# Module Values

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 : Type := positive.
Definition eq_block := peq.

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: Type :=
| Vundef: val
| Vint: int -> val
| Vlong: int64 -> val
| Vfloat: float -> val
| Vsingle: float32 -> 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.

# Operations over values

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 eq (x y: val): {x=y} + {x<>y}.
Proof.
decide equality.
apply Int.eq_dec.
apply Int64.eq_dec.
apply Float.eq_dec.
apply Float32.eq_dec.
apply Int.eq_dec.
apply eq_block.
Defined.
Global Opaque eq.

Definition has_type (v: val) (t: typ) : Prop :=
match v, t with
| Vundef, _ => True
| Vint _, Tint => True
| Vlong _, Tlong => True
| Vfloat _, Tfloat => True
| Vsingle _, Tsingle => True
| Vptr _ _, Tint => True
| (Vint _ | Vptr _ _ | Vsingle _), Tany32 => True
| _, Tany64 => 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.

Definition has_opttype (v: val) (ot: option typ) : Prop :=
match ot with
| None => v = Vundef
| Some t => has_type v t
end.

Lemma has_subtype:
forall ty1 ty2 v,
subtype ty1 ty2 = true -> has_type v ty1 -> has_type v ty2.
Proof.
intros. destruct ty1; destruct ty2; simpl in H; discriminate || assumption || idtac;
unfold has_type in *; destruct v; auto.
Qed.

Lemma has_subtype_list:
forall tyl1 tyl2 vl,
subtype_list tyl1 tyl2 = true -> has_type_list vl tyl1 -> has_type_list vl tyl2.
Proof.
induction tyl1; intros; destruct tyl2; try discriminate; destruct vl; try contradiction.
red; auto.
simpl in *. InvBooleans. destruct H0. split; auto. eapply has_subtype; eauto.
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.

Inductive bool_of_val: val -> bool -> Prop :=
| bool_of_val_int:
forall n, bool_of_val (Vint n) (negb (Int.eq n Int.zero)).

Arithmetic operations

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 negfs (v: val) : val :=
match v with
| Vsingle f => Vsingle (Float32.neg f)
| _ => Vundef
end.

Definition absfs (v: val) : val :=
match v with
| Vsingle f => Vsingle (Float32.abs f)
| _ => Vundef
end.

Definition maketotal (ov: option val) : val :=
match ov with Some v => v | None => Vundef end.

Definition intoffloat (v: val) : option val :=
match v with
| Vfloat f => option_map Vint (Float.to_int f)
| _ => None
end.

Definition intuoffloat (v: val) : option val :=
match v with
| Vfloat f => option_map Vint (Float.to_intu f)
| _ => None
end.

Definition floatofint (v: val) : option val :=
match v with
| Vint n => Some (Vfloat (Float.of_int n))
| _ => None
end.

Definition floatofintu (v: val) : option val :=
match v with
| Vint n => Some (Vfloat (Float.of_intu n))
| _ => None
end.

Definition intofsingle (v: val) : option val :=
match v with
| Vsingle f => option_map Vint (Float32.to_int f)
| _ => None
end.

Definition intuofsingle (v: val) : option val :=
match v with
| Vsingle f => option_map Vint (Float32.to_intu f)
| _ => None
end.

Definition singleofint (v: val) : option val :=
match v with
| Vint n => Some (Vsingle (Float32.of_int n))
| _ => None
end.

Definition singleofintu (v: val) : option val :=
match v with
| Vint n => Some (Vsingle (Float32.of_intu n))
| _ => None
end.

Definition negint (v: val) : val :=
match v with
| Vint n => Vint (Int.neg n)
| _ => Vundef
end.

Definition notint (v: val) : val :=
match v with
| Vint n => Vint (Int.not n)
| _ => Vundef
end.

Definition of_bool (b: bool): val := if b then Vtrue else Vfalse.

Definition boolval (v: val) : val :=
match v with
| Vint n => of_bool (negb (Int.eq n Int.zero))
| Vptr b ofs => Vtrue
| _ => 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 zero_ext (nbits: Z) (v: val) : val :=
match v with
| Vint n => Vint(Int.zero_ext nbits n)
| _ => Vundef
end.

Definition sign_ext (nbits: Z) (v: val) : val :=
match v with
| Vint n => Vint(Int.sign_ext nbits n)
| _ => Vundef
end.

Definition singleoffloat (v: val) : val :=
match v with
| Vfloat f => Vsingle (Float.to_single f)
| _ => Vundef
end.

Definition floatofsingle (v: val) : val :=
match v with
| Vsingle f => Vfloat (Float.of_single 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 eq_block 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 mulhs (v1 v2: val): val :=
match v1, v2 with
| Vint n1, Vint n2 => Vint(Int.mulhs n1 n2)
| _, _ => Vundef
end.

Definition mulhu (v1 v2: val): val :=
match v1, v2 with
| Vint n1, Vint n2 => Vint(Int.mulhu n1 n2)
| _, _ => Vundef
end.

Definition divs (v1 v2: val): option val :=
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.eq n2 Int.zero
|| Int.eq n1 (Int.repr Int.min_signed) && Int.eq n2 Int.mone
then None
else Some(Vint(Int.divs n1 n2))
| _, _ => None
end.

Definition mods (v1 v2: val): option val :=
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.eq n2 Int.zero
|| Int.eq n1 (Int.repr Int.min_signed) && Int.eq n2 Int.mone
then None
else Some(Vint(Int.mods n1 n2))
| _, _ => None
end.

Definition divu (v1 v2: val): option val :=
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.eq n2 Int.zero then None else Some(Vint(Int.divu n1 n2))
| _, _ => None
end.

Definition modu (v1 v2: val): option val :=
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.eq n2 Int.zero then None else Some(Vint(Int.modu n1 n2))
| _, _ => None
end.

Definition add_carry (v1 v2 cin: val): val :=
match v1, v2, cin with
| Vint n1, Vint n2, Vint c => Vint(Int.add_carry n1 n2 c)
| _, _, _ => Vundef
end.

Definition sub_overflow (v1 v2: val) : val :=
match v1, v2 with
| Vint n1, Vint n2 => Vint(Int.sub_overflow n1 n2 Int.zero)
| _, _ => Vundef
end.

