# Module Integers

Formalizations of machine integers modulo 2N.

Require Import Eqdep_dec Zquot Zwf.
Require Import Coqlib.
Require Archi.

# Comparisons

Inductive comparison : Type :=
| Ceq : comparison (* same *)
| Cne : comparison (* different *)
| Clt : comparison (* less than *)
| Cle : comparison (* less than or equal *)
| Cgt : comparison (* greater than *)
| Cge : comparison. (* greater than or equal *)

Definition negate_comparison (c: comparison): comparison :=
match c with
| Ceq => Cne
| Cne => Ceq
| Clt => Cge
| Cle => Cgt
| Cgt => Cle
| Cge => Clt
end.

Definition swap_comparison (c: comparison): comparison :=
match c with
| Ceq => Ceq
| Cne => Cne
| Clt => Cgt
| Cle => Cge
| Cgt => Clt
| Cge => Cle
end.

# Parameterization by the word size, in bits.

Module Type WORDSIZE.
Parameter wordsize: nat.
Axiom wordsize_not_zero: wordsize <> 0%nat.
End WORDSIZE.

Local Unset Elimination Schemes.
Local Unset Case Analysis Schemes.

Module Make(WS: WORDSIZE).

Definition wordsize: nat := WS.wordsize.
Definition zwordsize: Z := Z_of_nat wordsize.
Definition modulus : Z := two_power_nat wordsize.
Definition half_modulus : Z := modulus / 2.
Definition max_unsigned : Z := modulus - 1.
Definition max_signed : Z := half_modulus - 1.
Definition min_signed : Z := - half_modulus.

Remark wordsize_pos: zwordsize > 0.
Proof.
unfold zwordsize, wordsize. generalize WS.wordsize_not_zero. omega.
Qed.

Remark modulus_power: modulus = two_p zwordsize.
Proof.
unfold modulus. apply two_power_nat_two_p.
Qed.

Remark modulus_pos: modulus > 0.
Proof.
rewrite modulus_power. apply two_p_gt_ZERO. generalize wordsize_pos; omega.
Qed.

# Representation of machine integers

A machine integer (type int) is represented as a Coq arbitrary-precision integer (type Z) plus a proof that it is in the range 0 (included) to modulus (excluded).

Record int: Type := mkint { intval: Z; intrange: -1 < intval < modulus }.

Fast normalization modulo 2^wordsize

Fixpoint P_mod_two_p (p: positive) (n: nat) {struct n} : Z :=
match n with
| O => 0
| S m =>
match p with
| xH => 1
| xO q => Z.double (P_mod_two_p q m)
| xI q => Z.succ_double (P_mod_two_p q m)
end
end.

Definition Z_mod_modulus (x: Z) : Z :=
match x with
| Z0 => 0
| Zpos p => P_mod_two_p p wordsize
| Zneg p => let r := P_mod_two_p p wordsize in if zeq r 0 then 0 else modulus - r
end.

Lemma P_mod_two_p_range:
forall n p, 0 <= P_mod_two_p p n < two_power_nat n.
Proof.
induction n; simpl; intros.
- rewrite two_power_nat_O. omega.
- rewrite two_power_nat_S. destruct p.
+ generalize (IHn p). rewrite Z.succ_double_spec. omega.
+ generalize (IHn p). rewrite Z.double_spec. omega.
+ generalize (two_power_nat_pos n). omega.
Qed.

Lemma P_mod_two_p_eq:
forall n p, P_mod_two_p p n = (Zpos p) mod (two_power_nat n).
Proof.
assert (forall n p, exists y, Zpos p = y * two_power_nat n + P_mod_two_p p n).
{
induction n; simpl; intros.
- rewrite two_power_nat_O. exists (Zpos p). ring.
- rewrite two_power_nat_S. destruct p.
+ destruct (IHn p) as [y EQ]. exists y.
change (Zpos p~1) with (2 * Zpos p + 1). rewrite EQ.
rewrite Z.succ_double_spec. ring.
+ destruct (IHn p) as [y EQ]. exists y.
change (Zpos p~0) with (2 * Zpos p). rewrite EQ.
rewrite (Z.double_spec (P_mod_two_p p n)). ring.
+ exists 0; omega.
}
intros.
destruct (H n p) as [y EQ].
symmetry. apply Zmod_unique with y. auto. apply P_mod_two_p_range.
Qed.

Lemma Z_mod_modulus_range:
forall x, 0 <= Z_mod_modulus x < modulus.
Proof.
intros; unfold Z_mod_modulus.
destruct x.
- generalize modulus_pos; omega.
- apply P_mod_two_p_range.
- set (r := P_mod_two_p p wordsize).
assert (0 <= r < modulus) by apply P_mod_two_p_range.
destruct (zeq r 0).
+ generalize modulus_pos; omega.
+ omega.
Qed.

Lemma Z_mod_modulus_range':
forall x, -1 < Z_mod_modulus x < modulus.
Proof.
intros. generalize (Z_mod_modulus_range x); omega.
Qed.

Lemma Z_mod_modulus_eq:
forall x, Z_mod_modulus x = x mod modulus.
Proof.
intros. unfold Z_mod_modulus. destruct x.
- rewrite Zmod_0_l. auto.
- apply P_mod_two_p_eq.
- generalize (P_mod_two_p_range wordsize p) (P_mod_two_p_eq wordsize p).
fold modulus. intros A B.
exploit (Z_div_mod_eq (Zpos p) modulus). apply modulus_pos. intros C.
set (q := Zpos p / modulus) in *.
set (r := P_mod_two_p p wordsize) in *.
rewrite <- B in C.
change (Z.neg p) with (- (Z.pos p)). destruct (zeq r 0).
+ symmetry. apply Zmod_unique with (-q). rewrite C; rewrite e. ring.
generalize modulus_pos; omega.
+ symmetry. apply Zmod_unique with (-q - 1). rewrite C. ring.
omega.
Qed.

The unsigned and signed functions return the Coq integer corresponding to the given machine integer, interpreted as unsigned or signed respectively.

Definition unsigned (n: int) : Z := intval n.

Definition signed (n: int) : Z :=
let x := unsigned n in
if zlt x half_modulus then x else x - modulus.

Conversely, repr takes a Coq integer and returns the corresponding machine integer. The argument is treated modulo modulus.

Definition repr (x: Z) : int :=
mkint (Z_mod_modulus x) (Z_mod_modulus_range' x).

Definition zero := repr 0.
Definition one := repr 1.
Definition mone := repr (-1).
Definition iwordsize := repr zwordsize.

Lemma mkint_eq:
forall x y Px Py, x = y -> mkint x Px = mkint y Py.
Proof.
intros. subst y.
assert (forall (n m: Z) (P1 P2: n < m), P1 = P2).
{
unfold Zlt; intros.
apply eq_proofs_unicity.
intros c1 c2. destruct c1; destruct c2; (left; reflexivity) || (right; congruence).
}
destruct Px as [Px1 Px2]. destruct Py as [Py1 Py2].
rewrite (H _ _ Px1 Py1).
rewrite (H _ _ Px2 Py2).
reflexivity.
Qed.

Lemma eq_dec: forall (x y: int), {x = y} + {x <> y}.
Proof.
intros. destruct x; destruct y. destruct (zeq intval0 intval1).
left. apply mkint_eq. auto.
right. red; intro. injection H. exact n.
Defined.

# Arithmetic and logical operations over machine integers

Definition eq (x y: int) : bool :=
if zeq (unsigned x) (unsigned y) then true else false.
Definition lt (x y: int) : bool :=
if zlt (signed x) (signed y) then true else false.
Definition ltu (x y: int) : bool :=
if zlt (unsigned x) (unsigned y) then true else false.

Definition neg (x: int) : int := repr (- unsigned x).

Definition add (x y: int) : int :=
repr (unsigned x + unsigned y).
Definition sub (x y: int) : int :=
repr (unsigned x - unsigned y).
Definition mul (x y: int) : int :=
repr (unsigned x * unsigned y).

Definition divs (x y: int) : int :=
repr (Z.quot (signed x) (signed y)).
Definition mods (x y: int) : int :=
repr (Z.rem (signed x) (signed y)).

Definition divu (x y: int) : int :=
repr (unsigned x / unsigned y).
Definition modu (x y: int) : int :=
repr ((unsigned x) mod (unsigned y)).

Bitwise boolean operations.

Definition and (x y: int): int := repr (Z.land (unsigned x) (unsigned y)).
Definition or (x y: int): int := repr (Z.lor (unsigned x) (unsigned y)).
Definition xor (x y: int) : int := repr (Z.lxor (unsigned x) (unsigned y)).

Definition not (x: int) : int := xor x mone.

Shifts and rotates.

Definition shl (x y: int): int := repr (Z.shiftl (unsigned x) (unsigned y)).
Definition shru (x y: int): int := repr (Z.shiftr (unsigned x) (unsigned y)).
Definition shr (x y: int): int := repr (Z.shiftr (signed x) (unsigned y)).

Definition rol (x y: int) : int :=
let n := (unsigned y) mod zwordsize in
repr (Z.lor (Z.shiftl (unsigned x) n) (Z.shiftr (unsigned x) (zwordsize - n))).
Definition ror (x y: int) : int :=
let n := (unsigned y) mod zwordsize in
repr (Z.lor (Z.shiftr (unsigned x) n) (Z.shiftl (unsigned x) (zwordsize - n))).

Definition rolm (x a m: int): int := and (rol x a) m.

Viewed as signed divisions by powers of two, shrx rounds towards zero, while shr rounds towards minus infinity.

Definition shrx (x y: int): int :=
divs x (shl one y).

High half of full multiply.

Definition mulhu (x y: int): int := repr ((unsigned x * unsigned y) / modulus).
Definition mulhs (x y: int): int := repr ((signed x * signed y) / modulus).

Condition flags

Definition negative (x: int): int :=
if lt x zero then one else zero.

Definition add_carry (x y cin: int): int :=
if zlt (unsigned x + unsigned y + unsigned cin) modulus then zero else one.

Definition add_overflow (x y cin: int): int :=
let s := signed x + signed y + signed cin in
if zle min_signed s && zle s max_signed then zero else one.

Definition sub_borrow (x y bin: int): int :=
if zlt (unsigned x - unsigned y - unsigned bin) 0 then one else zero.

Definition sub_overflow (x y bin: int): int :=
let s := signed x - signed y - signed bin in
if zle min_signed s && zle s max_signed then zero else one.

shr_carry x y is 1 if x is negative and at least one 1 bit is shifted away.

Definition shr_carry (x y: int) : int :=
if lt x zero && negb (eq (and x (sub (shl one y) one)) zero)
then one else zero.

Zero and sign extensions

Definition Zshiftin (b: bool) (x: Z) : Z :=
if b then Z.succ_double x else Z.double x.

In pseudo-code:
```    Fixpoint Zzero_ext (n: Z) (x: Z) : Z :=
if zle n 0 then
0
else
Zshiftin (Z.odd x) (Zzero_ext (Z.pred n) (Z.div2 x)).
Fixpoint Zsign_ext (n: Z) (x: Z) : Z :=
if zle n 1 then
if Z.odd x then -1 else 0
else
Zshiftin (Z.odd x) (Zzero_ext (Z.pred n) (Z.div2 x)).```
We encode this nat-like recursion using the Z.iter iteration function, in order to make the Zzero_ext and Zsign_ext functions efficiently executable within Coq.

Definition Zzero_ext (n: Z) (x: Z) : Z :=
Z.iter n
(fun rec x => Zshiftin (Z.odd x) (rec (Z.div2 x)))
(fun x => 0)
x.

Definition Zsign_ext (n: Z) (x: Z) : Z :=
Z.iter (Z.pred n)
(fun rec x => Zshiftin (Z.odd x) (rec (Z.div2 x)))
(fun x => if Z.odd x then -1 else 0)
x.

Definition zero_ext (n: Z) (x: int) : int := repr (Zzero_ext n (unsigned x)).

Definition sign_ext (n: Z) (x: int) : int := repr (Zsign_ext n (unsigned x)).

Decomposition of a number as a sum of powers of two.

Fixpoint Z_one_bits (n: nat) (x: Z) (i: Z) {struct n}: list Z :=
match n with
| O => nil
| S m =>
if Z.odd x
then i :: Z_one_bits m (Z.div2 x) (i+1)
else Z_one_bits m (Z.div2 x) (i+1)
end.

Definition one_bits (x: int) : list int :=
List.map repr (Z_one_bits wordsize (unsigned x) 0).

Recognition of powers of two.

Definition is_power2 (x: int) : option int :=
match Z_one_bits wordsize (unsigned x) 0 with
| i :: nil => Some (repr i)
| _ => None
end.

Comparisons.

Definition cmp (c: comparison) (x y: int) : bool :=
match c with
| Ceq => eq x y
| Cne => negb (eq x y)
| Clt => lt x y
| Cle => negb (lt y x)
| Cgt => lt y x
| Cge => negb (lt x y)
end.

Definition cmpu (c: comparison) (x y: int) : bool :=
match c with
| Ceq => eq x y
| Cne => negb (eq x y)
| Clt => ltu x y
| Cle => negb (ltu y x)
| Cgt => ltu y x
| Cge => negb (ltu x y)
end.

Definition is_false (x: int) : Prop := x = zero.
Definition is_true (x: int) : Prop := x <> zero.
Definition notbool (x: int) : int := if eq x zero then one else zero.

x86-style extended division and modulus

Definition divmodu2 (nhi nlo: int) (d: int) : option (int * int) :=
if eq_dec d zero then None else
(let (q, r) := Z.div_eucl (unsigned nhi * modulus + unsigned nlo) (unsigned d) in
if zle q max_unsigned then Some(repr q, repr r) else None).

Definition divmods2 (nhi nlo: int) (d: int) : option (int * int) :=
if eq_dec d zero then None else
(let (q, r) := Z.quotrem (signed nhi * modulus + unsigned nlo) (signed d) in
if zle min_signed q && zle q max_signed then Some(repr q, repr r) else None).

# Properties of integers and integer arithmetic

## Properties of modulus, max_unsigned, etc.

Remark half_modulus_power:
half_modulus = two_p (zwordsize - 1).
Proof.
unfold half_modulus. rewrite modulus_power.
set (ws1 := zwordsize - 1).
replace (zwordsize) with (Zsucc ws1).
rewrite two_p_S. rewrite Zmult_comm. apply Z_div_mult. omega.
unfold ws1. generalize wordsize_pos; omega.
unfold ws1. omega.
Qed.

Remark half_modulus_modulus: modulus = 2 * half_modulus.
Proof.
rewrite half_modulus_power. rewrite modulus_power.
rewrite <- two_p_S. apply f_equal. omega.
generalize wordsize_pos; omega.
Qed.

Relative positions, from greatest to smallest:
```      max_unsigned
max_signed
2*wordsize-1
wordsize
0
min_signed```

Remark half_modulus_pos: half_modulus > 0.
Proof.
rewrite half_modulus_power. apply two_p_gt_ZERO. generalize wordsize_pos; omega.
Qed.

Remark min_signed_neg: min_signed < 0.
Proof.
unfold min_signed. generalize half_modulus_pos. omega.
Qed.

Remark max_signed_pos: max_signed >= 0.
Proof.
unfold max_signed. generalize half_modulus_pos. omega.
Qed.

Remark wordsize_max_unsigned: zwordsize <= max_unsigned.
Proof.
assert (zwordsize < modulus).
rewrite modulus_power. apply two_p_strict.
generalize wordsize_pos. omega.
unfold max_unsigned. omega.
Qed.

Remark two_wordsize_max_unsigned: 2 * zwordsize - 1 <= max_unsigned.
Proof.
assert (2 * zwordsize - 1 < modulus).
rewrite modulus_power. apply two_p_strict_2. generalize wordsize_pos; omega.
unfold max_unsigned; omega.
Qed.

Remark max_signed_unsigned: max_signed < max_unsigned.
Proof.
unfold max_signed, max_unsigned. rewrite half_modulus_modulus.
generalize half_modulus_pos. omega.
Qed.

Lemma unsigned_repr_eq:
forall x, unsigned (repr x) = Zmod x modulus.
Proof.
intros. simpl. apply Z_mod_modulus_eq.
Qed.

Lemma signed_repr_eq:
forall x, signed (repr x) = if zlt (Zmod x modulus) half_modulus then Zmod x modulus else Zmod x modulus - modulus.
Proof.
intros. unfold signed. rewrite unsigned_repr_eq. auto.
Qed.

## Modulo arithmetic

We define and state properties of equality and arithmetic modulo a positive integer.

Section EQ_MODULO.

Variable modul: Z.
Hypothesis modul_pos: modul > 0.

Definition eqmod (x y: Z) : Prop := exists k, x = k * modul + y.

Lemma eqmod_refl: forall x, eqmod x x.
Proof.
intros; red. exists 0. omega.
Qed.

Lemma eqmod_refl2: forall x y, x = y -> eqmod x y.
Proof.
intros. subst y. apply eqmod_refl.
Qed.

Lemma eqmod_sym: forall x y, eqmod x y -> eqmod y x.
Proof.
intros x y [k EQ]; red. exists (-k). subst x. ring.
Qed.

Lemma eqmod_trans: forall x y z, eqmod x y -> eqmod y z -> eqmod x z.
Proof.
intros x y z [k1 EQ1] [k2 EQ2]; red.
exists (k1 + k2). subst x; subst y. ring.
Qed.

Lemma eqmod_small_eq:
forall x y, eqmod x y -> 0 <= x < modul -> 0 <= y < modul -> x = y.
Proof.
intros x y [k EQ] I1 I2.
generalize (Zdiv_unique _ _ _ _ EQ I2). intro.
rewrite (Zdiv_small x modul I1) in H. subst k. omega.
Qed.

Lemma eqmod_mod_eq:
forall x y, eqmod x y -> x mod modul = y mod modul.
Proof.
intros x y [k EQ]. subst x.
rewrite Zplus_comm. apply Z_mod_plus. auto.
Qed.

Lemma eqmod_mod:
forall x, eqmod x (x mod modul).
Proof.
intros; red. exists (x / modul).
rewrite Zmult_comm. apply Z_div_mod_eq. auto.
Qed.

forall a b c d, eqmod a b -> eqmod c d -> eqmod (a + c) (b + d).
Proof.
intros a b c d [k1 EQ1] [k2 EQ2]; red.
subst a; subst c. exists (k1 + k2). ring.
Qed.

Lemma eqmod_neg:
forall x y, eqmod x y -> eqmod (-x) (-y).
Proof.
intros x y [k EQ]; red. exists (-k). rewrite EQ. ring.
Qed.

Lemma eqmod_sub:
forall a b c d, eqmod a b -> eqmod c d -> eqmod (a - c) (b - d).
Proof.
intros a b c d [k1 EQ1] [k2 EQ2]; red.
subst a; subst c. exists (k1 - k2). ring.
Qed.

Lemma eqmod_mult:
forall a b c d, eqmod a c -> eqmod b d -> eqmod (a * b) (c * d).
Proof.
intros a b c d [k1 EQ1] [k2 EQ2]; red.
subst a; subst b.
exists (k1 * k2 * modul + c * k2 + k1 * d).
ring.
Qed.

End EQ_MODULO.

Lemma eqmod_divides:
forall n m x y, eqmod n x y -> Zdivide m n -> eqmod m x y.
Proof.
intros. destruct H as [k1 EQ1]. destruct H0 as [k2 EQ2].
exists (k1*k2). rewrite <- Zmult_assoc. rewrite <- EQ2. auto.
Qed.

We then specialize these definitions to equality modulo 2wordsize.

Hint Resolve modulus_pos: ints.

Definition eqm := eqmod modulus.

Lemma eqm_refl: forall x, eqm x x.
Proof (eqmod_refl modulus).
Hint Resolve eqm_refl: ints.

Lemma eqm_refl2:
forall x y, x = y -> eqm x y.
Proof (eqmod_refl2 modulus).
Hint Resolve eqm_refl2: ints.

Lemma eqm_sym: forall x y, eqm x y -> eqm y x.
Proof (eqmod_sym modulus).
Hint Resolve eqm_sym: ints.

Lemma eqm_trans: forall x y z, eqm x y -> eqm y z -> eqm x z.
Proof (eqmod_trans modulus).
Hint Resolve eqm_trans: ints.

Lemma eqm_small_eq:
forall x y, eqm x y -> 0 <= x < modulus -> 0 <= y < modulus -> x = y.
Proof (eqmod_small_eq modulus).
Hint Resolve eqm_small_eq: ints.

forall a b c d, eqm a b -> eqm c d -> eqm (a + c) (b + d).

Lemma eqm_neg:
forall x y, eqm x y -> eqm (-x) (-y).
Proof (eqmod_neg modulus).
Hint Resolve eqm_neg: ints.

Lemma eqm_sub:
forall a b c d, eqm a b -> eqm c d -> eqm (a - c) (b - d).
Proof (eqmod_sub modulus).
Hint Resolve eqm_sub: ints.

Lemma eqm_mult:
forall a b c d, eqm a c -> eqm b d -> eqm (a * b) (c * d).
Proof (eqmod_mult modulus).
Hint Resolve eqm_mult: ints.

## Properties of the coercions between Z and int

Lemma eqm_samerepr: forall x y, eqm x y -> repr x = repr y.
Proof.
intros. unfold repr. apply mkint_eq.
rewrite !Z_mod_modulus_eq. apply eqmod_mod_eq. auto with ints. exact H.
Qed.

Lemma eqm_unsigned_repr:
forall z, eqm z (unsigned (repr z)).
Proof.
unfold eqm; intros. rewrite unsigned_repr_eq. apply eqmod_mod. auto with ints.
Qed.
Hint Resolve eqm_unsigned_repr: ints.

