diff options
Diffstat (limited to 'common/Values.v')
| -rw-r--r-- | common/Values.v | 162 |
1 files changed, 159 insertions, 3 deletions
diff --git a/common/Values.v b/common/Values.v index 59e6388f..6401ba52 100644 --- a/common/Values.v +++ b/common/Values.v @@ -149,6 +149,23 @@ Proof. auto. Defined. +Definition has_rettype (v: val) (r: rettype) : Prop := + match r, v with + | Tret t, _ => has_type v t + | Tint8signed, Vint n => n = Int.sign_ext 8 n + | Tint8unsigned, Vint n => n = Int.zero_ext 8 n + | Tint16signed, Vint n => n = Int.sign_ext 16 n + | Tint16unsigned, Vint n => n = Int.zero_ext 16 n + | _, Vundef => True + | _, _ => False + end. + +Lemma has_proj_rettype: forall v r, + has_rettype v r -> has_type v (proj_rettype r). +Proof. + destruct r; simpl; intros; auto; destruct v; try contradiction; exact I. +Qed. + (** Truth values. Non-zero integers are treated as [True]. The integer 0 (also used to represent the null pointer) is [False]. Other values are neither true nor false. *) @@ -783,6 +800,17 @@ Definition rolml (v: val) (amount: int) (mask: int64): val := | _ => Vundef end. +Definition zero_ext_l (nbits: Z) (v: val) : val := + match v with + | Vlong n => Vlong(Int64.zero_ext nbits n) + | _ => Vundef + end. + +Definition sign_ext_l (nbits: Z) (v: val) : val := + match v with + | Vlong n => Vlong(Int64.sign_ext nbits n) + | _ => Vundef + end. (** Comparisons *) @@ -992,10 +1020,24 @@ Definition load_result (chunk: memory_chunk) (v: val) := | _, _ => Vundef end. +Lemma load_result_rettype: + forall chunk v, has_rettype (load_result chunk v) (rettype_of_chunk chunk). +Proof. + intros. unfold has_rettype; destruct chunk; destruct v; simpl; auto. +- rewrite Int.sign_ext_idem by omega; auto. +- rewrite Int.zero_ext_idem by omega; auto. +- rewrite Int.sign_ext_idem by omega; auto. +- rewrite Int.zero_ext_idem by omega; auto. +- destruct Archi.ptr64 eqn:SF; simpl; auto. +- destruct Archi.ptr64 eqn:SF; simpl; auto. +- destruct Archi.ptr64 eqn:SF; simpl; auto. +Qed. + Lemma load_result_type: forall chunk v, has_type (load_result chunk v) (type_of_chunk chunk). Proof. - intros. unfold has_type; destruct chunk; destruct v; simpl; auto; destruct Archi.ptr64 eqn:SF; simpl; auto. + intros. rewrite <- proj_rettype_of_chunk. apply has_proj_rettype. + apply load_result_rettype. Qed. Lemma load_result_same: @@ -1428,6 +1470,60 @@ Proof. assert (32 < Int.max_unsigned) by reflexivity. omega. Qed. +Theorem shrx1_shr: + forall x z, + shrx x (Vint (Int.repr 1)) = Some z -> + z = shr (add x (shru x (Vint (Int.repr 31)))) (Vint (Int.repr 1)). +Proof. + intros. destruct x; simpl in H; try discriminate. + change (Int.ltu (Int.repr 1) (Int.repr 31)) with true in H; simpl in H. + inversion_clear H. + simpl. + change (Int.ltu (Int.repr 31) Int.iwordsize) with true; simpl. + change (Int.ltu (Int.repr 1) Int.iwordsize) with true; simpl. + f_equal. + rewrite Int.shrx1_shr by reflexivity. + reflexivity. +Qed. + +Theorem shrx_shr_3: + forall n x z, + shrx x (Vint n) = Some z -> + z = (if Int.eq n Int.zero then x else + if Int.eq n Int.one + then shr (add x (shru x (Vint (Int.repr 31)))) (Vint Int.one) + else shr (add x (shru (shr x (Vint (Int.repr 31))) + (Vint (Int.sub (Int.repr 32) n)))) + (Vint n)). +Proof. + intros. destruct x; simpl in H; try discriminate. + destruct (Int.ltu n (Int.repr 31)) eqn:LT; inv H. + exploit Int.ltu_inv; eauto. change (Int.unsigned (Int.repr 31)) with 31; intros LT'. + predSpec Int.eq Int.eq_spec n Int.zero. +- subst n. unfold Int.shrx. rewrite Int.shl_zero. unfold Int.divs. change (Int.signed Int.one) with 1. + rewrite Z.quot_1_r. rewrite Int.repr_signed; auto. +- predSpec Int.eq Int.eq_spec n Int.one. + * subst n. simpl. + change (Int.ltu (Int.repr 31) Int.iwordsize) with true. simpl. + change (Int.ltu Int.one Int.iwordsize) with true. simpl. + f_equal. + apply Int.shrx1_shr. + reflexivity. + * clear H0. + simpl. change (Int.ltu (Int.repr 31) Int.iwordsize) with true. simpl. + replace (Int.ltu (Int.sub (Int.repr 32) n) Int.iwordsize) with true. simpl. + replace (Int.ltu n Int.iwordsize) with true. + f_equal; apply Int.shrx_shr_2; assumption. + symmetry; apply zlt_true. change (Int.unsigned n < 32); omega. + symmetry; apply zlt_true. unfold Int.sub. change (Int.unsigned (Int.repr 32)) with 32. + assert (Int.unsigned n <> 0). + { red; intros; elim H. + rewrite <- (Int.repr_unsigned n), H0. auto. } + rewrite Int.unsigned_repr. + change (Int.unsigned Int.iwordsize) with 32; omega. + assert (32 < Int.max_unsigned) by reflexivity. omega. +Qed. + Theorem or_rolm: forall x n m1 m2, or (rolm x n m1) (rolm x n m2) = rolm x n (Int.or m1 m2). @@ -1687,6 +1783,58 @@ Proof. assert (64 < Int.max_unsigned) by reflexivity. omega. Qed. +Theorem shrxl1_shrl: + forall x z, + shrxl x (Vint (Int.repr 1)) = Some z -> + z = shrl (addl x (shrlu x (Vint (Int.repr 63)))) (Vint (Int.repr 1)). +Proof. + intros. destruct x; simpl in H; try discriminate. + change (Int.ltu (Int.repr 1) (Int.repr 63)) with true in H; simpl in H. + inversion_clear H. + simpl. + change (Int.ltu (Int.repr 63) Int64.iwordsize') with true; simpl. + change (Int.ltu (Int.repr 1) Int64.iwordsize') with true; simpl. + f_equal. + rewrite Int64.shrx'1_shr' by reflexivity. + reflexivity. +Qed. + +Theorem shrxl_shrl_3: + forall n x z, + shrxl x (Vint n) = Some z -> + z = (if Int.eq n Int.zero then x else + if Int.eq n Int.one + then shrl (addl x (shrlu x (Vint (Int.repr 63)))) (Vint Int.one) + else shrl (addl x (shrlu (shrl x (Vint (Int.repr 63))) + (Vint (Int.sub (Int.repr 64) n)))) + (Vint n)). +Proof. + intros. destruct x; simpl in H; try discriminate. + destruct (Int.ltu n (Int.repr 63)) eqn:LT; inv H. + exploit Int.ltu_inv; eauto. change (Int.unsigned (Int.repr 63)) with 63; intros LT'. + predSpec Int.eq Int.eq_spec n Int.zero. +- subst n. unfold Int64.shrx'. rewrite Int64.shl'_zero. unfold Int64.divs. change (Int64.signed Int64.one) with 1. + rewrite Z.quot_1_r. rewrite Int64.repr_signed; auto. +- predSpec Int.eq Int.eq_spec n Int.one. + * subst n. simpl. + change (Int.ltu (Int.repr 63) Int64.iwordsize') with true. simpl. + change (Int.ltu Int.one Int64.iwordsize') with true. simpl. + f_equal. + apply Int64.shrx'1_shr'. + reflexivity. + * clear H0. +simpl. change (Int.ltu (Int.repr 63) Int64.iwordsize') with true. simpl. + replace (Int.ltu (Int.sub (Int.repr 64) n) Int64.iwordsize') with true. simpl. + replace (Int.ltu n Int64.iwordsize') with true. + f_equal; apply Int64.shrx'_shr_2; assumption. + symmetry; apply zlt_true. change (Int.unsigned n < 64); omega. + symmetry; apply zlt_true. unfold Int.sub. change (Int.unsigned (Int.repr 64)) with 64. + assert (Int.unsigned n <> 0). { red; intros; elim H. rewrite <- (Int.repr_unsigned n), H0. auto. } + rewrite Int.unsigned_repr. + change (Int.unsigned Int64.iwordsize') with 64; omega. + assert (64 < Int.max_unsigned) by reflexivity. omega. +Qed. + Theorem negate_cmp_bool: forall c x y, cmp_bool (negate_comparison c) x y = option_map negb (cmp_bool c x y). Proof. @@ -1899,10 +2047,18 @@ Qed. Lemma zero_ext_and: forall n v, - 0 < n < Int.zwordsize -> + 0 <= n -> Val.zero_ext n v = Val.and v (Vint (Int.repr (two_p n - 1))). Proof. - intros. destruct v; simpl; auto. decEq. apply Int.zero_ext_and; auto. omega. + intros. destruct v; simpl; auto. decEq. apply Int.zero_ext_and; auto. +Qed. + +Lemma zero_ext_andl: + forall n v, + 0 <= n -> + Val.zero_ext_l n v = Val.andl v (Vlong (Int64.repr (two_p n - 1))). +Proof. + intros. destruct v; simpl; auto. decEq. apply Int64.zero_ext_and; auto. Qed. Lemma rolm_lt_zero: |