The Coq Proof Assistant

# 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.
Lemma Ptestbit_Pbit :
forall p n, Pos.testbit p (N.of_nat n) = Pos.testbit_nat p n.
Lemma Ntestbit_Nbit : forall a n, N.testbit a (N.of_nat n) = N.testbit_nat a n.
Lemma Pbit_Ptestbit :
forall p n, Pos.testbit_nat p (N.to_nat n) = Pos.testbit p n.
Lemma Nbit_Ntestbit :
forall a n, N.testbit_nat a (N.to_nat n) = N.testbit a n.
Equivalence of shifts, index in N or nat
Lemma Nshiftr_nat_S : forall a n,
N.shiftr_nat a (S n) = N.div2 (N.shiftr_nat a n).
Lemma Nshiftl_nat_S : forall a n,
N.shiftl_nat a (S n) = N.double (N.shiftl_nat a n).
Lemma Nshiftr_nat_equiv :
forall a n, N.shiftr_nat a (N.to_nat n) = N.shiftr a n.
Lemma Nshiftr_equiv_nat :
forall a n, N.shiftr a (N.of_nat n) = N.shiftr_nat a n.
Lemma Nshiftl_nat_equiv :
forall a n, N.shiftl_nat a (N.to_nat n) = N.shiftl a n.
Lemma Nshiftl_equiv_nat :
forall a n, N.shiftl a (N.of_nat n) = N.shiftl_nat a n.
Correctness proofs for shifts, nat version
Lemma Nshiftr_nat_spec : forall a n m,
N.testbit_nat (N.shiftr_nat a n) m = N.testbit_nat a (m+n).
Lemma Nshiftl_nat_spec_high : forall a n m, (n<=m)%nat ->
N.testbit_nat (N.shiftl_nat a n) m = N.testbit_nat a (m-n).
Lemma Nshiftl_nat_spec_low : forall a n m, (m<n)%nat ->
N.testbit_nat (N.shiftl_nat a n) m = false.
A left shift for positive numbers (used in BigN)
Lemma Pshiftl_nat_0 : forall p, Pos.shiftl_nat p 0 = p.
Lemma Pshiftl_nat_S :
forall p n, Pos.shiftl_nat p (S n) = xO (Pos.shiftl_nat p n).
Lemma Pshiftl_nat_N :
forall p n, Npos (Pos.shiftl_nat p n) = N.shiftl_nat (Npos p) n.
Lemma Pshiftl_nat_plus : forall n m p,
Pos.shiftl_nat p (m + n) = Pos.shiftl_nat (Pos.shiftl_nat p n) m.
Semantics of bitwise operations with respect to N.testbit_nat
Lemma Pxor_semantics p p' n :
N.testbit_nat (Pos.lxor p p') n = xorb (Pos.testbit_nat p n) (Pos.testbit_nat p' n).
Lemma Nxor_semantics a a' n :
N.testbit_nat (N.lxor a a') n = xorb (N.testbit_nat a n) (N.testbit_nat a' n).
Lemma Por_semantics p p' n :
Pos.testbit_nat (Pos.lor p p') n = (Pos.testbit_nat p n) || (Pos.testbit_nat p' n).
Lemma Nor_semantics a a' n :
N.testbit_nat (N.lor a a') n = (N.testbit_nat a n) || (N.testbit_nat a' n).
Lemma Pand_semantics p p' n :
N.testbit_nat (Pos.land p p') n = (Pos.testbit_nat p n) && (Pos.testbit_nat p' n).
Lemma Nand_semantics a a' n :
N.testbit_nat (N.land a a') n = (N.testbit_nat a n) && (N.testbit_nat a' n).
Lemma Pdiff_semantics p p' n :
N.testbit_nat (Pos.ldiff p p') n = (Pos.testbit_nat p n) && negb (Pos.testbit_nat p' n).
Lemma Ndiff_semantics a a' n :
N.testbit_nat (N.ldiff a a') n = (N.testbit_nat a n) && negb (N.testbit_nat a' n).
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.
If two numbers produce the same stream of bits, they are equal.
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
Lemma N2Bv_N2Bv_gen (a:N) : N2Bv a = N2Bv_gen_deprecated (N.size_nat a) a.
In fact, if k is large enough, N2Bv_gen k a contains all digits of a plus some zeros.
Lemma N2Bv_N2Bv_gen_above : forall (a:N)(k:nat),
N2Bv_gen_deprecated (N.size_nat a + k) a
= Vector.append (N2Bv a) (Bvect_false k).
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.