Lemma eqm_unsigned_repr_l:
forall a b, eqm a b -> eqm (unsigned (repr a)) b.
Proof.
intros. apply eqm_trans with a.
apply eqm_sym. apply eqm_unsigned_repr. auto.
Qed.
Hint Resolve eqm_unsigned_repr_l: ints.

Lemma eqm_unsigned_repr_r:
forall a b, eqm a b -> eqm a (unsigned (repr b)).
Proof.
intros. apply eqm_trans with b. auto.
apply eqm_unsigned_repr.
Qed.
Hint Resolve eqm_unsigned_repr_r: ints.

Lemma eqm_signed_unsigned:
forall x, eqm (signed x) (unsigned x).
Proof.
intros; red. unfold signed. set (y := unsigned x).
case (zlt y half_modulus); intro.
apply eqmod_refl. red; exists (-1); ring.
Qed.

Theorem unsigned_range:
forall i, 0 <= unsigned i < modulus.
Proof.
destruct i. simpl. omega.
Qed.
Hint Resolve unsigned_range: ints.

Theorem unsigned_range_2:
forall i, 0 <= unsigned i <= max_unsigned.
Proof.
intro; unfold max_unsigned.
generalize (unsigned_range i). omega.
Qed.
Hint Resolve unsigned_range_2: ints.

Theorem signed_range:
forall i, min_signed <= signed i <= max_signed.
Proof.
intros. unfold signed.
generalize (unsigned_range i). set (n := unsigned i). intros.
case (zlt n half_modulus); intro.
unfold max_signed. generalize min_signed_neg. omega.
unfold min_signed, max_signed.
rewrite half_modulus_modulus in *. omega.
Qed.

Theorem repr_unsigned:
forall i, repr (unsigned i) = i.
Proof.
destruct i; simpl. unfold repr. apply mkint_eq.
rewrite Z_mod_modulus_eq. apply Zmod_small; omega.
Qed.
Hint Resolve repr_unsigned: ints.

Lemma repr_signed:
forall i, repr (signed i) = i.
Proof.
intros. transitivity (repr (unsigned i)).
apply eqm_samerepr. apply eqm_signed_unsigned. auto with ints.
Qed.
Hint Resolve repr_signed: ints.

Opaque repr.

Lemma eqm_repr_eq: forall x y, eqm x (unsigned y) -> repr x = y.
Proof.
intros. rewrite <- (repr_unsigned y). apply eqm_samerepr; auto.
Qed.

Theorem unsigned_repr:
forall z, 0 <= z <= max_unsigned -> unsigned (repr z) = z.
Proof.
intros. rewrite unsigned_repr_eq.
apply Zmod_small. unfold max_unsigned in H. omega.
Qed.
Hint Resolve unsigned_repr: ints.

Theorem signed_repr:
forall z, min_signed <= z <= max_signed -> signed (repr z) = z.
Proof.
intros. unfold signed. destruct (zle 0 z).
replace (unsigned (repr z)) with z.
rewrite zlt_true. auto. unfold max_signed in H. omega.
symmetry. apply unsigned_repr. generalize max_signed_unsigned. omega.
pose (z' := z + modulus).
replace (repr z) with (repr z').
replace (unsigned (repr z')) with z'.
rewrite zlt_false. unfold z'. omega.
unfold z'. unfold min_signed in H.
rewrite half_modulus_modulus. omega.
symmetry. apply unsigned_repr.
unfold z', max_unsigned. unfold min_signed, max_signed in H.
rewrite half_modulus_modulus. omega.
apply eqm_samerepr. unfold z'; red. exists 1. omega.
Qed.

Theorem signed_eq_unsigned:
forall x, unsigned x <= max_signed -> signed x = unsigned x.
Proof.
intros. unfold signed. destruct (zlt (unsigned x) half_modulus).
auto. unfold max_signed in H. omegaContradiction.
Qed.

Theorem signed_positive:
forall x, signed x >= 0 <-> unsigned x <= max_signed.
Proof.
intros. unfold signed, max_signed.
generalize (unsigned_range x) half_modulus_modulus half_modulus_pos; intros.
destruct (zlt (unsigned x) half_modulus); omega.
Qed.

## Properties of zero, one, minus one

Theorem unsigned_zero: unsigned zero = 0.
Proof.
unfold zero; rewrite unsigned_repr_eq. apply Zmod_0_l.
Qed.

Theorem unsigned_one: unsigned one = 1.
Proof.
unfold one; rewrite unsigned_repr_eq. apply Zmod_small. split. omega.
unfold modulus. replace wordsize with (S(pred wordsize)).
rewrite two_power_nat_S. generalize (two_power_nat_pos (pred wordsize)).
omega.
generalize wordsize_pos. unfold zwordsize. omega.
Qed.

Theorem unsigned_mone: unsigned mone = modulus - 1.
Proof.
unfold mone; rewrite unsigned_repr_eq.
replace (-1) with ((modulus - 1) + (-1) * modulus).
rewrite Z_mod_plus_full. apply Zmod_small.
generalize modulus_pos. omega. omega.
Qed.

Theorem signed_zero: signed zero = 0.
Proof.
unfold signed. rewrite unsigned_zero. apply zlt_true. generalize half_modulus_pos; omega.
Qed.

Theorem signed_mone: signed mone = -1.
Proof.
unfold signed. rewrite unsigned_mone.
rewrite zlt_false. omega.
rewrite half_modulus_modulus. generalize half_modulus_pos. omega.
Qed.

Theorem one_not_zero: one <> zero.
Proof.
assert (unsigned one <> unsigned zero).
rewrite unsigned_one; rewrite unsigned_zero; congruence.
congruence.
Qed.

Theorem unsigned_repr_wordsize:
unsigned iwordsize = zwordsize.
Proof.
unfold iwordsize; rewrite unsigned_repr_eq. apply Zmod_small.
generalize wordsize_pos wordsize_max_unsigned; unfold max_unsigned; omega.
Qed.

## Properties of equality

Theorem eq_sym:
forall x y, eq x y = eq y x.
Proof.
intros; unfold eq. case (zeq (unsigned x) (unsigned y)); intro.
rewrite e. rewrite zeq_true. auto.
rewrite zeq_false. auto. auto.
Qed.

Theorem eq_spec: forall (x y: int), if eq x y then x = y else x <> y.
Proof.
intros; unfold eq. case (eq_dec x y); intro.
subst y. rewrite zeq_true. auto.
rewrite zeq_false. auto.
destruct x; destruct y.
simpl. red; intro. elim n. apply mkint_eq. auto.
Qed.

Theorem eq_true: forall x, eq x x = true.
Proof.
intros. generalize (eq_spec x x); case (eq x x); intros; congruence.
Qed.

Theorem eq_false: forall x y, x <> y -> eq x y = false.
Proof.
intros. generalize (eq_spec x y); case (eq x y); intros; congruence.
Qed.

Theorem eq_signed:
forall x y, eq x y = if zeq (signed x) (signed y) then true else false.
Proof.
intros. predSpec eq eq_spec x y.
subst x. rewrite zeq_true; auto.
destruct (zeq (signed x) (signed y)); auto.
elim H. rewrite <- (repr_signed x). rewrite <- (repr_signed y). congruence.
Qed.

Theorem add_unsigned: forall x y, add x y = repr (unsigned x + unsigned y).
Proof.
intros; reflexivity.
Qed.

Theorem add_signed: forall x y, add x y = repr (signed x + signed y).
Proof.
apply eqm_add; apply eqm_sym; apply eqm_signed_unsigned.
Qed.

Proof.
intros; unfold add. decEq. omega. Qed.

Proof.
rewrite Zplus_0_r. apply repr_unsigned.
Qed.

Proof.
Qed.

Proof.
set (x' := unsigned x).
set (y' := unsigned y).
set (z' := unsigned z).
apply eqm_samerepr.
apply eqm_trans with ((x' + y') + z').
auto with ints.
rewrite <- Zplus_assoc. auto with ints.
Qed.

Proof.
Qed.

Proof.
intros; unfold add, neg, zero. apply eqm_samerepr.
replace 0 with (unsigned x + (- (unsigned x))).
auto with ints. omega.
Qed.

forall x y,
unsigned (add x y) = unsigned x + unsigned y - unsigned (add_carry x y zero) * modulus.
Proof.
intros.
rewrite unsigned_repr_eq.
generalize (unsigned_range x) (unsigned_range y). intros.
destruct (zlt (unsigned x + unsigned y) modulus).
rewrite unsigned_zero. apply Zmod_unique with 0. omega. omega.
rewrite unsigned_one. apply Zmod_unique with 1. omega. omega.
Qed.

forall x y,
unsigned (add x y) = unsigned x + unsigned y
\/ unsigned (add x y) = unsigned x + unsigned y - modulus.
Proof.
rewrite unsigned_zero. rewrite Zplus_0_r.
destruct (zlt (unsigned x + unsigned y) modulus).
rewrite unsigned_zero. left; omega.
rewrite unsigned_one. right; omega.
Qed.

## Properties of negation

Theorem neg_repr: forall z, neg (repr z) = repr (-z).
Proof.
intros; unfold neg. apply eqm_samerepr. auto with ints.
Qed.

Theorem neg_zero: neg zero = zero.
Proof.
unfold neg. rewrite unsigned_zero. auto.
Qed.

Theorem neg_involutive: forall x, neg (neg x) = x.
Proof.
intros; unfold neg.
apply eqm_repr_eq. eapply eqm_trans. apply eqm_neg.
apply eqm_unsigned_repr_l. apply eqm_refl. apply eqm_refl2. omega.
Qed.

Proof.
intros; unfold neg, add. apply eqm_samerepr.
apply eqm_trans with (- (unsigned x + unsigned y)).
auto with ints.
replace (- (unsigned x + unsigned y))
with ((- unsigned x) + (- unsigned y)).
auto with ints. omega.
Qed.

## Properties of subtraction

Theorem sub_zero_l: forall x, sub x zero = x.
Proof.
intros; unfold sub. rewrite unsigned_zero.
replace (unsigned x - 0) with (unsigned x) by omega. apply repr_unsigned.
Qed.

Theorem sub_zero_r: forall x, sub zero x = neg x.
Proof.
intros; unfold sub, neg. rewrite unsigned_zero. auto.
Qed.

Theorem sub_add_opp: forall x y, sub x y = add x (neg y).
Proof.
intros; unfold sub, add, neg. apply eqm_samerepr.
Qed.

Theorem sub_idem: forall x, sub x x = zero.
Proof.
intros; unfold sub. unfold zero. decEq. omega.
Qed.

Theorem sub_add_l: forall x y z, sub (add x y) z = add (sub x z) y.
Proof.
Qed.

Theorem sub_add_r: forall x y z, sub x (add y z) = add (sub x z) (neg y).
Proof.
Qed.

Theorem sub_shifted:
forall x y z,
Proof.
rewrite (add_commut (neg y) (neg z)).
Qed.

Theorem sub_signed:
forall x y, sub x y = repr (signed x - signed y).
Proof.
intros. unfold sub. apply eqm_samerepr.
apply eqm_sub; apply eqm_sym; apply eqm_signed_unsigned.
Qed.

Theorem unsigned_sub_borrow:
forall x y,
unsigned (sub x y) = unsigned x - unsigned y + unsigned (sub_borrow x y zero) * modulus.
Proof.
intros.
unfold sub, sub_borrow. rewrite unsigned_zero. rewrite Zminus_0_r.
rewrite unsigned_repr_eq.
generalize (unsigned_range x) (unsigned_range y). intros.
destruct (zlt (unsigned x - unsigned y) 0).
rewrite unsigned_one. apply Zmod_unique with (-1). omega. omega.
rewrite unsigned_zero. apply Zmod_unique with 0. omega. omega.
Qed.

## Properties of multiplication

Theorem mul_commut: forall x y, mul x y = mul y x.
Proof.
intros; unfold mul. decEq. ring.
Qed.

Theorem mul_zero: forall x, mul x zero = zero.
Proof.
intros; unfold mul. rewrite unsigned_zero.
unfold zero. decEq. ring.
Qed.

Theorem mul_one: forall x, mul x one = x.
Proof.
intros; unfold mul. rewrite unsigned_one.
transitivity (repr (unsigned x)). decEq. ring.
apply repr_unsigned.
Qed.

Theorem mul_mone: forall x, mul x mone = neg x.
Proof.
intros; unfold mul, neg. rewrite unsigned_mone.
apply eqm_samerepr.
replace (-unsigned x) with (0 - unsigned x) by omega.
replace (unsigned x * (modulus - 1)) with (unsigned x * modulus - unsigned x) by ring.
apply eqm_sub. exists (unsigned x). omega. apply eqm_refl.
Qed.

Theorem mul_assoc: forall x y z, mul (mul x y) z = mul x (mul y z).
Proof.
intros; unfold mul.
set (x' := unsigned x).
set (y' := unsigned y).
set (z' := unsigned z).
apply eqm_samerepr. apply eqm_trans with ((x' * y') * z').
auto with ints.
rewrite <- Zmult_assoc. auto with ints.
Qed.

forall x y z, mul (add x y) z = add (mul x z) (mul y z).
Proof.
apply eqm_samerepr.
set (x' := unsigned x).
set (y' := unsigned y).
set (z' := unsigned z).
apply eqm_trans with ((x' + y') * z').
auto with ints.
replace ((x' + y') * z') with (x' * z' + y' * z').
auto with ints.
ring.
Qed.

forall x y z, mul x (add y z) = add (mul x y) (mul x z).
Proof.
decEq; apply mul_commut.
Qed.

Theorem neg_mul_distr_l:
forall x y, neg(mul x y) = mul (neg x) y.
Proof.
intros. unfold mul, neg.
set (x' := unsigned x). set (y' := unsigned y).
apply eqm_samerepr. apply eqm_trans with (- (x' * y')).
auto with ints.
replace (- (x' * y')) with ((-x') * y') by ring.
auto with ints.
Qed.

Theorem neg_mul_distr_r:
forall x y, neg(mul x y) = mul x (neg y).
Proof.
intros. rewrite (mul_commut x y). rewrite (mul_commut x (neg y)).
apply neg_mul_distr_l.
Qed.

Theorem mul_signed:
forall x y, mul x y = repr (signed x * signed y).
Proof.
intros; unfold mul. apply eqm_samerepr.
apply eqm_mult; apply eqm_sym; apply eqm_signed_unsigned.
Qed.

## Properties of division and modulus

Lemma modu_divu_Euclid:
forall x y, y <> zero -> x = add (mul (divu x y) y) (modu x y).
Proof.
intros. unfold add, mul, divu, modu.
transitivity (repr (unsigned x)). auto with ints.
apply eqm_samerepr.
set (x' := unsigned x). set (y' := unsigned y).
apply eqm_trans with ((x' / y') * y' + x' mod y').
apply eqm_refl2. rewrite Zmult_comm. apply Z_div_mod_eq.
generalize (unsigned_range y); intro.
assert (unsigned y <> 0). red; intro.
elim H. rewrite <- (repr_unsigned y). unfold zero. congruence.
unfold y'. omega.
auto with ints.
Qed.

Theorem modu_divu:
forall x y, y <> zero -> modu x y = sub x (mul (divu x y) y).
Proof.
intros.
assert (forall a b c, a = add b c -> c = sub a b).
intros. subst a. rewrite sub_add_l. rewrite sub_idem.
apply H0. apply modu_divu_Euclid. auto.
Qed.

Lemma mods_divs_Euclid:
forall x y, x = add (mul (divs x y) y) (mods x y).
Proof.
intros. unfold add, mul, divs, mods.
transitivity (repr (signed x)). auto with ints.
apply eqm_samerepr.
set (x' := signed x). set (y' := signed y).
apply eqm_trans with ((Z.quot x' y') * y' + Z.rem x' y').
apply eqm_refl2. rewrite Zmult_comm. apply Z.quot_rem'.
apply eqm_unsigned_repr_r. apply eqm_mult; auto with ints.
unfold y'. apply eqm_signed_unsigned.
Qed.

Theorem mods_divs:
forall x y, mods x y = sub x (mul (divs x y) y).
Proof.
intros.
assert (forall a b c, a = add b c -> c = sub a b).
intros. subst a. rewrite sub_add_l. rewrite sub_idem.
apply H. apply mods_divs_Euclid.
Qed.

Theorem divu_one:
forall x, divu x one = x.
Proof.
unfold divu; intros. rewrite unsigned_one. rewrite Zdiv_1_r. apply repr_unsigned.
Qed.

Theorem modu_one:
forall x, modu x one = zero.
Proof.
intros. rewrite modu_divu. rewrite divu_one. rewrite mul_one. apply sub_idem.
apply one_not_zero.
Qed.

Theorem divs_mone:
forall x, divs x mone = neg x.
Proof.
unfold divs, neg; intros.
rewrite signed_mone.
replace (Z.quot (signed x) (-1)) with (- (signed x)).
apply eqm_samerepr. apply eqm_neg. apply eqm_signed_unsigned.
set (x' := signed x).
set (one := 1).
change (-1) with (- one). rewrite Zquot_opp_r.
assert (Z.quot x' one = x').
symmetry. apply Zquot_unique_full with 0. red.
change (Z.abs one) with 1.
destruct (zle 0 x'). left. omega. right. omega.
unfold one; ring.
congruence.
Qed.

Theorem mods_mone:
forall x, mods x mone = zero.
Proof.
intros. rewrite mods_divs. rewrite divs_mone.
rewrite <- neg_mul_distr_l. rewrite mul_mone. rewrite neg_involutive. apply sub_idem.
Qed.

Theorem divmodu2_divu_modu:
forall n d,
d <> zero -> divmodu2 zero n d = Some (divu n d, modu n d).
Proof.
unfold divmodu2, divu, modu; intros.
rewrite dec_eq_false by auto.
set (N := unsigned zero * modulus + unsigned n).
assert (E1: unsigned n = N) by (unfold N; rewrite unsigned_zero; ring). rewrite ! E1.
set (D := unsigned d).
set (Q := N / D); set (R := N mod D).
assert (E2: Z.div_eucl N D = (Q, R)).
{ unfold Q, R, Z.div, Z.modulo. destruct (Z.div_eucl N D); auto. }
rewrite E2. rewrite zle_true. auto.
assert (unsigned d <> 0).
{ red; intros. elim H. rewrite <- (repr_unsigned d). rewrite H0; auto. }
assert (0 < D).
{ unfold D. generalize (unsigned_range d); intros. omega. }
assert (0 <= Q <= max_unsigned).
{ unfold Q. apply Zdiv_interval_2.
rewrite <- E1; apply unsigned_range_2.
omega. unfold max_unsigned; generalize modulus_pos; omega. omega. }
omega.
Qed.

Lemma unsigned_signed:
forall n, unsigned n = if lt n zero then signed n + modulus else signed n.
Proof.
intros. unfold lt. rewrite signed_zero. unfold signed.
generalize (unsigned_range n). rewrite half_modulus_modulus. intros.
destruct (zlt (unsigned n) half_modulus).
- rewrite zlt_false by omega. auto.
- rewrite zlt_true by omega. ring.
Qed.

Theorem divmods2_divs_mods:
forall n d,
d <> zero -> n <> repr min_signed \/ d <> mone ->
divmods2 (if lt n zero then mone else zero) n d = Some (divs n d, mods n d).
Proof.
unfold divmods2, divs, mods; intros.
rewrite dec_eq_false by auto.
set (N := signed (if lt n zero then mone else zero) * modulus + unsigned n).
set (D := signed d).
assert (D <> 0).
{ unfold D; red; intros. elim H. rewrite <- (repr_signed d). rewrite H1; auto. }
assert (N = signed n).
{ unfold N. rewrite unsigned_signed. destruct (lt n zero).
rewrite signed_mone. ring.
rewrite signed_zero. ring. }
set (Q := Z.quot N D); set (R := Z.rem N D).
assert (E2: Z.quotrem N D = (Q, R)).
{ unfold Q, R, Z.quot, Z.rem. destruct (Z.quotrem N D); auto. }
rewrite E2.
assert (min_signed <= N <= max_signed) by (rewrite H2; apply signed_range).
assert (min_signed <= Q <= max_signed).
{ unfold Q. destruct (zeq D 1); [ | destruct (zeq D (-1))].
- (* D = 1 *)
rewrite e. rewrite Z.quot_1_r; auto.
- (* D = -1 *)
rewrite e. change (-1) with (Z.opp 1). rewrite Z.quot_opp_r by omega.
rewrite Z.quot_1_r.
assert (N <> min_signed).
{ red; intros; destruct H0.
+ elim H0. rewrite <- (repr_signed n). rewrite <- H2. rewrite H4. auto.
+ elim H0. rewrite <- (repr_signed d). unfold D in e; rewrite e; auto. }
unfold min_signed, max_signed in *. omega.
- (* |D| > 1 *)
assert (Z.abs (Z.quot N D) < half_modulus).
{ rewrite <- Z.quot_abs by omega. apply Zquot_lt_upper_bound.
xomega. xomega.
apply Zle_lt_trans with (half_modulus * 1).
rewrite Z.mul_1_r. unfold min_signed, max_signed in H3; xomega.
apply Zmult_lt_compat_l. generalize half_modulus_pos; omega. xomega. }
rewrite Z.abs_lt in H4.
unfold min_signed, max_signed; omega.
}
unfold proj_sumbool; rewrite ! zle_true by omega; simpl.
unfold Q, R; rewrite H2; auto.
Qed.

## Properties of bit-level operations over Z

Remark Ztestbit_0: forall n, Z.testbit 0 n = false.
Proof Z.testbit_0_l.

Remark Ztestbit_1: forall n, Z.testbit 1 n = zeq n 0.
Proof.
intros. destruct n; simpl; auto.
Qed.

Remark Ztestbit_m1: forall n, 0 <= n -> Z.testbit (-1) n = true.
Proof.
intros. destruct n; simpl; auto.
Qed.

