Proofs that conversions between hexadecimal numbers and N are bijections

Module Unsigned.

Lemma of_to (n:N) : N.of_hex_uint (N.to_hex_uint n) = n.

Lemma to_of (d:uint) : N.to_hex_uint (N.of_hex_uint d) = unorm d.

Lemma to_uint_inj n n' : N.to_hex_uint n = N.to_hex_uint n' -> n = n'.

Lemma to_uint_surj d : exists p, N.to_hex_uint p = unorm d.

Lemma of_uint_norm d : N.of_hex_uint (unorm d) = N.of_hex_uint d.

Lemma of_inj d d' :
N.of_hex_uint d = N.of_hex_uint d' -> unorm d = unorm d'.

Lemma of_iff d d' : N.of_hex_uint d = N.of_hex_uint d' <-> unorm d = unorm d'.

End Unsigned.

Module Signed.

Lemma of_to (n:N) : N.of_hex_int (N.to_hex_int n) = Some n.

Lemma to_of (d:int)(n:N) : N.of_hex_int d = Some n -> N.to_hex_int n = norm d.

Lemma to_int_inj n n' : N.to_hex_int n = N.to_hex_int n' -> n = n'.

Lemma to_int_pos_surj d : exists n, N.to_hex_int n = norm (Pos d).

Lemma of_int_norm d : N.of_hex_int (norm d) = N.of_hex_int d.

Lemma of_inj_pos d d' :
N.of_hex_int (Pos d) = N.of_hex_int (Pos d') -> unorm d = unorm d'.

End Signed.