Definition negative (v: val) : val :=
match v with
| Vint n => Vint (Int.negative n)
| _ => 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.iwordsize
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.iwordsize
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.iwordsize
then Vint(Int.shr_carry n1 n2)
else Vundef
| _, _ => Vundef
end.

Definition shrx (v1 v2: val): option val :=
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.ltu n2 (Int.repr 31)
then Some(Vint(Int.shrx n1 n2))
else None
| _, _ => None
end.

Definition shru (v1 v2: val): val :=
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.ltu n2 Int.iwordsize
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 ror (v1 v2: val): val :=
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.ltu n2 Int.iwordsize
then Vint(Int.ror n1 n2)
else Vundef
| _, _ => 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 floatofwords (v1 v2: val) : val :=
match v1, v2 with
| Vint n1, Vint n2 => Vfloat (Float.from_words n1 n2)
| _, _ => Vundef
end.

Definition addfs (v1 v2: val): val :=
match v1, v2 with
| Vsingle f1, Vsingle f2 => Vsingle(Float32.add f1 f2)
| _, _ => Vundef
end.

Definition subfs (v1 v2: val): val :=
match v1, v2 with
| Vsingle f1, Vsingle f2 => Vsingle(Float32.sub f1 f2)
| _, _ => Vundef
end.

Definition mulfs (v1 v2: val): val :=
match v1, v2 with
| Vsingle f1, Vsingle f2 => Vsingle(Float32.mul f1 f2)
| _, _ => Vundef
end.

Definition divfs (v1 v2: val): val :=
match v1, v2 with
| Vsingle f1, Vsingle f2 => Vsingle(Float32.div f1 f2)
| _, _ => Vundef
end.

Operations on 64-bit integers

Definition longofwords (v1 v2: val) : val :=
match v1, v2 with
| Vint n1, Vint n2 => Vlong (Int64.ofwords n1 n2)
| _, _ => Vundef
end.

Definition loword (v: val) : val :=
match v with
| Vlong n => Vint (Int64.loword n)
| _ => Vundef
end.

Definition hiword (v: val) : val :=
match v with
| Vlong n => Vint (Int64.hiword n)
| _ => Vundef
end.

Definition negl (v: val) : val :=
match v with
| Vlong n => Vlong (Int64.neg n)
| _ => Vundef
end.

Definition notl (v: val) : val :=
match v with
| Vlong n => Vlong (Int64.not n)
| _ => Vundef
end.

Definition longofint (v: val) : val :=
match v with
| Vint n => Vlong (Int64.repr (Int.signed n))
| _ => Vundef
end.

Definition longofintu (v: val) : val :=
match v with
| Vint n => Vlong (Int64.repr (Int.unsigned n))
| _ => Vundef
end.

Definition longoffloat (v: val) : option val :=
match v with
| Vfloat f => option_map Vlong (Float.to_long f)
| _ => None
end.

Definition longuoffloat (v: val) : option val :=
match v with
| Vfloat f => option_map Vlong (Float.to_longu f)
| _ => None
end.

Definition longofsingle (v: val) : option val :=
match v with
| Vsingle f => option_map Vlong (Float32.to_long f)
| _ => None
end.

Definition longuofsingle (v: val) : option val :=
match v with
| Vsingle f => option_map Vlong (Float32.to_longu f)
| _ => None
end.

Definition floatoflong (v: val) : option val :=
match v with
| Vlong n => Some (Vfloat (Float.of_long n))
| _ => None
end.

Definition floatoflongu (v: val) : option val :=
match v with
| Vlong n => Some (Vfloat (Float.of_longu n))
| _ => None
end.

Definition singleoflong (v: val) : option val :=
match v with
| Vlong n => Some (Vsingle (Float32.of_long n))
| _ => None
end.

Definition singleoflongu (v: val) : option val :=
match v with
| Vlong n => Some (Vsingle (Float32.of_longu n))
| _ => None
end.

Definition addl (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vlong n2 => Vlong(Int64.add n1 n2)
| _, _ => Vundef
end.

Definition subl (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vlong n2 => Vlong(Int64.sub n1 n2)
| _, _ => Vundef
end.

Definition mull (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vlong n2 => Vlong(Int64.mul n1 n2)
| _, _ => Vundef
end.

Definition mull' (v1 v2: val): val :=
match v1, v2 with
| Vint n1, Vint n2 => Vlong(Int64.mul' n1 n2)
| _, _ => Vundef
end.

Definition divls (v1 v2: val): option val :=
match v1, v2 with
| Vlong n1, Vlong n2 =>
if Int64.eq n2 Int64.zero
|| Int64.eq n1 (Int64.repr Int64.min_signed) && Int64.eq n2 Int64.mone
then None
else Some(Vlong(Int64.divs n1 n2))
| _, _ => None
end.

Definition modls (v1 v2: val): option val :=
match v1, v2 with
| Vlong n1, Vlong n2 =>
if Int64.eq n2 Int64.zero
|| Int64.eq n1 (Int64.repr Int64.min_signed) && Int64.eq n2 Int64.mone
then None
else Some(Vlong(Int64.mods n1 n2))
| _, _ => None
end.

Definition divlu (v1 v2: val): option val :=
match v1, v2 with
| Vlong n1, Vlong n2 =>
if Int64.eq n2 Int64.zero then None else Some(Vlong(Int64.divu n1 n2))
| _, _ => None
end.

Definition modlu (v1 v2: val): option val :=
match v1, v2 with
| Vlong n1, Vlong n2 =>
if Int64.eq n2 Int64.zero then None else Some(Vlong(Int64.modu n1 n2))
| _, _ => None
end.

Definition andl (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vlong n2 => Vlong(Int64.and n1 n2)
| _, _ => Vundef
end.

Definition orl (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vlong n2 => Vlong(Int64.or n1 n2)
| _, _ => Vundef
end.

Definition xorl (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vlong n2 => Vlong(Int64.xor n1 n2)
| _, _ => Vundef
end.

Definition shll (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vint n2 =>
if Int.ltu n2 Int64.iwordsize'
then Vlong(Int64.shl' n1 n2)
else Vundef
| _, _ => Vundef
end.