Remark Zshiftin_spec:
forall b x, Zshiftin b x = 2 * x + (if b then 1 else 0).
Proof.
unfold Zshiftin; intros. destruct b.
- rewrite Z.succ_double_spec. omega.
- rewrite Z.double_spec. omega.
Qed.

Remark Zshiftin_inj:
forall b1 x1 b2 x2,
Zshiftin b1 x1 = Zshiftin b2 x2 -> b1 = b2 /\ x1 = x2.
Proof.
intros. rewrite !Zshiftin_spec in H.
destruct b1; destruct b2.
split; [auto|omega].
split; [auto|omega].
Qed.

Remark Zdecomp:
forall x, x = Zshiftin (Z.odd x) (Z.div2 x).
Proof.
intros. destruct x; simpl.
- auto.
- destruct p; auto.
- destruct p; auto. simpl. rewrite Pos.pred_double_succ. auto.
Qed.

Remark Ztestbit_shiftin:
forall b x n,
0 <= n ->
Z.testbit (Zshiftin b x) n = if zeq n 0 then b else Z.testbit x (Z.pred n).
Proof.
intros. rewrite Zshiftin_spec. destruct (zeq n 0).
- subst n. destruct b.
+ apply Z.testbit_odd_0.
+ rewrite Zplus_0_r. apply Z.testbit_even_0.
- assert (0 <= Z.pred n) by omega.
set (n' := Z.pred n) in *.
replace n with (Z.succ n') by (unfold n'; omega).
destruct b.
+ apply Z.testbit_odd_succ; auto.
+ rewrite Zplus_0_r. apply Z.testbit_even_succ; auto.
Qed.

Remark Ztestbit_shiftin_base:
forall b x, Z.testbit (Zshiftin b x) 0 = b.
Proof.
intros. rewrite Ztestbit_shiftin. apply zeq_true. omega.
Qed.

Remark Ztestbit_shiftin_succ:
forall b x n, 0 <= n -> Z.testbit (Zshiftin b x) (Z.succ n) = Z.testbit x n.
Proof.
intros. rewrite Ztestbit_shiftin. rewrite zeq_false. rewrite Z.pred_succ. auto.
omega. omega.
Qed.

Remark Ztestbit_eq:
forall n x, 0 <= n ->
Z.testbit x n = if zeq n 0 then Z.odd x else Z.testbit (Z.div2 x) (Z.pred n).
Proof.
intros. rewrite (Zdecomp x) at 1. apply Ztestbit_shiftin; auto.
Qed.

Remark Ztestbit_base:
forall x, Z.testbit x 0 = Z.odd x.
Proof.
intros. rewrite Ztestbit_eq. apply zeq_true. omega.
Qed.

Remark Ztestbit_succ:
forall n x, 0 <= n -> Z.testbit x (Z.succ n) = Z.testbit (Z.div2 x) n.
Proof.
intros. rewrite Ztestbit_eq. rewrite zeq_false. rewrite Z.pred_succ. auto.
omega. omega.
Qed.

Lemma eqmod_same_bits:
forall n x y,
(forall i, 0 <= i < Z.of_nat n -> Z.testbit x i = Z.testbit y i) ->
eqmod (two_power_nat n) x y.
Proof.
induction n; intros.
- change (two_power_nat 0) with 1. exists (x-y); ring.
- rewrite two_power_nat_S.
assert (eqmod (two_power_nat n) (Z.div2 x) (Z.div2 y)).
apply IHn. intros. rewrite <- !Ztestbit_succ. apply H. rewrite inj_S; omega.
omega. omega.
destruct H0 as [k EQ].
exists k. rewrite (Zdecomp x). rewrite (Zdecomp y).
replace (Z.odd y) with (Z.odd x).
rewrite EQ. rewrite !Zshiftin_spec. ring.
exploit (H 0). rewrite inj_S; omega.
rewrite !Ztestbit_base. auto.
Qed.

Lemma eqm_same_bits:
forall x y,
(forall i, 0 <= i < zwordsize -> Z.testbit x i = Z.testbit y i) ->
eqm x y.
Proof (eqmod_same_bits wordsize).

Lemma same_bits_eqmod:
forall n x y i,
eqmod (two_power_nat n) x y -> 0 <= i < Z.of_nat n ->
Z.testbit x i = Z.testbit y i.
Proof.
induction n; intros.
- rewrite inj_S in H0. rewrite two_power_nat_S in H.
rewrite !(Ztestbit_eq i); intuition.
destruct H as [k EQ].
assert (EQ': Zshiftin (Z.odd x) (Z.div2 x) =
Zshiftin (Z.odd y) (k * two_power_nat n + Z.div2 y)).
{
rewrite (Zdecomp x) in EQ. rewrite (Zdecomp y) in EQ.
rewrite EQ. rewrite !Zshiftin_spec. ring.
}
exploit Zshiftin_inj; eauto. intros [A B].
destruct (zeq i 0).
+ auto.
+ apply IHn. exists k; auto. omega.
Qed.

Lemma same_bits_eqm:
forall x y i,
eqm x y ->
0 <= i < zwordsize ->
Z.testbit x i = Z.testbit y i.
Proof (same_bits_eqmod wordsize).

Remark two_power_nat_infinity:
forall x, 0 <= x -> exists n, x < two_power_nat n.
Proof.
intros x0 POS0; pattern x0; apply natlike_ind; auto.
exists O. compute; auto.
intros. destruct H0 as [n LT]. exists (S n). rewrite two_power_nat_S.
generalize (two_power_nat_pos n). omega.
Qed.

Lemma equal_same_bits:
forall x y,
(forall i, 0 <= i -> Z.testbit x i = Z.testbit y i) ->
x = y.
Proof.
intros.
set (z := if zlt x y then y - x else x - y).
assert (0 <= z).
unfold z; destruct (zlt x y); omega.
exploit (two_power_nat_infinity z); auto. intros [n LT].
assert (eqmod (two_power_nat n) x y).
apply eqmod_same_bits. intros. apply H. tauto.
assert (eqmod (two_power_nat n) z 0).
unfold z. destruct (zlt x y).
replace 0 with (y - y) by omega. apply eqmod_sub. apply eqmod_refl. auto.
replace 0 with (x - x) by omega. apply eqmod_sub. apply eqmod_refl. apply eqmod_sym; auto.
assert (z = 0).
apply eqmod_small_eq with (two_power_nat n). auto. omega. generalize (two_power_nat_pos n); omega.
unfold z in H3. destruct (zlt x y); omega.
Qed.

Lemma Z_one_complement:
forall i, 0 <= i ->
forall x, Z.testbit (-x-1) i = negb (Z.testbit x i).
Proof.
intros i0 POS0. pattern i0. apply Zlt_0_ind; auto.
intros i IND POS x.
rewrite (Zdecomp x). set (y := Z.div2 x).
replace (- Zshiftin (Z.odd x) y - 1)
with (Zshiftin (negb (Z.odd x)) (- y - 1)).
rewrite !Ztestbit_shiftin; auto.
destruct (zeq i 0). auto. apply IND. omega.
rewrite !Zshiftin_spec. destruct (Z.odd x); simpl negb; ring.
Qed.

Lemma Ztestbit_above:
forall n x i,
0 <= x < two_power_nat n ->
i >= Z.of_nat n ->
Z.testbit x i = false.
Proof.
induction n; intros.
- change (two_power_nat 0) with 1 in H.
replace x with 0 by omega.
apply Z.testbit_0_l.
- rewrite inj_S in H0. rewrite Ztestbit_eq. rewrite zeq_false.
apply IHn. rewrite two_power_nat_S in H. rewrite (Zdecomp x) in H.
rewrite Zshiftin_spec in H. destruct (Z.odd x); omega.
omega. omega. omega.
Qed.

Lemma Ztestbit_above_neg:
forall n x i,
-two_power_nat n <= x < 0 ->
i >= Z.of_nat n ->
Z.testbit x i = true.
Proof.
intros. set (y := -x-1).
assert (Z.testbit y i = false).
apply Ztestbit_above with n.
unfold y; omega. auto.
unfold y in H1. rewrite Z_one_complement in H1.
change true with (negb false). rewrite <- H1. rewrite negb_involutive; auto.
omega.
Qed.

Lemma Zsign_bit:
forall n x,
0 <= x < two_power_nat (S n) ->
Z.testbit x (Z_of_nat n) = if zlt x (two_power_nat n) then false else true.
Proof.
induction n; intros.
- change (two_power_nat 1) with 2 in H.
assert (x = 0 \/ x = 1) by omega.
destruct H0; subst x; reflexivity.
- rewrite inj_S. rewrite Ztestbit_eq. rewrite zeq_false. rewrite Z.pred_succ.
rewrite IHn. rewrite two_power_nat_S.
destruct (zlt (Z.div2 x) (two_power_nat n)); rewrite (Zdecomp x); rewrite Zshiftin_spec.
rewrite zlt_true. auto. destruct (Z.odd x); omega.
rewrite zlt_false. auto. destruct (Z.odd x); omega.
rewrite (Zdecomp x) in H; rewrite Zshiftin_spec in H.
rewrite two_power_nat_S in H. destruct (Z.odd x); omega.
omega. omega.
Qed.

Lemma Zshiftin_ind:
forall (P: Z -> Prop),
P 0 ->
(forall b x, 0 <= x -> P x -> P (Zshiftin b x)) ->
forall x, 0 <= x -> P x.
Proof.
intros. destruct x.
- auto.
- induction p.
+ change (P (Zshiftin true (Z.pos p))). auto.
+ change (P (Zshiftin false (Z.pos p))). auto.
+ change (P (Zshiftin true 0)). apply H0. omega. auto.
- compute in H1. intuition congruence.
Qed.

Lemma Zshiftin_pos_ind:
forall (P: Z -> Prop),
P 1 ->
(forall b x, 0 < x -> P x -> P (Zshiftin b x)) ->
forall x, 0 < x -> P x.
Proof.
intros. destruct x; simpl in H1; try discriminate.
induction p.
+ change (P (Zshiftin true (Z.pos p))). auto.
+ change (P (Zshiftin false (Z.pos p))). auto.
+ auto.
Qed.

Lemma Ztestbit_le:
forall x y,
0 <= y ->
(forall i, 0 <= i -> Z.testbit x i = true -> Z.testbit y i = true) ->
x <= y.
Proof.
intros x y0 POS0; revert x; pattern y0; apply Zshiftin_ind; auto; intros.
- replace x with 0. omega. apply equal_same_bits; intros.
rewrite Ztestbit_0. destruct (Z.testbit x i) as [] eqn:E; auto.
exploit H; eauto. rewrite Ztestbit_0. auto.
- assert (Z.div2 x0 <= x).
{ apply H0. intros. exploit (H1 (Zsucc i)).
omega. rewrite Ztestbit_succ; auto. rewrite Ztestbit_shiftin_succ; auto.
}
rewrite (Zdecomp x0). rewrite !Zshiftin_spec.
destruct (Z.odd x0) as [] eqn:E1; destruct b as [] eqn:E2; try omega.
exploit (H1 0). omega. rewrite Ztestbit_base; auto.
rewrite Ztestbit_shiftin_base. congruence.
Qed.

## Bit-level reasoning over type int

Definition testbit (x: int) (i: Z) : bool := Z.testbit (unsigned x) i.

Lemma testbit_repr:
forall x i,
0 <= i < zwordsize ->
testbit (repr x) i = Z.testbit x i.
Proof.
intros. unfold testbit. apply same_bits_eqm; auto with ints.
Qed.

Lemma same_bits_eq:
forall x y,
(forall i, 0 <= i < zwordsize -> testbit x i = testbit y i) ->
x = y.
Proof.
intros. rewrite <- (repr_unsigned x). rewrite <- (repr_unsigned y).
apply eqm_samerepr. apply eqm_same_bits. auto.
Qed.

Lemma bits_above:
forall x i, i >= zwordsize -> testbit x i = false.
Proof.
intros. apply Ztestbit_above with wordsize; auto. apply unsigned_range.
Qed.

Lemma bits_zero:
forall i, testbit zero i = false.
Proof.
intros. unfold testbit. rewrite unsigned_zero. apply Ztestbit_0.
Qed.

Remark bits_one: forall n, testbit one n = zeq n 0.
Proof.
unfold testbit; intros. rewrite unsigned_one. apply Ztestbit_1.
Qed.

Lemma bits_mone:
forall i, 0 <= i < zwordsize -> testbit mone i = true.
Proof.
intros. unfold mone. rewrite testbit_repr; auto. apply Ztestbit_m1. omega.
Qed.

Hint Rewrite bits_zero bits_mone : ints.

Ltac bit_solve :=
intros; apply same_bits_eq; intros; autorewrite with ints; auto with bool.

Lemma sign_bit_of_unsigned:
forall x, testbit x (zwordsize - 1) = if zlt (unsigned x) half_modulus then false else true.
Proof.
intros. unfold testbit.
set (ws1 := pred wordsize).
assert (zwordsize - 1 = Z_of_nat ws1).
unfold zwordsize, ws1, wordsize.
destruct WS.wordsize as [] eqn:E.
elim WS.wordsize_not_zero; auto.
rewrite inj_S. simpl. omega.
assert (half_modulus = two_power_nat ws1).
rewrite two_power_nat_two_p. rewrite <- H. apply half_modulus_power.
rewrite H; rewrite H0.
apply Zsign_bit. rewrite two_power_nat_S. rewrite <- H0.
rewrite <- half_modulus_modulus. apply unsigned_range.
Qed.

Lemma bits_signed:
forall x i, 0 <= i ->
Z.testbit (signed x) i = testbit x (if zlt i zwordsize then i else zwordsize - 1).
Proof.
intros.
destruct (zlt i zwordsize).
- apply same_bits_eqm. apply eqm_signed_unsigned. omega.
- unfold signed. rewrite sign_bit_of_unsigned. destruct (zlt (unsigned x) half_modulus).
+ apply Ztestbit_above with wordsize. apply unsigned_range. auto.
+ apply Ztestbit_above_neg with wordsize.
fold modulus. generalize (unsigned_range x). omega. auto.
Qed.

Lemma bits_le:
forall x y,
(forall i, 0 <= i < zwordsize -> testbit x i = true -> testbit y i = true) ->
unsigned x <= unsigned y.
Proof.
intros. apply Ztestbit_le. generalize (unsigned_range y); omega.
intros. fold (testbit y i). destruct (zlt i zwordsize).
apply H. omega. auto.
fold (testbit x i) in H1. rewrite bits_above in H1; auto. congruence.
Qed.

## Properties of bitwise and, or, xor

Lemma bits_and:
forall x y i, 0 <= i < zwordsize ->
testbit (and x y) i = testbit x i && testbit y i.
Proof.
intros. unfold and. rewrite testbit_repr; auto. rewrite Z.land_spec; intuition.
Qed.

Lemma bits_or:
forall x y i, 0 <= i < zwordsize ->
testbit (or x y) i = testbit x i || testbit y i.
Proof.
intros. unfold or. rewrite testbit_repr; auto. rewrite Z.lor_spec; intuition.
Qed.

Lemma bits_xor:
forall x y i, 0 <= i < zwordsize ->
testbit (xor x y) i = xorb (testbit x i) (testbit y i).
Proof.
intros. unfold xor. rewrite testbit_repr; auto. rewrite Z.lxor_spec; intuition.
Qed.

Lemma bits_not:
forall x i, 0 <= i < zwordsize ->
testbit (not x) i = negb (testbit x i).
Proof.
intros. unfold not. rewrite bits_xor; auto. rewrite bits_mone; auto.
Qed.

Hint Rewrite bits_and bits_or bits_xor bits_not: ints.

Theorem and_commut: forall x y, and x y = and y x.
Proof.
bit_solve.
Qed.

Theorem and_assoc: forall x y z, and (and x y) z = and x (and y z).
Proof.
bit_solve.
Qed.

Theorem and_zero: forall x, and x zero = zero.
Proof.
bit_solve. apply andb_b_false.
Qed.

Corollary and_zero_l: forall x, and zero x = zero.
Proof.
intros. rewrite and_commut. apply and_zero.
Qed.

Theorem and_mone: forall x, and x mone = x.
Proof.
bit_solve. apply andb_b_true.
Qed.

Corollary and_mone_l: forall x, and mone x = x.
Proof.
intros. rewrite and_commut. apply and_mone.
Qed.

Theorem and_idem: forall x, and x x = x.
Proof.
bit_solve. destruct (testbit x i); auto.
Qed.

Theorem or_commut: forall x y, or x y = or y x.
Proof.
bit_solve.
Qed.

Theorem or_assoc: forall x y z, or (or x y) z = or x (or y z).
Proof.
bit_solve.
Qed.

Theorem or_zero: forall x, or x zero = x.
Proof.
bit_solve.
Qed.

Corollary or_zero_l: forall x, or zero x = x.
Proof.
intros. rewrite or_commut. apply or_zero.
Qed.

Theorem or_mone: forall x, or x mone = mone.
Proof.
bit_solve.
Qed.

Theorem or_idem: forall x, or x x = x.
Proof.
bit_solve. destruct (testbit x i); auto.
Qed.

Theorem and_or_distrib:
forall x y z,
and x (or y z) = or (and x y) (and x z).
Proof.
bit_solve. apply demorgan1.
Qed.

Corollary and_or_distrib_l:
forall x y z,
and (or x y) z = or (and x z) (and y z).
Proof.
intros. rewrite (and_commut (or x y)). rewrite and_or_distrib. f_equal; apply and_commut.
Qed.

Theorem or_and_distrib:
forall x y z,
or x (and y z) = and (or x y) (or x z).
Proof.
bit_solve. apply orb_andb_distrib_r.
Qed.

Corollary or_and_distrib_l:
forall x y z,
or (and x y) z = and (or x z) (or y z).
Proof.
intros. rewrite (or_commut (and x y)). rewrite or_and_distrib. f_equal; apply or_commut.
Qed.

Theorem and_or_absorb: forall x y, and x (or x y) = x.
Proof.
bit_solve.
assert (forall a b, a && (a || b) = a) by destr_bool.
auto.
Qed.

Theorem or_and_absorb: forall x y, or x (and x y) = x.
Proof.
bit_solve.
assert (forall a b, a || (a && b) = a) by destr_bool.
auto.
Qed.

Theorem xor_commut: forall x y, xor x y = xor y x.
Proof.
bit_solve. apply xorb_comm.
Qed.

Theorem xor_assoc: forall x y z, xor (xor x y) z = xor x (xor y z).
Proof.
bit_solve. apply xorb_assoc.
Qed.

Theorem xor_zero: forall x, xor x zero = x.
Proof.
bit_solve. apply xorb_false.
Qed.

Corollary xor_zero_l: forall x, xor zero x = x.
Proof.
intros. rewrite xor_commut. apply xor_zero.
Qed.

Theorem xor_idem: forall x, xor x x = zero.
Proof.
bit_solve. apply xorb_nilpotent.
Qed.

Theorem xor_zero_one: xor zero one = one.
Proof.
rewrite xor_commut. apply xor_zero. Qed.

Theorem xor_one_one: xor one one = zero.
Proof.
apply xor_idem. Qed.

Theorem xor_zero_equal: forall x y, xor x y = zero -> x = y.
Proof.
intros. apply same_bits_eq; intros.
assert (xorb (testbit x i) (testbit y i) = false).
rewrite <- bits_xor; auto. rewrite H. apply bits_zero.
destruct (testbit x i); destruct (testbit y i); reflexivity || discriminate.
Qed.

Theorem and_xor_distrib:
forall x y z,
and x (xor y z) = xor (and x y) (and x z).
Proof.
bit_solve.
assert (forall a b c, a && (xorb b c) = xorb (a && b) (a && c)) by destr_bool.
auto.
Qed.

Theorem and_le:
forall x y, unsigned (and x y) <= unsigned x.
Proof.
intros. apply bits_le; intros.
rewrite bits_and in H0; auto. rewrite andb_true_iff in H0. tauto.
Qed.

Theorem or_le:
forall x y, unsigned x <= unsigned (or x y).
Proof.
intros. apply bits_le; intros.
rewrite bits_or; auto. rewrite H0; auto.
Qed.

Properties of bitwise complement.

Theorem not_involutive:
forall (x: int), not (not x) = x.
Proof.
intros. unfold not. rewrite xor_assoc. rewrite xor_idem. apply xor_zero.
Qed.

Theorem not_zero:
not zero = mone.
Proof.
unfold not. rewrite xor_commut. apply xor_zero.
Qed.

Theorem not_mone:
not mone = zero.
Proof.
rewrite <- (not_involutive zero). symmetry. decEq. apply not_zero.
Qed.

Theorem not_or_and_not:
forall x y, not (or x y) = and (not x) (not y).
Proof.
bit_solve. apply negb_orb.
Qed.

Theorem not_and_or_not:
forall x y, not (and x y) = or (not x) (not y).
Proof.
bit_solve. apply negb_andb.
Qed.

Theorem and_not_self:
forall x, and x (not x) = zero.
Proof.
bit_solve.
Qed.

Theorem or_not_self:
forall x, or x (not x) = mone.
Proof.
bit_solve.
Qed.

Theorem xor_not_self:
forall x, xor x (not x) = mone.
Proof.
bit_solve. destruct (testbit x i); auto.
Qed.

Lemma unsigned_not:
forall x, unsigned (not x) = max_unsigned - unsigned x.
Proof.
intros. transitivity (unsigned (repr(-unsigned x - 1))).
f_equal. bit_solve. rewrite testbit_repr; auto. symmetry. apply Z_one_complement. omega.
rewrite unsigned_repr_eq. apply Zmod_unique with (-1).
unfold max_unsigned. omega.
generalize (unsigned_range x). unfold max_unsigned. omega.
Qed.

