# Library Coq.NArith.Ndigits

Require Import Bool Morphisms Setoid Bvector BinPos BinNat PeanoNat Pnat Nnat
Basics ByteVector.

Local Open Scope N_scope.
Local Open Scope program_scope.

This file is mostly obsolete, see directly BinNat now.
Compatibility names for some bitwise operations

Notation Pxor := Pos.lxor (only parsing).
Notation Nxor := N.lxor (only parsing).
Notation Pbit := Pos.testbit_nat (only parsing).
Notation Nbit := N.testbit_nat (only parsing).

Notation Nxor_eq := N.lxor_eq (only parsing).
Notation Nxor_comm := N.lxor_comm (only parsing).
Notation Nxor_assoc := N.lxor_assoc (only parsing).
Notation Nxor_neutral_left := N.lxor_0_l (only parsing).
Notation Nxor_neutral_right := N.lxor_0_r (only parsing).
Notation Nxor_nilpotent := N.lxor_nilpotent (only parsing).

Equivalence of bit-testing functions, either with index in N or in nat.
Equivalence of shifts, index in N or nat
Correctness proofs for shifts, nat version
A left shift for positive numbers (used in BigN)
Semantics of bitwise operations with respect to N.testbit_nat
Equality over functional streams of bits

Definition eqf (f g:nat -> bool) := forall n:nat, f n = g n.

#[global]
Program Instance eqf_equiv : Equivalence eqf.

Local Infix "==" := eqf (at level 70, no associativity).

If two numbers produce the same stream of bits, they are equal.

Local Notation Step H := (fun n => H (S n)).

Lemma Pbit_faithful_0 : forall p, ~(Pos.testbit_nat p == (fun _ => false)).

Lemma Pbit_faithful : forall p p', Pos.testbit_nat p == Pos.testbit_nat p' -> p = p'.

Lemma Nbit_faithful : forall n n', N.testbit_nat n == N.testbit_nat n' -> n = n'.

Lemma Nbit_faithful_iff : forall n n', N.testbit_nat n == N.testbit_nat n' <-> n = n'.

Local Close Scope N_scope.

Checking whether a number is odd, i.e. if its lower bit is set.

Notation Nbit0 := N.odd (only parsing).

Definition Nodd (n:N) := N.odd n = true.
Definition Neven (n:N) := N.odd n = false.

Lemma Nbit0_correct : forall n:N, N.testbit_nat n 0 = N.odd n.

Lemma Ndouble_bit0 : forall n:N, N.odd (N.double n) = false.

Lemma Ndouble_plus_one_bit0 :
forall n:N, N.odd (N.succ_double n) = true.

Lemma Ndiv2_double :
forall n:N, Neven n -> N.double (N.div2 n) = n.

Lemma Ndiv2_double_plus_one :
forall n:N, Nodd n -> N.succ_double (N.div2 n) = n.

Lemma Ndiv2_correct :
forall (a:N) (n:nat), N.testbit_nat (N.div2 a) n = N.testbit_nat a (S n).

Lemma Nxor_bit0 :
forall a a':N, N.odd (N.lxor a a') = xorb (N.odd a) (N.odd a').

Lemma Nxor_div2 :
forall a a':N, N.div2 (N.lxor a a') = N.lxor (N.div2 a) (N.div2 a').

Lemma Nneg_bit0 :
forall a a':N,
N.odd (N.lxor a a') = true -> N.odd a = negb (N.odd a').

Lemma Nneg_bit0_1 :
forall a a':N, N.lxor a a' = Npos 1 -> N.odd a = negb (N.odd a').

Lemma Nneg_bit0_2 :
forall (a a':N) (p:positive),
N.lxor a a' = Npos (xI p) -> N.odd a = negb (N.odd a').

Lemma Nsame_bit0 :
forall (a a':N) (p:positive),
N.lxor a a' = Npos (xO p) -> N.odd a = N.odd a'.

a lexicographic order on bits, starting from the lowest bit

Fixpoint Nless_aux (a a':N) (p:positive) : bool :=
match p with
| xO p' => Nless_aux (N.div2 a) (N.div2 a') p'
| _ => andb (negb (N.odd a)) (N.odd a')
end.

Definition Nless (a a':N) :=
match N.lxor a a' with
| N0 => false
| Npos p => Nless_aux a a' p
end.

Lemma Nbit0_less :
forall a a',
N.odd a = false -> N.odd a' = true -> Nless a a' = true.

Lemma Nbit0_gt :
forall a a',
N.odd a = true -> N.odd a' = false -> Nless a a' = false.

Lemma Nless_not_refl : forall a, Nless a a = false.

Lemma Nless_def_1 :
forall a a', Nless (N.double a) (N.double a') = Nless a a'.

Lemma Nless_def_2 :
forall a a',
Nless (N.succ_double a) (N.succ_double a') = Nless a a'.

Lemma Nless_def_3 :
forall a a', Nless (N.double a) (N.succ_double a') = true.

Lemma Nless_def_4 :
forall a a', Nless (N.succ_double a) (N.double a') = false.

Lemma Nless_z : forall a, Nless a N0 = false.

Lemma N0_less_1 :
forall a, Nless N0 a = true -> {p : positive | a = Npos p}.

Lemma N0_less_2 : forall a, Nless N0 a = false -> a = N0.

