - mul_match_aux e1 e2 - end. - -Definition mul (e1: expr) (e2: expr) := - match mul_match e1 e2 with - | mul_case1 n1 t2 => - mulimm n1 t2 - | mul_case2 t1 n2 => - mulimm n2 t1 - | mul_default e1 e2 => - Eop Omul (e1:::e2:::Enil) - end. - -(** ** Bitwise and, or, xor *) - -Definition andimm (n1: int) (e2: expr) := - if is_rlw_mask n1 - then rolm e2 Int.zero n1 - else Eop (Oandimm n1) (e2:::Enil). - -Definition and (e1: expr) (e2: expr) := - match mul_match e1 e2 with - | mul_case1 n1 t2 => - andimm n1 t2 - | mul_case2 t1 n2 => - andimm n2 t1 - | mul_default e1 e2 => - Eop Oand (e1:::e2:::Enil) - end. - -Definition same_expr_pure (e1 e2: expr) := - match e1, e2 with - | Evar v1, Evar v2 => if ident_eq v1 v2 then true else false - | _, _ => false - end. - -Inductive or_cases: forall (e1: expr) (e2: expr), Type := - | or_case1: - forall (amount1: int) (mask1: int) (t1: expr) - (amount2: int) (mask2: int) (t2: expr), - or_cases (Eop (Orolm amount1 mask1) (t1:::Enil)) - (Eop (Orolm amount2 mask2) (t2:::Enil)) - | or_default: - forall (e1: expr) (e2: expr), - or_cases e1 e2. - -Definition or_match (e1: expr) (e2: expr) := - match e1 as z1, e2 as z2 return or_cases z1 z2 with - | Eop (Orolm amount1 mask1) (t1:::Enil), - Eop (Orolm amount2 mask2) (t2:::Enil) => - or_case1 amount1 mask1 t1 amount2 mask2 t2 - | e1, e2 => - or_default e1 e2 - end. - -Definition or (e1: expr) (e2: expr) := - match or_match e1 e2 with - | or_case1 amount1 mask1 t1 amount2 mask2 t2 => - if Int.eq amount1 amount2 - && is_rlw_mask (Int.or mask1 mask2) - && same_expr_pure t1 t2 then - Eop (Orolm amount1 (Int.or mask1 mask2)) (t1:::Enil) - else if Int.eq amount1 Int.zero - && Int.eq mask1 (Int.not mask2) then - Eop (Oroli amount2 mask2) (t1:::t2:::Enil) - else if Int.eq amount2 Int.zero - && Int.eq mask2 (Int.not mask1) then - Eop (Oroli amount1 mask1) (t2:::t1:::Enil) - else - Eop Oor (e1:::e2:::Enil) - | or_default e1 e2 => - Eop Oor (e1:::e2:::Enil) - end. - -(** ** Integer division and modulus *) - -Definition divs (e1: expr) (e2: expr) := Eop Odiv (e1:::e2:::Enil). - -Definition mod_aux (divop: operation) (e1 e2: expr) := - Elet e1 - (Elet (lift e2) - (Eop Osub (Eletvar 1 ::: - Eop Omul (Eop divop (Eletvar 1 ::: Eletvar 0 ::: Enil) ::: - Eletvar 0 ::: - Enil) ::: - Enil))). - -Definition mods := mod_aux Odiv. - -Inductive divu_cases: forall (e2: expr), Type := - | divu_case1: - forall (n2: int), - divu_cases (Eop (Ointconst n2) Enil) - | divu_default: - forall (e2: expr), - divu_cases e2. - -Definition divu_match (e2: expr) := - match e2 as z1 return divu_cases z1 with - | Eop (Ointconst n2) Enil => - divu_case1 n2 - | e2 => - divu_default e2 - end. - -Definition divu (e1: expr) (e2: expr) := - match divu_match e2 with - | divu_case1 n2 => - match Int.is_power2 n2 with - | Some l2 => shruimm e1 l2 - | None => Eop Odivu (e1:::e2:::Enil) - end - | divu_default e2 => - Eop Odivu (e1:::e2:::Enil) - end. - -Definition modu (e1: expr) (e2: expr) := - match divu_match e2 with - | divu_case1 n2 => - match Int.is_power2 n2 with - | Some l2 => andimm (Int.sub n2 Int.one) e1 - | None => mod_aux Odivu e1 e2 - end - | divu_default e2 => - mod_aux Odivu e1 e2 - end. - -(** ** General shifts *) - -Inductive shift_cases: forall (e1: expr), Type := - | shift_case1: - forall (n2: int), - shift_cases (Eop (Ointconst n2) Enil) - | shift_default: - forall (e1: expr), - shift_cases e1. - -Definition shift_match (e1: expr) := - match e1 as z1 return shift_cases z1 with - | Eop (Ointconst n2) Enil => - shift_case1 n2 - | e1 => - shift_default e1 - end. - -Definition shl (e1: expr) (e2: expr) := - match shift_match e2 with - | shift_case1 n2 => - shlimm e1 n2 - | shift_default e2 => - Eop Oshl (e1:::e2:::Enil) - end. - -Definition shru (e1: expr) (e2: expr) := - match shift_match e2 with - | shift_case1 n2 => - shruimm e1 n2 - | shift_default e2 => - Eop Oshru (e1:::e2:::Enil) - end. - -(** ** Floating-point arithmetic *) - -Parameter use_fused_mul : unit -> bool. - -(* -Definition addf (e1: expr) (e2: expr) := - match e1, e2 with - | Eop Omulf (t1:::t2:::Enil), t3 => Eop Omuladdf (t1:::t2:::t3:::Enil) - | t1, Eop Omulf (t2:::t3:::Enil) => Elet t1 (Eop Omuladdf (t2:::t3:::Rvar 0:::Enil)) - | _, _ => Eop Oaddf (e1:::e2:::Enil) - end. -*) - -Inductive addf_cases: forall (e1: expr) (e2: expr), Type := - | addf_case1: - forall (t1: expr) (t2: expr) (t3: expr), - addf_cases (Eop Omulf (t1:::t2:::Enil)) (t3) - | addf_case2: - forall (t1: expr) (t2: expr) (t3: expr), - addf_cases (t1) (Eop Omulf (t2:::t3:::Enil)) - | addf_default: - forall (e1: expr) (e2: expr), - addf_cases e1 e2. - -Definition addf_match_aux (e1: expr) (e2: expr) := - match e2 as z2 return addf_cases e1 z2 with - | Eop Omulf (t2:::t3:::Enil) => - addf_case2 e1 t2 t3 - | e2 => - addf_default e1 e2 - end. - -Definition addf_match (e1: expr) (e2: expr) := - match e1 as z1 return addf_cases z1 e2 with - | Eop Omulf (t1:::t2:::Enil) => - addf_case1 t1 t2 e2 - | e1 => - addf_match_aux e1 e2 - end. - -Definition addf (e1: expr) (e2: expr) := - if use_fused_mul tt then - match addf_match e1 e2 with - | addf_case1 t1 t2 t3 => - Eop Omuladdf (t1:::t2:::t3:::Enil) - | addf_case2 t1 t2 t3 => - Eop Omuladdf (t2:::t3:::t1:::Enil) - | addf_default e1 e2 => - Eop Oaddf (e1:::e2:::Enil) - end - else Eop Oaddf (e1:::e2:::Enil). - -(* -Definition subf (e1: expr) (e2: expr) := - match e1, e2 with - | Eop Omulfloat (t1:::t2:::Enil), t3 => Eop Omulsubf (t1:::t2:::t3:::Enil) - | _, _ => Eop Osubf (e1:::e2:::Enil) - end. -*) - -Inductive subf_cases: forall (e1: expr) (e2: expr), Type := - | subf_case1: - forall (t1: expr) (t2: expr) (t3: expr), - subf_cases (Eop Omulf (t1:::t2:::Enil)) (t3) - | subf_default: - forall (e1: expr) (e2: expr), - subf_cases e1 e2. - -Definition subf_match (e1: expr) (e2: expr) := - match e1 as z1 return subf_cases z1 e2 with - | Eop Omulf (t1:::t2:::Enil) => - subf_case1 t1 t2 e2 - | e1 => - subf_default e1 e2 - end. - -Definition subf (e1: expr) (e2: expr) := - if use_fused_mul tt then - match subf_match e1 e2 with - | subf_case1 t1 t2 t3 => - Eop Omulsubf (t1:::t2:::t3:::Enil) - | subf_default e1 e2 => - Eop Osubf (e1:::e2:::Enil) - end - else Eop Osubf (e1:::e2:::Enil). - -(** ** Comparisons *) - -Inductive comp_cases: forall (e1: expr) (e2: expr), Type := - | comp_case1: - forall n1 t2, - comp_cases (Eop (Ointconst n1) Enil) (t2) - | comp_case2: - forall t1 n2, - comp_cases (t1) (Eop (Ointconst n2) Enil) - | comp_default: - forall (e1: expr) (e2: expr), - comp_cases e1 e2. - -Definition comp_match (e1: expr) (e2: expr) := - match e1 as z1, e2 as z2 return comp_cases z1 z2 with - | Eop (Ointconst n1) Enil, t2 => - comp_case1 n1 t2 - | t1, Eop (Ointconst n2) Enil => - comp_case2 t1 n2 - | e1, e2 => - comp_default e1 e2 - end. - -Definition comp (c: comparison) (e1: expr) (e2: expr) := - match comp_match e1 e2 with - | comp_case1 n1 t2 => - Eop (Ocmp (Ccompimm (swap_comparison c) n1)) (t2 ::: Enil) - | comp_case2 t1 n2 => - Eop (Ocmp (Ccompimm c n2)) (t1 ::: Enil) - | comp_default e1 e2 => - Eop (Ocmp (Ccomp c)) (e1 ::: e2 ::: Enil) - end. - -Definition compu (c: comparison) (e1: expr) (e2: expr) := - match comp_match e1 e2 with - | comp_case1 n1 t2 => - Eop (Ocmp (Ccompuimm (swap_comparison c) n1)) (t2 ::: Enil) - | comp_case2 t1 n2 => - Eop (Ocmp (Ccompuimm c n2)) (t1 ::: Enil) - | comp_default e1 e2 => - Eop (Ocmp (Ccompu c)) (e1 ::: e2 ::: Enil) - end. - -Definition compf (c: comparison) (e1: expr) (e2: expr) := - Eop (Ocmp (Ccompf c)) (e1 ::: e2 ::: Enil). - -(** ** Floating-point conversions *) - -Definition intoffloat (e: expr) := Eop Ointoffloat (e ::: Enil). - -Definition intuoffloat (e: expr) := - let f := Eop (Ofloatconst (Float.floatofintu Float.ox8000_0000)) Enil in - Elet e - (Econdition (CEcond (Ccompf Clt) (Eletvar O ::: f ::: Enil)) - (intoffloat (Eletvar O)) - (addimm Float.ox8000_0000 (intoffloat (subf (Eletvar O) f)))). - -Definition floatofintu (e: expr) := - subf (Eop Ofloatofwords (Eop (Ointconst Float.ox4330_0000) Enil ::: e ::: Enil)) - (Eop (Ofloatconst (Float.from_words Float.ox4330_0000 Int.zero)) Enil). - -Definition floatofint (e: expr) := - subf (Eop Ofloatofwords (Eop (Ointconst Float.ox4330_0000) Enil - ::: addimm Float.ox8000_0000 e ::: Enil)) - (Eop (Ofloatconst (Float.from_words Float.ox4330_0000 Float.ox8000_0000)) Enil). - -(** ** Other operators, not optimized. *) - -Definition cast8unsigned (e: expr) := Eop Ocast8unsigned (e ::: Enil). -Definition cast8signed (e: expr) := Eop Ocast8signed (e ::: Enil). -Definition cast16unsigned (e: expr) := Eop Ocast16unsigned (e ::: Enil). -Definition cast16signed (e: expr) := Eop Ocast16signed (e ::: Enil). -Definition singleoffloat (e: expr) := Eop Osingleoffloat (e ::: Enil). -Definition negint (e: expr) := Eop (Osubimm Int.zero) (e ::: Enil). -Definition negf (e: expr) := Eop Onegf (e ::: Enil). -Definition absf (e: expr) := Eop Oabsf (e ::: Enil). -Definition xor (e1 e2: expr) := Eop Oxor (e1 ::: e2 ::: Enil). -Definition shr (e1 e2: expr) := Eop Oshr (e1 ::: e2 ::: Enil). -Definition mulf (e1 e2: expr) := Eop Omulf (e1 ::: e2 ::: Enil). -Definition divf (e1 e2: expr) := Eop Odivf (e1 ::: e2 ::: Enil). - -(** ** Recognition of addressing modes for load and store operations *) - -(* -Definition addressing (e: expr) := - match e with - | Eop (Oaddrsymbol s n) Enil => (Aglobal s n, Enil) - | Eop (Oaddrstack n) Enil => (Ainstack n, Enil) - | Eop Oadd (Eop (Oaddrsymbol s n) Enil) e2 => (Abased(s, n), e2:::Enil) - | Eop (Oaddimm n) (e1:::Enil) => (Aindexed n, e1:::Enil) - | Eop Oadd (e1:::e2:::Enil) => (Aindexed2, e1:::e2:::Enil) - | _ => (Aindexed Int.zero, e:::Enil) - end. -*) - -Inductive addressing_cases: forall (e: expr), Type := - | addressing_case1: - forall (s: ident) (n: int), - addressing_cases (Eop (Oaddrsymbol s n) Enil) - | addressing_case2: - forall (n: int), - addressing_cases (Eop (Oaddrstack n) Enil) - | addressing_case3: - forall (s: ident) (n: int) (e2: expr), - addressing_cases - (Eop Oadd (Eop (Oaddrsymbol s n) Enil:::e2:::Enil)) - | addressing_case4: - forall (n: int) (e1: expr), - addressing_cases (Eop (Oaddimm n) (e1:::Enil)) - | addressing_case5: - forall (e1: expr) (e2: expr), - addressing_cases (Eop Oadd (e1:::e2:::Enil)) - | addressing_default: - forall (e: expr), - addressing_cases e. - -Definition addressing_match (e: expr) := - match e as z1 return addressing_cases z1 with - | Eop (Oaddrsymbol s n) Enil => - addressing_case1 s n - | Eop (Oaddrstack n) Enil => - addressing_case2 n - | Eop Oadd (Eop (Oaddrsymbol s n) Enil:::e2:::Enil) => - addressing_case3 s n e2 - | Eop (Oaddimm n) (e1:::Enil) => - addressing_case4 n e1 - | Eop Oadd (e1:::e2:::Enil) => - addressing_case5 e1 e2 - | e => - addressing_default e - end. - -Definition addressing (chunk: memory_chunk) (e: expr) := - match addressing_match e with - | addressing_case1 s n => - (Aglobal s n, Enil) - | addressing_case2 n => - (Ainstack n, Enil) - | addressing_case3 s n e2 => - (Abased s n, e2:::Enil) - | addressing_case4 n e1 => - (Aindexed n, e1:::Enil) - | addressing_case5 e1 e2 => - (Aindexed2, e1:::e2:::Enil) - | addressing_default e => - (Aindexed Int.zero, e:::Enil) - end. diff --git a/powerpc/SelectOp.vp b/powerpc/SelectOp.vp new file mode 100644 index 0000000..40c9011 --- /dev/null +++ b/powerpc/SelectOp.vp @@ -0,0 +1,432 @@ +(* *********************************************************************) +(* *) +(* 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. *) +(* *) +(* *********************************************************************) + +(** Instruction selection for operators *) + +(** The instruction selection pass recognizes opportunities for using + combined arithmetic and logical operations and addressing modes + offered by the target processor. For instance, the expression [x + 1] + can take advantage of the "immediate add" instruction of the processor, + and on the PowerPC, the expression [(x >> 6) & 0xFF] can be turned + into a "rotate and mask" instruction. + + This file defines functions for building CminorSel expressions and + statements, especially expressions consisting of operator + applications. These functions examine their arguments to choose + cheaper forms of operators whenever possible. + + For instance, [add e1 e2] will return a CminorSel expression semantically + equivalent to [Eop Oadd (e1 ::: e2 ::: Enil)], but will use a + [Oaddimm] operator if one of the arguments is an integer constant, + or suppress the addition altogether if one of the arguments is the + null integer. In passing, we perform operator reassociation + ([(e + c1) * c2] becomes [(e * c2) + (c1 * c2)]) and a small amount + of constant propagation. + + On top of the "smart constructor" functions defined below, + module [Selection] implements the actual instruction selection pass. +*) + +Require Import Coqlib. +Require Import Maps. +Require Import AST. +Require Import Integers. +Require Import Floats. +Require Import Values. +Require Import Memory. +Require Import Globalenvs. +Require Cminor. +Require Import Op. +Require Import CminorSel. + +Open Local Scope cminorsel_scope. + +(** ** Constants **) + +Definition addrsymbol (id: ident) (ofs: int) := + Eop (Oaddrsymbol id ofs) Enil. + +Definition addrstack (ofs: int) := + Eop (Oaddrstack ofs) Enil. + +(** ** Integer logical negation *) + +Nondetfunction notint (e: expr) := + match e with + | Eop (Ointconst n) Enil => Eop (Ointconst (Int.not n)) Enil + | Eop Oand (t1:::t2:::Enil) => Eop Onand (t1:::t2:::Enil) + | Eop Oor (t1:::t2:::Enil) => Eop Onor (t1:::t2:::Enil) + | Eop Oxor (t1:::t2:::Enil) => Eop Onxor (t1:::t2:::Enil) + | _ => Elet e (Eop Onor (Eletvar O ::: Eletvar O ::: Enil)) + end. + +(** ** Boolean negation *) + +Fixpoint notbool (e: expr) {struct e} : expr := + let default := Eop (Ocmp (Ccompuimm Ceq Int.zero)) (e ::: Enil) in + match e with + | Eop (Ointconst n) Enil => + Eop (Ointconst (if Int.eq n Int.zero then Int.one else Int.zero)) Enil + | Eop (Ocmp cond) args => + Eop (Ocmp (negate_condition cond)) args + | Econdition e1 e2 e3 => + Econdition e1 (notbool e2) (notbool e3) + | _ => + default + end. + +(** ** Integer addition and pointer addition *) + +Nondetfunction addimm (n: int) (e: expr) := + if Int.eq n Int.zero then e else + match e with + | Eop (Ointconst m) Enil => Eop (Ointconst(Int.add n m)) Enil + | Eop (Oaddrsymbol s m) Enil => Eop (Oaddrsymbol s (Int.add n m)) Enil + | Eop (Oaddrstack m) Enil => Eop (Oaddrstack (Int.add n m)) Enil + | Eop (Oaddimm m) (t ::: Enil) => Eop (Oaddimm(Int.add n m)) (t ::: Enil) + | _ => Eop (Oaddimm n) (e ::: Enil) + end. + +Nondetfunction add (e1: expr) (e2: expr) := + match e1, e2 with + | Eop (Ointconst n1) Enil, t2 => + addimm n1 t2 + | Eop (Oaddimm n1) (t1:::Enil), Eop (Oaddimm n2) (t2:::Enil) => + addimm (Int.add n1 n2) (Eop Oadd (t1:::t2:::Enil)) + | Eop (Oaddimm n1) (t1:::Enil), t2 => + addimm n1 (Eop Oadd (t1:::t2:::Enil)) + | Eop (Oaddrsymbol s n1) Enil, Eop (Oaddimm n2) (t2:::Enil) => + Eop Oadd (Eop (Oaddrsymbol s (Int.add n1 n2)) Enil ::: t2 ::: Enil) + | Eop (Oaddrstack n1) Enil, Eop (Oaddimm n2) (t2:::Enil) => + Eop Oadd (Eop (Oaddrstack (Int.add n1 n2)) Enil ::: t2 ::: Enil) + | t1, Eop (Ointconst n2) Enil => + addimm n2 t1 + | t1, Eop (Oaddimm n2) (t2:::Enil) => + addimm n2 (Eop Oadd (t1:::t2:::Enil)) + | _, _ => + Eop Oadd (e1:::e2:::Enil) + end. + +(** ** Integer and pointer subtraction *) + +Nondetfunction sub (e1: expr) (e2: expr) := + match e1, e2 with + | t1, Eop (Ointconst n2) Enil => + addimm (Int.neg n2) t1 + | Eop (Oaddimm n1) (t1:::Enil), Eop (Oaddimm n2) (t2:::Enil) => + addimm (Int.sub n1 n2) (Eop Osub (t1:::t2:::Enil)) + | Eop (Oaddimm n1) (t1:::Enil), t2 => + addimm n1 (Eop Osub (t1:::t2:::Enil)) + | t1, Eop (Oaddimm n2) (t2:::Enil) => + addimm (Int.neg n2) (Eop Osub (t1:::t2:::Enil)) + | _, _ => + Eop Osub (e1:::e2:::Enil) + end. + +Definition negint (e: expr) := Eop (Osubimm Int.zero) (e ::: Enil). + +(** ** Rotates and immediate shifts *) + +Nondetfunction rolm (e1: expr) (amount2: int) (mask2: int) := + match e1 with + | Eop (Ointconst n1) Enil => + Eop (Ointconst(Int.and (Int.rol n1 amount2) mask2)) Enil + | Eop (Orolm amount1 mask1) (t1:::Enil) => + Eop (Orolm (Int.modu (Int.add amount1 amount2) Int.iwordsize) + (Int.and (Int.rol mask1 