Theorem not_neg:
forall x, not x = add (neg x) mone.
Proof.
bit_solve.
rewrite <- (repr_unsigned x) at 1. unfold add.
rewrite !testbit_repr; auto.
transitivity (Z.testbit (-unsigned x - 1) i).
symmetry. apply Z_one_complement. omega.
apply same_bits_eqm; auto.
replace (-unsigned x - 1) with (-unsigned x + (-1)) by omega.
unfold neg. apply eqm_unsigned_repr.
rewrite unsigned_mone. exists (-1). ring.
Qed.

Theorem neg_not:
forall x, neg x = add (not x) one.
Proof.
apply eqm_samerepr. rewrite unsigned_mone. rewrite unsigned_one.
exists (-1). ring.
Qed.

forall x y, sub x y = add (add x (not y)) one.
Proof.
Qed.

forall x y b,
b = zero \/ b = one ->
sub (sub x y) b = add (add x (not y)) (xor b one).
Proof.
rewrite <- neg_not. rewrite <- sub_add_opp. destruct H; subst b.
rewrite xor_zero_l. rewrite sub_zero_l. auto.
rewrite xor_idem. rewrite sub_idem. auto.
Qed.

forall x y b,
b = zero \/ b = one ->
sub_borrow x y b = xor (add_carry x (not y) (xor b one)) one.
Proof.
intros. unfold sub_borrow, add_carry. rewrite unsigned_not.
replace (unsigned (xor b one)) with (1 - unsigned b).
destruct (zlt (unsigned x - unsigned y - unsigned b)).
rewrite zlt_true. rewrite xor_zero_l; auto.
unfold max_unsigned; omega.
rewrite zlt_false. rewrite xor_idem; auto.
unfold max_unsigned; omega.
destruct H; subst b.
rewrite xor_zero_l. rewrite unsigned_one, unsigned_zero; auto.
rewrite xor_idem. rewrite unsigned_one, unsigned_zero; auto.
Qed.

Connections between add and bitwise logical operations.

forall i, 0 <= i ->
forall x y,
(forall j, 0 <= j <= i -> Z.testbit x j && Z.testbit y j = false) ->
Z.testbit (x + y) i = Z.testbit x i || Z.testbit y i.
Proof.
intros i0 POS0. pattern i0. apply Zlt_0_ind; auto.
intros i IND POS x y EXCL.
rewrite (Zdecomp x) in *. rewrite (Zdecomp y) in *.
transitivity (Z.testbit (Zshiftin (Z.odd x || Z.odd y) (Z.div2 x + Z.div2 y)) i).
- f_equal. rewrite !Zshiftin_spec.
exploit (EXCL 0). omega. rewrite !Ztestbit_shiftin_base. intros.
Opaque Z.mul.
destruct (Z.odd x); destruct (Z.odd y); simpl in *; discriminate || ring.
- rewrite !Ztestbit_shiftin; auto.
destruct (zeq i 0).
+ auto.
+ apply IND. omega. intros.
exploit (EXCL (Z.succ j)). omega.
rewrite !Ztestbit_shiftin_succ. auto.
omega. omega.
Qed.

forall x y,
and x y = zero ->
add x y = or x y.
Proof.
bit_solve. unfold add. rewrite testbit_repr; auto.
intros.
assert (testbit (and x y) j = testbit zero j) by congruence.
autorewrite with ints in H2. assumption. omega.
Qed.

Theorem xor_is_or:
forall x y, and x y = zero -> xor x y = or x y.
Proof.
bit_solve.
assert (testbit (and x y) i = testbit zero i) by congruence.
autorewrite with ints in H1; auto.
destruct (testbit x i); destruct (testbit y i); simpl in *; congruence.
Qed.

forall x y,
and x y = zero ->
add x y = xor x y.
Proof.
intros. rewrite xor_is_or; auto. apply add_is_or; auto.
Qed.

forall x y z,
and y z = zero ->
add (and x y) (and x z) = and x (or y z).
Proof.
rewrite and_or_distrib; auto.
rewrite (and_commut x y).
rewrite and_assoc.
repeat rewrite <- (and_assoc x).
rewrite (and_commut (and x x)).
rewrite <- and_assoc.
rewrite H. rewrite and_commut. apply and_zero.
Qed.

## Properties of shifts

Lemma bits_shl:
forall x y i,
0 <= i < zwordsize ->
testbit (shl x y) i =
if zlt i (unsigned y) then false else testbit x (i - unsigned y).
Proof.
intros. unfold shl. rewrite testbit_repr; auto.
destruct (zlt i (unsigned y)).
apply Z.shiftl_spec_low. auto.
apply Z.shiftl_spec_high. omega. omega.
Qed.

Lemma bits_shru:
forall x y i,
0 <= i < zwordsize ->
testbit (shru x y) i =
if zlt (i + unsigned y) zwordsize then testbit x (i + unsigned y) else false.
Proof.
intros. unfold shru. rewrite testbit_repr; auto.
rewrite Z.shiftr_spec. fold (testbit x (i + unsigned y)).
destruct (zlt (i + unsigned y) zwordsize).
auto.
apply bits_above; auto.
omega.
Qed.

Lemma bits_shr:
forall x y i,
0 <= i < zwordsize ->
testbit (shr x y) i =
testbit x (if zlt (i + unsigned y) zwordsize then i + unsigned y else zwordsize - 1).
Proof.
intros. unfold shr. rewrite testbit_repr; auto.
rewrite Z.shiftr_spec. apply bits_signed.
generalize (unsigned_range y); omega.
omega.
Qed.

Hint Rewrite bits_shl bits_shru bits_shr: ints.

Theorem shl_zero: forall x, shl x zero = x.
Proof.
bit_solve. rewrite unsigned_zero. rewrite zlt_false. f_equal; omega. omega.
Qed.

Lemma bitwise_binop_shl:
forall f f' x y n,
(forall x y i, 0 <= i < zwordsize -> testbit (f x y) i = f' (testbit x i) (testbit y i)) ->
f' false false = false ->
f (shl x n) (shl y n) = shl (f x y) n.
Proof.
intros. apply same_bits_eq; intros.
rewrite H; auto. rewrite !bits_shl; auto.
destruct (zlt i (unsigned n)); auto.
rewrite H; auto. generalize (unsigned_range n); omega.
Qed.

Theorem and_shl:
forall x y n,
and (shl x n) (shl y n) = shl (and x y) n.
Proof.
intros. apply bitwise_binop_shl with andb. exact bits_and. auto.
Qed.

Theorem or_shl:
forall x y n,
or (shl x n) (shl y n) = shl (or x y) n.
Proof.
intros. apply bitwise_binop_shl with orb. exact bits_or. auto.
Qed.

Theorem xor_shl:
forall x y n,
xor (shl x n) (shl y n) = shl (xor x y) n.
Proof.
intros. apply bitwise_binop_shl with xorb. exact bits_xor. auto.
Qed.

Lemma ltu_inv:
forall x y, ltu x y = true -> 0 <= unsigned x < unsigned y.
Proof.
unfold ltu; intros. destruct (zlt (unsigned x) (unsigned y)).
split; auto. generalize (unsigned_range x); omega.
discriminate.
Qed.

Lemma ltu_iwordsize_inv:
forall x, ltu x iwordsize = true -> 0 <= unsigned x < zwordsize.
Proof.
intros. generalize (ltu_inv _ _ H). rewrite unsigned_repr_wordsize. auto.
Qed.

Theorem shl_shl:
forall x y z,
ltu y iwordsize = true ->
ltu z iwordsize = true ->
ltu (add y z) iwordsize = true ->
shl (shl x y) z = shl x (add y z).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros.
assert (unsigned (add y z) = unsigned y + unsigned z).
generalize two_wordsize_max_unsigned; omega.
apply same_bits_eq; intros.
rewrite bits_shl; auto.
destruct (zlt i (unsigned z)).
- rewrite bits_shl; auto. rewrite zlt_true. auto. omega.
- rewrite bits_shl. destruct (zlt (i - unsigned z) (unsigned y)).
+ rewrite bits_shl; auto. rewrite zlt_true. auto. omega.
+ rewrite bits_shl; auto. rewrite zlt_false. f_equal. omega. omega.
+ omega.
Qed.

Theorem shru_zero: forall x, shru x zero = x.
Proof.
bit_solve. rewrite unsigned_zero. rewrite zlt_true. f_equal; omega. omega.
Qed.

Lemma bitwise_binop_shru:
forall f f' x y n,
(forall x y i, 0 <= i < zwordsize -> testbit (f x y) i = f' (testbit x i) (testbit y i)) ->
f' false false = false ->
f (shru x n) (shru y n) = shru (f x y) n.
Proof.
intros. apply same_bits_eq; intros.
rewrite H; auto. rewrite !bits_shru; auto.
destruct (zlt (i + unsigned n) zwordsize); auto.
rewrite H; auto. generalize (unsigned_range n); omega.
Qed.

Theorem and_shru:
forall x y n,
and (shru x n) (shru y n) = shru (and x y) n.
Proof.
intros. apply bitwise_binop_shru with andb; auto. exact bits_and.
Qed.

Theorem or_shru:
forall x y n,
or (shru x n) (shru y n) = shru (or x y) n.
Proof.
intros. apply bitwise_binop_shru with orb; auto. exact bits_or.
Qed.

Theorem xor_shru:
forall x y n,
xor (shru x n) (shru y n) = shru (xor x y) n.
Proof.
intros. apply bitwise_binop_shru with xorb; auto. exact bits_xor.
Qed.

Theorem shru_shru:
forall x y z,
ltu y iwordsize = true ->
ltu z iwordsize = true ->
ltu (add y z) iwordsize = true ->
shru (shru x y) z = shru x (add y z).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros.
assert (unsigned (add y z) = unsigned y + unsigned z).
generalize two_wordsize_max_unsigned; omega.
apply same_bits_eq; intros.
rewrite bits_shru; auto.
destruct (zlt (i + unsigned z) zwordsize).
- rewrite bits_shru. destruct (zlt (i + unsigned z + unsigned y) zwordsize).
+ rewrite bits_shru; auto. rewrite zlt_true. f_equal. omega. omega.
+ rewrite bits_shru; auto. rewrite zlt_false. auto. omega.
+ omega.
- rewrite bits_shru; auto. rewrite zlt_false. auto. omega.
Qed.

Theorem shr_zero: forall x, shr x zero = x.
Proof.
bit_solve. rewrite unsigned_zero. rewrite zlt_true. f_equal; omega. omega.
Qed.

Lemma bitwise_binop_shr:
forall f f' x y n,
(forall x y i, 0 <= i < zwordsize -> testbit (f x y) i = f' (testbit x i) (testbit y i)) ->
f (shr x n) (shr y n) = shr (f x y) n.
Proof.
intros. apply same_bits_eq; intros.
rewrite H; auto. rewrite !bits_shr; auto.
rewrite H; auto.
destruct (zlt (i + unsigned n) zwordsize).
generalize (unsigned_range n); omega.
omega.
Qed.

Theorem and_shr:
forall x y n,
and (shr x n) (shr y n) = shr (and x y) n.
Proof.
intros. apply bitwise_binop_shr with andb. exact bits_and.
Qed.

Theorem or_shr:
forall x y n,
or (shr x n) (shr y n) = shr (or x y) n.
Proof.
intros. apply bitwise_binop_shr with orb. exact bits_or.
Qed.

Theorem xor_shr:
forall x y n,
xor (shr x n) (shr y n) = shr (xor x y) n.
Proof.
intros. apply bitwise_binop_shr with xorb. exact bits_xor.
Qed.

Theorem shr_shr:
forall x y z,
ltu y iwordsize = true ->
ltu z iwordsize = true ->
ltu (add y z) iwordsize = true ->
shr (shr x y) z = shr x (add y z).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros.
assert (unsigned (add y z) = unsigned y + unsigned z).
generalize two_wordsize_max_unsigned; omega.
apply same_bits_eq; intros.
rewrite !bits_shr; auto. f_equal.
destruct (zlt (i + unsigned z) zwordsize).
rewrite H4. replace (i + (unsigned y + unsigned z)) with (i + unsigned z + unsigned y) by omega. auto.
rewrite (zlt_false _ (i + unsigned (add y z))).
destruct (zlt (zwordsize - 1 + unsigned y) zwordsize); omega.
omega.
destruct (zlt (i + unsigned z) zwordsize); omega.
Qed.

Theorem and_shr_shru:
forall x y z,
and (shr x z) (shru y z) = shru (and x y) z.
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_and; auto. rewrite bits_shr; auto. rewrite !bits_shru; auto.
destruct (zlt (i + unsigned z) zwordsize).
- rewrite bits_and; auto. generalize (unsigned_range z); omega.
- apply andb_false_r.
Qed.

Theorem shr_and_shru_and:
forall x y z,
shru (shl z y) y = z ->
and (shr x y) z = and (shru x y) z.
Proof.
intros.
rewrite <- H.
rewrite and_shru. rewrite and_shr_shru. auto.
Qed.

Theorem shru_lt_zero:
forall x,
shru x (repr (zwordsize - 1)) = if lt x zero then one else zero.
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_shru; auto.
rewrite unsigned_repr.
destruct (zeq i 0).
subst i. rewrite Zplus_0_l. rewrite zlt_true.
rewrite sign_bit_of_unsigned.
unfold lt. rewrite signed_zero. unfold signed.
destruct (zlt (unsigned x) half_modulus).
rewrite zlt_false. auto. generalize (unsigned_range x); omega.
rewrite zlt_true. unfold one; rewrite testbit_repr; auto.
generalize (unsigned_range x); omega.
omega.
rewrite zlt_false.
unfold testbit. rewrite Ztestbit_eq. rewrite zeq_false.
destruct (lt x zero).
rewrite unsigned_one. simpl Z.div2. rewrite Z.testbit_0_l; auto.
rewrite unsigned_zero. simpl Z.div2. rewrite Z.testbit_0_l; auto.
auto. omega. omega.
generalize wordsize_max_unsigned; omega.
Qed.

Theorem shr_lt_zero:
forall x,
shr x (repr (zwordsize - 1)) = if lt x zero then mone else zero.
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_shr; auto.
rewrite unsigned_repr.
transitivity (testbit x (zwordsize - 1)).
f_equal. destruct (zlt (i + (zwordsize - 1)) zwordsize); omega.
rewrite sign_bit_of_unsigned.
unfold lt. rewrite signed_zero. unfold signed.
destruct (zlt (unsigned x) half_modulus).
rewrite zlt_false. rewrite bits_zero; auto. generalize (unsigned_range x); omega.
rewrite zlt_true. rewrite bits_mone; auto. generalize (unsigned_range x); omega.
generalize wordsize_max_unsigned; omega.
Qed.

## Properties of rotations

Lemma bits_rol:
forall x y i,
0 <= i < zwordsize ->
testbit (rol x y) i = testbit x ((i - unsigned y) mod zwordsize).
Proof.
intros. unfold rol.
exploit (Z_div_mod_eq (unsigned y) zwordsize). apply wordsize_pos.
set (j := unsigned y mod zwordsize). set (k := unsigned y / zwordsize).
intros EQ.
exploit (Z_mod_lt (unsigned y) zwordsize). apply wordsize_pos.
fold j. intros RANGE.
rewrite testbit_repr; auto.
rewrite Z.lor_spec. rewrite Z.shiftr_spec. 2: omega.
destruct (zlt i j).
- rewrite Z.shiftl_spec_low; auto. simpl.
unfold testbit. f_equal.
symmetry. apply Zmod_unique with (-k - 1).
rewrite EQ. ring.
omega.
- rewrite Z.shiftl_spec_high.
fold (testbit x (i + (zwordsize - j))).
rewrite bits_above. rewrite orb_false_r.
fold (testbit x (i - j)).
f_equal. symmetry. apply Zmod_unique with (-k).
rewrite EQ. ring.
omega. omega. omega. omega.
Qed.

Lemma bits_ror:
forall x y i,
0 <= i < zwordsize ->
testbit (ror x y) i = testbit x ((i + unsigned y) mod zwordsize).
Proof.
intros. unfold ror.
exploit (Z_div_mod_eq (unsigned y) zwordsize). apply wordsize_pos.
set (j := unsigned y mod zwordsize). set (k := unsigned y / zwordsize).
intros EQ.
exploit (Z_mod_lt (unsigned y) zwordsize). apply wordsize_pos.
fold j. intros RANGE.
rewrite testbit_repr; auto.
rewrite Z.lor_spec. rewrite Z.shiftr_spec. 2: omega.
destruct (zlt (i + j) zwordsize).
- rewrite Z.shiftl_spec_low; auto. rewrite orb_false_r.
unfold testbit. f_equal.
symmetry. apply Zmod_unique with k.
rewrite EQ. ring.
omega. omega.
- rewrite Z.shiftl_spec_high.
fold (testbit x (i + j)).
rewrite bits_above. simpl.
unfold testbit. f_equal.
symmetry. apply Zmod_unique with (k + 1).
rewrite EQ. ring.
omega. omega. omega. omega.
Qed.

Hint Rewrite bits_rol bits_ror: ints.

Theorem shl_rolm:
forall x n,
ltu n iwordsize = true ->
shl x n = rolm x n (shl mone n).
Proof.
intros. generalize (ltu_inv _ _ H). rewrite unsigned_repr_wordsize; intros.
unfold rolm. apply same_bits_eq; intros.
rewrite bits_and; auto. rewrite !bits_shl; auto. rewrite bits_rol; auto.
destruct (zlt i (unsigned n)).
- rewrite andb_false_r; auto.
- generalize (unsigned_range n); intros.
rewrite bits_mone. rewrite andb_true_r. f_equal.
symmetry. apply Zmod_small. omega.
omega.
Qed.

Theorem shru_rolm:
forall x n,
ltu n iwordsize = true ->
shru x n = rolm x (sub iwordsize n) (shru mone n).
Proof.
intros. generalize (ltu_inv _ _ H). rewrite unsigned_repr_wordsize; intros.
unfold rolm. apply same_bits_eq; intros.
rewrite bits_and; auto. rewrite !bits_shru; auto. rewrite bits_rol; auto.
destruct (zlt (i + unsigned n) zwordsize).
- generalize (unsigned_range n); intros.
rewrite bits_mone. rewrite andb_true_r. f_equal.
unfold sub. rewrite unsigned_repr. rewrite unsigned_repr_wordsize.
symmetry. apply Zmod_unique with (-1). ring. omega.
rewrite unsigned_repr_wordsize. generalize wordsize_max_unsigned. omega.
omega.
- rewrite andb_false_r; auto.
Qed.

Theorem rol_zero:
forall x,
rol x zero = x.
Proof.
bit_solve. f_equal. rewrite unsigned_zero. rewrite Zminus_0_r.
apply Zmod_small; auto.
Qed.

Lemma bitwise_binop_rol:
forall f f' x y n,
(forall x y i, 0 <= i < zwordsize -> testbit (f x y) i = f' (testbit x i) (testbit y i)) ->
rol (f x y) n = f (rol x n) (rol y n).
Proof.
intros. apply same_bits_eq; intros.
rewrite H; auto. rewrite !bits_rol; auto. rewrite H; auto.
apply Z_mod_lt. apply wordsize_pos.
Qed.

Theorem rol_and:
forall x y n,
rol (and x y) n = and (rol x n) (rol y n).
Proof.
intros. apply bitwise_binop_rol with andb. exact bits_and.
Qed.

Theorem rol_or:
forall x y n,
rol (or x y) n = or (rol x n) (rol y n).
Proof.
intros. apply bitwise_binop_rol with orb. exact bits_or.
Qed.

Theorem rol_xor:
forall x y n,
rol (xor x y) n = xor (rol x n) (rol y n).
Proof.
intros. apply bitwise_binop_rol with xorb. exact bits_xor.
Qed.

Theorem rol_rol:
forall x n m,
Zdivide zwordsize modulus ->
rol (rol x n) m = rol x (modu (add n m) iwordsize).
Proof.
bit_solve. f_equal. apply eqmod_mod_eq. apply wordsize_pos.
set (M := unsigned m); set (N := unsigned n).
apply eqmod_trans with (i - M - N).
apply eqmod_sub.
apply eqmod_sym. apply eqmod_mod. apply wordsize_pos.
apply eqmod_refl.
replace (i - M - N) with (i - (M + N)) by omega.
apply eqmod_sub.
apply eqmod_refl.
apply eqmod_trans with (Zmod (unsigned n + unsigned m) zwordsize).
replace (M + N) with (N + M) by omega. apply eqmod_mod. apply wordsize_pos.
unfold modu, add. fold M; fold N. rewrite unsigned_repr_wordsize.
assert (forall a, eqmod zwordsize a (unsigned (repr a))).
intros. eapply eqmod_divides. apply eqm_unsigned_repr. assumption.
eapply eqmod_trans. 2: apply H1.
apply eqmod_refl2. apply eqmod_mod_eq. apply wordsize_pos. auto.
apply Z_mod_lt. apply wordsize_pos.
Qed.

Theorem rolm_zero:
forall x m,
rolm x zero m = and x m.
Proof.
intros. unfold rolm. rewrite rol_zero. auto.
Qed.

Theorem rolm_rolm:
forall x n1 m1 n2 m2,
Zdivide zwordsize modulus ->
rolm (rolm x n1 m1) n2 m2 =
rolm x (modu (add n1 n2) iwordsize)
(and (rol m1 n2) m2).
Proof.
intros.
unfold rolm. rewrite rol_and. rewrite and_assoc.
rewrite rol_rol. reflexivity. auto.
Qed.

Theorem or_rolm:
forall x n m1 m2,
or (rolm x n m1) (rolm x n m2) = rolm x n (or m1 m2).
Proof.
intros; unfold rolm. symmetry. apply and_or_distrib.
Qed.

Theorem ror_rol:
forall x y,
ltu y iwordsize = true ->
ror x y = rol x (sub iwordsize y).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H); intros.
apply same_bits_eq; intros.
rewrite bits_ror; auto. rewrite bits_rol; auto. f_equal.
unfold sub. rewrite unsigned_repr. rewrite unsigned_repr_wordsize.
apply eqmod_mod_eq. apply wordsize_pos. exists 1. ring.
rewrite unsigned_repr_wordsize.
generalize wordsize_pos; generalize wordsize_max_unsigned; omega.
Qed.

