diff options
Diffstat (limited to 'backend/Values.v')
| -rw-r--r-- | backend/Values.v | 888 |
1 files changed, 0 insertions, 888 deletions
diff --git a/backend/Values.v b/backend/Values.v deleted file mode 100644 index aa59e045..00000000 --- a/backend/Values.v +++ /dev/null @@ -1,888 +0,0 @@ -(** This module defines the type of values that is used in the dynamic - semantics of all our intermediate languages. *) - -Require Import Coqlib. -Require Import AST. -Require Import Integers. -Require Import Floats. - -Definition block : Set := Z. -Definition eq_block := zeq. - -(** A value is either: -- a machine integer; -- a floating-point number; -- a pointer: a pair of a memory address and an integer offset with respect - to this address; -- the [Vundef] value denoting an arbitrary bit pattern, such as the - value of an uninitialized variable. -*) - -Inductive val: Set := - | Vundef: val - | Vint: int -> val - | Vfloat: float -> val - | Vptr: block -> int -> val. - -Definition Vzero: val := Vint Int.zero. -Definition Vone: val := Vint Int.one. -Definition Vmone: val := Vint Int.mone. - -Definition Vtrue: val := Vint Int.one. -Definition Vfalse: val := Vint Int.zero. - -(** The module [Val] defines a number of arithmetic and logical operations - over type [val]. Most of these operations are straightforward extensions - of the corresponding integer or floating-point operations. *) - -Module Val. - -Definition of_bool (b: bool): val := if b then Vtrue else Vfalse. - -Definition has_type (v: val) (t: typ) : Prop := - match v, t with - | Vundef, _ => True - | Vint _, Tint => True - | Vfloat _, Tfloat => True - | Vptr _ _, Tint => True - | _, _ => False - end. - -Fixpoint has_type_list (vl: list val) (tl: list typ) {struct vl} : Prop := - match vl, tl with - | nil, nil => True - | v1 :: vs, t1 :: ts => has_type v1 t1 /\ has_type_list vs ts - | _, _ => False - end. - -(** Truth values. Pointers and non-zero integers are treated as [True]. - The integer 0 (also used to represent the null pointer) is [False]. - [Vundef] and floats are neither true nor false. *) - -Definition is_true (v: val) : Prop := - match v with - | Vint n => n <> Int.zero - | Vptr b ofs => True - | _ => False - end. - -Definition is_false (v: val) : Prop := - match v with - | Vint n => n = Int.zero - | _ => False - end. - -Inductive bool_of_val: val -> bool -> Prop := - | bool_of_val_int_true: - forall n, n <> Int.zero -> bool_of_val (Vint n) true - | bool_of_val_int_false: - bool_of_val (Vint Int.zero) false - | bool_of_val_ptr: - forall b ofs, bool_of_val (Vptr b ofs) true. - -Definition neg (v: val) : val := - match v with - | Vint n => Vint (Int.neg n) - | _ => Vundef - end. - -Definition negf (v: val) : val := - match v with - | Vfloat f => Vfloat (Float.neg f) - | _ => Vundef - end. - -Definition absf (v: val) : val := - match v with - | Vfloat f => Vfloat (Float.abs f) - | _ => Vundef - end. - -Definition intoffloat (v: val) : val := - match v with - | Vfloat f => Vint (Float.intoffloat f) - | _ => Vundef - end. - -Definition floatofint (v: val) : val := - match v with - | Vint n => Vfloat (Float.floatofint n) - | _ => Vundef - end. - -Definition floatofintu (v: val) : val := - match v with - | Vint n => Vfloat (Float.floatofintu n) - | _ => Vundef - end. - -Definition notint (v: val) : val := - match v with - | Vint n => Vint (Int.xor n Int.mone) - | _ => Vundef - end. - -Definition notbool (v: val) : val := - match v with - | Vint n => of_bool (Int.eq n Int.zero) - | Vptr b ofs => Vfalse - | _ => Vundef - end. - -Definition cast8signed (v: val) : val := - match v with - | Vint n => Vint(Int.cast8signed n) - | _ => Vundef - end. - -Definition cast8unsigned (v: val) : val := - match v with - | Vint n => Vint(Int.cast8unsigned n) - | _ => Vundef - end. - -Definition cast16signed (v: val) : val := - match v with - | Vint n => Vint(Int.cast16signed n) - | _ => Vundef - end. - -Definition cast16unsigned (v: val) : val := - match v with - | Vint n => Vint(Int.cast16unsigned n) - | _ => Vundef - end. - -Definition singleoffloat (v: val) : val := - match v with - | Vfloat f => Vfloat(Float.singleoffloat f) - | _ => Vundef - end. - -Definition add (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => Vint(Int.add n1 n2) - | Vptr b1 ofs1, Vint n2 => Vptr b1 (Int.add ofs1 n2) - | Vint n1, Vptr b2 ofs2 => Vptr b2 (Int.add ofs2 n1) - | _, _ => Vundef - end. - -Definition sub (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => Vint(Int.sub n1 n2) - | Vptr b1 ofs1, Vint n2 => Vptr b1 (Int.sub ofs1 n2) - | Vptr b1 ofs1, Vptr b2 ofs2 => - if zeq b1 b2 then Vint(Int.sub ofs1 ofs2) else Vundef - | _, _ => Vundef - end. - -Definition mul (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => Vint(Int.mul n1 n2) - | _, _ => Vundef - end. - -Definition divs (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.eq n2 Int.zero then Vundef else Vint(Int.divs n1 n2) - | _, _ => Vundef - end. - -Definition mods (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.eq n2 Int.zero then Vundef else Vint(Int.mods n1 n2) - | _, _ => Vundef - end. - -Definition divu (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.eq n2 Int.zero then Vundef else Vint(Int.divu n1 n2) - | _, _ => Vundef - end. - -Definition modu (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.eq n2 Int.zero then Vundef else Vint(Int.modu n1 n2) - | _, _ => Vundef - end. - -Definition and (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => Vint(Int.and n1 n2) - | _, _ => Vundef - end. - -Definition or (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => Vint(Int.or n1 n2) - | _, _ => Vundef - end. - -Definition xor (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => Vint(Int.xor n1 n2) - | _, _ => Vundef - end. - -Definition shl (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.ltu n2 (Int.repr 32) - then Vint(Int.shl n1 n2) - else Vundef - | _, _ => Vundef - end. - -Definition shr (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.ltu n2 (Int.repr 32) - then Vint(Int.shr n1 n2) - else Vundef - | _, _ => Vundef - end. - -Definition shr_carry (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.ltu n2 (Int.repr 32) - then Vint(Int.shr_carry n1 n2) - else Vundef - | _, _ => Vundef - end. - -Definition shrx (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.ltu n2 (Int.repr 32) - then Vint(Int.shrx n1 n2) - else Vundef - | _, _ => Vundef - end. - -Definition shru (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - if Int.ltu n2 (Int.repr 32) - then Vint(Int.shru n1 n2) - else Vundef - | _, _ => Vundef - end. - -Definition rolm (v: val) (amount mask: int): val := - match v with - | Vint n => Vint(Int.rolm n amount mask) - | _ => Vundef - end. - -Definition addf (v1 v2: val): val := - match v1, v2 with - | Vfloat f1, Vfloat f2 => Vfloat(Float.add f1 f2) - | _, _ => Vundef - end. - -Definition subf (v1 v2: val): val := - match v1, v2 with - | Vfloat f1, Vfloat f2 => Vfloat(Float.sub f1 f2) - | _, _ => Vundef - end. - -Definition mulf (v1 v2: val): val := - match v1, v2 with - | Vfloat f1, Vfloat f2 => Vfloat(Float.mul f1 f2) - | _, _ => Vundef - end. - -Definition divf (v1 v2: val): val := - match v1, v2 with - | Vfloat f1, Vfloat f2 => Vfloat(Float.div f1 f2) - | _, _ => Vundef - end. - -Definition cmp_mismatch (c: comparison): val := - match c with - | Ceq => Vfalse - | Cne => Vtrue - | _ => Vundef - end. - -Definition cmp (c: comparison) (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => of_bool (Int.cmp c n1 n2) - | Vint n1, Vptr b2 ofs2 => - if Int.eq n1 Int.zero then cmp_mismatch c else Vundef - | Vptr b1 ofs1, Vptr b2 ofs2 => - if zeq b1 b2 - then of_bool (Int.cmp c ofs1 ofs2) - else cmp_mismatch c - | Vptr b1 ofs1, Vint n2 => - if Int.eq n2 Int.zero then cmp_mismatch c else Vundef - | _, _ => Vundef - end. - -Definition cmpu (c: comparison) (v1 v2: val): val := - match v1, v2 with - | Vint n1, Vint n2 => - of_bool (Int.cmpu c n1 n2) - | Vint n1, Vptr b2 ofs2 => - if Int.eq n1 Int.zero then cmp_mismatch c else Vundef - | Vptr b1 ofs1, Vptr b2 ofs2 => - if zeq b1 b2 - then of_bool (Int.cmpu c ofs1 ofs2) - else cmp_mismatch c - | Vptr b1 ofs1, Vint n2 => - if Int.eq n2 Int.zero then cmp_mismatch c else Vundef - | _, _ => Vundef - end. - -Definition cmpf (c: comparison) (v1 v2: val): val := - match v1, v2 with - | Vfloat f1, Vfloat f2 => of_bool (Float.cmp c f1 f2) - | _, _ => Vundef - end. - -(** [load_result] is used in the memory model (library [Mem]) - to post-process the results of a memory read. For instance, - consider storing the integer value [0xFFF] on 1 byte at a - given address, and reading it back. If it is read back with - chunk [Mint8unsigned], zero-extension must be performed, resulting - in [0xFF]. If it is read back as a [Mint8signed], sign-extension - is performed and [0xFFFFFFFF] is returned. Type mismatches - (e.g. reading back a float as a [Mint32]) read back as [Vundef]. *) - -Definition load_result (chunk: memory_chunk) (v: val) := - match chunk, v with - | Mint8signed, Vint n => Vint (Int.cast8signed n) - | Mint8unsigned, Vint n => Vint (Int.cast8unsigned n) - | Mint16signed, Vint n => Vint (Int.cast16signed n) - | Mint16unsigned, Vint n => Vint (Int.cast16unsigned n) - | Mint32, Vint n => Vint n - | Mint32, Vptr b ofs => Vptr b ofs - | Mfloat32, Vfloat f => Vfloat(Float.singleoffloat f) - | Mfloat64, Vfloat f => Vfloat f - | _, _ => Vundef - end. - -(** Theorems on arithmetic operations. *) - -Theorem cast8unsigned_and: - forall x, cast8unsigned x = and x (Vint(Int.repr 255)). -Proof. - destruct x; simpl; auto. decEq. apply Int.cast8unsigned_and. -Qed. - -Theorem cast16unsigned_and: - forall x, cast16unsigned x = and x (Vint(Int.repr 65535)). -Proof. - destruct x; simpl; auto. decEq. apply Int.cast16unsigned_and. -Qed. - -Theorem istrue_not_isfalse: - forall v, is_false v -> is_true (notbool v). -Proof. - destruct v; simpl; try contradiction. - intros. subst i. simpl. discriminate. -Qed. - -Theorem isfalse_not_istrue: - forall v, is_true v -> is_false (notbool v). -Proof. - destruct