Definition shrl (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vint n2 =>
if Int.ltu n2 Int64.iwordsize'
then Vlong(Int64.shr' n1 n2)
else Vundef
| _, _ => Vundef
end.

Definition shrlu (v1 v2: val): val :=
match v1, v2 with
| Vlong n1, Vint n2 =>
if Int.ltu n2 Int64.iwordsize'
then Vlong(Int64.shru' n1 n2)
else Vundef
| _, _ => Vundef
end.

Comparisons

Section COMPARISONS.

Variable valid_ptr: block -> Z -> bool.
Let weak_valid_ptr (b: block) (ofs: Z) := valid_ptr b ofs || valid_ptr b (ofs - 1).

Definition cmp_bool (c: comparison) (v1 v2: val): option bool :=
match v1, v2 with
| Vint n1, Vint n2 => Some (Int.cmp c n1 n2)
| _, _ => None
end.

Definition cmp_different_blocks (c: comparison): option bool :=
match c with
| Ceq => Some false
| Cne => Some true
| _ => None
end.

Definition cmpu_bool (c: comparison) (v1 v2: val): option bool :=
match v1, v2 with
| Vint n1, Vint n2 =>
Some (Int.cmpu c n1 n2)
| Vint n1, Vptr b2 ofs2 =>
if Int.eq n1 Int.zero && weak_valid_ptr b2 (Int.unsigned ofs2)
then cmp_different_blocks c
else None
| Vptr b1 ofs1, Vptr b2 ofs2 =>
if eq_block b1 b2 then
if weak_valid_ptr b1 (Int.unsigned ofs1)
&& weak_valid_ptr b2 (Int.unsigned ofs2)
then Some (Int.cmpu c ofs1 ofs2)
else None
else
if valid_ptr b1 (Int.unsigned ofs1)
&& valid_ptr b2 (Int.unsigned ofs2)
then cmp_different_blocks c
else None
| Vptr b1 ofs1, Vint n2 =>
if Int.eq n2 Int.zero && weak_valid_ptr b1 (Int.unsigned ofs1)
then cmp_different_blocks c
else None
| _, _ => None
end.

Definition cmpf_bool (c: comparison) (v1 v2: val): option bool :=
match v1, v2 with
| Vfloat f1, Vfloat f2 => Some (Float.cmp c f1 f2)
| _, _ => None
end.

Definition cmpfs_bool (c: comparison) (v1 v2: val): option bool :=
match v1, v2 with
| Vsingle f1, Vsingle f2 => Some (Float32.cmp c f1 f2)
| _, _ => None
end.

Definition cmpl_bool (c: comparison) (v1 v2: val): option bool :=
match v1, v2 with
| Vlong n1, Vlong n2 => Some (Int64.cmp c n1 n2)
| _, _ => None
end.

Definition cmplu_bool (c: comparison) (v1 v2: val): option bool :=
match v1, v2 with
| Vlong n1, Vlong n2 => Some (Int64.cmpu c n1 n2)
| _, _ => None
end.

Definition of_optbool (ob: option bool): val :=
match ob with Some true => Vtrue | Some false => Vfalse | None => Vundef end.

Definition cmp (c: comparison) (v1 v2: val): val :=
of_optbool (cmp_bool c v1 v2).

Definition cmpu (c: comparison) (v1 v2: val): val :=
of_optbool (cmpu_bool c v1 v2).

Definition cmpf (c: comparison) (v1 v2: val): val :=
of_optbool (cmpf_bool c v1 v2).

Definition cmpfs (c: comparison) (v1 v2: val): val :=
of_optbool (cmpfs_bool c v1 v2).

Definition cmpl (c: comparison) (v1 v2: val): option val :=
option_map of_bool (cmpl_bool c v1 v2).

Definition cmplu (c: comparison) (v1 v2: val): option val :=
option_map of_bool (cmplu_bool c v1 v2).

match v with
| Vint n => Some (Int.eq (Int.and n mask) Int.zero)
| _ => None
end.

End COMPARISONS.

load_result reflects the effect of storing a value with a given memory chunk, then reading it back with the same chunk. Depending on the chunk and the type of the value, some normalization occurs. 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.

Definition load_result (chunk: memory_chunk) (v: val) :=
match chunk, v with
| Mint8signed, Vint n => Vint (Int.sign_ext 8 n)
| Mint8unsigned, Vint n => Vint (Int.zero_ext 8 n)
| Mint16signed, Vint n => Vint (Int.sign_ext 16 n)
| Mint16unsigned, Vint n => Vint (Int.zero_ext 16 n)
| Mint32, Vint n => Vint n
| Mint32, Vptr b ofs => Vptr b ofs
| Mint64, Vlong n => Vlong n
| Mfloat32, Vsingle f => Vsingle f
| Mfloat64, Vfloat f => Vfloat f
| Many32, (Vint _ | Vptr _ _ | Vsingle _) => v
| Many64, _ => v
| _, _ => Vundef
end.

forall chunk v, has_type (load_result chunk v) (type_of_chunk chunk).
Proof.
intros. destruct chunk; destruct v; simpl; auto.
Qed.

forall v ty, has_type v ty -> load_result (chunk_of_type ty) v = v.
Proof.
unfold has_type; intros. destruct v; destruct ty; try contradiction; auto.
Qed.

Theorems on arithmetic operations.

Theorem cast8unsigned_and:
forall x, zero_ext 8 x = and x (Vint(Int.repr 255)).
Proof.
destruct x; simpl; auto. decEq.
change 255 with (two_p 8 - 1). apply Int.zero_ext_and. omega.
Qed.

Theorem cast16unsigned_and:
forall x, zero_ext 16 x = and x (Vint(Int.repr 65535)).
Proof.
destruct x; simpl; auto. decEq.
change 65535 with (two_p 16 - 1). apply Int.zero_ext_and. omega.
Qed.

Theorem bool_of_val_of_bool:
forall b1 b2, bool_of_val (of_bool b1) b2 -> b1 = b2.
Proof.
intros. destruct b1; simpl in H; inv H; auto.
Qed.