Theorem ror_rol_neg:
forall x y, (zwordsize | modulus) -> ror x y = rol x (neg y).
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_ror by auto. rewrite bits_rol by auto.
f_equal. apply eqmod_mod_eq. omega.
apply eqmod_trans with (i - (- unsigned y)).
apply eqmod_refl2; omega.
apply eqmod_sub. apply eqmod_refl.
apply eqmod_divides with modulus.
apply eqm_unsigned_repr. auto.
Qed.

Theorem or_ror:
forall x y z,
ltu y iwordsize = true ->
ltu z iwordsize = true ->
add y z = iwordsize ->
ror x z = or (shl x y) (shru x z).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros.
unfold ror, or, shl, shru. apply same_bits_eq; intros.
rewrite !testbit_repr; auto.
rewrite !Z.lor_spec. rewrite orb_comm. f_equal; apply same_bits_eqm; auto.
- apply eqm_unsigned_repr_r. apply eqm_refl2. f_equal.
rewrite Zmod_small; auto.
assert (unsigned (add y z) = zwordsize).
rewrite H1. apply unsigned_repr_wordsize.
unfold add in H5. rewrite unsigned_repr in H5.
omega.
generalize two_wordsize_max_unsigned; omega.
- apply eqm_unsigned_repr_r. apply eqm_refl2. f_equal.
apply Zmod_small; auto.
Qed.

## Properties of Z_one_bits and is_power2.

Fixpoint powerserie (l: list Z): Z :=
match l with
| nil => 0
| x :: xs => two_p x + powerserie xs
end.

Lemma Z_one_bits_powerserie:
forall x, 0 <= x < modulus -> x = powerserie (Z_one_bits wordsize x 0).
Proof.
assert (forall n x i,
0 <= i ->
0 <= x < two_power_nat n ->
x * two_p i = powerserie (Z_one_bits n x i)).
{
induction n; intros.
simpl. rewrite two_power_nat_O in H0.
assert (x = 0) by omega. subst x. omega.
rewrite two_power_nat_S in H0. simpl Z_one_bits.
rewrite (Zdecomp x) in H0. rewrite Zshiftin_spec in H0.
assert (EQ: Z.div2 x * two_p (i + 1) = powerserie (Z_one_bits n (Z.div2 x) (i + 1))).
apply IHn. omega.
destruct (Z.odd x); omega.
rewrite two_p_is_exp in EQ. change (two_p 1) with 2 in EQ.
rewrite (Zdecomp x) at 1. rewrite Zshiftin_spec.
destruct (Z.odd x); simpl powerserie; rewrite <- EQ; ring.
omega. omega.
}
intros. rewrite <- H. change (two_p 0) with 1. omega.
omega. exact H0.
Qed.

Lemma Z_one_bits_range:
forall x i, In i (Z_one_bits wordsize x 0) -> 0 <= i < zwordsize.
Proof.
assert (forall n x i j,
In j (Z_one_bits n x i) -> i <= j < i + Z_of_nat n).
{
induction n; simpl In.
tauto.
intros x i j. rewrite inj_S.
assert (In j (Z_one_bits n (Z.div2 x) (i + 1)) -> i <= j < i + Z.succ (Z.of_nat n)).
intros. exploit IHn; eauto. omega.
destruct (Z.odd x); simpl.
intros [A|B]. subst j. omega. auto.
auto.
}
intros. generalize (H wordsize x 0 i H0). fold zwordsize; omega.
Qed.

Lemma is_power2_rng:
forall n logn,
is_power2 n = Some logn ->
0 <= unsigned logn < zwordsize.
Proof.
intros n logn. unfold is_power2.
generalize (Z_one_bits_range (unsigned n)).
destruct (Z_one_bits wordsize (unsigned n) 0).
intros; discriminate.
destruct l.
intros. injection H0; intro; subst logn; clear H0.
assert (0 <= z < zwordsize).
apply H. auto with coqlib.
rewrite unsigned_repr. auto. generalize wordsize_max_unsigned; omega.
intros; discriminate.
Qed.

Theorem is_power2_range:
forall n logn,
is_power2 n = Some logn -> ltu logn iwordsize = true.
Proof.
intros. unfold ltu. rewrite unsigned_repr_wordsize.
apply zlt_true. generalize (is_power2_rng _ _ H). tauto.
Qed.

Lemma is_power2_correct:
forall n logn,
is_power2 n = Some logn ->
unsigned n = two_p (unsigned logn).
Proof.
intros n logn. unfold is_power2.
generalize (Z_one_bits_powerserie (unsigned n) (unsigned_range n)).
generalize (Z_one_bits_range (unsigned n)).
destruct (Z_one_bits wordsize (unsigned n) 0).
intros; discriminate.
destruct l.
intros. simpl in H0. injection H1; intros; subst logn; clear H1.
rewrite unsigned_repr. replace (two_p z) with (two_p z + 0).
auto. omega. elim (H z); intros.
generalize wordsize_max_unsigned; omega.
auto with coqlib.
intros; discriminate.
Qed.

Remark two_p_range:
forall n,
0 <= n < zwordsize ->
0 <= two_p n <= max_unsigned.
Proof.
intros. split.
assert (two_p n > 0). apply two_p_gt_ZERO. omega. omega.
generalize (two_p_monotone_strict _ _ H).
unfold zwordsize; rewrite <- two_power_nat_two_p.
unfold max_unsigned, modulus. omega.
Qed.

Remark Z_one_bits_zero:
forall n i, Z_one_bits n 0 i = nil.
Proof.
induction n; intros; simpl; auto.
Qed.

Remark Z_one_bits_two_p:
forall n x i,
0 <= x < Z_of_nat n ->
Z_one_bits n (two_p x) i = (i + x) :: nil.
Proof.
induction n; intros; simpl. simpl in H. omegaContradiction.
rewrite inj_S in H.
assert (x = 0 \/ 0 < x) by omega. destruct H0.
subst x; simpl. decEq. omega. apply Z_one_bits_zero.
assert (Z.odd (two_p x) = false /\ Z.div2 (two_p x) = two_p (x-1)).
apply Zshiftin_inj. rewrite <- Zdecomp. rewrite !Zshiftin_spec.
rewrite <- two_p_S. rewrite Zplus_0_r. f_equal; omega. omega.
destruct H1 as [A B]; rewrite A; rewrite B.
rewrite IHn. f_equal; omega. omega.
Qed.

Lemma is_power2_two_p:
forall n, 0 <= n < zwordsize ->
is_power2 (repr (two_p n)) = Some (repr n).
Proof.
intros. unfold is_power2. rewrite unsigned_repr.
rewrite Z_one_bits_two_p. auto. auto.
apply two_p_range. auto.
Qed.

## Relation between bitwise operations and multiplications / divisions by powers of 2

Left shifts and multiplications by powers of 2.

Lemma Zshiftl_mul_two_p:
forall x n, 0 <= n -> Z.shiftl x n = x * two_p n.
Proof.
intros. destruct n; simpl.
- omega.
- pattern p. apply Pos.peano_ind.
+ change (two_power_pos 1) with 2. simpl. ring.
+ intros. rewrite Pos.iter_succ. rewrite H0.
rewrite Pplus_one_succ_l. rewrite two_power_pos_is_exp.
change (two_power_pos 1) with 2. ring.
- compute in H. congruence.
Qed.

Lemma shl_mul_two_p:
forall x y,
shl x y = mul x (repr (two_p (unsigned y))).
Proof.
intros. unfold shl, mul. apply eqm_samerepr.
rewrite Zshiftl_mul_two_p. auto with ints.
generalize (unsigned_range y); omega.
Qed.

Theorem shl_mul:
forall x y,
shl x y = mul x (shl one y).
Proof.
intros.
assert (shl one y = repr (two_p (unsigned y))).
{
rewrite shl_mul_two_p. rewrite mul_commut. rewrite mul_one. auto.
}
rewrite H. apply shl_mul_two_p.
Qed.

Theorem mul_pow2:
forall x n logn,
is_power2 n = Some logn ->
mul x n = shl x logn.
Proof.
intros. generalize (is_power2_correct n logn H); intro.
rewrite shl_mul_two_p. rewrite <- H0. rewrite repr_unsigned.
auto.
Qed.

forall x y n,
0 <= n < zwordsize ->
unsigned y < two_p n ->
or (shl x (repr n)) y = repr(unsigned x * two_p n + unsigned y).
Proof.
rewrite shl_mul_two_p. unfold mul. apply eqm_unsigned_repr_l.
apply eqm_mult; auto with ints. apply eqm_unsigned_repr_l.
apply eqm_refl2. rewrite unsigned_repr. auto.
generalize wordsize_max_unsigned; omega.
- bit_solve.
rewrite unsigned_repr.
destruct (zlt i n).
+ auto.
+ replace (testbit y i) with false. apply andb_false_r.
symmetry. unfold testbit.
assert (EQ: Z.of_nat (Z.to_nat n) = n) by (apply Z2Nat.id; omega).
apply Ztestbit_above with (Z.to_nat n).
rewrite <- EQ in H0. rewrite <- two_power_nat_two_p in H0.
generalize (unsigned_range y); omega.
rewrite EQ; auto.
+ generalize wordsize_max_unsigned; omega.
Qed.

Unsigned right shifts and unsigned divisions by powers of 2.

Lemma Zshiftr_div_two_p:
forall x n, 0 <= n -> Z.shiftr x n = x / two_p n.
Proof.
intros. destruct n; unfold Z.shiftr; simpl.
- rewrite Zdiv_1_r. auto.
- pattern p. apply Pos.peano_ind.
+ change (two_power_pos 1) with 2. simpl. apply Zdiv2_div.
+ intros. rewrite Pos.iter_succ. rewrite H0.
rewrite Pplus_one_succ_l. rewrite two_power_pos_is_exp.
change (two_power_pos 1) with 2.
rewrite Zdiv2_div. rewrite Zmult_comm. apply Zdiv_Zdiv.
rewrite two_power_pos_nat. apply two_power_nat_pos. omega.
- compute in H. congruence.
Qed.

Lemma shru_div_two_p:
forall x y,
shru x y = repr (unsigned x / two_p (unsigned y)).
Proof.
intros. unfold shru.
rewrite Zshiftr_div_two_p. auto.
generalize (unsigned_range y); omega.
Qed.

Theorem divu_pow2:
forall x n logn,
is_power2 n = Some logn ->
divu x n = shru x logn.
Proof.
intros. generalize (is_power2_correct n logn H). intro.
symmetry. unfold divu. rewrite H0. apply shru_div_two_p.
Qed.

Signed right shifts and signed divisions by powers of 2.

Lemma shr_div_two_p:
forall x y,
shr x y = repr (signed x / two_p (unsigned y)).
Proof.
intros. unfold shr.
rewrite Zshiftr_div_two_p. auto.
generalize (unsigned_range y); omega.
Qed.

Theorem divs_pow2:
forall x n logn,
is_power2 n = Some logn ->
divs x n = shrx x logn.
Proof.
intros. generalize (is_power2_correct _ _ H); intro.
unfold shrx. rewrite shl_mul_two_p.
rewrite mul_commut. rewrite mul_one.
rewrite <- H0. rewrite repr_unsigned. auto.
Qed.

Unsigned modulus over 2^n is masking with 2^n-1.

Lemma Ztestbit_mod_two_p:
forall n x i,
0 <= n -> 0 <= i ->
Z.testbit (x mod (two_p n)) i = if zlt i n then Z.testbit x i else false.
Proof.
intros n0 x i N0POS. revert x i; pattern n0; apply natlike_ind; auto.
- intros. change (two_p 0) with 1. rewrite Zmod_1_r. rewrite Z.testbit_0_l.
rewrite zlt_false; auto. omega.
- intros. rewrite two_p_S; auto.
replace (x0 mod (2 * two_p x))
with (Zshiftin (Z.odd x0) (Z.div2 x0 mod two_p x)).
rewrite Ztestbit_shiftin; auto. rewrite (Ztestbit_eq i x0); auto. destruct (zeq i 0).
+ rewrite zlt_true; auto. omega.
+ rewrite H0. destruct (zlt (Z.pred i) x).
* rewrite zlt_true; auto. omega.
* rewrite zlt_false; auto. omega.
* omega.
+ rewrite (Zdecomp x0) at 3. set (x1 := Z.div2 x0). symmetry.
apply Zmod_unique with (x1 / two_p x).
rewrite !Zshiftin_spec. rewrite Zplus_assoc. f_equal.
transitivity (2 * (two_p x * (x1 / two_p x) + x1 mod two_p x)).
f_equal. apply Z_div_mod_eq. apply two_p_gt_ZERO; auto.
ring.
rewrite Zshiftin_spec. exploit (Z_mod_lt x1 (two_p x)). apply two_p_gt_ZERO; auto.
destruct (Z.odd x0); omega.
Qed.

Corollary Ztestbit_two_p_m1:
forall n i, 0 <= n -> 0 <= i ->
Z.testbit (two_p n - 1) i = if zlt i n then true else false.
Proof.
intros. replace (two_p n - 1) with ((-1) mod (two_p n)).
rewrite Ztestbit_mod_two_p; auto. destruct (zlt i n); auto. apply Ztestbit_m1; auto.
apply Zmod_unique with (-1). ring.
exploit (two_p_gt_ZERO n). auto. omega.
Qed.

Theorem modu_and:
forall x n logn,
is_power2 n = Some logn ->
modu x n = and x (sub n one).
Proof.
intros. generalize (is_power2_correct _ _ H); intro.
generalize (is_power2_rng _ _ H); intro.
apply same_bits_eq; intros.
rewrite bits_and; auto.
unfold sub. rewrite testbit_repr; auto.
rewrite H0. rewrite unsigned_one.
unfold modu. rewrite testbit_repr; auto. rewrite H0.
rewrite Ztestbit_mod_two_p. rewrite Ztestbit_two_p_m1.
destruct (zlt i (unsigned logn)).
rewrite andb_true_r; auto.
rewrite andb_false_r; auto.
tauto. tauto. tauto. tauto.
Qed.

## Properties of shrx (signed division by a power of 2)

Lemma Zquot_Zdiv:
forall x y,
y > 0 ->
Z.quot x y = if zlt x 0 then (x + y - 1) / y else x / y.
Proof.
intros. destruct (zlt x 0).
- symmetry. apply Zquot_unique_full with ((x + y - 1) mod y - (y - 1)).
+ red. right; split. omega.
exploit (Z_mod_lt (x + y - 1) y); auto.
rewrite Z.abs_eq. omega. omega.
+ transitivity ((y * ((x + y - 1) / y) + (x + y - 1) mod y) - (y-1)).
rewrite <- Z_div_mod_eq. ring. auto. ring.
- apply Zquot_Zdiv_pos; omega.
Qed.

Theorem shrx_shr:
forall x y,
ltu y (repr (zwordsize - 1)) = true ->
shrx x y = shr (if lt x zero then add x (sub (shl one y) one) else x) y.
Proof.
intros.
set (uy := unsigned y).
assert (0 <= uy < zwordsize - 1).
generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto.
generalize wordsize_pos wordsize_max_unsigned; omega.
rewrite shr_div_two_p. unfold shrx. unfold divs.
assert (shl one y = repr (two_p uy)).
transitivity (mul one (repr (two_p uy))).
symmetry. apply mul_pow2. replace y with (repr uy).
apply is_power2_two_p. omega. apply repr_unsigned.
rewrite mul_commut. apply mul_one.
assert (two_p uy > 0). apply two_p_gt_ZERO. omega.
assert (two_p uy < half_modulus).
rewrite half_modulus_power.
apply two_p_monotone_strict. auto.
assert (two_p uy < modulus).
rewrite modulus_power. apply two_p_monotone_strict. omega.
assert (unsigned (shl one y) = two_p uy).
rewrite H1. apply unsigned_repr. unfold max_unsigned. omega.
assert (signed (shl one y) = two_p uy).
rewrite H1. apply signed_repr.
unfold max_signed. generalize min_signed_neg. omega.
rewrite H6.
rewrite Zquot_Zdiv; auto.
unfold lt. rewrite signed_zero.
destruct (zlt (signed x) 0); auto.
assert (signed (sub (shl one y) one) = two_p uy - 1).
unfold sub. rewrite H5. rewrite unsigned_one.
apply signed_repr.
generalize min_signed_neg. unfold max_signed. omega.
rewrite H7. rewrite signed_repr. f_equal. f_equal. omega.
generalize (signed_range x). intros.
assert (two_p uy - 1 <= max_signed). unfold max_signed. omega. omega.
Qed.

Theorem shrx_shr_2:
forall x y,
ltu y (repr (zwordsize - 1)) = true ->
shrx x y = shr (add x (shru (shr x (repr (zwordsize - 1))) (sub iwordsize y))) y.
Proof.
intros.
rewrite shrx_shr by auto. f_equal.
rewrite shr_lt_zero. destruct (lt x zero).
- set (uy := unsigned y).
generalize (unsigned_range y); fold uy; intros.
assert (0 <= uy < zwordsize - 1).
generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto.
generalize wordsize_pos wordsize_max_unsigned; omega.
assert (two_p uy < modulus).
rewrite modulus_power. apply two_p_monotone_strict. omega.
f_equal. rewrite shl_mul_two_p. fold uy. rewrite mul_commut. rewrite mul_one.
unfold sub. rewrite unsigned_one. rewrite unsigned_repr.
rewrite unsigned_repr_wordsize. fold uy.
apply same_bits_eq; intros. rewrite bits_shru by auto.
rewrite testbit_repr by auto. rewrite Ztestbit_two_p_m1 by omega.
rewrite unsigned_repr by (generalize wordsize_max_unsigned; omega).
destruct (zlt i uy).
rewrite zlt_true by omega. rewrite bits_mone by omega. auto.
rewrite zlt_false by omega. auto.
assert (two_p uy > 0) by (apply two_p_gt_ZERO; omega). unfold max_unsigned; omega.
- replace (shru zero (sub iwordsize y)) with zero.
bit_solve. destruct (zlt (i + unsigned (sub iwordsize y)) zwordsize); auto.
Qed.

Lemma Zdiv_shift:
forall x y, y > 0 ->
(x + (y - 1)) / y = x / y + if zeq (Zmod x y) 0 then 0 else 1.
Proof.
intros. generalize (Z_div_mod_eq x y H). generalize (Z_mod_lt x y H).
set (q := x / y). set (r := x mod y). intros.
destruct (zeq r 0).
apply Zdiv_unique with (y - 1). rewrite H1. rewrite e. ring. omega.
apply Zdiv_unique with (r - 1). rewrite H1. ring. omega.
Qed.

Theorem shrx_carry:
forall x y,
ltu y (repr (zwordsize - 1)) = true ->
shrx x y = add (shr x y) (shr_carry x y).
Proof.
intros. rewrite shrx_shr; auto. unfold shr_carry.
unfold lt. set (sx := signed x). rewrite signed_zero.
destruct (zlt sx 0); simpl.
set (uy := unsigned y).
assert (0 <= uy < zwordsize - 1).
generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto.
generalize wordsize_pos wordsize_max_unsigned; omega.
assert (shl one y = repr (two_p uy)).
rewrite shl_mul_two_p. rewrite mul_commut. apply mul_one.
assert (and x (sub (shl one y) one) = modu x (repr (two_p uy))).
symmetry. rewrite H1. apply modu_and with (logn := y).
rewrite is_power2_two_p. unfold uy. rewrite repr_unsigned. auto.
omega.
rewrite H2. rewrite H1.
repeat rewrite shr_div_two_p. fold sx. fold uy.
assert (two_p uy > 0). apply two_p_gt_ZERO. omega.
assert (two_p uy < modulus).
rewrite modulus_power. apply two_p_monotone_strict. omega.
assert (two_p uy < half_modulus).
rewrite half_modulus_power.
apply two_p_monotone_strict. auto.
assert (two_p uy < modulus).
rewrite modulus_power. apply two_p_monotone_strict. omega.
assert (sub (repr (two_p uy)) one = repr (two_p uy - 1)).
unfold sub. apply eqm_samerepr. apply eqm_sub. apply eqm_sym; apply eqm_unsigned_repr.
rewrite unsigned_one. apply eqm_refl.
rewrite H7. rewrite add_signed. fold sx.
rewrite (signed_repr (two_p uy - 1)). rewrite signed_repr.
unfold modu. rewrite unsigned_repr.
unfold eq. rewrite unsigned_zero. rewrite unsigned_repr.
assert (unsigned x mod two_p uy = sx mod two_p uy).
apply eqmod_mod_eq; auto. apply eqmod_divides with modulus.
fold eqm. unfold sx. apply eqm_sym. apply eqm_signed_unsigned.
unfold modulus. rewrite two_power_nat_two_p.
exists (two_p (zwordsize - uy)). rewrite <- two_p_is_exp.
f_equal. fold zwordsize; omega. omega. omega.
rewrite H8. rewrite Zdiv_shift; auto.
apply eqm_unsigned_repr.
destruct (zeq (sx mod two_p uy) 0); simpl.
rewrite unsigned_zero. apply eqm_refl.
rewrite unsigned_one. apply eqm_refl.
generalize (Z_mod_lt (unsigned x) (two_p uy) H3). unfold max_unsigned. omega.
unfold max_unsigned; omega.
generalize (signed_range x). fold sx. intros. split. omega. unfold max_signed. omega.
generalize min_signed_neg. unfold max_signed. omega.
Qed.

Connections between shr and shru.

Lemma shr_shru_positive:
forall x y,
signed x >= 0 ->
shr x y = shru x y.
Proof.
intros.
rewrite shr_div_two_p. rewrite shru_div_two_p.
rewrite signed_eq_unsigned. auto. apply signed_positive. auto.
Qed.