amount2) mask2)) + (t1:::Enil) + | Eop (Oandimm mask1) (t1:::Enil) => + Eop (Orolm (Int.modu amount2 Int.iwordsize) + (Int.and (Int.rol mask1 amount2) mask2)) + (t1:::Enil) + | _ => + Eop (Orolm amount2 mask2) (e1:::Enil) + end. + +Definition shlimm (e1: expr) (n2: int) := + if Int.eq n2 Int.zero then + e1 + else if Int.ltu n2 Int.iwordsize then + rolm e1 n2 (Int.shl Int.mone n2) + else + Eop Oshl (e1:::Eop (Ointconst n2) Enil:::Enil). + +Definition shrimm (e1: expr) (n2: int) := + if Int.eq n2 Int.zero then + e1 + else + Eop (Oshrimm n2) (e1:::Enil). + +Definition shruimm (e1: expr) (n2: int) := + if Int.eq n2 Int.zero then + e1 + else if Int.ltu n2 Int.iwordsize then + rolm e1 (Int.sub Int.iwordsize n2) (Int.shru Int.mone n2) + else + Eop Oshru (e1:::Eop (Ointconst n2) Enil:::Enil). + +(** ** Integer multiply *) + +Definition mulimm_base (n1: int) (e2: expr) := + match Int.one_bits n1 with + | i :: nil => + shlimm e2 i + | i :: j :: nil => + Elet e2 + (Eop Oadd (shlimm (Eletvar 0) i ::: + shlimm (Eletvar 0) j ::: Enil)) + | _ => + Eop (Omulimm n1) (e2:::Enil) + end. + +Nondetfunction mulimm (n1: int) (e2: expr) := + if Int.eq n1 Int.zero then Eop (Ointconst Int.zero) Enil + else if Int.eq n1 Int.one then e2 + else match e2 with + | Eop (Ointconst n2) Enil => Eop (Ointconst(Int.mul n1 n2)) Enil + | Eop (Oaddimm n2) (t2:::Enil) => addimm (Int.mul n1 n2) (mulimm_base n1 t2) + | _ => mulimm_base n1 e2 + end. + +Nondetfunction mul (e1: expr) (e2: expr) := + match e1, e2 with + | Eop (Ointconst n1) Enil, t2 => mulimm n1 t2 + | t1, Eop (Ointconst n2) Enil => mulimm n2 t1 + | _, _ => Eop Omul (e1:::e2:::Enil) + end. + +(** ** Bitwise and, or, xor *) + +Nondetfunction andimm (n1: int) (e2: expr) := + match e2 with + | Eop (Ointconst n2) Enil => + Eop (Ointconst (Int.and n1 n2)) Enil + | Eop (Oandimm n2) (t2:::Enil) => + Eop (Oandimm (Int.and n1 n2)) (t2:::Enil) + | Eop (Orolm amount2 mask2) (t2:::Enil) => + Eop (Orolm amount2 (Int.and n1 mask2)) (t2:::Enil) + | Eop (Oshrimm amount) (t2:::Enil) => + if Int.eq (Int.shru (Int.shl n1 amount) amount) n1 + && Int.ltu amount Int.iwordsize + then rolm t2 (Int.sub Int.iwordsize amount) + (Int.and (Int.shru Int.mone amount) n1) + else Eop (Oandimm n1) (e2:::Enil) + | _ => + Eop (Oandimm n1) (e2:::Enil) + end. + +Nondetfunction and (e1: expr) (e2: expr) := + match e1, e2 with + | Eop (Ointconst n1) Enil, t2 => andimm n1 t2 + | t1, Eop (Ointconst n2) Enil => andimm n2 t1 + | _, _ => Eop Oand (e1:::e2:::Enil) + end. + +Definition same_expr_pure (e1 e2: expr) := + match e1, e2 with + | Evar v1, Evar v2 => if ident_eq v1 v2 then true else false + | _, _ => false + end. + +Nondetfunction orimm (n1: int) (e2: expr) := + match e2 with + | Eop (Ointconst n2) Enil => Eop (Ointconst (Int.or n1 n2)) Enil + | Eop (Oorimm n2) (t2:::Enil) => Eop (Oorimm (Int.or n1 n2)) (t2:::Enil) + | _ => Eop (Oorimm n1) (e2:::Enil) + end. + +Nondetfunction or (e1: expr) (e2: expr) := + match e1, e2 with + | Eop (Orolm amount1 mask1) (t1:::Enil), Eop (Orolm amount2 mask2) (t2:::Enil) => + if Int.eq amount1 amount2 && same_expr_pure t1 t2 + then Eop (Orolm amount1 (Int.or mask1 mask2)) (t1:::Enil) + else Eop Oor (e1:::e2:::Enil) + | Eop (Oandimm mask1) (t1:::Enil), Eop (Orolm amount2 mask2) (t2:::Enil) => + if Int.eq mask1 (Int.not mask2) && is_rlw_mask mask2 + then Eop (Oroli amount2 mask2) (t1:::t2:::Enil) + else Eop Oor (e1:::e2:::Enil) + | Eop (Orolm amount1 mask1) (t1:::Enil), Eop (Oandimm mask2) (t2:::Enil) => + if Int.eq mask2 (Int.not mask1) && is_rlw_mask mask1 + then Eop (Oroli amount1 mask1) (t2:::t1:::Enil) + else Eop Oor (e1:::e2:::Enil) + | Eop (Ointconst n1) Enil, t2 => orimm n1 t2 + | t1, Eop (Ointconst n2) Enil => orimm n2 t1 + | _, _ => Eop Oor (e1:::e2:::Enil) + end. + +Nondetfunction xorimm (n1: int) (e2: expr) := + match e2 with + | Eop (Ointconst n2) Enil => Eop (Ointconst (Int.xor n1 n2)) Enil + | Eop (Oxorimm n2) (t2:::Enil) => Eop (Oxorimm (Int.xor n1 n2)) (t2:::Enil) + | _ => Eop (Oxorimm n1) (e2:::Enil) + end. + +Nondetfunction xor (e1: expr) (e2: expr) := + match e1, e2 with + | Eop (Ointconst n1) Enil, t2 => xorimm n1 t2 + | t1, Eop (Ointconst n2) Enil => xorimm n2 t1 + | _, _ => Eop Oxor (e1:::e2:::Enil) + end. + +(** ** Integer division and modulus *) + +Definition divs (e1: expr) (e2: expr) := Eop Odiv (e1:::e2:::Enil). + +Definition mod_aux (divop: operation) (e1 e2: expr) := + Elet e1 + (Elet (lift e2) + (Eop Osub (Eletvar 1 ::: + Eop Omul (Eop divop (Eletvar 1 ::: Eletvar 0 ::: Enil) ::: + Eletvar 0 ::: + Enil) ::: + Enil))). + +Definition mods := mod_aux Odiv. + +Definition divuimm (e: expr) (n: int) := + match Int.is_power2 n with + | Some l => shruimm e l + | None => Eop Odivu (e ::: Eop (Ointconst n) Enil ::: Enil) + end. + +Nondetfunction divu (e1: expr) (e2: expr) := + match e2 with + | Eop (Ointconst n2) Enil => divuimm e1 n2 + | _ => Eop Odivu (e1:::e2:::Enil) + end. + +Definition moduimm (e: expr) (n: int) := + match Int.is_power2 n with + | Some l => andimm (Int.sub n Int.one) e + | None => mod_aux Odivu e (Eop (Ointconst n) Enil) + end. + +Nondetfunction modu (e1: expr) (e2: expr) := + match e2 with + | Eop (Ointconst n2) Enil => moduimm e1 n2 + | _ => mod_aux Odivu e1 e2 + end. + +(** ** General shifts *) + +Nondetfunction shl (e1: expr) (e2: expr) := + match e2 with + | Eop (Ointconst n2) Enil => shlimm e1 n2 + | _ => Eop Oshl (e1:::e2:::Enil) + end. + +Nondetfunction shr (e1: expr) (e2: expr) := + match e2 with + | Eop (Ointconst n2) Enil => shrimm e1 n2 + | _ => Eop Oshr (e1:::e2:::Enil) + end. + +Nondetfunction shru (e1: expr) (e2: expr) := + match e2 with + | Eop (Ointconst n2) Enil => shruimm e1 n2 + | _ => Eop Oshru (e1:::e2:::Enil) + end. + +(** ** Floating-point arithmetic *) + +Definition negf (e: expr) := Eop Onegf (e ::: Enil). +Definition absf (e: expr) := Eop Oabsf (e ::: Enil). + +Parameter use_fused_mul : unit -> bool. + +Nondetfunction addf (e1: expr) (e2: expr) := + if negb(use_fused_mul tt) then Eop Oaddf (e1:::e2:::Enil) else + match e1, e2 with + | Eop Omulf (t1:::t2:::Enil), t3 => Eop Omuladdf (t1:::t2:::t3:::Enil) + | t1, Eop Omulf (t2:::t3:::Enil) => Eop Omuladdf (t2:::t3:::t1:::Enil) + | _, _ => Eop Oaddf (e1:::e2:::Enil) + end. + +Nondetfunction subf (e1: expr) (e2: expr) := + if negb(use_fused_mul tt) then Eop Osubf (e1:::e2:::Enil) else + match e1 with + | Eop Omulf (t1:::t2:::Enil) => Eop Omulsubf (t1:::t2:::e2:::Enil) + | _ => Eop Osubf (e1:::e2:::Enil) + end. + +Definition mulf (e1 e2: expr) := Eop Omulf (e1 ::: e2 ::: Enil). +Definition divf (e1 e2: expr) := Eop Odivf (e1 ::: e2 ::: Enil). + +(** ** Comparisons *) + +Nondetfunction comp (c: comparison) (e1: expr) (e2: expr) := + match e1, e2 with + | Eop (Ointconst n1) Enil, t2 => + Eop (Ocmp (Ccompimm (swap_comparison c) n1)) (t2 ::: Enil) + | t1, Eop (Ointconst n2) Enil => + Eop (Ocmp (Ccompimm c n2)) (t1 ::: Enil) + | _, _ => + Eop (Ocmp (Ccomp c)) (e1 ::: e2 ::: Enil) + end. + +Nondetfunction compu (c: comparison) (e1: expr) (e2: expr) := + match e1, e2 with + | Eop (Ointconst n1) Enil, t2 => + Eop (Ocmp (Ccompuimm (swap_comparison c) n1)) (t2 ::: Enil) + | t1, Eop (Ointconst n2) Enil => + Eop (Ocmp (Ccompuimm c n2)) (t1 ::: Enil) + | _, _ => + Eop (Ocmp (Ccompu c)) (e1 ::: e2 ::: Enil) + end. + +Definition compf (c: comparison) (e1: expr) (e2: expr) := + Eop (Ocmp (Ccompf c)) (e1 ::: e2 ::: Enil). + +(** ** Integer conversions *) + +Definition cast8unsigned (e: expr) := andimm (Int.repr 255) e. + +Definition cast8signed (e: expr) := Eop Ocast8signed (e ::: Enil). + +Definition cast16unsigned (e: expr) := andimm (Int.repr 65535) e. + +Definition cast16signed (e: expr) := Eop Ocast16signed (e ::: Enil). + +(** ** Floating-point conversions *) + +Definition intoffloat (e: expr) := Eop Ointoffloat (e ::: Enil). + +Definition intuoffloat (e: expr) := + let f := Eop (Ofloatconst (Float.floatofintu Float.ox8000_0000)) Enil in + Elet e + (Econdition (CEcond (Ccompf Clt) (Eletvar O ::: f ::: Enil)) + (intoffloat (Eletvar O)) + (addimm Float.ox8000_0000 (intoffloat (subf (Eletvar O) f)))). + +Definition floatofintu (e: expr) := + subf (Eop Ofloatofwords (Eop (Ointconst Float.ox4330_0000) Enil ::: e ::: Enil)) + (Eop (Ofloatconst (Float.from_words Float.ox4330_0000 Int.zero)) Enil). + +Definition floatofint (e: expr) := + subf (Eop Ofloatofwords (Eop (Ointconst Float.ox4330_0000) Enil + ::: addimm Float.ox8000_0000 e ::: Enil)) + (Eop (Ofloatconst (Float.from_words Float.ox4330_0000 Float.ox8000_0000)) Enil). + +Definition singleoffloat (e: expr) := Eop Osingleoffloat (e ::: Enil). + +(** ** Recognition of addressing modes for load and store operations *) + +Nondetfunction addressing (chunk: memory_chunk) (e: expr) := + match e with + | Eop (Oaddrsymbol s n) Enil => (Aglobal s n, Enil) + | Eop (Oaddrstack n) Enil => (Ainstack n, Enil) + | Eop Oadd (Eop (Oaddrsymbol s n) Enil ::: e2 ::: Enil) => (Abased s n, e2:::Enil) + | Eop (Oaddimm n) (e1:::Enil) => (Aindexed n, e1:::Enil) + | Eop Oadd (e1:::e2:::Enil) => (Aindexed2, e1:::e2:::Enil) + | _ => (Aindexed Int.zero, e:::Enil) + end. + diff --git a/powerpc/SelectOpproof.v b/powerpc/SelectOpproof.v index b23e5a5..8ad9807 100644 --- a/powerpc/SelectOpproof.v +++ b/powerpc/SelectOpproof.v @@ -44,8 +44,6 @@ Variable m: mem. Ltac EvalOp := eapply eval_Eop; eauto with evalexpr. -Ltac TrivialOp cstr := unfold cstr; intros; EvalOp. - Ltac InvEval1 := match goal with | [ H: (eval_expr _ _ _ _ _ (Eop _ Enil) _) |- _ ] => @@ -78,14 +76,19 @@ Ltac InvEval2 := 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 *) (** 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 integer value [Vint n], then [SelectOp.notint e] - evaluates to a value [Vint (Int.not n)] which is indeed the integer - negation of the value of [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 @@ -95,405 +98,286 @@ Ltac InvEval := InvEval1; InvEval2; InvEval2. - 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] and [Float] modules. + 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 b, - Genv.find_symbol ge id = Some b -> - eval_expr ge sp e m le (addrsymbol id ofs) (Vptr b ofs). + forall le id ofs, + exists v, eval_expr ge sp e m le (addrsymbol id ofs) v /\ Val.lessdef (symbol_address ge id ofs) v. Proof. - intros. unfold addrsymbol. econstructor. constructor. - simpl. rewrite H. auto. + intros. unfold addrsymbol. econstructor; split. + EvalOp. simpl; eauto. + auto. Qed. Theorem eval_addrstack: - forall le ofs b n, - sp = Vptr b n -> - eval_expr ge sp e m le (addrstack ofs) (Vptr b (Int.add n ofs)). + forall le ofs, + exists v, eval_expr ge sp e m le (addrstack ofs) v /\ Val.lessdef (Val.add sp (Vint ofs)) v. Proof. - intros. unfold addrstack. econstructor. constructor. - simpl. unfold offset_sp. rewrite H. auto. + intros. unfold addrstack. econstructor; split. + EvalOp. simpl; eauto. + auto. Qed. -Theorem eval_notint: - forall le a x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (notint a) (Vint (Int.not x)). -Proof. - unfold notint; intros until x; case (notint_match a); intros; InvEval. - EvalOp. - EvalOp. simpl. congruence. - EvalOp. simpl. congruence. - EvalOp. simpl. congruence. +Theorem eval_notint: unary_constructor_sound notint Val.notint. +Proof. + unfold notint; red; intros until x; case (notint_match a); intros; InvEval. + TrivialExists. + subst. TrivialExists. + subst. TrivialExists. + subst. TrivialExists. + econstructor; split; eauto. eapply eval_Elet. eexact H. eapply eval_Eop. eapply eval_Econs. apply eval_Eletvar. simpl. reflexivity. eapply eval_Econs. apply eval_Eletvar. simpl. reflexivity. apply eval_Enil. - simpl. rewrite Int.or_idem. auto. -Qed. - -Lemma eval_notbool_base: - forall le a v b, - eval_expr ge sp e m le a v -> - Val.bool_of_val v b -> - eval_expr ge sp e m le (notbool_base a) (Val.of_bool (negb b)). -Proof. - TrivialOp notbool_base. simpl. - inv H0. - rewrite Int.eq_false; auto. - rewrite Int.eq_true; auto. - reflexivity. -Qed. - -Hint Resolve Val.bool_of_true_val Val.bool_of_false_val - Val.bool_of_true_val_inv Val.bool_of_false_val_inv: valboolof. - -Theorem eval_notbool: - forall le a v b, - eval_expr ge sp e m le a v -> - Val.bool_of_val v b -> - eval_expr ge sp e m le (notbool a) (Val.of_bool (negb b)). -Proof. - induction a; simpl; intros; try (eapply eval_notbool_base; eauto). - destruct o; try (eapply eval_notbool_base; eauto). - - destruct e0. InvEval. - inv H0. rewrite Int.eq_false; auto. - simpl; eauto with evalexpr. - rewrite Int.eq_true; simpl; eauto with evalexpr. - eapply eval_notbool_base; eauto. - - inv H. eapply eval_Eop; eauto. - simpl. assert (eval_condition c vl m = Some b). - generalize H6. simpl. - case (eval_condition c vl m); intros. - destruct b0; inv H1; inversion H0; auto; congruence. - congruence. - rewrite (Op.eval_negate_condition _ _ _ H). - destruct b; reflexivity. - - inv H. eapply eval_Econdition; eauto. - destruct v1; eauto. + simpl. destruct x; simpl; auto. rewrite Int.or_idem. auto. +Qed. + +Theorem eval_notbool: unary_constructor_sound notbool Val.notbool. +Proof. + assert (DFL: + forall le a x, + eval_expr ge sp e m le a x -> + exists v, eval_expr ge sp e m le (Eop (Ocmp (Ccompuimm Ceq Int.zero)) (a ::: Enil)) v + /\ Val.lessdef (Val.notbool x) v). + intros. TrivialExists. simpl. destruct x; simpl; auto. + + red. induction a; simpl; intros; eauto. destruct o; eauto. +(* intconst *) + destruct e0; eauto. InvEval. TrivialExists. simpl. destruct (Int.eq i Int.zero); auto. +(* cmp *) + inv H. simpl in H5. + destruct (eval_condition c vl m) as []_eqn. + TrivialExists. simpl. rewrite (eval_negate_condition _ _ _ Heqo). destruct b; inv H5; auto. + inv H5. simpl. + destruct (eval_condition (negate_condition c) vl m) as []_eqn. + destruct b; [exists Vtrue | exists Vfalse]; split; auto; EvalOp; simpl. rewrite Heqo0; auto. rewrite Heqo0; auto. + exists Vundef; split; auto; EvalOp; simpl. rewrite Heqo0; auto. +(* condition *) + inv H. destruct v1. + exploit IHa1; eauto. intros [v [A B]]. exists v; split; auto. eapply eval_Econdition; eauto. + exploit IHa2; eauto. intros [v [A B]]. exists v; split; auto. eapply eval_Econdition; eauto. Qed. Theorem eval_addimm: - forall le n a x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (addimm n a) (Vint (Int.add x n)). -Proof. - unfold addimm; intros until x. - generalize (Int.eq_spec n Int.zero). case (Int.eq n Int.zero); intro. - subst n. rewrite Int.add_zero. auto. - case (addimm_match a); intros; InvEval; EvalOp; simpl. + 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 Int.add_zero. auto. + case (addimm_match a); intros; InvEval; simpl; TrivialExists; simpl. rewrite Int.add_commut. auto. - destruct (Genv.find_symbol ge s); discriminate. - destruct sp; simpl in H1; discriminate. - subst x. rewrite Int.add_assoc. decEq; decEq; decEq. apply Int.add_commut. + unfold symbol_address. destruct (Genv.find_symbol ge s); simpl; auto. rewrite Int.add_commut; auto. + rewrite Val.add_assoc. rewrite Int.add_commut. auto. + subst x. rewrite Val.add_assoc. rewrite Int.add_commut. auto. Qed. -Theorem eval_addimm_ptr: - forall le n a b ofs, - eval_expr ge sp e m le a (Vptr b ofs) -> - eval_expr ge sp e m le (addimm n a) (Vptr b (Int.add ofs n)). -Proof. - unfold addimm; intros until ofs. - generalize (Int.eq_spec n Int.zero). case (Int.eq n Int.zero); intro. - subst n. rewrite Int.add_zero. auto. - case (addimm_match a); intros; InvEval; EvalOp; simpl. - destruct (Genv.find_symbol ge s). - rewrite Int.add_commut. congruence. - discriminate. - destruct sp; simpl in H1; try discriminate. - inv H1. simpl. decEq. decEq. - rewrite Int.add_assoc. decEq. apply Int.add_commut. - subst. rewrite (Int.add_commut n m0). rewrite Int.add_assoc. auto. -Qed. - -Theorem eval_add: - forall le a b x y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (add a b) (Vint (Int.add x y)). +Theorem eval_add: binary_constructor_sound add Val.add. Proof. - intros until y. + red; intros until y. unfold add; case (add_match a b); intros; InvEval. - rewrite Int.add_commut. apply eval_addimm. auto. - replace (Int.add x y) with (Int.add (Int.add i0 i) (Int.add n1 n2)). - apply eval_addimm. EvalOp. - subst x; subst y. - repeat rewrite Int.add_assoc. decEq. apply Int.add_permut. - replace (Int.add x y) with (Int.add (Int.add i y) n1). - apply eval_addimm. EvalOp. - subst x. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - apply eval_addimm. auto. - replace (Int.add x y) with (Int.add (Int.add x i) n2). - apply eval_addimm. EvalOp. - subst y. rewrite Int.add_assoc. auto. - destruct (Genv.find_symbol ge s); inv H0. - destruct sp; simpl in H0; inv H0. - EvalOp. -Qed. - -Theorem eval_add_ptr: - forall le a b p x y, - eval_expr ge sp e m le a (Vptr p x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (add a b) (Vptr p (Int.add x y)). -Proof. - intros until y. unfold add; case (add_match a b); intros; InvEval. - replace (Int.add x y) with (Int.add (Int.add i0 i) (Int.add n1 n2)). - apply eval_addimm_ptr. subst b0. EvalOp. - subst x; subst y. - repeat rewrite Int.add_assoc. decEq. apply Int.add_permut. - replace (Int.add x y) with (Int.add (Int.add i y) n1). - apply eval_addimm_ptr. subst b0. EvalOp. - subst x. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - apply eval_addimm_ptr. auto. - replace (Int.add x y) with (Int.add (Int.add x i) n2). - apply eval_addimm_ptr. EvalOp. - subst y. rewrite Int.add_assoc. auto. - revert H0. case_eq (Genv.find_symbol ge s); intros; inv H1. - EvalOp. constructor. EvalOp. simpl. rewrite H0; eauto. - constructor. eauto. constructor. - simpl. decEq. decEq. rewrite Int.add_assoc. decEq. apply Int.add_commut. - destruct sp; simpl in H0; inv H0. - EvalOp. constructor. EvalOp. simpl. eauto. constructor. eauto. constructor. - simpl. decEq. decEq. repeat rewrite Int.add_assoc. - decEq. decEq. apply Int.add_commut. - EvalOp. -Qed. - -Theorem eval_add_ptr_2: - forall le a b x p y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vptr p y) -> - eval_expr ge sp e m le (add a b) (Vptr p (Int.add y x)). -Proof. - intros until y. unfold add; case (add_match a b); intros; InvEval. - apply eval_addimm_ptr. auto. - replace (Int.add y x) with (Int.add (Int.add i i0) (Int.add n1 n2)). - apply eval_addimm_ptr. subst b0. EvalOp. - subst x; subst y. - repeat rewrite Int.add_assoc. decEq. - rewrite (Int.add_commut n1 n2). apply Int.add_permut. - replace (Int.add y x) with (Int.add (Int.add y i) n1). - apply eval_addimm_ptr. EvalOp. - subst x. repeat rewrite Int.add_assoc. auto. - replace (Int.add y x) with (Int.add (Int.add i x) n2). - apply eval_addimm_ptr. EvalOp. subst b0; reflexivity. - subst y. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - destruct (Genv.find_symbol ge s); inv H0. - destruct sp; simpl in H0; inv H0. - EvalOp. -Qed. - -Theorem eval_sub: - forall le a b x y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (sub a b) (Vint (Int.sub x y)). -Proof. - intros until y. + rewrite Val.add_commut. apply eval_addimm; 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_commut v1). rewrite <- Val.add_assoc. decEq; decEq. + unfold symbol_address. destruct (Genv.find_symbol ge s); auto. + subst. TrivialExists. + econstructor. EvalOp. simpl. reflexivity. econstructor. eauto. constructor. + simpl. repeat rewrite Val.add_assoc. decEq; decEq. + rewrite Val.add_commut. rewrite Val.add_permut. auto. + apply eval_addimm; auto. + subst. rewrite <- Val.add_assoc. apply eval_addimm. EvalOp. + 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 Int.sub_add_opp. - apply eval_addimm. assumption. - replace (Int.sub x y) with (Int.add (Int.sub i0 i) (Int.sub n1 n2)). - apply eval_addimm. EvalOp. - subst x; subst y. - repeat rewrite Int.sub_add_opp. - repeat rewrite Int.add_assoc. decEq. - rewrite Int.add_permut. decEq. symmetry. apply Int.neg_add_distr. - replace (Int.sub x y) with (Int.add (Int.sub i y) n1). - apply eval_addimm. EvalOp. - subst x. rewrite Int.sub_add_l. auto. - replace (Int.sub x y) with (Int.add (Int.sub x i) (Int.neg n2)). - apply eval_addimm. EvalOp. - subst y. rewrite (Int.add_commut i n2). symmetry. apply Int.sub_add_r. - EvalOp. + rewrite Val.sub_add_opp. apply eval_addimm; 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_sub_ptr_int: - forall le a b p x y, - eval_expr ge sp e m le a (Vptr p x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (sub a b) (Vptr p (Int.sub x y)). +Theorem eval_negint: unary_constructor_sound negint (fun v => Val.sub Vzero v). Proof. - intros until y. - unfold sub; case (sub_match a b); intros; InvEval. - rewrite Int.sub_add_opp. - apply eval_addimm_ptr. assumption. - subst b0. replace (Int.sub x y) with (Int.add (Int.sub i0 i) (Int.sub n1 n2)). - apply eval_addimm_ptr. EvalOp. - subst x; subst y. - repeat rewrite Int.sub_add_opp. - repeat rewrite Int.add_assoc. decEq. - rewrite Int.add_permut. decEq. symmetry. apply Int.neg_add_distr. - subst b0. replace (Int.sub x y) with (Int.add (Int.sub i y) n1). - apply eval_addimm_ptr. EvalOp. - subst x. rewrite Int.sub_add_l. auto. - replace (Int.sub x y) with (Int.add (Int.sub x i) (Int.neg n2)). - apply eval_addimm_ptr. EvalOp. - subst y. rewrite (Int.add_commut i n2). symmetry. apply Int.sub_add_r. - EvalOp. + red; intros. unfold negint. TrivialExists. Qed. -Theorem eval_sub_ptr_ptr: - forall le a b p x y, - eval_expr ge sp e m le a (Vptr p x) -> - eval_expr ge sp e m le b (Vptr p y) -> - eval_expr ge sp e m le (sub a b) (Vint (Int.sub x y)). +Lemma eval_rolm: + forall amount mask, + unary_constructor_sound (fun a => rolm a amount mask) + (fun x => Val.rolm x amount mask). Proof. - intros until y. - unfold sub; case (sub_match a b); intros; InvEval. - replace (Int.sub x y) with (Int.add (Int.sub i0 i) (Int.sub n1 n2)). - apply eval_addimm. EvalOp. - simpl; unfold eq_block. subst b0; subst b1; rewrite zeq_true. auto. - subst x; subst y. - repeat rewrite Int.sub_add_opp. - repeat rewrite Int.add_assoc. decEq. - rewrite Int.add_permut. decEq. symmetry. apply Int.neg_add_distr. - subst b0. replace (Int.sub x y) with (Int.add (Int.sub i y) n1). - apply eval_addimm. EvalOp. - simpl. unfold eq_block. rewrite zeq_true. auto. - subst x. rewrite Int.sub_add_l. auto. - subst b0. replace (Int.sub x y) with (Int.add (Int.sub x i) (Int.neg n2)). - apply eval_addimm. EvalOp. - simpl. unfold eq_block. rewrite zeq_true. auto. - subst y. rewrite (Int.add_commut i n2). symmetry. apply Int.sub_add_r. - EvalOp. simpl. unfold eq_block. rewrite zeq_true. auto. + 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. -Lemma eval_rolm: - forall le a amount mask x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (rolm a amount mask) (Vint (Int.rolm x amount mask)). -Proof. - intros until x. unfold rolm; case (rolm_match a); intros; InvEval. - eauto with evalexpr. - case (is_rlw_mask (Int.and (Int.rol mask1 amount) mask)). - EvalOp. simpl. subst x. - decEq. decEq. - symmetry. apply Int.rolm_rolm. apply int_wordsize_divides_modulus. - EvalOp. econstructor. EvalOp. simpl. rewrite H. reflexivity. constructor. auto. - EvalOp. +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) as []_eqn. + rewrite Val.shl_rolm; auto. apply eval_rolm; auto. + TrivialExists. econstructor. eauto. econstructor. EvalOp. simpl; eauto. constructor. auto. Qed. -Theorem eval_shlimm: - forall le a n x, - eval_expr ge sp e m le a (Vint x) -> - Int.ltu n Int.iwordsize = true -> - eval_expr ge sp e m le (shlimm a n) (Vint (Int.shl x n)). +Theorem eval_shrimm: + forall n, unary_constructor_sound (fun a => shrimm a n) + (fun x => Val.shr x (Vint n)). Proof. - intros. unfold shlimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intro. - subst n. rewrite Int.shl_zero. auto. - rewrite H0. - replace (Int.shl x n) with (Int.rolm x n (Int.shl Int.mone n)). - apply eval_rolm. auto. symmetry. apply Int.shl_rolm. exact H0. + red; intros. unfold shrimm. + predSpec Int.eq Int.eq_spec n Int.zero. + subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.shr_zero; auto. + TrivialExists. Qed. Theorem eval_shruimm: - forall le a n x, - eval_expr ge sp e m le a (Vint x) -> - Int.ltu n Int.iwordsize = true -> - eval_expr ge sp e m le (shruimm a n) (Vint (Int.shru x n)). + forall n, unary_constructor_sound (fun a => shruimm a n) + (fun x => Val.shru x (Vint n)). Proof. - intros. unfold shruimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intro. - subst n. rewrite Int.shru_zero. auto. - rewrite H0. - replace (Int.shru x n) with (Int.rolm x (Int.sub Int.iwordsize