Theorem bool_of_val_of_optbool:
forall ob b, bool_of_val (of_optbool ob) b -> ob = Some b.
Proof.
intros. destruct ob; simpl in H.
destruct b0; simpl in H; inv H; auto.
inv H.
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_negb_3:
forall ob, of_optbool (option_map negb ob) = notbool (of_optbool ob).
Proof.
destruct ob; auto. destruct b; auto.
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 notbool_idem4:
forall ob, notbool (notbool (of_optbool ob)) = of_optbool ob.
Proof.
destruct ob; auto. destruct b; auto.
Qed.

Proof.
destruct x; destruct y; simpl; auto.
Qed.

Proof.
destruct x; destruct y; destruct z; simpl; auto.
Qed.

Proof.
Qed.

Proof.
Qed.

Theorem neg_zero: neg Vzero = Vzero.
Proof.
reflexivity.
Qed.

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_opp_add: forall x y, sub x (Vint (Int.neg y)) = add x (Vint y).
Proof.
destruct x; auto; rewrite Int.sub_add_opp; rewrite Int.neg_involutive; auto.
Qed.

forall v1 v2 i, sub (add v1 (Vint i)) v2 = add (sub v1 v2) (Vint i).
Proof.
destruct v1; destruct v2; intros; simpl; auto.
case (eq_block b b0); intro. rewrite Int.sub_add_l. auto. reflexivity.
Qed.

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.
case (eq_block b b0); intro. simpl. decEq.
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.

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.
Qed.

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.
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 Int.zwordsize.
rewrite (Int.is_power2_range _ _ H). decEq. apply Int.mul_pow2. auto.
Qed.

Theorem mods_divs:
forall x y z,
mods x y = Some z -> exists v, divs x y = Some v /\ z = sub x (mul v y).
Proof.
intros. destruct x; destruct y; simpl in *; try discriminate.
destruct (Int.eq i0 Int.zero
|| Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H.
exists (Vint (Int.divs i i0)); split; auto.
simpl. rewrite Int.mods_divs. auto.
Qed.

Theorem modu_divu:
forall x y z,
modu x y = Some z -> exists v, divu x y = Some v /\ z = sub x (mul v y).
Proof.
intros. destruct x; destruct y; simpl in *; try discriminate.
destruct (Int.eq i0 Int.zero) eqn:?; inv H.
exists (Vint (Int.divu i i0)); split; auto.
simpl. rewrite Int.modu_divu. auto.
generalize (Int.eq_spec i0 Int.zero). rewrite Heqb; auto.
Qed.

Theorem divs_pow2:
forall x n logn y,
Int.is_power2 n = Some logn -> Int.ltu logn (Int.repr 31) = true ->
divs x (Vint n) = Some y ->
shrx x (Vint logn) = Some y.
Proof.
intros; destruct x; simpl in H1; inv H1.
destruct (Int.eq n Int.zero
|| Int.eq i (Int.repr Int.min_signed) && Int.eq n Int.mone); inv H3.
simpl. rewrite H0. decEq. decEq. symmetry. apply Int.divs_pow2. auto.
Qed.

Theorem divu_pow2:
forall x n logn y,
Int.is_power2 n = Some logn ->
divu x (Vint n) = Some y ->
shru x (Vint logn) = y.
Proof.
intros; destruct x; simpl in H0; inv H0.
destruct (Int.eq n Int.zero); inv H2.
simpl.
rewrite (Int.is_power2_range _ _ H).
decEq. symmetry. apply Int.divu_pow2. auto.
Qed.

Theorem modu_pow2:
forall x n logn y,
Int.is_power2 n = Some logn ->
modu x (Vint n) = Some y ->
and x (Vint (Int.sub n Int.one)) = y.
Proof.
intros; destruct x; simpl in H0; inv H0.
destruct (Int.eq n Int.zero); inv H2.
simpl. decEq. symmetry. 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 not_xor: forall x, notint x = xor x (Vint Int.mone).
Proof.
destruct x; simpl; auto.
Qed.

Theorem shl_mul: forall x y, mul x (shl Vone y) = shl x y.
Proof.
destruct x; destruct y; simpl; auto.
case (Int.ltu i0 Int.iwordsize); auto.
decEq. symmetry. apply Int.shl_mul.
Qed.

Theorem shl_rolm:
forall x n,
Int.ltu n Int.iwordsize = 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.iwordsize = true ->
shru x (Vint n) = rolm x (Int.sub Int.iwordsize 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 z,
shrx x y = Some z ->
add (shr x y) (shr_carry x y) = z.
Proof.
intros. destruct x; destruct y; simpl in H; inv H.
destruct (Int.ltu i0 (Int.repr 31)) eqn:?; inv H1.
exploit Int.ltu_inv; eauto. change (Int.unsigned (Int.repr 31)) with 31. intros.
assert (Int.ltu i0 Int.iwordsize = true).
unfold Int.ltu. apply zlt_true. change (Int.unsigned Int.iwordsize) with 32. omega.
simpl. rewrite H0. simpl. decEq. rewrite Int.shrx_carry; auto.
Qed.

Theorem shrx_shr:
forall x y z,
shrx x y = Some z ->
exists p, exists q,
x = Vint p /\ y = Vint q /\
z = shr (if Int.lt p Int.zero then add x (Vint (Int.sub (Int.shl Int.one q) Int.one)) else x) (Vint q).
Proof.
intros. destruct x; destruct y; simpl in H; inv H.
destruct (Int.ltu i0 (Int.repr 31)) eqn:?; inv H1.
exploit Int.ltu_inv; eauto. change (Int.unsigned (Int.repr 31)) with 31. intros.
assert (Int.ltu i0 Int.iwordsize = true).
unfold Int.ltu. apply zlt_true. change (Int.unsigned Int.iwordsize) with 32. omega.
exists i; exists i0; intuition.
rewrite Int.shrx_shr; auto. destruct (Int.lt i Int.zero); simpl; rewrite H0; auto.
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.modu (Int.add n1 n2) Int.iwordsize)
(Int.and (Int.rol m1 n2) m2).
Proof.
intros; destruct x; simpl; auto.
decEq.
apply Int.rolm_rolm. apply int_wordsize_divides_modulus.
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 negate_cmp_bool:
forall c x y, cmp_bool (negate_comparison c) x y = option_map negb (cmp_bool c x y).
Proof.
destruct x; destruct y; simpl; auto. rewrite Int.negate_cmp. auto.
Qed.