Lemma and_positive:
forall x y, signed y >= 0 -> signed (and x y) >= 0.
Proof.
intros.
assert (unsigned y < half_modulus). rewrite signed_positive in H. unfold max_signed in H; omega.
generalize (sign_bit_of_unsigned y). rewrite zlt_true; auto. intros A.
generalize (sign_bit_of_unsigned (and x y)). rewrite bits_and. rewrite A.
rewrite andb_false_r. unfold signed.
destruct (zlt (unsigned (and x y)) half_modulus).
intros. generalize (unsigned_range (and x y)); omega.
congruence.
generalize wordsize_pos; omega.
Qed.

Theorem shr_and_is_shru_and:
forall x y z,
lt y zero = false -> shr (and x y) z = shru (and x y) z.
Proof.
intros. apply shr_shru_positive. apply and_positive.
unfold lt in H. rewrite signed_zero in H. destruct (zlt (signed y) 0). congruence. auto.
Qed.

## Properties of integer zero extension and sign extension.

Lemma Ziter_base:
forall (A: Type) n (f: A -> A) x, n <= 0 -> Z.iter n f x = x.
Proof.
intros. unfold Z.iter. destruct n; auto. compute in H. elim H; auto.
Qed.

Lemma Ziter_succ:
forall (A: Type) n (f: A -> A) x,
0 <= n -> Z.iter (Z.succ n) f x = f (Z.iter n f x).
Proof.
intros. destruct n; simpl.
- auto.
- compute in H. elim H; auto.
Qed.

Lemma Znatlike_ind:
forall (P: Z -> Prop),
(forall n, n <= 0 -> P n) ->
(forall n, 0 <= n -> P n -> P (Z.succ n)) ->
forall n, P n.
Proof.
intros. destruct (zle 0 n).
apply natlike_ind; auto. apply H; omega.
apply H. omega.
Qed.

Lemma Zzero_ext_spec:
forall n x i, 0 <= i ->
Z.testbit (Zzero_ext n x) i = if zlt i n then Z.testbit x i else false.
Proof.
unfold Zzero_ext. induction n using Znatlike_ind.
- intros. rewrite Ziter_base; auto.
rewrite zlt_false. rewrite Ztestbit_0; auto. omega.
- intros. rewrite Ziter_succ; auto.
rewrite Ztestbit_shiftin; auto.
rewrite (Ztestbit_eq i x); auto.
destruct (zeq i 0).
+ subst i. rewrite zlt_true; auto. omega.
+ rewrite IHn. destruct (zlt (Z.pred i) n).
rewrite zlt_true; auto. omega.
rewrite zlt_false; auto. omega.
omega.
Qed.

Lemma bits_zero_ext:
forall n x i, 0 <= i ->
testbit (zero_ext n x) i = if zlt i n then testbit x i else false.
Proof.
intros. unfold zero_ext. destruct (zlt i zwordsize).
rewrite testbit_repr; auto. rewrite Zzero_ext_spec. auto. auto.
rewrite !bits_above; auto. destruct (zlt i n); auto.
Qed.

Lemma Zsign_ext_spec:
forall n x i, 0 <= i -> 0 < n ->
Z.testbit (Zsign_ext n x) i = Z.testbit x (if zlt i n then i else n - 1).
Proof.
intros n0 x i I0 N0.
revert x i I0. pattern n0. apply Zlt_lower_bound_ind with (z := 1).
- unfold Zsign_ext. intros.
destruct (zeq x 1).
+ subst x; simpl.
replace (if zlt i 1 then i else 0) with 0.
rewrite Ztestbit_base.
destruct (Z.odd x0).
apply Ztestbit_m1; auto.
apply Ztestbit_0.
destruct (zlt i 1); omega.
+ set (x1 := Z.pred x). replace x1 with (Z.succ (Z.pred x1)).
rewrite Ziter_succ. rewrite Ztestbit_shiftin.
destruct (zeq i 0).
* subst i. rewrite zlt_true. rewrite Ztestbit_base; auto. omega.
* rewrite H. unfold x1. destruct (zlt (Z.pred i) (Z.pred x)).
rewrite zlt_true. rewrite (Ztestbit_eq i x0); auto. rewrite zeq_false; auto. omega.
rewrite zlt_false. rewrite (Ztestbit_eq (x - 1) x0). rewrite zeq_false; auto.
omega. omega. omega. unfold x1; omega. omega.
* omega.
* unfold x1; omega.
* omega.
- omega.
Qed.

Lemma bits_sign_ext:
forall n x i, 0 <= i < zwordsize -> 0 < n ->
testbit (sign_ext n x) i = testbit x (if zlt i n then i else n - 1).
Proof.
intros. unfold sign_ext.
rewrite testbit_repr; auto. rewrite Zsign_ext_spec. destruct (zlt i n); auto.
omega. auto.
Qed.

Hint Rewrite bits_zero_ext bits_sign_ext: ints.

Theorem zero_ext_above:
forall n x, n >= zwordsize -> zero_ext n x = x.
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_zero_ext. apply zlt_true. omega. omega.
Qed.

Theorem sign_ext_above:
forall n x, n >= zwordsize -> sign_ext n x = x.
Proof.
intros. apply same_bits_eq; intros.
unfold sign_ext; rewrite testbit_repr; auto.
rewrite Zsign_ext_spec. rewrite zlt_true. auto. omega. omega. omega.
Qed.

Theorem zero_ext_and:
forall n x, 0 <= n -> zero_ext n x = and x (repr (two_p n - 1)).
Proof.
bit_solve. rewrite testbit_repr; auto. rewrite Ztestbit_two_p_m1; intuition.
destruct (zlt i n).
rewrite andb_true_r; auto.
rewrite andb_false_r; auto.
tauto.
Qed.

Theorem zero_ext_mod:
forall n x, 0 <= n < zwordsize ->
unsigned (zero_ext n x) = Zmod (unsigned x) (two_p n).
Proof.
intros. apply equal_same_bits. intros.
rewrite Ztestbit_mod_two_p; auto.
fold (testbit (zero_ext n x) i).
destruct (zlt i zwordsize).
rewrite bits_zero_ext; auto.
rewrite bits_above. rewrite zlt_false; auto. omega. omega.
omega.
Qed.

Theorem zero_ext_widen:
forall x n n', 0 <= n <= n' ->
zero_ext n' (zero_ext n x) = zero_ext n x.
Proof.
bit_solve. destruct (zlt i n).
apply zlt_true. omega.
destruct (zlt i n'); auto.
tauto. tauto.
Qed.

Theorem sign_ext_widen:
forall x n n', 0 < n <= n' ->
sign_ext n' (sign_ext n x) = sign_ext n x.
Proof.
intros. destruct (zlt n' zwordsize).
bit_solve. destruct (zlt i n').
auto.
rewrite (zlt_false _ i n).
destruct (zlt (n' - 1) n); f_equal; omega.
omega. omega.
destruct (zlt i n'); omega.
omega. omega.
apply sign_ext_above; auto.
Qed.

Theorem sign_zero_ext_widen:
forall x n n', 0 <= n < n' ->
sign_ext n' (zero_ext n x) = zero_ext n x.
Proof.
intros. destruct (zlt n' zwordsize).
bit_solve.
destruct (zlt i n').
auto.
rewrite !zlt_false. auto. omega. omega. omega.
destruct (zlt i n'); omega.
omega.
apply sign_ext_above; auto.
Qed.

Theorem zero_ext_narrow:
forall x n n', 0 <= n <= n' ->
zero_ext n (zero_ext n' x) = zero_ext n x.
Proof.
bit_solve. destruct (zlt i n).
apply zlt_true. omega.
auto.
omega. omega. omega.
Qed.

Theorem sign_ext_narrow:
forall x n n', 0 < n <= n' ->
sign_ext n (sign_ext n' x) = sign_ext n x.
Proof.
intros. destruct (zlt n zwordsize).
bit_solve. destruct (zlt i n); f_equal; apply zlt_true; omega.
omega.
destruct (zlt i n); omega.
omega. omega.
rewrite (sign_ext_above n'). auto. omega.
Qed.

Theorem zero_sign_ext_narrow:
forall x n n', 0 < n <= n' ->
zero_ext n (sign_ext n' x) = zero_ext n x.
Proof.
intros. destruct (zlt n' zwordsize).
bit_solve.
destruct (zlt i n); auto.
rewrite zlt_true; auto. omega.
omega. omega. omega.
rewrite sign_ext_above; auto.
Qed.

Theorem zero_ext_idem:
forall n x, 0 <= n -> zero_ext n (zero_ext n x) = zero_ext n x.
Proof.
intros. apply zero_ext_widen. omega.
Qed.

Theorem sign_ext_idem:
forall n x, 0 < n -> sign_ext n (sign_ext n x) = sign_ext n x.
Proof.
intros. apply sign_ext_widen. omega.
Qed.

Theorem sign_ext_zero_ext:
forall n x, 0 < n -> sign_ext n (zero_ext n x) = sign_ext n x.
Proof.
intros. destruct (zlt n zwordsize).
bit_solve.
destruct (zlt i n).
rewrite zlt_true; auto.
rewrite zlt_true; auto. omega.
destruct (zlt i n); omega.
rewrite zero_ext_above; auto.
Qed.

Theorem zero_ext_sign_ext:
forall n x, 0 < n -> zero_ext n (sign_ext n x) = zero_ext n x.
Proof.
intros. apply zero_sign_ext_narrow. omega.
Qed.

Theorem sign_ext_equal_if_zero_equal:
forall n x y, 0 < n ->
zero_ext n x = zero_ext n y ->
sign_ext n x = sign_ext n y.
Proof.
intros. rewrite <- (sign_ext_zero_ext n x H).
rewrite <- (sign_ext_zero_ext n y H). congruence.
Qed.

Theorem zero_ext_shru_shl:
forall n x,
0 < n < zwordsize ->
let y := repr (zwordsize - n) in
zero_ext n x = shru (shl x y) y.
Proof.
intros.
assert (unsigned y = zwordsize - n).
unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. omega.
apply same_bits_eq; intros.
rewrite bits_zero_ext.
rewrite bits_shru; auto.
destruct (zlt i n).
rewrite zlt_true. rewrite bits_shl. rewrite zlt_false. f_equal. omega.
omega. omega. omega.
rewrite zlt_false. auto. omega.
omega.
Qed.

Theorem sign_ext_shr_shl:
forall n x,
0 < n < zwordsize ->
let y := repr (zwordsize - n) in
sign_ext n x = shr (shl x y) y.
Proof.
intros.
assert (unsigned y = zwordsize - n).
unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. omega.
apply same_bits_eq; intros.
rewrite bits_sign_ext.
rewrite bits_shr; auto.
destruct (zlt i n).
rewrite zlt_true. rewrite bits_shl. rewrite zlt_false. f_equal. omega.
omega. omega. omega.
rewrite zlt_false. rewrite bits_shl. rewrite zlt_false. f_equal. omega.
omega. omega. omega. omega. omega.
Qed.

zero_ext n x is the unique integer congruent to x modulo 2^n in the range 0...2^n-1.

Lemma zero_ext_range:
forall n x, 0 <= n < zwordsize -> 0 <= unsigned (zero_ext n x) < two_p n.
Proof.
intros. rewrite zero_ext_mod; auto. apply Z_mod_lt. apply two_p_gt_ZERO. omega.
Qed.

Lemma eqmod_zero_ext:
forall n x, 0 <= n < zwordsize -> eqmod (two_p n) (unsigned (zero_ext n x)) (unsigned x).
Proof.
intros. rewrite zero_ext_mod; auto. apply eqmod_sym. apply eqmod_mod.
apply two_p_gt_ZERO. omega.
Qed.

sign_ext n x is the unique integer congruent to x modulo 2^n in the range -2^(n-1)...2^(n-1) - 1.

Lemma sign_ext_range:
forall n x, 0 < n < zwordsize -> -two_p (n-1) <= signed (sign_ext n x) < two_p (n-1).
Proof.
intros. rewrite sign_ext_shr_shl; auto.
set (X := shl x (repr (zwordsize - n))).
assert (two_p (n - 1) > 0) by (apply two_p_gt_ZERO; omega).
assert (unsigned (repr (zwordsize - n)) = zwordsize - n).
apply unsigned_repr.
split. omega. generalize wordsize_max_unsigned; omega.
rewrite shr_div_two_p.
rewrite signed_repr.
rewrite H1.
apply Zdiv_interval_1.
omega. omega. apply two_p_gt_ZERO; omega.
replace (- two_p (n - 1) * two_p (zwordsize - n))
with (- (two_p (n - 1) * two_p (zwordsize - n))) by ring.
rewrite <- two_p_is_exp.
replace (n - 1 + (zwordsize - n)) with (zwordsize - 1) by omega.
rewrite <- half_modulus_power.
generalize (signed_range X). unfold min_signed, max_signed. omega.
omega. omega.
apply Zdiv_interval_2. apply signed_range.
generalize min_signed_neg; omega.
generalize max_signed_pos; omega.
rewrite H1. apply two_p_gt_ZERO. omega.
Qed.

Lemma eqmod_sign_ext':
forall n x, 0 < n < zwordsize ->
eqmod (two_p n) (unsigned (sign_ext n x)) (unsigned x).
Proof.
intros.
set (N := Z.to_nat n).
assert (Z.of_nat N = n) by (apply Z2Nat.id; omega).
rewrite <- H0. rewrite <- two_power_nat_two_p.
apply eqmod_same_bits; intros.
rewrite H0 in H1. rewrite H0.
fold (testbit (sign_ext n x) i). rewrite bits_sign_ext.
rewrite zlt_true. auto. omega. omega. omega.
Qed.

Lemma eqmod_sign_ext:
forall n x, 0 < n < zwordsize ->
eqmod (two_p n) (signed (sign_ext n x)) (unsigned x).
Proof.
intros. apply eqmod_trans with (unsigned (sign_ext n x)).
apply eqmod_divides with modulus. apply eqm_signed_unsigned.
exists (two_p (zwordsize - n)).
unfold modulus. rewrite two_power_nat_two_p. fold zwordsize.
rewrite <- two_p_is_exp. f_equal. omega. omega. omega.
apply eqmod_sign_ext'; auto.
Qed.

## Properties of one_bits (decomposition in sum of powers of two)

Theorem one_bits_range:
forall x i, In i (one_bits x) -> ltu i iwordsize = true.
Proof.
assert (A: forall p, 0 <= p < zwordsize -> ltu (repr p) iwordsize = true).
intros. unfold ltu, iwordsize. apply zlt_true.
repeat rewrite unsigned_repr. tauto.
generalize wordsize_max_unsigned; omega.
generalize wordsize_max_unsigned; omega.
unfold one_bits. intros.
destruct (list_in_map_inv _ _ _ H) as [i0 [EQ IN]].
subst i. apply A. apply Z_one_bits_range with (unsigned x); auto.
Qed.

Fixpoint int_of_one_bits (l: list int) : int :=
match l with
| nil => zero
| a :: b => add (shl one a) (int_of_one_bits b)
end.

Theorem one_bits_decomp:
forall x, x = int_of_one_bits (one_bits x).
Proof.
intros.
transitivity (repr (powerserie (Z_one_bits wordsize (unsigned x) 0))).
transitivity (repr (unsigned x)).
auto with ints. decEq. apply Z_one_bits_powerserie.
auto with ints.
unfold one_bits.
generalize (Z_one_bits_range (unsigned x)).
generalize (Z_one_bits wordsize (unsigned x) 0).
induction l.
intros; reflexivity.
intros; simpl. rewrite <- IHl. unfold add. apply eqm_samerepr.
apply eqm_add. rewrite shl_mul_two_p. rewrite mul_commut.
rewrite mul_one. apply eqm_unsigned_repr_r.
rewrite unsigned_repr. auto with ints.
generalize (H a (in_eq _ _)). generalize wordsize_max_unsigned. omega.
auto with ints.
intros; apply H; auto with coqlib.
Qed.

## Properties of comparisons

Theorem negate_cmp:
forall c x y, cmp (negate_comparison c) x y = negb (cmp c x y).
Proof.
intros. destruct c; simpl; try rewrite negb_elim; auto.
Qed.

Theorem negate_cmpu:
forall c x y, cmpu (negate_comparison c) x y = negb (cmpu c x y).
Proof.
intros. destruct c; simpl; try rewrite negb_elim; auto.
Qed.

Theorem swap_cmp:
forall c x y, cmp (swap_comparison c) x y = cmp c y x.
Proof.
intros. destruct c; simpl; auto. apply eq_sym. decEq. apply eq_sym.
Qed.

Theorem swap_cmpu:
forall c x y, cmpu (swap_comparison c) x y = cmpu c y x.
Proof.
intros. destruct c; simpl; auto. apply eq_sym. decEq. apply eq_sym.
Qed.

Lemma translate_eq:
forall x y d,
Proof.
intros. unfold eq. case (zeq (unsigned x) (unsigned y)); intro.
unfold add. rewrite e. apply zeq_true.
apply zeq_false. unfold add. red; intro. apply n.
apply eqm_small_eq; auto with ints.
replace (unsigned x) with ((unsigned x + unsigned d) - unsigned d).
replace (unsigned y) with ((unsigned y + unsigned d) - unsigned d).
apply eqm_sub. apply eqm_trans with (unsigned (repr (unsigned x + unsigned d))).
eauto with ints. apply eqm_trans with (unsigned (repr (unsigned y + unsigned d))).
eauto with ints. eauto with ints. eauto with ints.
omega. omega.
Qed.

Lemma translate_ltu:
forall x y d,
0 <= unsigned x + unsigned d <= max_unsigned ->
0 <= unsigned y + unsigned d <= max_unsigned ->
Proof.
repeat rewrite unsigned_repr; auto. case (zlt (unsigned x) (unsigned y)); intro.
apply zlt_true. omega.
apply zlt_false. omega.
Qed.

Theorem translate_cmpu:
forall c x y d,
0 <= unsigned x + unsigned d <= max_unsigned ->
0 <= unsigned y + unsigned d <= max_unsigned ->
cmpu c (add x d) (add y d) = cmpu c x y.
Proof.
intros. unfold cmpu.
rewrite translate_eq. repeat rewrite translate_ltu; auto.
Qed.

Lemma translate_lt:
forall x y d,
min_signed <= signed x + signed d <= max_signed ->
min_signed <= signed y + signed d <= max_signed ->
Proof.
intros. repeat rewrite add_signed. unfold lt.
repeat rewrite signed_repr; auto. case (zlt (signed x) (signed y)); intro.
apply zlt_true. omega.
apply zlt_false. omega.
Qed.

Theorem translate_cmp:
forall c x y d,
min_signed <= signed x + signed d <= max_signed ->
min_signed <= signed y + signed d <= max_signed ->
cmp c (add x d) (add y d) = cmp c x y.
Proof.
intros. unfold cmp.
rewrite translate_eq. repeat rewrite translate_lt; auto.
Qed.

Theorem notbool_isfalse_istrue:
forall x, is_false x -> is_true (notbool x).
Proof.
unfold is_false, is_true, notbool; intros; subst x.
rewrite eq_true. apply one_not_zero.
Qed.

Theorem notbool_istrue_isfalse:
forall x, is_true x -> is_false (notbool x).
Proof.
unfold is_false, is_true, notbool; intros.
generalize (eq_spec x zero). case (eq x zero); intro.
Qed.

Theorem ltu_range_test:
forall x y,
ltu x y = true -> unsigned y <= max_signed ->
0 <= signed x < unsigned y.
Proof.
intros.
unfold ltu in H. destruct (zlt (unsigned x) (unsigned y)); try discriminate.
rewrite signed_eq_unsigned.
generalize (unsigned_range x). omega. omega.
Qed.

Theorem lt_sub_overflow:
forall x y,
xor (sub_overflow x y zero) (negative (sub x y)) = if lt x y then one else zero.
Proof.
intros. unfold negative, sub_overflow, lt. rewrite sub_signed.
rewrite signed_zero. rewrite Zminus_0_r.
generalize (signed_range x) (signed_range y).
set (X := signed x); set (Y := signed y). intros RX RY.
unfold min_signed, max_signed in *.
generalize half_modulus_pos half_modulus_modulus; intros HM MM.
destruct (zle 0 (X - Y)).
- unfold proj_sumbool at 1; rewrite zle_true at 1 by omega. simpl.
rewrite (zlt_false _ X) by omega.
destruct (zlt (X - Y) half_modulus).
+ unfold proj_sumbool; rewrite zle_true by omega.
rewrite signed_repr. rewrite zlt_false by omega. apply xor_idem.
unfold min_signed, max_signed; omega.
+ unfold proj_sumbool; rewrite zle_false by omega.
replace (signed (repr (X - Y))) with (X - Y - modulus).
rewrite zlt_true by omega. apply xor_idem.
rewrite signed_repr_eq. replace ((X - Y) mod modulus) with (X - Y).
rewrite zlt_false; auto.
symmetry. apply Zmod_unique with 0; omega.
- unfold proj_sumbool at 2. rewrite zle_true at 1 by omega. rewrite andb_true_r.
rewrite (zlt_true _ X) by omega.
destruct (zlt (X - Y) (-half_modulus)).
+ unfold proj_sumbool; rewrite zle_false by omega.
replace (signed (repr (X - Y))) with (X - Y + modulus).
rewrite zlt_false by omega. apply xor_zero.
rewrite signed_repr_eq. replace ((X - Y) mod modulus) with (X - Y + modulus).
rewrite zlt_true by omega; auto.
symmetry. apply Zmod_unique with (-1); omega.
+ unfold proj_sumbool; rewrite zle_true by omega.
rewrite signed_repr. rewrite zlt_true by omega. apply xor_zero_l.
unfold min_signed, max_signed; omega.
Qed.

