# Conversion between hexadecimal numbers and Coq strings

Pretty straightforward, which is precisely the point of the Hexadecimal.int datatype. The only catch is Hexadecimal.Nil : we could choose to convert it as "" or as "0". In the first case, it is awkward to consider "" (or "-") as a number, while in the second case we don't have a perfect bijection. Since the second variant is implemented thanks to the first one, we provide both.
Hexadecimal digits are lower case ('a'..'f'). We ignore upper case digits ('A'..'F') for the sake of simplicity.

Local Open Scope string_scope.

Parsing one char

Definition uint_of_char (a:ascii)(d:option uint) :=
match d with
| None => None
| Some d =>
match a with
| "0" => Some (D0 d)
| "1" => Some (D1 d)
| "2" => Some (D2 d)
| "3" => Some (D3 d)
| "4" => Some (D4 d)
| "5" => Some (D5 d)
| "6" => Some (D6 d)
| "7" => Some (D7 d)
| "8" => Some (D8 d)
| "9" => Some (D9 d)
| "a" => Some (Da d)
| "b" => Some (Db d)
| "c" => Some (Dc d)
| "d" => Some (Dd d)
| "e" => Some (De d)
| "f" => Some (Df d)
| _ => None
end
end%char.

Lemma uint_of_char_spec c d d' :
uint_of_char c (Some d) = Some d' ->
(c = "0" /\ d' = D0 d \/
c = "1" /\ d' = D1 d \/
c = "2" /\ d' = D2 d \/
c = "3" /\ d' = D3 d \/
c = "4" /\ d' = D4 d \/
c = "5" /\ d' = D5 d \/
c = "6" /\ d' = D6 d \/
c = "7" /\ d' = D7 d \/
c = "8" /\ d' = D8 d \/
c = "9" /\ d' = D9 d \/
c = "a" /\ d' = Da d \/
c = "b" /\ d' = Db d \/
c = "c" /\ d' = Dc d \/
c = "d" /\ d' = Dd d \/
c = "e" /\ d' = De d \/
c = "f" /\ d' = Df d)%char.

Hexadecimal/String conversion where Nil is ""

Module NilEmpty.

Fixpoint string_of_uint (d:uint) :=
match d with
| Nil => EmptyString
| D0 d => String "0" (string_of_uint d)
| D1 d => String "1" (string_of_uint d)
| D2 d => String "2" (string_of_uint d)
| D3 d => String "3" (string_of_uint d)
| D4 d => String "4" (string_of_uint d)
| D5 d => String "5" (string_of_uint d)
| D6 d => String "6" (string_of_uint d)
| D7 d => String "7" (string_of_uint d)
| D8 d => String "8" (string_of_uint d)
| D9 d => String "9" (string_of_uint d)
| Da d => String "a" (string_of_uint d)
| Db d => String "b" (string_of_uint d)
| Dc d => String "c" (string_of_uint d)
| Dd d => String "d" (string_of_uint d)
| De d => String "e" (string_of_uint d)
| Df d => String "f" (string_of_uint d)
end.

Fixpoint uint_of_string s :=
match s with
| EmptyString => Some Nil
| String a s => uint_of_char a (uint_of_string s)
end.

Definition string_of_int (d:int) :=
match d with
| Pos d => string_of_uint d
| Neg d => String "-" (string_of_uint d)
end.

Definition int_of_string s :=
match s with
| EmptyString => Some (Pos Nil)
| String a s' =>
if Ascii.eqb a "-" then option_map Neg (uint_of_string s')
else option_map Pos (uint_of_string s)
end.

Corresponding proofs

Lemma usu d :
uint_of_string (string_of_uint d) = Some d.

Lemma sus s d :
uint_of_string s = Some d -> string_of_uint d = s.

Lemma isi d : int_of_string (string_of_int d) = Some d.

Lemma sis s d :
int_of_string s = Some d -> string_of_int d = s.

End NilEmpty.

Hexadecimal/String conversions where Nil is "0"

Module NilZero.

Definition string_of_uint (d:uint) :=
match d with
| Nil => "0"
| _ => NilEmpty.string_of_uint d
end.

Definition uint_of_string s :=
match s with
| EmptyString => None
| _ => NilEmpty.uint_of_string s
end.

Definition string_of_int (d:int) :=
match d with
| Pos d => string_of_uint d
| Neg d => String "-" (string_of_uint d)
end.

Definition int_of_string s :=
match s with
| EmptyString => None
| String a s' =>
if Ascii.eqb a "-" then option_map Neg (uint_of_string s')
else option_map Pos (uint_of_string s)
end.

Corresponding proofs

Lemma uint_of_string_nonnil s : uint_of_string s <> Some Nil.

Lemma sus s d :
uint_of_string s = Some d -> string_of_uint d = s.

Lemma usu d :
d<>Nil -> uint_of_string (string_of_uint d) = Some d.

Lemma usu_nil :
uint_of_string (string_of_uint Nil) = Some Hexadecimal.zero.

Lemma usu_gen d :
uint_of_string (string_of_uint d) = Some d \/
uint_of_string (string_of_uint d) = Some Hexadecimal.zero.

Lemma isi d :
d<>Pos Nil -> d<>Neg Nil ->
int_of_string (string_of_int d) = Some d.

Lemma isi_posnil :
int_of_string (string_of_int (Pos Nil)) = Some (Pos Hexadecimal.zero).

Warning! (-0) won't parse (compatibility with the behavior of Z).

Lemma isi_negnil :
int_of_string (string_of_int (Neg Nil)) = Some (Neg (D0 Nil)).

Lemma sis s d :
int_of_string s = Some d -> string_of_int d = s.

End NilZero.