Theorem negate_cmpu_bool:
forall valid_ptr c x y,
cmpu_bool valid_ptr (negate_comparison c) x y = option_map negb (cmpu_bool valid_ptr c x y).
Proof.
assert (forall c,
cmp_different_blocks (negate_comparison c) = option_map negb (cmp_different_blocks c)).
destruct c; auto.
destruct x; destruct y; simpl; auto.
rewrite Int.negate_cmpu. auto.
destruct (Int.eq i Int.zero && (valid_ptr b (Int.unsigned i0) || valid_ptr b (Int.unsigned i0 - 1))); auto.
destruct (Int.eq i0 Int.zero && (valid_ptr b (Int.unsigned i) || valid_ptr b (Int.unsigned i - 1))); auto.
destruct (eq_block b b0).
destruct ((valid_ptr b (Int.unsigned i) || valid_ptr b (Int.unsigned i - 1)) &&
(valid_ptr b0 (Int.unsigned i0) || valid_ptr b0 (Int.unsigned i0 - 1))).
rewrite Int.negate_cmpu. auto.
auto.
destruct (valid_ptr b (Int.unsigned i) && valid_ptr b0 (Int.unsigned i0)); auto.
Qed.

Lemma not_of_optbool:
forall ob, of_optbool (option_map negb ob) = notbool (of_optbool ob).
Proof.
destruct ob; auto. destruct b; auto.
Qed.

Theorem negate_cmp:
forall c x y,
cmp (negate_comparison c) x y = notbool (cmp c x y).
Proof.
intros. unfold cmp. rewrite negate_cmp_bool. apply not_of_optbool.
Qed.

Theorem negate_cmpu:
forall valid_ptr c x y,
cmpu valid_ptr (negate_comparison c) x y =
notbool (cmpu valid_ptr c x y).
Proof.
intros. unfold cmpu. rewrite negate_cmpu_bool. apply not_of_optbool.
Qed.

Theorem swap_cmp_bool:
forall c x y,
cmp_bool (swap_comparison c) x y = cmp_bool c y x.
Proof.
destruct x; destruct y; simpl; auto. rewrite Int.swap_cmp. auto.
Qed.

Theorem swap_cmpu_bool:
forall valid_ptr c x y,
cmpu_bool valid_ptr (swap_comparison c) x y =
cmpu_bool valid_ptr c y x.
Proof.
assert (forall c, cmp_different_blocks (swap_comparison c) = cmp_different_blocks c).
destruct c; auto.
destruct x; destruct y; simpl; auto.
rewrite Int.swap_cmpu. auto.
destruct (Int.eq i Int.zero && (valid_ptr b (Int.unsigned i0) || valid_ptr b (Int.unsigned i0 - 1))); auto.
destruct (Int.eq i0 Int.zero && (valid_ptr b (Int.unsigned i) || valid_ptr b (Int.unsigned i - 1))); auto.
destruct (eq_block b b0); subst.
rewrite dec_eq_true.
destruct (valid_ptr b0 (Int.unsigned i) || valid_ptr b0 (Int.unsigned i - 1));
destruct (valid_ptr b0 (Int.unsigned i0) || valid_ptr b0 (Int.unsigned i0 - 1));
simpl; auto.
rewrite Int.swap_cmpu. auto.
rewrite dec_eq_false by auto.
destruct (valid_ptr b (Int.unsigned i));
destruct (valid_ptr b0 (Int.unsigned i0)); simpl; auto.
Qed.

Theorem negate_cmpf_eq:
forall v1 v2, notbool (cmpf Cne v1 v2) = cmpf Ceq v1 v2.
Proof.
destruct v1; destruct v2; auto. unfold cmpf, cmpf_bool.
rewrite Float.cmp_ne_eq. destruct (Float.cmp Ceq f f0); auto.
Qed.

Theorem negate_cmpf_ne:
forall v1 v2, notbool (cmpf Ceq v1 v2) = cmpf Cne v1 v2.
Proof.
destruct v1; destruct v2; auto. unfold cmpf, cmpf_bool.
rewrite Float.cmp_ne_eq. destruct (Float.cmp Ceq f f0); auto.
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; auto. unfold cmpf, cmpf_bool.
rewrite Float.cmp_le_lt_eq.
destruct (Float.cmp Clt f f0); destruct (Float.cmp Ceq f f0); auto.
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; auto. unfold cmpf, cmpf_bool.
rewrite Float.cmp_ge_gt_eq.
destruct (Float.cmp Cgt f f0); destruct (Float.cmp Ceq f f0); auto.
Qed.

Theorem cmp_ne_0_optbool:
forall ob, cmp Cne (of_optbool ob) (Vint Int.zero) = of_optbool ob.
Proof.
intros. destruct ob; simpl; auto. destruct b; auto.
Qed.

Theorem cmp_eq_1_optbool:
forall ob, cmp Ceq (of_optbool ob) (Vint Int.one) = of_optbool ob.
Proof.
intros. destruct ob; simpl; auto. destruct b; auto.
Qed.

Theorem cmp_eq_0_optbool:
forall ob, cmp Ceq (of_optbool ob) (Vint Int.zero) = of_optbool (option_map negb ob).
Proof.
intros. destruct ob; simpl; auto. destruct b; auto.
Qed.

Theorem cmp_ne_1_optbool:
forall ob, cmp Cne (of_optbool ob) (Vint Int.one) = of_optbool (option_map negb ob).
Proof.
intros. destruct ob; simpl; auto. destruct b; auto.
Qed.

Theorem cmpu_ne_0_optbool:
forall valid_ptr ob,
cmpu valid_ptr Cne (of_optbool ob) (Vint Int.zero) = of_optbool ob.
Proof.
intros. destruct ob; simpl; auto. destruct b; auto.
Qed.

Theorem cmpu_eq_1_optbool:
forall valid_ptr ob,
cmpu valid_ptr Ceq (of_optbool ob) (Vint Int.one) = of_optbool ob.
Proof.
intros. destruct ob; simpl; auto. destruct b; auto.
Qed.