Lemma signed_eq:
forall x y, eq x y = zeq (signed x) (signed y).
Proof.
intros. unfold eq. unfold proj_sumbool.
destruct (zeq (unsigned x) (unsigned y));
destruct (zeq (signed x) (signed y)); auto.
elim n. unfold signed. rewrite e; auto.
elim n. apply eqm_small_eq; auto with ints.
eapply eqm_trans. apply eqm_sym. apply eqm_signed_unsigned.
rewrite e. apply eqm_signed_unsigned.
Qed.

Lemma not_lt:
forall x y, negb (lt y x) = (lt x y || eq x y).
Proof.
intros. unfold lt. rewrite signed_eq. unfold proj_sumbool.
destruct (zlt (signed y) (signed x)).
rewrite zlt_false. rewrite zeq_false. auto. omega. omega.
destruct (zeq (signed x) (signed y)).
rewrite zlt_false. auto. omega.
rewrite zlt_true. auto. omega.
Qed.

Lemma lt_not:
forall x y, lt y x = negb (lt x y) && negb (eq x y).
Proof.
intros. rewrite <- negb_orb. rewrite <- not_lt. rewrite negb_involutive. auto.
Qed.

Lemma not_ltu:
forall x y, negb (ltu y x) = (ltu x y || eq x y).
Proof.
intros. unfold ltu, eq.
destruct (zlt (unsigned y) (unsigned x)).
rewrite zlt_false. rewrite zeq_false. auto. omega. omega.
destruct (zeq (unsigned x) (unsigned y)).
rewrite zlt_false. auto. omega.
rewrite zlt_true. auto. omega.
Qed.

Lemma ltu_not:
forall x y, ltu y x = negb (ltu x y) && negb (eq x y).
Proof.
intros. rewrite <- negb_orb. rewrite <- not_ltu. rewrite negb_involutive. auto.
Qed.

Non-overlapping test

Definition no_overlap (ofs1: int) (sz1: Z) (ofs2: int) (sz2: Z) : bool :=
let x1 := unsigned ofs1 in let x2 := unsigned ofs2 in
zlt (x1 + sz1) modulus && zlt (x2 + sz2) modulus
&& (zle (x1 + sz1) x2 || zle (x2 + sz2) x1).

Lemma no_overlap_sound:
forall ofs1 sz1 ofs2 sz2 base,
sz1 > 0 -> sz2 > 0 -> no_overlap ofs1 sz1 ofs2 sz2 = true ->
\/ unsigned (add base ofs2) + sz2 <= unsigned (add base ofs1).
Proof.
intros.
destruct (andb_prop _ _ H1). clear H1.
destruct (andb_prop _ _ H2). clear H2.
apply proj_sumbool_true in H1.
apply proj_sumbool_true in H4.
assert (unsigned ofs1 + sz1 <= unsigned ofs2 \/ unsigned ofs2 + sz2 <= unsigned ofs1).
destruct (orb_prop _ _ H3). left. eapply proj_sumbool_true; eauto. right. eapply proj_sumbool_true; eauto.
clear H3.
generalize (unsigned_range ofs1) (unsigned_range ofs2). intros P Q.
intros [C|C] [D|D]; omega.
Qed.

Size of integers, in bits.

Definition Zsize (x: Z) : Z :=
match x with
| Zpos p => Zpos (Pos.size p)
| _ => 0
end.

Definition size (x: int) : Z := Zsize (unsigned x).

Remark Zsize_pos: forall x, 0 <= Zsize x.
Proof.
destruct x; simpl. omega. compute; intuition congruence. omega.
Qed.

Remark Zsize_pos': forall x, 0 < x -> 0 < Zsize x.
Proof.
destruct x; simpl; intros; try discriminate. compute; auto.
Qed.

Lemma Zsize_shiftin:
forall b x, 0 < x -> Zsize (Zshiftin b x) = Zsucc (Zsize x).
Proof.
intros. destruct x; compute in H; try discriminate.
destruct b.
change (Zshiftin true (Zpos p)) with (Zpos (p~1)).
change (Zshiftin false (Zpos p)) with (Zpos (p~0)).
Qed.

Lemma Ztestbit_size_1:
forall x, 0 < x -> Z.testbit x (Zpred (Zsize x)) = true.
Proof.
intros x0 POS0; pattern x0; apply Zshiftin_pos_ind; auto.
intros. rewrite Zsize_shiftin; auto.
replace (Z.pred (Z.succ (Zsize x))) with (Z.succ (Z.pred (Zsize x))) by omega.
rewrite Ztestbit_shiftin_succ. auto. generalize (Zsize_pos' x H); omega.
Qed.

Lemma Ztestbit_size_2:
forall x, 0 <= x -> forall i, i >= Zsize x -> Z.testbit x i = false.
Proof.
intros x0 POS0. destruct (zeq x0 0).
- subst x0; intros. apply Ztestbit_0.
- pattern x0; apply Zshiftin_pos_ind.
+ simpl. intros. change 1 with (Zshiftin true 0). rewrite Ztestbit_shiftin.
rewrite zeq_false. apply Ztestbit_0. omega. omega.
+ intros. rewrite Zsize_shiftin in H1; auto.
generalize (Zsize_pos' _ H); intros.
rewrite Ztestbit_shiftin. rewrite zeq_false. apply H0. omega.
omega. omega.
+ omega.
Qed.

Lemma Zsize_interval_1:
forall x, 0 <= x -> 0 <= x < two_p (Zsize x).
Proof.
intros.
assert (x = x mod (two_p (Zsize x))).
apply equal_same_bits; intros.
rewrite Ztestbit_mod_two_p; auto.
destruct (zlt i (Zsize x)). auto. apply Ztestbit_size_2; auto.
apply Zsize_pos; auto.
rewrite H0 at 1. rewrite H0 at 3. apply Z_mod_lt. apply two_p_gt_ZERO. apply Zsize_pos; auto.
Qed.

Lemma Zsize_interval_2:
forall x n, 0 <= n -> 0 <= x < two_p n -> n >= Zsize x.
Proof.
intros. set (N := Z.to_nat n).
assert (Z.of_nat N = n) by (apply Z2Nat.id; auto).
rewrite <- H1 in H0. rewrite <- two_power_nat_two_p in H0.
destruct (zeq x 0).
subst x; simpl; omega.
destruct (zlt n (Zsize x)); auto.
exploit (Ztestbit_above N x (Zpred (Zsize x))). auto. omega.
rewrite Ztestbit_size_1. congruence. omega.
Qed.

Lemma Zsize_monotone:
forall x y, 0 <= x <= y -> Zsize x <= Zsize y.
Proof.
intros. apply Zge_le. apply Zsize_interval_2. apply Zsize_pos.
exploit (Zsize_interval_1 y). omega.
omega.
Qed.

Theorem size_zero: size zero = 0.
Proof.
unfold size; rewrite unsigned_zero; auto.
Qed.

Theorem bits_size_1:
forall x, x = zero \/ testbit x (Zpred (size x)) = true.
Proof.
intros. destruct (zeq (unsigned x) 0).
left. rewrite <- (repr_unsigned x). rewrite e; auto.
right. apply Ztestbit_size_1. generalize (unsigned_range x); omega.
Qed.

Theorem bits_size_2:
forall x i, size x <= i -> testbit x i = false.
Proof.
intros. apply Ztestbit_size_2. generalize (unsigned_range x); omega.
fold (size x); omega.
Qed.

Theorem size_range:
forall x, 0 <= size x <= zwordsize.
Proof.
intros; split. apply Zsize_pos.
destruct (bits_size_1 x).
subst x; unfold size; rewrite unsigned_zero; simpl. generalize wordsize_pos; omega.
destruct (zle (size x) zwordsize); auto.
rewrite bits_above in H. congruence. omega.
Qed.

Theorem bits_size_3:
forall x n,
0 <= n ->
(forall i, n <= i < zwordsize -> testbit x i = false) ->
size x <= n.
Proof.
intros. destruct (zle (size x) n). auto.
destruct (bits_size_1 x).
subst x. unfold size; rewrite unsigned_zero; assumption.
rewrite (H0 (Z.pred (size x))) in H1. congruence.
generalize (size_range x); omega.
Qed.

Theorem bits_size_4:
forall x n,
0 <= n ->
testbit x (Zpred n) = true ->
(forall i, n <= i < zwordsize -> testbit x i = false) ->
size x = n.
Proof.
intros.
assert (size x <= n).
apply bits_size_3; auto.
destruct (zlt (size x) n).
rewrite bits_size_2 in H0. congruence. omega.
omega.
Qed.

Theorem size_interval_1:
forall x, 0 <= unsigned x < two_p (size x).
Proof.
intros; apply Zsize_interval_1. generalize (unsigned_range x); omega.
Qed.

Theorem size_interval_2:
forall x n, 0 <= n -> 0 <= unsigned x < two_p n -> n >= size x.
Proof.
intros. apply Zsize_interval_2; auto.
Qed.

Theorem size_and:
forall a b, size (and a b) <= Z.min (size a) (size b).
Proof.
intros.
assert (0 <= Z.min (size a) (size b)).
generalize (size_range a) (size_range b). zify; omega.
apply bits_size_3. auto. intros.
rewrite bits_and. zify. subst z z0. destruct H1.
rewrite (bits_size_2 a). auto. omega.
rewrite (bits_size_2 b). apply andb_false_r. omega.
omega.
Qed.

Corollary and_interval:
forall a b, 0 <= unsigned (and a b) < two_p (Z.min (size a) (size b)).
Proof.
intros.
generalize (size_interval_1 (and a b)); intros.
assert (two_p (size (and a b)) <= two_p (Z.min (size a) (size b))).
apply two_p_monotone. split. generalize (size_range (and a b)); omega.
apply size_and.
omega.
Qed.

Theorem size_or:
forall a b, size (or a b) = Z.max (size a) (size b).
Proof.
intros. generalize (size_range a) (size_range b); intros.
destruct (bits_size_1 a).
subst a. rewrite size_zero. rewrite or_zero_l. zify; omega.
destruct (bits_size_1 b).
subst b. rewrite size_zero. rewrite or_zero. zify; omega.
zify. destruct H3 as [[P Q] | [P Q]]; subst.
apply bits_size_4. tauto. rewrite bits_or. rewrite H2. apply orb_true_r.
omega.
intros. rewrite bits_or. rewrite !bits_size_2. auto. omega. omega. omega.
apply bits_size_4. tauto. rewrite bits_or. rewrite H1. apply orb_true_l.
destruct (zeq (size a) 0). unfold testbit in H1. rewrite Z.testbit_neg_r in H1.
congruence. omega. omega.
intros. rewrite bits_or. rewrite !bits_size_2. auto. omega. omega. omega.
Qed.

Corollary or_interval:
forall a b, 0 <= unsigned (or a b) < two_p (Z.max (size a) (size b)).
Proof.
intros. rewrite <- size_or. apply size_interval_1.
Qed.

Theorem size_xor:
forall a b, size (xor a b) <= Z.max (size a) (size b).
Proof.
intros.
assert (0 <= Z.max (size a) (size b)).
generalize (size_range a) (size_range b). zify; omega.
apply bits_size_3. auto. intros.
rewrite bits_xor. rewrite !bits_size_2. auto.
zify; omega.
zify; omega.
omega.
Qed.

Corollary xor_interval:
forall a b, 0 <= unsigned (xor a b) < two_p (Z.max (size a) (size b)).
Proof.
intros.
generalize (size_interval_1 (xor a b)); intros.
assert (two_p (size (xor a b)) <= two_p (Z.max (size a) (size b))).
apply two_p_monotone. split. generalize (size_range (xor a b)); omega.
apply size_xor.
omega.
Qed.

End Make.

# Specialization to integers of size 8, 32, and 64 bits

Module Wordsize_32.
Definition wordsize := 32%nat.
Remark wordsize_not_zero: wordsize <> 0%nat.
Proof.
unfold wordsize; congruence. Qed.
End Wordsize_32.

Strategy opaque [Wordsize_32.wordsize].

Module Int := Make(Wordsize_32).

Strategy 0 [Wordsize_32.wordsize].

Notation int := Int.int.

Remark int_wordsize_divides_modulus:
Zdivide (Z_of_nat Int.wordsize) Int.modulus.
Proof.
exists (two_p (32-5)); reflexivity.
Qed.

Module Wordsize_8.
Definition wordsize := 8%nat.
Remark wordsize_not_zero: wordsize <> 0%nat.
Proof.
unfold wordsize; congruence. Qed.
End Wordsize_8.

Strategy opaque [Wordsize_8.wordsize].

Module Byte := Make(Wordsize_8).

Strategy 0 [Wordsize_8.wordsize].

Notation byte := Byte.int.

Module Wordsize_64.
Definition wordsize := 64%nat.
Remark wordsize_not_zero: wordsize <> 0%nat.
Proof.
unfold wordsize; congruence. Qed.
End Wordsize_64.

Strategy opaque [Wordsize_64.wordsize].

Module Int64.

Include Make(Wordsize_64).

Shifts with amount given as a 32-bit integer

Definition iwordsize': Int.int := Int.repr zwordsize.

Definition shl' (x: int) (y: Int.int): int :=
repr (Z.shiftl (unsigned x) (Int.unsigned y)).
Definition shru' (x: int) (y: Int.int): int :=
repr (Z.shiftr (unsigned x) (Int.unsigned y)).
Definition shr' (x: int) (y: Int.int): int :=
repr (Z.shiftr (signed x) (Int.unsigned y)).
Definition shrx' (x: int) (y: Int.int): int :=
divs x (shl' one y).

Lemma bits_shl':
forall x y i,
0 <= i < zwordsize ->
testbit (shl' x y) i =
if zlt i (Int.unsigned y) then false else testbit x (i - Int.unsigned y).
Proof.
intros. unfold shl'. rewrite testbit_repr; auto.
destruct (zlt i (Int.unsigned y)).
apply Z.shiftl_spec_low. auto.
apply Z.shiftl_spec_high. omega. omega.
Qed.

Lemma bits_shru':
forall x y i,
0 <= i < zwordsize ->
testbit (shru' x y) i =
if zlt (i + Int.unsigned y) zwordsize then testbit x (i + Int.unsigned y) else false.
Proof.
intros. unfold shru'. rewrite testbit_repr; auto.
rewrite Z.shiftr_spec. fold (testbit x (i + Int.unsigned y)).
destruct (zlt (i + Int.unsigned y) zwordsize).
auto.
apply bits_above; auto.
omega.
Qed.

Lemma bits_shr':
forall x y i,
0 <= i < zwordsize ->
testbit (shr' x y) i =
testbit x (if zlt (i + Int.unsigned y) zwordsize then i + Int.unsigned y else zwordsize - 1).
Proof.
intros. unfold shr'. rewrite testbit_repr; auto.
rewrite Z.shiftr_spec. apply bits_signed.
generalize (Int.unsigned_range y); omega.
omega.
Qed.

Lemma shl'_mul_two_p:
forall x y,
shl' x y = mul x (repr (two_p (Int.unsigned y))).
Proof.
intros. unfold shl', mul. apply eqm_samerepr.
rewrite Zshiftl_mul_two_p. apply eqm_mult. apply eqm_refl. apply eqm_unsigned_repr.
generalize (Int.unsigned_range y); omega.
Qed.

Lemma shl'_one_two_p:
forall y, shl' one y = repr (two_p (Int.unsigned y)).
Proof.
intros. rewrite shl'_mul_two_p. rewrite mul_commut. rewrite mul_one. auto.
Qed.

Theorem shl'_mul:
forall x y,
shl' x y = mul x (shl' one y).
Proof.
intros. rewrite shl'_one_two_p. apply shl'_mul_two_p.
Qed.

Theorem shrx'_zero:
forall x, shrx' x Int.zero = x.
Proof.
intros. unfold shrx'. rewrite shl'_one_two_p. unfold divs.
change (signed (repr (two_p (Int.unsigned Int.zero)))) with 1.
rewrite Z.quot_1_r. apply repr_signed.
Qed.

Theorem shrx'_shr_2:
forall x y,
Int.ltu y (Int.repr 63) = true ->
shrx' x y = shr' (add x (shru' (shr' x (Int.repr 63)) (Int.sub (Int.repr 64) y))) y.
Proof.
intros.
set (z := repr (Int.unsigned y)).
apply Int.ltu_inv in H. change (Int.unsigned (Int.repr 63)) with 63 in H.
assert (N1: 63 < max_unsigned) by reflexivity.
assert (N2: 63 < Int.max_unsigned) by reflexivity.
assert (A: unsigned z = Int.unsigned y).
{ unfold z; apply unsigned_repr; omega. }
assert (B: unsigned (sub (repr 64) z) = Int.unsigned (Int.sub (Int.repr 64) y)).
{ unfold z. unfold sub, Int.sub.
change (unsigned (repr 64)) with 64.
change (Int.unsigned (Int.repr 64)) with 64.
rewrite (unsigned_repr (Int.unsigned y)) by omega.
rewrite unsigned_repr, Int.unsigned_repr by omega.
auto. }
unfold shrx', shr', shru', shl'.
rewrite <- A.
change (Int.unsigned (Int.repr 63)) with (unsigned (repr 63)).
rewrite <- B.
apply shrx_shr_2.
unfold ltu. apply zlt_true. change (unsigned z < 63). rewrite A; omega.
Qed.

Remark int_ltu_2_inv:
forall y z,
Int.ltu y iwordsize' = true ->
Int.ltu z iwordsize' = true ->
Int.unsigned (Int.add y z) <= Int.unsigned iwordsize' ->
let y' := repr (Int.unsigned y) in
let z' := repr (Int.unsigned z) in
Int.unsigned y = unsigned y'
/\ Int.unsigned z = unsigned z'
/\ ltu y' iwordsize = true
/\ ltu z' iwordsize = true
Proof.
intros. apply Int.ltu_inv in H. apply Int.ltu_inv in H0.
change (Int.unsigned iwordsize') with 64 in *.
assert (128 < max_unsigned) by reflexivity.
assert (128 < Int.max_unsigned) by reflexivity.
assert (Y: unsigned y' = Int.unsigned y) by (apply unsigned_repr; omega).
assert (Z: unsigned z' = Int.unsigned z) by (apply unsigned_repr; omega).
{ unfold Int.add. rewrite Int.unsigned_repr by omega.
unfold add. rewrite unsigned_repr by omega. congruence. }
intuition auto.
apply zlt_true. rewrite Y; auto.
apply zlt_true. rewrite Z; auto.
rewrite P. rewrite repr_unsigned. auto.
Qed.

Theorem or_ror':
forall x y z,
Int.ltu y iwordsize' = true ->
Int.ltu z iwordsize' = true ->
Int.add y z = iwordsize' ->
ror x (repr (Int.unsigned z)) = or (shl' x y) (shru' x z).
Proof.
intros. destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. rewrite H1; omega.
replace (shl' x y) with (shl x (repr (Int.unsigned y))).
replace (shru' x z) with (shru x (repr (Int.unsigned z))).
apply or_ror; auto. rewrite F, H1. reflexivity.
unfold shru, shru'; rewrite <- B; auto.
unfold shl, shl'; rewrite <- A; auto.
Qed.

Theorem shl'_shl':
forall x y z,
Int.ltu y iwordsize' = true ->
Int.ltu z iwordsize' = true ->
Int.ltu (Int.add y z) iwordsize' = true ->
shl' (shl' x y) z = shl' x (Int.add y z).
Proof.
intros. apply Int.ltu_inv in H1.
destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. omega.
set (y' := repr (Int.unsigned y)) in *.
set (z' := repr (Int.unsigned z)) in *.
replace (shl' x y) with (shl x y').
replace (shl' (shl x y') z) with (shl (shl x y') z').
replace (shl' x (Int.add y z)) with (shl x (add y' z')).
apply shl_shl; auto. apply zlt_true. rewrite <- E.
change (unsigned iwordsize) with zwordsize. tauto.
unfold shl, shl'. rewrite E; auto.
unfold shl at 1, shl'. rewrite <- B; auto.
unfold shl, shl'; rewrite <- A; auto.
Qed.

Theorem shru'_shru':
forall x y z,
Int.ltu y iwordsize' = true ->
Int.ltu z iwordsize' = true ->
Int.ltu (Int.add y z) iwordsize' = true ->
shru' (shru' x y) z = shru' x (Int.add y z).
Proof.
intros. apply Int.ltu_inv in H1.
destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. omega.
set (y' := repr (Int.unsigned y)) in *.
set (z' := repr (Int.unsigned z)) in *.
replace (shru' x y) with (shru x y').
replace (shru' (shru x y') z) with (shru (shru x y') z').
replace (shru' x (Int.add y z)) with (shru x (add y' z')).
apply shru_shru; auto. apply zlt_true. rewrite <- E.
change (unsigned iwordsize) with zwordsize. tauto.
unfold shru, shru'. rewrite E; auto.
unfold shru at 1, shru'. rewrite <- B; auto.
unfold shru, shru'; rewrite <- A; auto.
Qed.

Theorem shr'_shr':
forall x y z,
Int.ltu y iwordsize' = true ->
Int.ltu z iwordsize' = true ->
Int.ltu (Int.add y z) iwordsize' = true ->
shr' (shr' x y) z = shr' x (Int.add y z).
Proof.
intros. apply Int.ltu_inv in H1.
destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. omega.
set (y' := repr (Int.unsigned y)) in *.
set (z' := repr (Int.unsigned z)) in *.
replace (shr' x y) with (shr x y').
replace (shr' (shr x y') z) with (shr (shr x y') z').
replace (shr' x (Int.add y z)) with (shr x (add y' z')).
apply shr_shr; auto. apply zlt_true. rewrite <- E.
change (unsigned iwordsize) with zwordsize. tauto.
unfold shr, shr'. rewrite E; auto.
unfold shr at 1, shr'. rewrite <- B; auto.
unfold shr, shr'; rewrite <- A; auto.
Qed.