v; simpl; try contradiction. - intros. generalize (Int.eq_spec i Int.zero). - case (Int.eq i Int.zero); intro. - contradiction. simpl. auto. - auto. -Qed. - -Theorem bool_of_true_val: - forall v, is_true v -> bool_of_val v true. -Proof. - intro. destruct v; simpl; intros; try contradiction. - constructor; auto. constructor. -Qed. - -Theorem bool_of_true_val2: - forall v, bool_of_val v true -> is_true v. -Proof. - intros. inversion H; simpl; auto. -Qed. - -Theorem bool_of_true_val_inv: - forall v b, is_true v -> bool_of_val v b -> b = true. -Proof. - intros. inversion H0; subst v b; simpl in H; auto. -Qed. - -Theorem bool_of_false_val: - forall v, is_false v -> bool_of_val v false. -Proof. - intro. destruct v; simpl; intros; try contradiction. - subst i; constructor. -Qed. - -Theorem bool_of_false_val2: - forall v, bool_of_val v false -> is_false v. -Proof. - intros. inversion H; simpl; auto. -Qed. - -Theorem bool_of_false_val_inv: - forall v b, is_false v -> bool_of_val v b -> b = false. -Proof. - intros. inversion H0; subst v b; simpl in H. - congruence. auto. contradiction. -Qed. - -Theorem notbool_negb_1: - forall b, of_bool (negb b) = notbool (of_bool b). -Proof. - destruct b; reflexivity. -Qed. - -Theorem notbool_negb_2: - forall b, of_bool b = notbool (of_bool (negb b)). -Proof. - destruct b; reflexivity. -Qed. - -Theorem notbool_idem2: - forall b, notbool(notbool(of_bool b)) = of_bool b. -Proof. - destruct b; reflexivity. -Qed. - -Theorem notbool_idem3: - forall x, notbool(notbool(notbool x)) = notbool x. -Proof. - destruct x; simpl; auto. - case (Int.eq i Int.zero); reflexivity. -Qed. - -Theorem add_commut: forall x y, add x y = add y x. -Proof. - destruct x; destruct y; simpl; auto. - decEq. apply Int.add_commut. -Qed. - -Theorem add_assoc: forall x y z, add (add x y) z = add x (add y z). -Proof. - destruct x; destruct y; destruct z; simpl; auto. - rewrite Int.add_assoc; auto. - rewrite Int.add_assoc; auto. - decEq. decEq. apply Int.add_commut. - decEq. rewrite Int.add_commut. rewrite <- Int.add_assoc. - decEq. apply Int.add_commut. - decEq. rewrite Int.add_assoc. auto. -Qed. - -Theorem add_permut: forall x y z, add x (add y z) = add y (add x z). -Proof. - intros. rewrite (add_commut y z). rewrite <- add_assoc. apply add_commut. -Qed. - -Theorem add_permut_4: - forall x y z t, add (add x y) (add z t) = add (add x z) (add y t). -Proof. - intros. rewrite add_permut. rewrite add_assoc. - rewrite add_permut. symmetry. apply add_assoc. -Qed. - -Theorem neg_zero: neg Vzero = Vzero. -Proof. - reflexivity. -Qed. - -Theorem neg_add_distr: forall x y, neg(add x y) = add (neg x) (neg y). -Proof. - destruct x; destruct y; simpl; auto. decEq. apply Int.neg_add_distr. -Qed. - -Theorem sub_zero_r: forall x, sub Vzero x = neg x. -Proof. - destruct x; simpl; auto. -Qed. - -Theorem sub_add_opp: forall x y, sub x (Vint y) = add x (Vint (Int.neg y)). -Proof. - destruct x; intro y; simpl; auto; rewrite Int.sub_add_opp; auto. -Qed. - -Theorem sub_add_l: - forall v1 v2 i, sub (add v1 (Vint i)) v2 = add (sub v1 v2) (Vint i). -Proof. - destruct v1; destruct v2; intros; simpl; auto. - rewrite Int.sub_add_l. auto. - rewrite Int.sub_add_l. auto. - case (zeq b b0); intro. rewrite Int.sub_add_l. auto. reflexivity. -Qed. - -Theorem sub_add_r: - forall v1 v2 i, sub v1 (add v2 (Vint i)) = add (sub v1 v2) (Vint (Int.neg