# Int63 numbers defines indeed a cyclic structure : Z/(2^31)Z

Author: Arnaud Spiwack (+ Pierre Letouzey)
Require Import CyclicAxioms.
Require Export ZArith.
Require Export Int63.
Import Zpow_facts.
Import Utf8.
Import Lia.

Local Open Scope int63_scope.
{2 Operators }

Definition Pdigits := Eval compute in P_of_succ_nat (size - 1).

Fixpoint positive_to_int_rec (n:nat) (p:positive) :=
match n, p with
| O, _ => (Npos p, 0)
| S n, xH => (0%N, 1)
| S n, xO p =>
let (N,i) := positive_to_int_rec n p in
(N, i << 1)
| S n, xI p =>
let (N,i) := positive_to_int_rec n p in
(N, (i << 1) + 1)
end.

Definition positive_to_int := positive_to_int_rec size.

Definition mulc_WW x y :=
let (h, l) := mulc x y in
if is_zero h then
if is_zero l then W0
else WW h l
else WW h l.
Notation "n '*c' m" := (mulc_WW n m) (at level 40, no associativity) : int63_scope.

Definition pos_mod p x :=
if p <= digits then
let p := digits - p in
(x << p) >> p
else x.

Notation pos_mod_int := pos_mod.

Import ZnZ.

Instance int_ops : ZnZ.Ops int :=
{|
digits := Pdigits;
zdigits := Int63.digits;
to_Z := Int63.to_Z;
of_pos := positive_to_int;
tail0 := Int63.tail0;
zero := 0;
one := 1;
minus_one := Int63.max_int;
compare := Int63.compare;
eq0 := Int63.is_zero;
opp_c := Int63.oppc;
opp := Int63.opp;
opp_carry := Int63.oppcarry;
succ_c := Int63.succc;
succ := Int63.succ;
pred_c := Int63.predc;
sub_c := Int63.subc;
sub_carry_c := Int63.subcarryc;
pred := Int63.pred;
sub := Int63.sub;
sub_carry := Int63.subcarry;
mul_c := mulc_WW;
mul := Int63.mul;
square_c := fun x => mulc_WW x x;
div21 := diveucl_21;
div_gt := diveucl;
div := diveucl;
modulo_gt := Int63.mod;
modulo := Int63.mod;
gcd_gt := Int63.gcd;
gcd := Int63.gcd;
pos_mod := pos_mod_int;
is_even := Int63.is_even;
sqrt2 := Int63.sqrt2;
sqrt := Int63.sqrt;
ZnZ.lor := Int63.lor;
ZnZ.land := Int63.land;
ZnZ.lxor := Int63.lxor
|}.

Local Open Scope Z_scope.

Lemma is_zero_spec_aux : forall x : int, is_zero x = true -> φ x = 0%Z.

Lemma positive_to_int_spec :
forall p : positive,
Zpos p =
Z_of_N (fst (positive_to_int p)) * wB + to_Z (snd (positive_to_int p)).

Lemma mulc_WW_spec :
forall x y, Φ ( x *c y ) = φ x * φ y.

Lemma squarec_spec :
forall x,
Φ(x *c x) = φ x * φ x.

Lemma diveucl_spec_aux : forall a b, 0 < φ b ->
let (q,r) := diveucl a b in
φ a = φ q * φ b + φ r /\
0 <= φ r < φ b.

Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n ->
((a * 2 ^ (n - p)) mod (2^n) / 2 ^ (n - p)) mod (2^n) =
a mod 2 ^ p.

Lemma div_le_0 : forall p x, 0 <= x -> 0 <= x / 2 ^ p.

Lemma div_lt : forall p x y, 0 <= x < y -> x / 2^p < y.

Lemma P (A B C: Prop) :
A → (BC) → (AB) → C.

Lemma shift_unshift_mod_3:
forall n p a : Z,
0 <= p <= n ->
(a * 2 ^ (n - p)) mod 2 ^ n / 2 ^ (n - p) = a mod 2 ^ p.

Lemma pos_mod_spec w p : φ(pos_mod p w) = φ(w) mod (2 ^ φ(p)).

{2 Specification and proof}
Global Instance int_specs : ZnZ.Specs int_ops := {
spec_to_Z := to_Z_bounded;
spec_of_pos := positive_to_int_spec;
spec_zdigits := refl_equal _;
spec_more_than_1_digit:= refl_equal _;
spec_0 := to_Z_0;
spec_1 := to_Z_1;
spec_m1 := refl_equal _;
spec_compare := compare_spec;
spec_eq0 := is_zero_spec_aux;
spec_opp_c := oppc_spec;
spec_opp := opp_spec;
spec_opp_carry := oppcarry_spec;
spec_succ_c := succc_spec;
spec_succ := succ_spec;
spec_pred_c := predc_spec;
spec_sub_c := subc_spec;
spec_sub_carry_c := subcarryc_spec;
spec_pred := pred_spec;
spec_sub := sub_spec;
spec_sub_carry := subcarry_spec;
spec_mul_c := mulc_WW_spec;
spec_mul := mul_spec;
spec_square_c := squarec_spec;
spec_div21 := diveucl_21_spec_aux;
spec_div_gt := fun a b _ => diveucl_spec_aux a b;
spec_div := diveucl_spec_aux;
spec_modulo_gt := fun a b _ _ => mod_spec a b;
spec_modulo := fun a b _ => mod_spec a b;
spec_gcd_gt := fun a b _ => gcd_spec a b;
spec_gcd := gcd_spec;
spec_tail00 := tail00_spec;
spec_tail0 := tail0_spec;