n) (Int.shru Int.mone n)). - apply eval_rolm. auto. symmetry. apply Int.shru_rolm. exact H0. + 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) as []_eqn. + rewrite Val.shru_rolm; auto. apply eval_rolm; auto. + TrivialExists. econstructor. eauto. econstructor. EvalOp. simpl; eauto. constructor. auto. Qed. Lemma eval_mulimm_base: - forall le a n x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (mulimm_base n a) (Vint (Int.mul x n)). + forall n, unary_constructor_sound (mulimm_base n) (fun x => Val.mul x (Vint n)). Proof. - intros; unfold mulimm_base. + intros; red; intros; unfold mulimm_base. generalize (Int.one_bits_decomp n). generalize (Int.one_bits_range n). - change (Z_of_nat Int.wordsize) with 32. destruct (Int.one_bits n). - intros. EvalOp. + intros. TrivialExists. destruct l. intros. rewrite H1. simpl. - rewrite Int.add_zero. rewrite <- Int.shl_mul. - apply eval_shlimm. auto. auto with coqlib. + 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. apply eval_Elet with (Vint x). auto. - rewrite H1. simpl. rewrite Int.add_zero. - rewrite Int.mul_add_distr_r. - rewrite <- Int.shl_mul. - rewrite <- Int.shl_mul. - EvalOp. eapply eval_Econs. - apply eval_shlimm. apply eval_Eletvar. simpl. reflexivity. - auto with coqlib. - eapply eval_Econs. - apply eval_shlimm. apply eval_Eletvar. simpl. reflexivity. - auto with coqlib. - auto with evalexpr. - reflexivity. - intros. EvalOp. + intros. 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 le a n x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (mulimm n a) (Vint (Int.mul x n)). -Proof. - intros until x; unfold mulimm. - generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intro. - subst n. rewrite Int.mul_zero. - intro. eapply eval_Elet; eauto with evalexpr. - generalize (Int.eq_spec n Int.one); case (Int.eq n Int.one); intro. - subst n. rewrite Int.mul_one. auto. + 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. - EvalOp. rewrite Int.mul_commut. reflexivity. - replace (Int.mul x n) with (Int.add (Int.mul i n) (Int.mul n n2)). - apply eval_addimm. apply eval_mulimm_base. auto. - subst x. rewrite Int.mul_add_distr_l. decEq. apply Int.mul_commut. - apply eval_mulimm_base. assumption. + 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: - forall le a b x y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (mul a b) (Vint (Int.mul x y)). +Theorem eval_mul: binary_constructor_sound mul Val.mul. Proof. - intros until y. + red; intros until y. unfold mul; case (mul_match a b); intros; InvEval. - rewrite Int.mul_commut. apply eval_mulimm. auto. + rewrite Val.mul_commut. apply eval_mulimm. auto. apply eval_mulimm. auto. - EvalOp. + TrivialExists. Qed. Theorem eval_andimm: - forall le n a x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (andimm n a) (Vint (Int.and x n)). -Proof. - intros. unfold andimm. case (is_rlw_mask n). - rewrite <- Int.rolm_zero. apply eval_rolm; auto. - EvalOp. + forall n, unary_constructor_sound (andimm n) (fun x => Val.and x (Vint n)). +Proof. + intros; red; intros until x. unfold andimm. case (andimm_match a); intros. + InvEval. TrivialExists. simpl. rewrite Int.and_commut; auto. + 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) as []_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. + TrivialExists. Qed. -Theorem eval_and: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (and a b) (Vint (Int.and x y)). +Theorem eval_orimm: + forall n, unary_constructor_sound (orimm n) (fun x => Val.or x (Vint n)). Proof. - intros until y; unfold and; case (mul_match a b); intros; InvEval. - rewrite Int.and_commut. apply eval_andimm; auto. - apply eval_andimm; auto. - EvalOp. + intros; red; intros until x. + unfold orimm. 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: @@ -511,59 +395,71 @@ Proof. discriminate. Qed. -Lemma eval_or: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (or a b) (Vint (Int.or x y)). -Proof. - intros until y; unfold or; case (or_match a b); intros; InvEval. - caseEq (Int.eq amount1 amount2 - && is_rlw_mask (Int.or mask1 mask2) - && same_expr_pure t1 t2); intro. - destruct (andb_prop _ _ H1). destruct (andb_prop _ _ H4). - generalize (Int.eq_spec amount1 amount2). rewrite H6. intro. subst amount2. - exploit eval_same_expr; eauto. intros [EQ1 EQ2]. inv EQ1. inv EQ2. - simpl. EvalOp. simpl. rewrite Int.or_rolm. auto. - caseEq (Int.eq amount1 Int.zero && Int.eq mask1 (Int.not mask2)); intro. - destruct (andb_prop _ _ H4). - generalize (Int.eq_spec amount1 Int.zero). rewrite H5. intro. - generalize (Int.eq_spec mask1 (Int.not mask2)). rewrite H6. intro. - subst. rewrite Int.rolm_zero. EvalOp. - caseEq (Int.eq amount2 Int.zero && Int.eq mask2 (Int.not mask1)); intro. - destruct (andb_prop _ _ H5). - generalize (Int.eq_spec amount2 Int.zero). rewrite H6. intro. - generalize (Int.eq_spec mask2 (Int.not mask1)). rewrite H7. intro. - subst. rewrite Int.rolm_zero. rewrite Int.or_commut. EvalOp. - simpl. apply eval_Eop with (Vint x :: Vint y :: nil). - econstructor. EvalOp. simpl. congruence. - econstructor. EvalOp. simpl. congruence. constructor. auto. - EvalOp. +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) as []_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) as []_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) as []_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. +(* 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. destruct (xorimm_match a); intros; InvEval. + TrivialExists. simpl. rewrite Int.xor_commut; auto. + subst. rewrite Val.xor_assoc. simpl. rewrite Int.xor_commut. TrivialExists. + 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. + TrivialExists. Qed. Theorem eval_divs: - forall le a b x y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - y <> Int.zero -> - eval_expr ge sp e m le (divs a b) (Vint (Int.divs 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 -> + Val.divs x y = Some z -> + exists v, eval_expr ge sp e m le (divs a b) v /\ Val.lessdef z v. Proof. - TrivialOp divs. simpl. - predSpec Int.eq Int.eq_spec y Int.zero. contradiction. auto. + intros. unfold divs. exists z; split. EvalOp. auto. Qed. Lemma eval_mod_aux: forall divop semdivop, - (forall sp x y m, - y <> Int.zero -> - eval_operation ge sp divop (Vint x :: Vint y :: nil) m = - Some (Vint (semdivop x y))) -> - forall le a b x y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - y <> Int.zero -> - eval_expr ge sp e m le (mod_aux divop a b) - (Vint (Int.sub x (Int.mul (semdivop x y) y))). + (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. @@ -575,7 +471,7 @@ Proof. eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity. eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity. apply eval_Enil. - apply H. assumption. + rewrite H. eauto. eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity. apply eval_Enil. simpl; reflexivity. apply eval_Enil. @@ -583,374 +479,273 @@ Proof. Qed. Theorem eval_mods: - forall le a b x y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - y <> Int.zero -> - eval_expr ge sp e m le (mods a b) (Vint (Int.mods 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 -> + Val.mods x y = Some z -> + exists v, eval_expr ge sp e m le (mods a b) v /\ Val.lessdef z v. Proof. intros; unfold mods. - rewrite Int.mods_divs. - eapply eval_mod_aux; eauto. - intros. simpl. predSpec Int.eq Int.eq_spec y0 Int.zero. - contradiction. auto. + exploit Val.mods_divs; eauto. intros [v [A B]]. + subst. econstructor; split; eauto. + apply eval_mod_aux with (semdivop := Val.divs); auto. Qed. -Lemma eval_divu_base: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - y <> Int.zero -> - eval_expr ge sp e m le (Eop Odivu (a ::: b ::: Enil)) (Vint (Int.divu x y)). +Theorem eval_divuimm: + forall le n a x z, + eval_expr ge sp e m le a x -> + Val.divu x (Vint n) = Some z -> + exists v, eval_expr ge sp e m le (divuimm a n) v /\ Val.lessdef z v. Proof. - intros. EvalOp. simpl. - predSpec Int.eq Int.eq_spec y Int.zero. contradiction. auto. + intros; unfold divuimm. + destruct (Int.is_power2 n) as []_eqn. + replace z with (Val.shru x (Vint i)). apply eval_shruimm; auto. + eapply