Powers of two with exponents given as 32-bit ints

Definition one_bits' (x: int) : list Int.int :=
List.map Int.repr (Z_one_bits wordsize (unsigned x) 0).

Definition is_power2' (x: int) : option Int.int :=
match Z_one_bits wordsize (unsigned x) 0 with
| i :: nil => Some (Int.repr i)
| _ => None
end.

Theorem one_bits'_range:
forall x i, In i (one_bits' x) -> Int.ltu i iwordsize' = true.
Proof.
intros.
destruct (list_in_map_inv _ _ _ H) as [i0 [EQ IN]].
exploit Z_one_bits_range; eauto. intros R.
unfold Int.ltu. rewrite EQ. rewrite Int.unsigned_repr.
change (Int.unsigned iwordsize') with zwordsize. apply zlt_true. omega.
assert (zwordsize < Int.max_unsigned) by reflexivity. omega.
Qed.

Fixpoint int_of_one_bits' (l: list Int.int) : int :=
match l with
| nil => zero
| a :: b => add (shl' one a) (int_of_one_bits' b)
end.

Theorem one_bits'_decomp:
forall x, x = int_of_one_bits' (one_bits' x).
Proof.
assert (REC: forall l,
(forall i, In i l -> 0 <= i < zwordsize) ->
int_of_one_bits' (List.map Int.repr l) = repr (powerserie l)).
{ induction l; simpl; intros.
- auto.
- rewrite IHl by eauto. apply eqm_samerepr; apply eqm_add.
+ rewrite shl'_one_two_p. rewrite Int.unsigned_repr. apply eqm_sym; apply eqm_unsigned_repr.
exploit (H a). auto. assert (zwordsize < Int.max_unsigned) by reflexivity. omega.
+ apply eqm_sym; apply eqm_unsigned_repr.
}
intros. rewrite <- (repr_unsigned x) at 1. unfold one_bits'. rewrite REC.
rewrite <- Z_one_bits_powerserie. auto. apply unsigned_range.
apply Z_one_bits_range.
Qed.

Lemma is_power2'_rng:
forall n logn,
is_power2' n = Some logn ->
0 <= Int.unsigned logn < zwordsize.
Proof.
unfold is_power2'; intros n logn P2.
destruct (Z_one_bits wordsize (unsigned n) 0) as [ | i [ | ? ?]] eqn:B; inv P2.
assert (0 <= i < zwordsize).
{ apply Z_one_bits_range with (unsigned n). rewrite B; auto with coqlib. }
rewrite Int.unsigned_repr. auto.
assert (zwordsize < Int.max_unsigned) by reflexivity.
omega.
Qed.

Theorem is_power2'_range:
forall n logn,
is_power2' n = Some logn -> Int.ltu logn iwordsize' = true.
Proof.
intros. unfold Int.ltu. change (Int.unsigned iwordsize') with zwordsize.
apply zlt_true. generalize (is_power2'_rng _ _ H). tauto.
Qed.

Lemma is_power2'_correct:
forall n logn,
is_power2' n = Some logn ->
unsigned n = two_p (Int.unsigned logn).
Proof.
unfold is_power2'; intros.
destruct (Z_one_bits wordsize (unsigned n) 0) as [ | i [ | ? ?]] eqn:B; inv H.
rewrite (Z_one_bits_powerserie (unsigned n)) by (apply unsigned_range).
rewrite Int.unsigned_repr. rewrite B; simpl. omega.
assert (0 <= i < zwordsize).
{ apply Z_one_bits_range with (unsigned n). rewrite B; auto with coqlib. }
assert (zwordsize < Int.max_unsigned) by reflexivity.
omega.
Qed.

Theorem mul_pow2':
forall x n logn,
is_power2' n = Some logn ->
mul x n = shl' x logn.
Proof.
intros. rewrite shl'_mul. f_equal. rewrite shl'_one_two_p.
rewrite <- (repr_unsigned n). f_equal. apply is_power2'_correct; auto.
Qed.

Theorem divu_pow2':
forall x n logn,
is_power2' n = Some logn ->
divu x n = shru' x logn.
Proof.
intros. generalize (is_power2'_correct n logn H). intro.
symmetry. unfold divu. rewrite H0. unfold shru'. rewrite Zshiftr_div_two_p. auto.
eapply is_power2'_rng; eauto.
Qed.

Decomposing 64-bit ints as pairs of 32-bit ints

Definition loword (n: int) : Int.int := Int.repr (unsigned n).

Definition hiword (n: int) : Int.int := Int.repr (unsigned (shru n (repr Int.zwordsize))).

Definition ofwords (hi lo: Int.int) : int :=
or (shl (repr (Int.unsigned hi)) (repr Int.zwordsize)) (repr (Int.unsigned lo)).

Lemma bits_loword:
forall n i, 0 <= i < Int.zwordsize -> Int.testbit (loword n) i = testbit n i.
Proof.
intros. unfold loword. rewrite Int.testbit_repr; auto.
Qed.

Lemma bits_hiword:
forall n i, 0 <= i < Int.zwordsize -> Int.testbit (hiword n) i = testbit n (i + Int.zwordsize).
Proof.
intros. unfold hiword. rewrite Int.testbit_repr; auto.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
fold (testbit (shru n (repr Int.zwordsize)) i). rewrite bits_shru.
change (unsigned (repr Int.zwordsize)) with Int.zwordsize.
apply zlt_true. omega. omega.
Qed.

Lemma bits_ofwords:
forall hi lo i, 0 <= i < zwordsize ->
testbit (ofwords hi lo) i =
if zlt i Int.zwordsize then Int.testbit lo i else Int.testbit hi (i - Int.zwordsize).
Proof.
intros. unfold ofwords. rewrite bits_or; auto. rewrite bits_shl; auto.
change (unsigned (repr Int.zwordsize)) with Int.zwordsize.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
destruct (zlt i Int.zwordsize).
rewrite testbit_repr; auto.
rewrite !testbit_repr; auto.
fold (Int.testbit lo i). rewrite Int.bits_above. apply orb_false_r. auto.
omega.
Qed.

Lemma lo_ofwords:
forall hi lo, loword (ofwords hi lo) = lo.
Proof.
intros. apply Int.same_bits_eq; intros.
rewrite bits_loword; auto. rewrite bits_ofwords. apply zlt_true. omega.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity. omega.
Qed.

Lemma hi_ofwords:
forall hi lo, hiword (ofwords hi lo) = hi.
Proof.
intros. apply Int.same_bits_eq; intros.
rewrite bits_hiword; auto. rewrite bits_ofwords.
rewrite zlt_false. f_equal. omega. omega.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity. omega.
Qed.

Lemma ofwords_recompose:
forall n, ofwords (hiword n) (loword n) = n.
Proof.
intros. apply same_bits_eq; intros. rewrite bits_ofwords; auto.
destruct (zlt i Int.zwordsize).
apply bits_loword. omega.
rewrite bits_hiword. f_equal. omega.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity. omega.
Qed.

forall lo hi, ofwords hi lo = repr (Int.unsigned hi * two_p 32 + Int.unsigned lo).
Proof.
apply eqm_samerepr. apply eqm_add. apply eqm_mult.
apply eqm_sym; apply eqm_unsigned_repr.
apply eqm_refl.
apply eqm_sym; apply eqm_unsigned_repr.
change Int.zwordsize with 32; change zwordsize with 64; omega.
rewrite unsigned_repr. generalize (Int.unsigned_range lo). intros [A B]. exact B.
assert (Int.max_unsigned < max_unsigned) by (compute; auto).
generalize (Int.unsigned_range_2 lo); omega.
Qed.

forall lo hi, unsigned (ofwords hi lo) = Int.unsigned hi * two_p 32 + Int.unsigned lo.
Proof.
generalize (Int.unsigned_range hi) (Int.unsigned_range lo).
change (two_p 32) with Int.modulus.
change Int.modulus with 4294967296.
change max_unsigned with 18446744073709551615.
omega.
Qed.

Remark eqm_mul_2p32:
forall x y, Int.eqm x y -> eqm (x * two_p 32) (y * two_p 32).
Proof.
intros. destruct H as [k EQ]. exists k. rewrite EQ.
change Int.modulus with (two_p 32).
change modulus with (two_p 32 * two_p 32).
ring.
Qed.

forall lo hi, signed (ofwords hi lo) = Int.signed hi * two_p 32 + Int.unsigned lo.
Proof.
replace (repr (Int.unsigned hi * two_p 32 + Int.unsigned lo))
with (repr (Int.signed hi * two_p 32 + Int.unsigned lo)).
apply signed_repr.
generalize (Int.signed_range hi) (Int.unsigned_range lo).
change (two_p 32) with Int.modulus.
change min_signed with (Int.min_signed * Int.modulus).
change max_signed with (Int.max_signed * Int.modulus + Int.modulus - 1).
change Int.modulus with 4294967296.
omega.
apply eqm_samerepr. apply eqm_add. apply eqm_mul_2p32. apply Int.eqm_signed_unsigned. apply eqm_refl.
Qed.

Expressing 64-bit operations in terms of 32-bit operations

Lemma decompose_bitwise_binop:
forall f f64 f32 xh xl yh yl,
(forall x y i, 0 <= i < zwordsize -> testbit (f64 x y) i = f (testbit x i) (testbit y i)) ->
(forall x y i, 0 <= i < Int.zwordsize -> Int.testbit (f32 x y) i = f (Int.testbit x i) (Int.testbit y i)) ->
f64 (ofwords xh xl) (ofwords yh yl) = ofwords (f32 xh yh) (f32 xl yl).
Proof.
intros. apply Int64.same_bits_eq; intros.
rewrite H by auto. rewrite ! bits_ofwords by auto.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
destruct (zlt i Int.zwordsize); rewrite H0 by omega; auto.
Qed.

Lemma decompose_and:
forall xh xl yh yl,
and (ofwords xh xl) (ofwords yh yl) = ofwords (Int.and xh yh) (Int.and xl yl).
Proof.
intros. apply decompose_bitwise_binop with andb.
apply bits_and. apply Int.bits_and.
Qed.

Lemma decompose_or:
forall xh xl yh yl,
or (ofwords xh xl) (ofwords yh yl) = ofwords (Int.or xh yh) (Int.or xl yl).
Proof.
intros. apply decompose_bitwise_binop with orb.
apply bits_or. apply Int.bits_or.
Qed.

Lemma decompose_xor:
forall xh xl yh yl,
xor (ofwords xh xl) (ofwords yh yl) = ofwords (Int.xor xh yh) (Int.xor xl yl).
Proof.
intros. apply decompose_bitwise_binop with xorb.
apply bits_xor. apply Int.bits_xor.
Qed.

Lemma decompose_not:
forall xh xl,
not (ofwords xh xl) = ofwords (Int.not xh) (Int.not xl).
Proof.
intros. unfold not, Int.not. rewrite <- decompose_xor. f_equal.
apply (Int64.eq_spec mone (ofwords Int.mone Int.mone)).
Qed.

Lemma decompose_shl_1:
forall xh xl y,
0 <= Int.unsigned y < Int.zwordsize ->
shl' (ofwords xh xl) y =
ofwords (Int.or (Int.shl xh y) (Int.shru xl (Int.sub Int.iwordsize y)))
(Int.shl xl y).
Proof.
intros.
assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; omega. }
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
apply Int64.same_bits_eq; intros.
rewrite bits_shl' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize). rewrite Int.bits_shl by omega.
destruct (zlt i (Int.unsigned y)). auto.
rewrite bits_ofwords by omega. rewrite zlt_true by omega. auto.
rewrite zlt_false by omega. rewrite bits_ofwords by omega.
rewrite Int.bits_or by omega. rewrite Int.bits_shl by omega.
rewrite Int.bits_shru by omega. rewrite H0.
destruct (zlt (i - Int.unsigned y) (Int.zwordsize)).
rewrite zlt_true by omega. rewrite zlt_true by omega.
rewrite orb_false_l. f_equal. omega.
rewrite zlt_false by omega. rewrite zlt_false by omega.
rewrite orb_false_r. f_equal. omega.
Qed.

Lemma decompose_shl_2:
forall xh xl y,
Int.zwordsize <= Int.unsigned y < zwordsize ->
shl' (ofwords xh xl) y =
ofwords (Int.shl xl (Int.sub y Int.iwordsize)) Int.zero.
Proof.
intros.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). omega. }
apply Int64.same_bits_eq; intros.
rewrite bits_shl' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize). rewrite zlt_true by omega. apply Int.bits_zero.
rewrite Int.bits_shl by omega.
destruct (zlt i (Int.unsigned y)).
rewrite zlt_true by omega. auto.
rewrite zlt_false by omega.
rewrite bits_ofwords by omega. rewrite zlt_true by omega. f_equal. omega.
Qed.

Lemma decompose_shru_1:
forall xh xl y,
0 <= Int.unsigned y < Int.zwordsize ->
shru' (ofwords xh xl) y =
ofwords (Int.shru xh y)
(Int.or (Int.shru xl y) (Int.shl xh (Int.sub Int.iwordsize y))).
Proof.
intros.
assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; omega. }
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
apply Int64.same_bits_eq; intros.
rewrite bits_shru' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize).
rewrite zlt_true by omega.
rewrite bits_ofwords by omega.
rewrite Int.bits_or by omega. rewrite Int.bits_shl by omega.
rewrite Int.bits_shru by omega. rewrite H0.
destruct (zlt (i + Int.unsigned y) (Int.zwordsize)).
rewrite zlt_true by omega.
rewrite orb_false_r. auto.
rewrite zlt_false by omega.
rewrite orb_false_l. f_equal. omega.
rewrite Int.bits_shru by omega.
destruct (zlt (i + Int.unsigned y) zwordsize).
rewrite bits_ofwords by omega.
rewrite zlt_true by omega. rewrite zlt_false by omega. f_equal. omega.
rewrite zlt_false by omega. auto.
Qed.

Lemma decompose_shru_2:
forall xh xl y,
Int.zwordsize <= Int.unsigned y < zwordsize ->
shru' (ofwords xh xl) y =
ofwords Int.zero (Int.shru xh (Int.sub y Int.iwordsize)).
Proof.
intros.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). omega. }
apply Int64.same_bits_eq; intros.
rewrite bits_shru' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize).
rewrite Int.bits_shru by omega. rewrite H1.
destruct (zlt (i + Int.unsigned y) zwordsize).
rewrite zlt_true by omega. rewrite bits_ofwords by omega.
rewrite zlt_false by omega. f_equal; omega.
rewrite zlt_false by omega. auto.
rewrite zlt_false by omega. apply Int.bits_zero.
Qed.

Lemma decompose_shr_1:
forall xh xl y,
0 <= Int.unsigned y < Int.zwordsize ->
shr' (ofwords xh xl) y =
ofwords (Int.shr xh y)
(Int.or (Int.shru xl y) (Int.shl xh (Int.sub Int.iwordsize y))).
Proof.
intros.
assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; omega. }
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
apply Int64.same_bits_eq; intros.
rewrite bits_shr' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize).
rewrite zlt_true by omega.
rewrite bits_ofwords by omega.
rewrite Int.bits_or by omega. rewrite Int.bits_shl by omega.
rewrite Int.bits_shru by omega. rewrite H0.
destruct (zlt (i + Int.unsigned y) (Int.zwordsize)).
rewrite zlt_true by omega.
rewrite orb_false_r. auto.
rewrite zlt_false by omega.
rewrite orb_false_l. f_equal. omega.
rewrite Int.bits_shr by omega.
destruct (zlt (i + Int.unsigned y) zwordsize).
rewrite bits_ofwords by omega.
rewrite zlt_true by omega. rewrite zlt_false by omega. f_equal. omega.
rewrite zlt_false by omega. rewrite bits_ofwords by omega.
rewrite zlt_false by omega. f_equal.
Qed.

Lemma decompose_shr_2:
forall xh xl y,
Int.zwordsize <= Int.unsigned y < zwordsize ->
shr' (ofwords xh xl) y =
ofwords (Int.shr xh (Int.sub Int.iwordsize Int.one))
(Int.shr xh (Int.sub y Int.iwordsize)).
Proof.
intros.
assert (zwordsize = 2 * Int.zwordsize) by reflexivity.
assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). omega. }
apply Int64.same_bits_eq; intros.
rewrite bits_shr' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize).
rewrite Int.bits_shr by omega. rewrite H1.
destruct (zlt (i + Int.unsigned y) zwordsize).
rewrite zlt_true by omega. rewrite bits_ofwords by omega.
rewrite zlt_false by omega. f_equal; omega.
rewrite zlt_false by omega. rewrite bits_ofwords by omega.
rewrite zlt_false by omega. auto.
rewrite Int.bits_shr by omega.
change (Int.unsigned (Int.sub Int.iwordsize Int.one)) with (Int.zwordsize - 1).
destruct (zlt (i + Int.unsigned y) zwordsize);
rewrite bits_ofwords by omega.
symmetry. rewrite zlt_false by omega. f_equal.
destruct (zlt (i - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); omega.
symmetry. rewrite zlt_false by omega. f_equal.
destruct (zlt (i - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); omega.
Qed.

forall xh xl yh yl,
add (ofwords xh xl) (ofwords yh yl) =
Proof.
apply eqm_samerepr.
set (cc := Int.add_carry xl yl Int.zero).
set (Xl := Int.unsigned xl); set (Xh := Int.unsigned xh);
set (Yl := Int.unsigned yl); set (Yh := Int.unsigned yh).
change Int.modulus with (two_p 32).
replace (Xh * two_p 32 + Xl + (Yh * two_p 32 + Yl))
with ((Xh + Yh) * two_p 32 + (Xl + Yl)) by ring.
(Xl + Yl - Int.unsigned cc * two_p 32))
with ((Int.unsigned (Int.add (Int.add xh yh) cc) - Int.unsigned cc) * two_p 32
+ (Xl + Yl)) by ring.
apply eqm_add. 2: apply eqm_refl. apply eqm_mul_2p32.
replace (Xh + Yh) with ((Xh + Yh + Int.unsigned cc) - Int.unsigned cc) by ring.
apply Int.eqm_sub. 2: apply Int.eqm_refl.
apply Int.eqm_unsigned_repr_l. apply Int.eqm_add. 2: apply Int.eqm_refl.
apply Int.eqm_unsigned_repr_l. apply Int.eqm_refl.
Qed.

Lemma decompose_sub:
forall xh xl yh yl,
sub (ofwords xh xl) (ofwords yh yl) =
ofwords (Int.sub (Int.sub xh yh) (Int.sub_borrow xl yl Int.zero))
(Int.sub xl yl).
Proof.
apply eqm_samerepr.
rewrite ! ofwords_add'. rewrite (Int.unsigned_sub_borrow xl yl).
set (bb := Int.sub_borrow xl yl Int.zero).
set (Xl := Int.unsigned xl); set (Xh := Int.unsigned xh);
set (Yl := Int.unsigned yl); set (Yh := Int.unsigned yh).
change Int.modulus with (two_p 32).
replace (Xh * two_p 32 + Xl - (Yh * two_p 32 + Yl))
with ((Xh - Yh) * two_p 32 + (Xl - Yl)) by ring.
replace (Int.unsigned (Int.sub (Int.sub xh yh) bb) * two_p 32 +
(Xl - Yl + Int.unsigned bb * two_p 32))
with ((Int.unsigned (Int.sub (Int.sub xh yh) bb) + Int.unsigned bb) * two_p 32
+ (Xl - Yl)) by ring.
apply eqm_add. 2: apply eqm_refl. apply eqm_mul_2p32.
replace (Xh - Yh) with ((Xh - Yh - Int.unsigned bb) + Int.unsigned bb) by ring.
apply Int.eqm_unsigned_repr_l. apply Int.eqm_add. 2: apply Int.eqm_refl.
apply Int.eqm_unsigned_repr_l. apply Int.eqm_refl.
Qed.

Lemma decompose_sub':
forall xh xl yh yl,
sub (ofwords xh xl) (ofwords yh yl) =
(Int.sub xl yl).
Proof.
intros. rewrite decompose_sub. f_equal.
rewrite Int.sub_add_not_3. rewrite Int.xor_assoc. rewrite Int.xor_idem.
rewrite Int.xor_zero. auto.
destruct (zlt (Int.unsigned xl + Int.unsigned (Int.not yl) + Int.unsigned Int.one) Int.modulus);
compute; [right|left]; apply Int.mkint_eq; auto.
Qed.

Definition mul' (x y: Int.int) : int := repr (Int.unsigned x * Int.unsigned y).

Lemma mul'_mulhu:
forall x y, mul' x y = ofwords (Int.mulhu x y) (Int.mul x y).
Proof.
intros.
rewrite ofwords_add. unfold mul', Int.mulhu, Int.mul.
set (p := Int.unsigned x * Int.unsigned y).
set (ph := p / Int.modulus). set (pl := p mod Int.modulus).
transitivity (repr (ph * Int.modulus + pl)).
- f_equal. rewrite Zmult_comm. apply Z_div_mod_eq. apply Int.modulus_pos.
- apply eqm_samerepr. apply eqm_add. apply eqm_mul_2p32. auto with ints.
rewrite Int.unsigned_repr_eq. apply eqm_refl.
Qed.

Lemma decompose_mul:
forall xh xl yh yl,
mul (ofwords xh xl) (ofwords yh yl) =
ofwords (Int.add (Int.add (hiword (mul' xl yl)) (Int.mul xl yh)) (Int.mul xh yl))
(loword (mul' xl yl)).
Proof.
intros.
set (pl := loword (mul' xl yl)); set (ph := hiword (mul' xl yl)).
assert (EQ0: unsigned (mul' xl yl) = Int.unsigned ph * two_p 32 + Int.unsigned pl).
{ rewrite <- (ofwords_recompose (mul' xl yl)). apply ofwords_add'. }