Lemma Nless_trans :
forall a a' a'',
Nless a a' = true -> Nless a' a'' = true -> Nless a a'' = true.

Lemma Nless_total :
forall a a', {Nless a a' = true} + {Nless a' a = true} + {a = a'}.

Number of digits in a number

Notation Nsize := N.size_nat (only parsing).

conversions between N and bit vectors.

Fixpoint P2Bv (p:positive) : Bvector (Pos.size_nat p) :=
match p return Bvector (Pos.size_nat p) with
| xH => Bvect_true 1%nat
| xO p => Bcons false (Pos.size_nat p) (P2Bv p)
| xI p => Bcons true (Pos.size_nat p) (P2Bv p)
end.

Definition N2Bv (n:N) : Bvector (N.size_nat n) :=
match n as n0 return Bvector (N.size_nat n0) with
| N0 => Bnil
| Npos p => P2Bv p
end.

Fixpoint P2Bv_sized (m : nat) (p : positive) : Bvector m :=
match m with
| O => []
| S m =>
match p with
| xI p => true :: P2Bv_sized m p
| xO p => false :: P2Bv_sized m p
| xH => true :: Bvect_false m
end
end.

Definition N2Bv_sized (m : nat) (n : N) : Bvector m :=
match n with
| N0 => Bvect_false m
| Npos p => P2Bv_sized m p
end.

Definition N2ByteV_sized (m : nat) : N -> ByteVector m :=
of_Bvector N2Bv_sized (m * 8).

Fixpoint Bv2N (n:nat)(bv:Bvector n) : N :=
match bv with
| Vector.nil _ => N0
| Vector.cons _ false n bv => N.double (Bv2N n bv)
| Vector.cons _ true n bv => N.succ_double (Bv2N n bv)
end.

Arguments Bv2N {n} bv, n bv.

Definition ByteV2N {n : nat} : ByteVector n -> N :=
Bv2N to_Bvector.

Lemma Bv2N_N2Bv : forall n, Bv2N _ (N2Bv n) = n.

The opposite composition is not so simple: if the considered bit vector has some zeros on its right, they will disappear during the return Bv2N translation:

Lemma Bv2N_Nsize : forall n (bv:Bvector n), N.size_nat (Bv2N n bv) <= n.

In the previous lemma, we can only replace the inequality by an equality whenever the highest bit is non-null.

Lemma Bv2N_Nsize_1 : forall n (bv:Bvector (S n)),
Bsign _ bv = true <->
N.size_nat (Bv2N _ bv) = (S n).

Lemma Bv2N_upper_bound (n : nat) (bv : Bvector n) :
(Bv2N bv < N.shiftl_nat 1 n)%N.

Corollary ByteV2N_upper_bound (n : nat) (v : ByteVector n) :
(ByteV2N v < N.shiftl_nat 1 (n * 8))%N.

To state nonetheless a second result about composition of conversions, we define a conversion on a given number of bits :

Fixpoint N2Bv_gen_deprecated (n:nat)(a:N) : Bvector n :=
match n return Bvector n with
| 0 => Bnil
| S n => match a with
| N0 => Bvect_false (S n)
| Npos xH => Bcons true _ (Bvect_false n)
| Npos (xO p) => Bcons false _ (N2Bv_gen_deprecated n (Npos p))
| Npos (xI p) => Bcons true _ (N2Bv_gen_deprecated n (Npos p))
end
end.

#[deprecated(since = "8.9.0", note = "Use N2Bv_sized instead.")]
Notation N2Bv_gen := N2Bv_gen_deprecated (only parsing).

The first N2Bv is then a special case of N2Bv_gen
In fact, if k is large enough, N2Bv_gen k a contains all digits of a plus some zeros.
Here comes now the second composition result.

Lemma N2Bv_Bv2N : forall n (bv:Bvector n),
N2Bv_gen_deprecated n (Bv2N n bv) = bv.

accessing some precise bits.

Lemma Nbit0_Blow : forall n, forall (bv:Bvector (S n)),
N.odd (Bv2N _ bv) = Blow _ bv.

Notation Bnth := (@Vector.nth_order bool).

Lemma Bnth_Nbit : forall n (bv:Bvector n) p (H:p<n),
Bnth bv H = N.testbit_nat (Bv2N _ bv) p.

Lemma Nbit_Nsize : forall n p, N.size_nat n <= p -> N.testbit_nat n p = false.

Lemma Nbit_Bth: forall n p (H:p < N.size_nat n),
N.testbit_nat n p = Bnth (N2Bv n) H.

Binary bitwise operations are the same in the two worlds.

Lemma Nxor_BVxor : forall n (bv bv' : Bvector n),
Bv2N _ (BVxor _ bv bv') = N.lxor (Bv2N _ bv) (Bv2N _ bv').

Lemma Nand_BVand : forall n (bv bv' : Bvector n),
Bv2N _ (BVand _ bv bv') = N.land (Bv2N _ bv) (Bv2N _ bv').

Lemma N2Bv_sized_Nsize (n : N) :
N2Bv_sized (N.size_nat n) n = N2Bv n.

Lemma N2Bv_sized_Bv2N (n : nat) (v : Bvector n) :
N2Bv_sized n (Bv2N n v) = v.

Lemma N2Bv_N2Bv_sized_above (a : N) (k : nat) :
N2Bv_sized (N.size_nat a + k) a = N2Bv a ++ Bvect_false k.