Val.divu_pow2; eauto. + TrivialExists. + econstructor. eauto. econstructor. EvalOp. simpl; eauto. constructor. auto. Qed. Theorem eval_divu: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - y <> Int.zero -> - eval_expr ge sp e m le (divu a b) (Vint (Int.divu x y)). -Proof. - intros until y. - unfold divu; case (divu_match b); intros; InvEval. - caseEq (Int.is_power2 y). - intros. rewrite (Int.divu_pow2 x y i H0). - apply eval_shruimm. auto. - apply Int.is_power2_range with y. auto. - intros. apply eval_divu_base. auto. EvalOp. auto. - eapply eval_divu_base; eauto. + forall le a x b 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 a b) v /\ Val.lessdef z v. +Proof. + intros until z. unfold divu; destruct (divu_match b); intros; InvEval. + eapply eval_divuimm; eauto. + TrivialExists. +Qed. + +Theorem eval_moduimm: + forall le n a x z, + eval_expr ge sp e m le a x -> + Val.modu x (Vint n) = Some z -> + exists v, eval_expr ge sp e m le (moduimm a n) v /\ Val.lessdef z v. +Proof. + intros; unfold moduimm. + destruct (Int.is_power2 n) as []_eqn. + replace z with (Val.and x (Vint (Int.sub n Int.one))). apply eval_andimm; auto. + eapply Val.modu_pow2; eauto. + exploit Val.modu_divu; eauto. intros [v [A B]]. + subst. econstructor; split; eauto. + apply eval_mod_aux with (semdivop := Val.divu); auto. + EvalOp. Qed. Theorem eval_modu: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - y <> Int.zero -> - eval_expr ge sp e m le (modu a b) (Vint (Int.modu x y)). + forall le a x b 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 a b) v /\ Val.lessdef z v. Proof. - intros until y; unfold modu; case (divu_match b); intros; InvEval. - caseEq (Int.is_power2 y). - intros. rewrite (Int.modu_and x y i H0). apply eval_andimm. auto. - intro. rewrite Int.modu_divu. eapply eval_mod_aux. - intros. simpl. predSpec Int.eq Int.eq_spec y0 Int.zero. - contradiction. auto. - auto. EvalOp. auto. auto. - rewrite Int.modu_divu. eapply eval_mod_aux. - intros. simpl. predSpec Int.eq Int.eq_spec y0 Int.zero. - contradiction. auto. auto. auto. auto. auto. + intros until y; unfold modu; case (modu_match b); intros; InvEval. + eapply eval_moduimm; eauto. + 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_shl: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - Int.ltu y Int.iwordsize = true -> - eval_expr ge sp e m le (shl a b) (Vint (Int.shl x y)). +Theorem eval_shl: binary_constructor_sound shl Val.shl. Proof. - intros until y; unfold shl; case (shift_match b); intros. + red; intros until y; unfold shl; case (shl_match b); intros. InvEval. apply eval_shlimm; auto. - EvalOp. simpl. rewrite H1. auto. + TrivialExists. Qed. -Theorem eval_shru: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - Int.ltu y Int.iwordsize = true -> - eval_expr ge sp e m le (shru a b) (Vint (Int.shru x y)). +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. - intros until y; unfold shru; case (shift_match b); intros. + red; intros until y; unfold shru; case (shru_match b); intros. InvEval. apply eval_shruimm; auto. - EvalOp. simpl. rewrite H1. auto. + TrivialExists. Qed. -Theorem eval_addf: - forall le a x b y, - eval_expr ge sp e m le a (Vfloat x) -> - eval_expr ge sp e m le b (Vfloat y) -> - eval_expr ge sp e m le (addf a b) (Vfloat (Float.add x y)). +Theorem eval_negf: unary_constructor_sound negf Val.negf. Proof. - intros until y; unfold addf. - destruct (use_fused_mul tt). + 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 until y; unfold addf. + destruct (use_fused_mul tt); simpl. case (addf_match a b); intros; InvEval. - EvalOp. simpl. congruence. - EvalOp. simpl. rewrite Float.addf_commut. congruence. - EvalOp. - intros. EvalOp. + TrivialExists. simpl. congruence. + TrivialExists. simpl. rewrite Val.addf_commut. congruence. + intros. TrivialExists. + intros. TrivialExists. Qed. -Theorem eval_subf: - forall le a x b y, - eval_expr ge sp e m le a (Vfloat x) -> - eval_expr ge sp e m le b (Vfloat y) -> - eval_expr ge sp e m le (subf a b) (Vfloat (Float.sub x y)). -Proof. - intros until y; unfold subf. - destruct (use_fused_mul tt). - case (subf_match a b); intros. - InvEval. EvalOp. simpl. congruence. - EvalOp. - intros. EvalOp. +Theorem eval_subf: binary_constructor_sound subf Val.subf. +Proof. + red; intros until y; unfold subf. + destruct (use_fused_mul tt); simpl. + case (subf_match a); intros; InvEval. + TrivialExists. simpl. congruence. + TrivialExists. + intros. TrivialExists. Qed. -Theorem eval_cast8signed: - forall le a v, - eval_expr ge sp e m le a v -> - eval_expr ge sp e m le (cast8signed a) (Val.sign_ext 8 v). -Proof. TrivialOp cast8signed. Qed. +Theorem eval_mulf: binary_constructor_sound mulf Val.mulf. +Proof. + red; intros; TrivialExists. +Qed. -Theorem eval_cast8unsigned: - forall le a v, - eval_expr ge sp e m le a v -> - eval_expr ge sp e m le (cast8unsigned a) (Val.zero_ext 8 v). -Proof. TrivialOp cast8unsigned. Qed. +Theorem eval_divf: binary_constructor_sound divf Val.divf. +Proof. + red; intros; TrivialExists. +Qed. -Theorem eval_cast16signed: - forall le a v, - eval_expr ge sp e m le a v -> - eval_expr ge sp e m le (cast16signed a) (Val.sign_ext 16 v). -Proof. TrivialOp cast16signed. Qed. +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. + TrivialExists. simpl. rewrite Val.swap_cmp_bool. auto. + TrivialExists. + TrivialExists. +Qed. -Theorem eval_cast16unsigned: - forall le a v, - eval_expr ge sp e m le a v -> - eval_expr ge sp e m le (cast16unsigned a) (Val.zero_ext 16 v). -Proof. TrivialOp cast16unsigned. 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. + TrivialExists. simpl. rewrite Val.swap_cmpu_bool. auto. + TrivialExists. + TrivialExists. +Qed. -Theorem eval_singleoffloat: - forall le a v, - eval_expr ge sp e m le a v -> - eval_expr ge sp e m le (singleoffloat a) (Val.singleoffloat v). -Proof. TrivialOp singleoffloat. Qed. +Theorem eval_compf: + forall c, binary_constructor_sound (compf c) (Val.cmpf c). +Proof. + intros; red; intros. unfold compf. TrivialExists. +Qed. -Theorem eval_comp: - forall le c a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (comp c a b) (Val.of_bool(Int.cmp c x y)). -Proof. - intros until y. - unfold comp; case (comp_match a b); intros; InvEval. - EvalOp. simpl. rewrite Int.swap_cmp. destruct (Int.cmp c x y); reflexivity. - EvalOp. simpl. destruct (Int.cmp c x y); reflexivity. - EvalOp. simpl. destruct (Int.cmp c x y); reflexivity. -Qed. - -Theorem eval_compu_int: - forall le c a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (compu c a b) (Val.of_bool(Int.cmpu c x y)). -Proof. - intros until y. - unfold compu; case (comp_match a b); intros; InvEval. - EvalOp. simpl. rewrite Int.swap_cmpu. destruct (Int.cmpu c x y); reflexivity. - EvalOp. simpl. destruct (Int.cmpu c x y); reflexivity. - EvalOp. simpl. destruct (Int.cmpu c x y); reflexivity. -Qed. - -Remark eval_compare_null_transf: - forall c x v, - Cminor.eval_compare_null c x = Some v -> - match eval_compare_null c x with - | Some true => Some Vtrue - | Some false => Some Vfalse - | None => None (A:=val) - end = Some v. -Proof. - unfold Cminor.eval_compare_null, eval_compare_null; intros. - destruct (Int.eq x Int.zero); try discriminate. - destruct c; try discriminate; auto. -Qed. - -Theorem eval_compu_ptr_int: - forall le c a x1 x2 b y v, - eval_expr ge sp e m le a (Vptr x1 x2) -> - eval_expr ge sp e m le b (Vint y) -> - Cminor.eval_compare_null c y = Some v -> - eval_expr ge sp e m le (compu c a b) v. -Proof. - intros until v. - unfold compu; case (comp_match a b); intros; InvEval. - EvalOp. simpl. apply eval_compare_null_transf; auto. - EvalOp. simpl. apply eval_compare_null_transf; auto. -Qed. - -Remark eval_compare_null_swap: - forall c x, - Cminor.eval_compare_null (swap_comparison c) x = - Cminor.eval_compare_null c x. -Proof. - intros. unfold Cminor.eval_compare_null. - destruct (Int.eq x Int.zero). destruct c; auto. auto. -Qed. - -Theorem eval_compu_int_ptr: - forall le c a x b y1 y2 v, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vptr y1 y2) -> - Cminor.eval_compare_null c x = Some v -> - eval_expr ge sp e m le (compu c a b) v. -Proof. - intros until v. - unfold compu; case (comp_match a