The Coq Proof Assistant

# Library Coq.Numbers.HexadecimalZ

# 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.
Lemma to_of (d:int) : Z.to_hex_int (Z.of_hex_int d) = norm d.
Some consequences
Lemma to_int_inj n n' : Z.to_hex_int n = Z.to_hex_int n' -> n = n'.
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.
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'.
Various lemmas
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).
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).
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).
Lemma of_hex_int_double d :
Z.of_hex_int (double d) = Z.double (Z.of_hex_int d).
Lemma double_to_hex_int n :
double (Z.to_hex_int n) = Z.to_hex_int (Z.double n).
Lemma nztail_to_hex_uint_pow16 n :
Hexadecimal.nztail (Pos.to_hex_uint (Nat.iter n (Pos.mul 16) 1%positive))
= (D1 Nil, n).