Timings for HexadecimalZ.v
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(** * HexadecimalZ
Proofs that conversions between hexadecimal numbers and [Z]
are bijections. *)
Require Import Hexadecimal HexadecimalFacts HexadecimalPos HexadecimalN ZArith.
Lemma of_to (z:Z) : Z.of_hex_int (Z.to_hex_int z) = z.
rewrite HexadecimalPos.Unsigned.of_to.
rewrite HexadecimalPos.Unsigned.of_to.
Lemma to_of (d:int) : Z.to_hex_int (Z.of_hex_int d) = norm d.
destruct d; simpl; unfold Z.to_hex_int, Z.of_hex_uint.
rewrite <- (HexadecimalN.Unsigned.to_of d).
now destruct (Pos.of_hex_uint d).
destruct (Pos.of_hex_uint d) eqn:Hd; simpl; f_equal.
generalize (HexadecimalPos.Unsigned.to_of d).
assert (Hp := HexadecimalPos.Unsigned.to_of d).
destruct (nzhead d); trivial.
generalize (HexadecimalPos.Unsigned.of_to p).
Lemma to_int_inj n n' : Z.to_hex_int n = Z.to_hex_int n' -> n = n'.
now rewrite <- (of_to n), <- (of_to n'), EQ.
Lemma to_int_surj d : exists n, Z.to_hex_int n = norm d.
Lemma of_int_norm d : Z.of_hex_int (norm d) = Z.of_hex_int d.
unfold Z.of_hex_int, Z.of_hex_uint.
now rewrite HexadecimalPos.Unsigned.of_uint_norm.
destruct (nzhead d) eqn:H;
[ induction d; simpl; auto; discriminate |
destruct (nzhead_nonzero _ _ H) | .. ];
f_equal; f_equal; apply HexadecimalPos.Unsigned.of_iff;
unfold unorm; now rewrite H.
Lemma of_inj d d' :
Z.of_hex_int d = Z.of_hex_int d' -> norm d = norm d'.
Lemma of_iff d d' : Z.of_hex_int d = Z.of_hex_int d' <-> norm d = norm d'.
rewrite <- of_int_norm, E.
Lemma of_hex_uint_iter_D0 d n :
Z.of_hex_uint (app d (Nat.iter n D0 Nil))
= Nat.iter n (Z.mul 0x10) (Z.of_hex_uint d).
rewrite <-(rev_rev (app _ _)), <-(of_list_to_list (rev (app _ _))).
rewrite rev_spec, app_spec, List.rev_app_distr.
rewrite <-!rev_spec, <-app_spec, of_list_to_list.
unfold Z.of_hex_uint; rewrite Unsigned.of_lu_rev.
unfold app; rewrite Unsigned.of_lu_revapp, !rev_rev.
rewrite <-!Unsigned.of_lu_rev, !rev_rev.
assert (H' : Pos.of_hex_uint (Nat.iter n D0 Nil) = 0%N).
now induction n; [|rewrite Unsigned.nat_iter_S].
rewrite H', N.add_0_l; clear H'.
induction n; [now simpl; rewrite N.mul_1_r|].
rewrite !Unsigned.nat_iter_S, <-IHn.
simpl Unsigned.usize; rewrite N.pow_succ_r'.
rewrite !N2Z.inj_mul; simpl Z.of_N; ring.
Lemma of_hex_int_iter_D0 d n :
Z.of_hex_int (app_int d (Nat.iter n D0 Nil))
= Nat.iter n (Z.mul 0x10) (Z.of_hex_int d).
case d; clear d; intro d; simpl.
now rewrite of_hex_uint_iter_D0.
rewrite of_hex_uint_iter_D0; induction n; [now simpl|].
rewrite !Unsigned.nat_iter_S, <-IHn; ring.
Definition double d :=
match d with
| Pos u => Pos (Unsigned.double u)
| Neg u => Neg (Unsigned.double u)
end.
Lemma double_norm d : double (norm d) = norm (double d).
now simpl; rewrite Unsigned.double_unorm.
simpl; rewrite <-Unsigned.double_nzhead.
case (uint_eq_dec (nzhead d) Nil); intro Hnzd.
assert (H : Unsigned.double (nzhead d) <> Nil).
intro H; apply Hnzd, rev_nil_inv.
now generalize (rev_nil_inv _ H); case rev.
set (r := Unsigned.double _).
set (m := match r with Nil => Pos zero | _ => _ end).
now unfold m; clear m; revert H; case r.
rewrite H'; unfold r; clear m r H H'.
now revert Hnzd; case nzhead.
Lemma of_hex_int_double d :
Z.of_hex_int (double d) = Z.double (Z.of_hex_int d).
now destruct d; simpl; unfold Z.of_hex_uint;
rewrite Unsigned.of_hex_uint_double; case Pos.of_hex_uint.
Lemma double_to_hex_int n :
double (Z.to_hex_int n) = Z.to_hex_int (Z.double n).
now rewrite <-(of_to n), <-of_hex_int_double, !to_of, double_norm.
Lemma nztail_to_hex_uint_pow16 n :
Hexadecimal.nztail (Pos.to_hex_uint (Nat.iter n (Pos.mul 16) 1%positive))
= (D1 Nil, n).
case n as [|n]; [now simpl|].
rewrite <-(Nat2Pos.id (S n)); [|now simpl].
generalize (Pos.of_nat (S n)); clear n; intro p.
induction (Pos.to_nat p); [now simpl|].
rewrite Unsigned.nat_iter_S.
change (Pos.to_little_hex_uint _)
with (Unsigned.to_lu (16 * N.pos (Nat.iter n (Pos.mul 16) 1%positive))).
rewrite Unsigned.to_lhex_tenfold.
revert IHn; unfold Pos.to_hex_uint.
unfold Hexadecimal.nztail; rewrite !rev_rev; simpl.
set (f'' := _ (Pos.to_little_hex_uint _)).
now case f''; intros r n' H; inversion H.