b); intros; InvEval. - EvalOp. simpl. apply eval_compare_null_transf. - rewrite eval_compare_null_swap; auto. - EvalOp. simpl. apply eval_compare_null_transf. auto. -Qed. - -Theorem eval_compu_ptr_ptr: - forall le c a x1 x2 b y1 y2, - eval_expr ge sp e m le a (Vptr x1 x2) -> - eval_expr ge sp e m le b (Vptr y1 y2) -> - Mem.valid_pointer m x1 (Int.unsigned x2) - && Mem.valid_pointer m y1 (Int.unsigned y2) = true -> - x1 = y1 -> - eval_expr ge sp e m le (compu c a b) (Val.of_bool(Int.cmpu c x2 y2)). -Proof. - intros until y2. - unfold compu; case (comp_match a b); intros; InvEval. - EvalOp. simpl. rewrite H1. subst y1. rewrite dec_eq_true. - destruct (Int.cmpu c x2 y2); reflexivity. -Qed. - -Theorem eval_compu_ptr_ptr_2: - forall le c a x1 x2 b y1 y2 v, - eval_expr ge sp e m le a (Vptr x1 x2) -> - eval_expr ge sp e m le b (Vptr y1 y2) -> - Mem.valid_pointer m x1 (Int.unsigned x2) - && Mem.valid_pointer m y1 (Int.unsigned y2) = true -> - x1 <> y1 -> - Cminor.eval_compare_mismatch c = Some v -> - eval_expr ge sp e m le (compu c a b) v. -Proof. - intros until y2. - unfold compu; case (comp_match a b); intros; InvEval. - EvalOp. simpl. rewrite H1. rewrite dec_eq_false; auto. - destruct c; simpl in H3; inv H3; auto. + +Theorem eval_cast8signed: unary_constructor_sound cast8signed (Val.sign_ext 8). +Proof. + red; intros. unfold cast8signed. TrivialExists. Qed. -Theorem eval_compf: - forall le c a x b y, - eval_expr ge sp e m le a (Vfloat x) -> - eval_expr ge sp e m le b (Vfloat y) -> - eval_expr ge sp e m le (compf c a b) (Val.of_bool(Float.cmp c x y)). +Theorem eval_cast8unsigned: unary_constructor_sound cast8unsigned (Val.zero_ext 8). Proof. - intros. unfold compf. EvalOp. simpl. - destruct (Float.cmp c x y); reflexivity. + red; intros. unfold cast8unsigned. + rewrite Val.zero_ext_and. apply eval_andimm; auto. compute; auto. Qed. -Theorem eval_negint: - forall le a x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (negint a) (Vint (Int.neg x)). -Proof. intros; unfold negint; EvalOp. Qed. +Theorem eval_cast16signed: unary_constructor_sound cast16signed (Val.sign_ext 16). +Proof. + red; intros. unfold cast16signed. TrivialExists. +Qed. -Theorem eval_negf: - forall le a x, - eval_expr ge sp e m le a (Vfloat x) -> - eval_expr ge sp e m le (negf a) (Vfloat (Float.neg x)). -Proof. intros; unfold negf; EvalOp. 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. compute; auto. +Qed. -Theorem eval_absf: - forall le a x, - eval_expr ge sp e m le a (Vfloat x) -> - eval_expr ge sp e m le (absf a) (Vfloat (Float.abs x)). -Proof. intros; unfold absf; EvalOp. Qed. +Theorem eval_singleoffloat: unary_constructor_sound singleoffloat Val.singleoffloat. +Proof. + red; intros. unfold singleoffloat. TrivialExists. +Qed. Theorem eval_intoffloat: - forall le a x n, - eval_expr ge sp e m le a (Vfloat x) -> - Float.intoffloat x = Some n -> - eval_expr ge sp e m le (intoffloat a) (Vint n). + 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; EvalOp. simpl. rewrite H0; auto. + intros; unfold intoffloat. TrivialExists. Qed. Theorem eval_intuoffloat: - forall le a x n, - eval_expr ge sp e m le a (Vfloat x) -> - Float.intuoffloat x = Some n -> - eval_expr ge sp e m le (intuoffloat a) (Vint n). -Proof. - intros. unfold intuoffloat. - econstructor. eauto. + 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.intuoffloat f) as [n|]_eqn; simpl in H0; inv H0. + exists (Vint n); split; auto. unfold intuoffloat. set (im := Int.repr Int.half_modulus). set (fm := Float.floatofintu im). - assert (eval_expr ge sp e m (Vfloat x :: le) (Eletvar O) (Vfloat x)). + assert (eval_expr ge sp e m (Vfloat f :: le) (Eletvar O) (Vfloat f)). constructor. auto. - apply eval_Econdition with (v1 := Float.cmp Clt x fm). + econstructor. eauto. + apply eval_Econdition with (v1 := Float.cmp Clt f fm). econstructor. constructor. eauto. constructor. EvalOp. simpl; eauto. constructor. simpl. auto. - caseEq (Float.cmp Clt x fm); intros. + destruct (Float.cmp Clt f fm) as []_eqn. exploit Float.intuoffloat_intoffloat_1; eauto. intro EQ. EvalOp. simpl. rewrite EQ; auto. exploit Float.intuoffloat_intoffloat_2; eauto. intro EQ. - replace n with (Int.add (Int.sub n Float.ox8000_0000) Float.ox8000_0000). - apply eval_addimm. eapply eval_intoffloat; eauto. - apply eval_subf; auto. EvalOp. - rewrite Int.sub_add_opp. rewrite Int.add_assoc. apply Int.add_zero. + set (t1 := Eop (Ofloatconst (Float.floatofintu Float.ox8000_0000)) Enil). + set (t2 := subf (Eletvar 0) t1). + set (t3 := intoffloat t2). + exploit (eval_subf (Vfloat f :: le) (Eletvar 0) (Vfloat f) t1). + auto. unfold t1; EvalOp. simpl; eauto. + fold t2. intros [v2 [A2 B2]]. simpl in B2. inv B2. + exploit (eval_addimm Float.ox8000_0000 (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 Float.ox8000_0000)) in A4. + rewrite Int.add_neg_zero in A4. + rewrite Int.add_zero in A4. + auto. Qed. Theorem eval_floatofint: - forall le a x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (floatofint a) (Vfloat (Float.floatofint x)). -Proof. - intros. unfold floatofint. rewrite Float.floatofint_from_words. - apply eval_subf. - EvalOp. constructor. EvalOp. simpl; eauto. - constructor. apply eval_addimm. eauto. constructor. - simpl. auto. - EvalOp. + 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. destruct x; simpl in H0; inv H0. + exists (Vfloat (Float.floatofint i)); split; auto. + unfold floatofint. + 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.floatofint_from_words. auto. Qed. Theorem eval_floatofintu: - forall le a x, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le (floatofintu a) (Vfloat (Float.floatofintu x)). -Proof. - intros. unfold floatofintu. rewrite Float.floatofintu_from_words. - apply eval_subf. - EvalOp. constructor. EvalOp. simpl; eauto. - constructor. eauto. constructor. - simpl. auto. - EvalOp. + 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. destruct x; simpl in H0; inv H0. + exists (Vfloat (Float.floatofintu i)); split; auto. + unfold floatofintu. + 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.floatofintu_from_words. auto. Qed. -Theorem eval_xor: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - eval_expr ge sp e m le (xor a b) (Vint (Int.xor x y)). -Proof. intros; unfold xor; EvalOp. Qed. - -Theorem eval_shr: - forall le a x b y, - eval_expr ge sp e m le a (Vint x) -> - eval_expr ge sp e m le b (Vint y) -> - Int.ltu y Int.iwordsize = true -> - eval_expr ge sp e m le (shr a b) (Vint (Int.shr x y)). -Proof. intros; unfold shr; EvalOp. simpl. rewrite H1. auto. Qed. - -Theorem eval_mulf: - forall le a x b y, - eval_expr ge sp e m le a (Vfloat x) -> - eval_expr ge sp e m le b (Vfloat y) -> - eval_expr ge sp e m le (mulf a b) (Vfloat (Float.mul x y)). -Proof. intros; unfold mulf; EvalOp. Qed. - -Theorem eval_divf: - forall le a x b y, - eval_expr ge sp e m le a (Vfloat x) -> - eval_expr ge sp e m le b (Vfloat y) -> - eval_expr ge sp e m le (divf a b) (Vfloat (Float.div x y)). -Proof. intros; unfold divf; EvalOp. Qed. - Theorem eval_addressing: forall le chunk a v b ofs, eval_expr ge sp e m le a v -> @@ -964,18 +759,11 @@ Proof. intros until v. unfold addressing; case (addressing_match a); intros; InvEval. exists (@nil val). split. eauto with evalexpr. simpl. auto. exists (@nil val). split. eauto with evalexpr. simpl. auto. - destruct (Genv.find_symbol ge s); congruence. - exists (Vint i0 :: nil). split. eauto with evalexpr. - simpl. destruct (Genv.find_symbol ge s). congruence. discriminate. - exists (Vptr b0 i :: nil). split. eauto with evalexpr. - simpl. congruence. - exists (Vint i :: Vptr b0 i0 :: nil). - split. eauto with evalexpr. simpl. - congruence. - exists (Vptr b0 i :: Vint i0 :: nil). - split. eauto with evalexpr. simpl. congruence. - exists (v :: nil). split. eauto with evalexpr. - subst v. simpl. rewrite Int.add_zero. auto. + exists (v0 :: nil). split. eauto with evalexpr. simpl. congruence. + exists (v1 :: nil). split. eauto with evalexpr. simpl. congruence. + exists (v1 :: v0 :: nil). split. eauto with evalexpr. simpl. congruence. + exists (v :: nil). split. eauto with evalexpr. subst v. simpl. + rewrite Int.add_zero. auto. Qed. End CMCONSTR. |