Theorem cmpu_eq_0_optbool:
forall valid_ptr ob,
cmpu valid_ptr Ceq (of_optbool ob) (Vint Int.zero) = of_optbool (option_map negb ob).
Proof.
intros. destruct ob; simpl; auto. destruct b; auto.
Qed.

Theorem cmpu_ne_1_optbool:
forall valid_ptr ob,
cmpu valid_ptr Cne (of_optbool ob) (Vint Int.one) = of_optbool (option_map negb ob).
Proof.
intros. destruct ob; simpl; auto. destruct b; auto.
Qed.

Lemma zero_ext_and:
forall n v,
0 < n < Int.zwordsize ->
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.
Qed.

Lemma rolm_lt_zero:
forall v, rolm v Int.one Int.one = cmp Clt v (Vint Int.zero).
Proof.
intros. unfold cmp, cmp_bool; destruct v; simpl; auto.
transitivity (Vint (Int.shru i (Int.repr (Int.zwordsize - 1)))).
decEq. symmetry. rewrite Int.shru_rolm. auto. auto.
rewrite Int.shru_lt_zero. destruct (Int.lt i Int.zero); auto.
Qed.

Lemma rolm_ge_zero:
forall v,
xor (rolm v Int.one Int.one) (Vint Int.one) = cmp Cge v (Vint Int.zero).
Proof.
intros. rewrite rolm_lt_zero. destruct v; simpl; auto.
unfold cmp; simpl. destruct (Int.lt i Int.zero); auto.
Qed.

The ``is less defined'' relation between values. A value is less defined than itself, and Vundef is less defined than any value.

Inductive lessdef: val -> val -> Prop :=
| lessdef_refl: forall v, lessdef v v
| lessdef_undef: forall v, lessdef Vundef v.

Lemma lessdef_same:
forall v1 v2, v1 = v2 -> lessdef v1 v2.
Proof.
intros. subst v2. constructor.
Qed.

Lemma lessdef_trans:
forall v1 v2 v3, lessdef v1 v2 -> lessdef v2 v3 -> lessdef v1 v3.
Proof.
intros. inv H. auto. constructor.
Qed.

Inductive lessdef_list: list val -> list val -> Prop :=
| lessdef_list_nil:
lessdef_list nil nil
| lessdef_list_cons:
forall v1 v2 vl1 vl2,
lessdef v1 v2 -> lessdef_list vl1 vl2 ->
lessdef_list (v1 :: vl1) (v2 :: vl2).

Hint Resolve lessdef_refl lessdef_undef lessdef_list_nil lessdef_list_cons.

Lemma lessdef_list_inv:
forall vl1 vl2, lessdef_list vl1 vl2 -> vl1 = vl2 \/ In Vundef vl1.
Proof.
induction 1; simpl.
tauto.
inv H. destruct IHlessdef_list.
left; congruence. tauto. tauto.
Qed.

Lemma lessdef_list_trans:
forall vl1 vl2, lessdef_list vl1 vl2 -> forall vl3, lessdef_list vl2 vl3 -> lessdef_list vl1 vl3.
Proof.
induction 1; intros vl3 LD; inv LD; constructor; eauto using lessdef_trans.
Qed.

Compatibility of operations with the lessdef relation.

forall chunk v1 v2,
Proof.
intros. inv H. auto. destruct chunk; simpl; auto.
Qed.

Lemma zero_ext_lessdef:
forall n v1 v2, lessdef v1 v2 -> lessdef (zero_ext n v1) (zero_ext n v2).
Proof.
intros; inv H; simpl; auto.
Qed.

Lemma sign_ext_lessdef:
forall n v1 v2, lessdef v1 v2 -> lessdef (sign_ext n v1) (sign_ext n v2).
Proof.
intros; inv H; simpl; auto.
Qed.

Lemma singleoffloat_lessdef:
forall v1 v2, lessdef v1 v2 -> lessdef (singleoffloat v1) (singleoffloat v2).
Proof.
intros; inv H; simpl; auto.
Qed.

forall v1 v1' v2 v2',
lessdef v1 v1' -> lessdef v2 v2' -> lessdef (add v1 v2) (add v1' v2').
Proof.
intros. inv H. inv H0. auto. destruct v1'; simpl; auto. simpl; auto.
Qed.

Lemma cmpu_bool_lessdef:
forall valid_ptr valid_ptr' c v1 v1' v2 v2' b,
(forall b ofs, valid_ptr b ofs = true -> valid_ptr' b ofs = true) ->
lessdef v1 v1' -> lessdef v2 v2' ->
cmpu_bool valid_ptr c v1 v2 = Some b ->
cmpu_bool valid_ptr' c v1' v2' = Some b.
Proof.
intros.
assert (A: forall b ofs, valid_ptr b ofs || valid_ptr b (ofs - 1) = true ->
valid_ptr' b ofs || valid_ptr' b (ofs - 1) = true).
{ intros until ofs. rewrite ! orb_true_iff. intuition. }
destruct v1; simpl in H2; try discriminate;
destruct v2; simpl in H2; try discriminate;
inv H0; inv H1; simpl; auto.
destruct (Int.eq i Int.zero && (valid_ptr b0 (Int.unsigned i0) || valid_ptr b0 (Int.unsigned i0 - 1))) eqn:E; try discriminate.
InvBooleans. rewrite H0, A by auto. auto.
destruct (Int.eq i0 Int.zero && (valid_ptr b0 (Int.unsigned i) || valid_ptr b0 (Int.unsigned i - 1))) eqn:E; try discriminate.
InvBooleans. rewrite H0, A by auto. auto.
destruct (eq_block b0 b1).
destruct (valid_ptr b0 (Int.unsigned i) || valid_ptr b0 (Int.unsigned i - 1)) eqn:?; try discriminate.
destruct (valid_ptr b1 (Int.unsigned i0) || valid_ptr b1 (Int.unsigned i0 - 1)) eqn:?; try discriminate.
erewrite ! A by eauto. auto.
destruct (valid_ptr b0 (Int.unsigned i)) eqn:?; try discriminate.
destruct (valid_ptr b1 (Int.unsigned i0)) eqn:?; try discriminate.
erewrite ! H by eauto. auto.
Qed.