i)). -Proof. - destruct v1; destruct v2; intros; simpl; auto. - rewrite Int.sub_add_r. auto. - repeat rewrite Int.sub_add_opp. decEq. - repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. - decEq. repeat rewrite Int.sub_add_opp. - rewrite Int.add_assoc. decEq. apply Int.neg_add_distr. - case (zeq b b0); intro. simpl. decEq. - repeat rewrite Int.sub_add_opp. rewrite Int.add_assoc. decEq. - apply Int.neg_add_distr. - reflexivity. -Qed. - -Theorem mul_commut: forall x y, mul x y = mul y x. -Proof. - destruct x; destruct y; simpl; auto. decEq. apply Int.mul_commut. -Qed. - -Theorem mul_assoc: forall x y z, mul (mul x y) z = mul x (mul y z). -Proof. - destruct x; destruct y; destruct z; simpl; auto. - decEq. apply Int.mul_assoc. -Qed. - -Theorem mul_add_distr_l: - forall x y z, mul (add x y) z = add (mul x z) (mul y z). -Proof. - destruct x; destruct y; destruct z; simpl; auto. - decEq. apply Int.mul_add_distr_l. -Qed. - - -Theorem mul_add_distr_r: - forall x y z, mul x (add y z) = add (mul x y) (mul x z). -Proof. - destruct x; destruct y; destruct z; simpl; auto. - decEq. apply Int.mul_add_distr_r. -Qed. - -Theorem mul_pow2: - forall x n logn, - Int.is_power2 n = Some logn -> - mul x (Vint n) = shl x (Vint logn). -Proof. - intros; destruct x; simpl; auto. - change 32 with (Z_of_nat wordsize). - rewrite (Int.is_power2_range _ _ H). decEq. apply Int.mul_pow2. auto. -Qed. - -Theorem mods_divs: - forall x y, mods x y = sub x (mul (divs x y) y). -Proof. - destruct x; destruct y; simpl; auto. - case (Int.eq i0 Int.zero); simpl. auto. decEq. apply Int.mods_divs. -Qed. - -Theorem modu_divu: - forall x y, modu x y = sub x (mul (divu x y) y). -Proof. - destruct x; destruct y; simpl; auto. - generalize (Int.eq_spec i0 Int.zero); - case (Int.eq i0 Int.zero); simpl. auto. - intro. decEq. apply Int.modu_divu. auto. -Qed. - -Theorem divs_pow2: - forall x n logn, - Int.is_power2 n = Some logn -> - divs x (Vint n) = shrx x (Vint logn). -Proof. - intros; destruct x; simpl; auto. - change 32 with (Z_of_nat wordsize). - rewrite (Int.is_power2_range _ _ H). - generalize (Int.eq_spec n Int.zero); - case (Int.eq n Int.zero); intro. - subst n. compute in H. discriminate. - decEq. apply Int.divs_pow2. auto. -Qed. - -Theorem divu_pow2: - forall x n logn, - Int.is_power2 n = Some logn -> - divu x (Vint n) = shru x (Vint logn). -Proof. - intros; destruct x; simpl; auto. - change 32 with (Z_of_nat wordsize). - rewrite (Int.is_power2_range _ _ H). - generalize (Int.eq_spec n Int.zero); - case (Int.eq n Int.zero); intro. - subst n. compute in H. discriminate. - decEq. apply Int.divu_pow2. auto. -Qed. - -Theorem modu_pow2: - forall x n logn, - Int.is_power2 n = Some logn -> - modu x (Vint n) = and x (Vint (Int.sub n Int.one)). -Proof. - intros; destruct x; simpl; auto. - generalize (Int.eq_spec n Int.zero); - case (Int.eq n Int.zero); intro. - subst n. compute in H. discriminate. - decEq. eapply Int.modu_and; eauto. -Qed. - -Theorem and_commut: forall x y, and x y = and y x. -Proof. - destruct x; destruct y; simpl; auto. decEq. apply Int.and_commut. -Qed. - -Theorem and_assoc: forall x y z, and (and x y) z = and x (and y z). -Proof. - destruct x; destruct y; destruct z; simpl; auto. - decEq. apply Int.and_assoc. -Qed. - -Theorem or_commut: forall x y, or x y = or y x. -Proof. - destruct x; destruct y; simpl; auto. decEq. apply Int.or_commut. -Qed. - -Theorem or_assoc: forall x y z, or (or x y) z = or x (or y z). -Proof. - destruct x; destruct y; destruct z; simpl; auto. - decEq. apply Int.or_assoc. -Qed. - -Theorem xor_commut: forall x y, xor x y = xor y x. -Proof. - destruct x; destruct y; simpl; auto. decEq. apply Int.xor_commut. -Qed. - -Theorem xor_assoc: forall x y z, xor (xor x y) z = xor x (xor y z). -Proof. - destruct x; destruct y; destruct z; simpl; auto. - decEq. apply Int.xor_assoc. -Qed. - -Theorem shl_mul: forall x y, Val.mul x (Val.shl Vone y) = Val.shl x y. -Proof. - destruct x; destruct y; simpl; auto. - case (Int.ltu i0 (Int.repr 32)); auto. - decEq. symmetry. apply Int.shl_mul. -Qed. - -Theorem shl_rolm: - forall x n, - Int.ltu n (Int.repr 32) = true -> - shl x (Vint n) = rolm x n (Int.shl Int.mone n). -Proof. - intros; destruct x; simpl; auto. - rewrite H. decEq. apply Int.shl_rolm. exact H. -Qed. - -Theorem shru_rolm: - forall x n, - Int.ltu n (Int.repr 32) = true -> - shru x (Vint n) = rolm x (Int.sub (Int.repr 32) n) (Int.shru Int.mone n). -Proof. - intros; destruct x; simpl; auto. - rewrite H. decEq. apply Int.shru_rolm. exact H. -Qed. - -Theorem shrx_carry: - forall x y, - add (shr x y) (shr_carry x y) = shrx x y. -Proof. - destruct x; destruct y; simpl; auto. - case (Int.ltu i0 (Int.repr 32)); auto. - simpl. decEq. apply Int.shrx_carry. -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). -Proof. - intros; destruct x; simpl; auto. - decEq. apply Int.or_rolm. -Qed. - -Theorem rolm_rolm: - forall x n1 m1 n2 m2, - rolm (rolm x n1 m1) n2 m2 = - rolm x (Int.and (Int.add n1 n2) (Int.repr 31)) - (Int.and (Int.rol m1 n2) m2). -Proof. - intros; destruct x; simpl; auto. - decEq. - replace (Int.and (Int.add n1 n2) (Int.repr 31)) - with (Int.modu (Int.add n1 n2) (Int.repr 32)). - apply Int.rolm_rolm. - change (Int.repr 31) with (Int.sub (Int.repr 32) Int.one). - apply Int.modu_and with (Int.repr 5). reflexivity. -Qed. - -Theorem rolm_zero: - forall x m, - rolm x Int.zero m = and x (Vint m). -Proof. - intros; destruct x; simpl; auto. decEq. apply Int.rolm_zero. -Qed. - -Theorem addf_commut: forall x y, addf x y = addf y x. -Proof. - destruct x; destruct y; simpl; auto. decEq. apply Float.addf_commut. -Qed. - -Lemma negate_cmp_mismatch: - forall c, - cmp_mismatch (negate_comparison c) = notbool(cmp_mismatch c). -Proof. - destruct c; reflexivity. -Qed. - -Theorem negate_cmp: - forall c x y, - cmp (negate_comparison c) x y = notbool (cmp c x y). -Proof. - destruct x; destruct y; simpl; auto. - rewrite Int.negate_cmp. apply notbool_negb_1. - case (Int.eq i Int.zero). apply negate_cmp_mismatch. reflexivity. - case (Int.eq i0 Int.zero). apply negate_cmp_mismatch. reflexivity. - case (zeq b b0); intro. - rewrite Int.negate_cmp. apply notbool_negb_1. - apply negate_cmp_mismatch. -Qed. - -Theorem negate_cmpu: - forall c x y, - cmpu (negate_comparison c) x y = notbool (cmpu c x y). -Proof. - destruct x; destruct y; simpl; auto. - rewrite Int.negate_cmpu. apply notbool_negb_1. - case (Int.eq i Int.zero). apply negate_cmp_mismatch. reflexivity. - case (Int.eq i0 Int.zero). apply negate_cmp_mismatch. reflexivity. - case (zeq b b0); intro. - rewrite Int.negate_cmpu. apply notbool_negb_1. - apply negate_cmp_mismatch. -Qed. - -Lemma swap_cmp_mismatch: - forall c, cmp_mismatch (swap_comparison c) = cmp_mismatch