Lemma of_optbool_lessdef:
forall ob ob',
(forall b, ob = Some b -> ob' = Some b) ->
lessdef (of_optbool ob) (of_optbool ob').
Proof.
intros. destruct ob; simpl; auto. rewrite (H b); auto.
Qed.

Lemma longofwords_lessdef:
forall v1 v2 v1' v2',
lessdef v1 v1' -> lessdef v2 v2' -> lessdef (longofwords v1 v2) (longofwords v1' v2').
Proof.
intros. unfold longofwords. inv H; auto. inv H0; auto. destruct v1'; auto.
Qed.

Lemma loword_lessdef:
forall v v', lessdef v v' -> lessdef (loword v) (loword v').
Proof.
intros. inv H; auto.
Qed.

Lemma hiword_lessdef:
forall v v', lessdef v v' -> lessdef (hiword v) (hiword v').
Proof.
intros. inv H; auto.
Qed.

# Values and memory injections

A memory injection f is a function from addresses to either None or Some of an address and an offset. It defines a correspondence between the blocks of two memory states m1 and m2:
• if f b = None, the block b of m1 has no equivalent in m2;
• if f b = Some(b', ofs), the block b of m2 corresponds to a sub-block at offset ofs of the block b' in m2.

Definition meminj : Type := block -> option (block * Z).

A memory injection defines a relation between values that is the identity relation, except for pointer values which are shifted as prescribed by the memory injection. Moreover, Vundef values inject into any other value.

Inductive inject (mi: meminj): val -> val -> Prop :=
| inject_int:
forall i, inject mi (Vint i) (Vint i)
| inject_long:
forall i, inject mi (Vlong i) (Vlong i)
| inject_float:
forall f, inject mi (Vfloat f) (Vfloat f)
| inject_single:
forall f, inject mi (Vsingle f) (Vsingle f)
| inject_ptr:
forall b1 ofs1 b2 ofs2 delta,
mi b1 = Some (b2, delta) ->
ofs2 = Int.add ofs1 (Int.repr delta) ->
inject mi (Vptr b1 ofs1) (Vptr b2 ofs2)
| val_inject_undef: forall v,
inject mi Vundef v.

Hint Constructors inject.

Inductive inject_list (mi: meminj): list val -> list val-> Prop:=
| inject_list_nil :
inject_list mi nil nil
| inject_list_cons : forall v v' vl vl' ,
inject mi v v' -> inject_list mi vl vl'->
inject_list mi (v :: vl) (v' :: vl').

Hint Resolve inject_list_nil inject_list_cons.

Section VAL_INJ_OPS.

Variable f: meminj.

forall chunk v1 v2,
inject f v1 v2 ->
Proof.
intros. inv H; destruct chunk; simpl; econstructor; eauto.
Qed.

forall v1 v1' v2 v2',
inject f v1 v1' ->
inject f v2 v2' ->
Proof.
intros. inv H; inv H0; simpl; econstructor; eauto.
Qed.

Remark sub_inject:
forall v1 v1' v2 v2',
inject f v1 v1' ->
inject f v2 v2' ->
inject f (Val.sub v1 v2) (Val.sub v1' v2').
Proof.
intros. inv H; inv H0; simpl; auto.
destruct (eq_block b1 b0); auto. subst. rewrite H1 in H. inv H. rewrite dec_eq_true.
rewrite Int.sub_shifted. auto.
Qed.

Lemma cmp_bool_inject:
forall c v1 v2 v1' v2' b,
inject f v1 v1' ->
inject f v2 v2' ->
Val.cmp_bool c v1 v2 = Some b ->
Val.cmp_bool c v1' v2' = Some b.
Proof.
intros. inv H; simpl in H1; try discriminate; inv H0; simpl in H1; try discriminate; simpl; auto.
Qed.

Variable (valid_ptr1 valid_ptr2 : block -> Z -> bool).

Let weak_valid_ptr1 b ofs := valid_ptr1 b ofs || valid_ptr1 b (ofs - 1).
Let weak_valid_ptr2 b ofs := valid_ptr2 b ofs || valid_ptr2 b (ofs - 1).

Hypothesis valid_ptr_inj:
forall b1 ofs b2 delta,
f b1 = Some(b2, delta) ->
valid_ptr1 b1 (Int.unsigned ofs) = true ->
valid_ptr2 b2 (Int.unsigned (Int.add ofs (Int.repr delta))) = true.

Hypothesis weak_valid_ptr_inj:
forall b1 ofs b2 delta,
f b1 = Some(b2, delta) ->
weak_valid_ptr1 b1 (Int.unsigned ofs) = true ->
weak_valid_ptr2 b2 (Int.unsigned (Int.add ofs (Int.repr delta))) = true.

Hypothesis weak_valid_ptr_no_overflow:
forall b1 ofs b2 delta,
f b1 = Some(b2, delta) ->
weak_valid_ptr1 b1 (Int.unsigned ofs) = true ->
0 <= Int.unsigned ofs + Int.unsigned (Int.repr delta) <= Int.max_unsigned.

Hypothesis valid_different_ptrs_inj:
forall b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,
b1 <> b2 ->
valid_ptr1 b1 (Int.unsigned ofs1) = true ->
valid_ptr1 b2 (Int.unsigned ofs2) = true ->
f b1 = Some (b1', delta1) ->
f b2 = Some (b2', delta2) ->
b1' <> b2' \/

Lemma cmpu_bool_inject:
forall c v1 v2 v1' v2' b,
inject f v1 v1' ->
inject f v2 v2' ->
Val.cmpu_bool valid_ptr1 c v1 v2 = Some b ->
Val.cmpu_bool valid_ptr2 c v1' v2' = Some b.
Proof.
intros. inv H; simpl in H1; try discriminate; inv H0; simpl in H1; try discriminate; simpl; auto.
- fold (weak_valid_ptr1 b1 (Int.unsigned ofs1)) in H1.
fold (weak_valid_ptr2 b2 (Int.unsigned (Int.add ofs1 (Int.repr delta)))).
destruct (Int.eq i Int.zero); auto.
destruct (weak_valid_ptr1 b1 (Int.unsigned ofs1)) eqn:E; try discriminate.
erewrite weak_valid_ptr_inj by eauto. auto.
- fold (weak_valid_ptr1 b1 (Int.unsigned ofs1)) in H1.
fold (weak_valid_ptr2 b2 (Int.unsigned (Int.add ofs1 (Int.repr delta)))).
destruct (Int.eq i Int.zero); auto.
destruct (weak_valid_ptr1 b1 (Int.unsigned ofs1)) eqn:E; try discriminate.
erewrite weak_valid_ptr_inj by eauto. auto.
- fold (weak_valid_ptr1 b1 (Int.unsigned ofs1)) in H1.
fold (weak_valid_ptr1 b0 (Int.unsigned ofs0)) in H1.
fold (weak_valid_ptr2 b2 (Int.unsigned (Int.add ofs1 (Int.repr delta)))).
fold (weak_valid_ptr2 b3 (Int.unsigned (Int.add ofs0 (Int.repr delta0)))).
destruct (eq_block b1 b0); subst.
rewrite H in H2. inv H2. rewrite dec_eq_true.
destruct (weak_valid_ptr1 b0 (Int.unsigned ofs1)) eqn:?; try discriminate.
destruct (weak_valid_ptr1 b0 (Int.unsigned ofs0)) eqn:?; try discriminate.
erewrite !weak_valid_ptr_inj by eauto. simpl.
rewrite <-H1. simpl. decEq. apply Int.translate_cmpu; eauto.
destruct (valid_ptr1 b1 (Int.unsigned ofs1)) eqn:?; try discriminate.
destruct (valid_ptr1 b0 (Int.unsigned ofs0)) eqn:?; try discriminate.
destruct (eq_block b2 b3); subst.
assert (valid_ptr_implies: forall b ofs, valid_ptr1 b ofs = true -> weak_valid_ptr1 b ofs = true).
intros. unfold weak_valid_ptr1. rewrite H0; auto.
erewrite !weak_valid_ptr_inj by eauto using valid_ptr_implies. simpl.
exploit valid_different_ptrs_inj; eauto. intros [?|?]; [congruence |].
destruct c; simpl in H1; inv H1.
simpl; decEq. rewrite Int.eq_false; auto. congruence.
simpl; decEq. rewrite Int.eq_false; auto. congruence.
now erewrite !valid_ptr_inj by eauto.
Qed.

Lemma longofwords_inject:
forall v1 v2 v1' v2',
inject f v1 v1' -> inject f v2 v2' -> inject f (Val.longofwords v1 v2) (Val.longofwords v1' v2').
Proof.
intros. unfold Val.longofwords. inv H; auto. inv H0; auto.
Qed.

Lemma loword_inject:
forall v v', inject f v v' -> inject f (Val.loword v) (Val.loword v').
Proof.
intros. unfold Val.loword; inv H; auto.
Qed.

Lemma hiword_inject:
forall v v', inject f v v' -> inject f (Val.hiword v) (Val.hiword v').
Proof.
intros. unfold Val.hiword; inv H; auto.
Qed.

End VAL_INJ_OPS.

End Val.

Notation meminj := Val.meminj.

Monotone evolution of a memory injection.

Definition inject_incr (f1 f2: meminj) : Prop :=
forall b b' delta, f1 b = Some(b', delta) -> f2 b = Some(b', delta).

Lemma inject_incr_refl :
forall f , inject_incr f f .
Proof.
unfold inject_incr. auto. Qed.

Lemma inject_incr_trans :
forall f1 f2 f3,
inject_incr f1 f2 -> inject_incr f2 f3 -> inject_incr f1 f3 .
Proof.
unfold inject_incr; intros. eauto.
Qed.

Lemma val_inject_incr:
forall f1 f2 v v',
inject_incr f1 f2 ->
Val.inject f1 v v' ->
Val.inject f2 v v'.
Proof.
intros. inv H0; eauto.
Qed.

Lemma val_inject_list_incr:
forall f1 f2 vl vl' ,
inject_incr f1 f2 -> Val.inject_list f1 vl vl' ->
Val.inject_list f2 vl vl'.
Proof.
induction vl; intros; inv H0. auto.
constructor. eapply val_inject_incr; eauto. auto.
Qed.

Hint Resolve inject_incr_refl val_inject_incr val_inject_list_incr.

Lemma val_inject_lessdef:
forall v1 v2, Val.lessdef v1 v2 <-> Val.inject (fun b => Some(b, 0)) v1 v2.
Proof.
intros; split; intros.
inv H; auto. destruct v2; econstructor; eauto. rewrite Int.add_zero; auto.
inv H; auto. inv H0. rewrite Int.add_zero; auto.
Qed.

Lemma val_inject_list_lessdef:
forall vl1 vl2, Val.lessdef_list vl1 vl2 <-> Val.inject_list (fun b => Some(b, 0)) vl1 vl2.
Proof.
intros; split.
induction 1; constructor; auto. apply val_inject_lessdef; auto.
induction 1; constructor; auto. apply val_inject_lessdef; auto.
Qed.

The identity injection gives rise to the "less defined than" relation.

Definition inject_id : meminj := fun b => Some(b, 0).

Lemma val_inject_id:
forall v1 v2,
Val.inject inject_id v1 v2 <-> Val.lessdef v1 v2.
Proof.
intros; split; intros.
inv H; auto.
unfold inject_id in H0. inv H0. rewrite Int.add_zero. constructor.
inv H. destruct v2; econstructor. unfold inject_id; reflexivity. rewrite Int.add_zero; auto.
constructor.
Qed.

Composing two memory injections

Definition compose_meminj (f f': meminj) : meminj :=
fun b =>
match f b with
| None => None
| Some(b', delta) =>
match f' b' with
| None => None
| Some(b'', delta') => Some(b'', delta + delta')
end
end.

Lemma val_inject_compose:
forall f f' v1 v2 v3,
Val.inject f v1 v2 -> Val.inject f' v2 v3 ->
Val.inject (compose_meminj f f') v1 v3.
Proof.
intros. inv H; auto; inv H0; auto. econstructor.
unfold compose_meminj; rewrite H1; rewrite H3; eauto.