c. -Proof. - destruct c; reflexivity. -Qed. - -Theorem swap_cmp: - forall c x y, - cmp (swap_comparison c) x y = cmp c y x. -Proof. - destruct x; destruct y; simpl; auto. - rewrite Int.swap_cmp. auto. - case (Int.eq i Int.zero). apply swap_cmp_mismatch. auto. - case (Int.eq i0 Int.zero). apply swap_cmp_mismatch. auto. - case (zeq b b0); intro. - subst b0. rewrite zeq_true. rewrite Int.swap_cmp. auto. - rewrite zeq_false. apply swap_cmp_mismatch. auto. -Qed. - -Theorem swap_cmpu: - forall c x y, - cmpu (swap_comparison c) x y = cmpu c y x. -Proof. - destruct x; destruct y; simpl; auto. - rewrite Int.swap_cmpu. auto. - case (Int.eq i Int.zero). apply swap_cmp_mismatch. auto. - case (Int.eq i0 Int.zero). apply swap_cmp_mismatch. auto. - case (zeq b b0); intro. - subst b0. rewrite zeq_true. rewrite Int.swap_cmpu. auto. - rewrite zeq_false. apply swap_cmp_mismatch. auto. -Qed. - -Theorem negate_cmpf_eq: - forall v1 v2, notbool (cmpf Cne v1 v2) = cmpf Ceq v1 v2. -Proof. - destruct v1; destruct v2; simpl; auto. - rewrite Float.cmp_ne_eq. rewrite notbool_negb_1. - apply notbool_idem2. -Qed. - -Lemma or_of_bool: - forall b1 b2, or (of_bool b1) (of_bool b2) = of_bool (b1 || b2). -Proof. - destruct b1; destruct b2; reflexivity. -Qed. - -Theorem cmpf_le: - forall v1 v2, cmpf Cle v1 v2 = or (cmpf Clt v1 v2) (cmpf Ceq v1 v2). -Proof. - destruct v1; destruct v2; simpl; auto. - rewrite or_of_bool. decEq. apply Float.cmp_le_lt_eq. -Qed. - -Theorem cmpf_ge: - forall v1 v2, cmpf Cge v1 v2 = or (cmpf Cgt v1 v2) (cmpf Ceq v1 v2). -Proof. - destruct v1; destruct v2; simpl; auto. - rewrite or_of_bool. decEq. apply Float.cmp_ge_gt_eq. -Qed. - -Definition is_bool (v: val) := - v = Vundef \/ v = Vtrue \/ v = Vfalse. - -Lemma of_bool_is_bool: - forall b, is_bool (of_bool b). -Proof. - destruct b; unfold is_bool; simpl; tauto. -Qed. - -Lemma undef_is_bool: is_bool Vundef. -Proof. - unfold is_bool; tauto. -Qed. - -Lemma cmp_mismatch_is_bool: - forall c, is_bool (cmp_mismatch c). -Proof. - destruct c; simpl; unfold is_bool; tauto. -Qed. - -Lemma cmp_is_bool: - forall c v1 v2, is_bool (cmp c v1 v2). -Proof. - destruct v1; destruct v2; simpl; try apply undef_is_bool. - apply of_bool_is_bool. - case (Int.eq i Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool. - case (Int.eq i0 Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool. - case (zeq b b0); intro. apply of_bool_is_bool. apply cmp_mismatch_is_bool. -Qed. - -Lemma cmpu_is_bool: - forall c v1 v2, is_bool (cmpu c v1 v2). -Proof. - destruct v1; destruct v2; simpl; try apply undef_is_bool. - apply of_bool_is_bool. - case (Int.eq i Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool. - case (Int.eq i0 Int.zero). apply cmp_mismatch_is_bool. apply undef_is_bool. - case (zeq b b0); intro. apply of_bool_is_bool. apply cmp_mismatch_is_bool. -Qed. - -Lemma cmpf_is_bool: - forall c v1 v2, is_bool (cmpf c v1 v2). -Proof. - destruct v1; destruct v2; simpl; - apply undef_is_bool || apply of_bool_is_bool. -Qed. - -Lemma notbool_is_bool: - forall v, is_bool (notbool v). -Proof. - destruct v; simpl. - apply undef_is_bool. apply of_bool_is_bool. - apply undef_is_bool. unfold is_bool; tauto. -Qed. - -Lemma notbool_xor: - forall v, is_bool v -> v = xor (notbool v) Vone. -Proof. - intros. elim H; intro. - subst v. reflexivity. - elim H0; intro; subst v; reflexivity. -Qed. - -End Val. |