Library Coq.Init.Decimal
Decimal numbers
Inductive uint :=
| Nil
| D0 (_:uint)
| D1 (_:uint)
| D2 (_:uint)
| D3 (_:uint)
| D4 (_:uint)
| D5 (_:uint)
| D6 (_:uint)
| D7 (_:uint)
| D8 (_:uint)
| D9 (_:uint).
Nil is the number terminator. Taken alone, it behaves as zero,
    but rather use D0 Nil instead, since this form will be denoted
    as 0, while Nil will be printed as Nil. 
For signed integers, we use two constructors Pos and Neg. 
Inductive int := Pos (d:uint) | Neg (d:uint).
Delimit Scope uint_scope with uint.
Delimit Scope int_scope with int.
This representation favors simplicity over canonicity.
    For normalizing numbers, we need to remove head zero digits,
    and choose our canonical representation of 0 (here D0 Nil
    for unsigned numbers and Pos (D0 Nil) for signed numbers). 
 
 nzhead removes all head zero digits 
unorm : normalization of unsigned integers 
norm : normalization of signed integers 
Definition norm d :=
match d with
| Pos d => Pos (unorm d)
| Neg d =>
match nzhead d with
| Nil => Pos zero
| d => Neg d
end
end.
A few easy operations. For more advanced computations, use the conversions
    with other Coq numeral datatypes (e.g. Z) and the operations on them. 
For conversions with binary numbers, it is easier to operate
    on little-endian numbers. 
Fixpoint revapp (d d' : uint) :=
match d with
| Nil => d'
| D0 d => revapp d (D0 d')
| D1 d => revapp d (D1 d')
| D2 d => revapp d (D2 d')
| D3 d => revapp d (D3 d')
| D4 d => revapp d (D4 d')
| D5 d => revapp d (D5 d')
| D6 d => revapp d (D6 d')
| D7 d => revapp d (D7 d')
| D8 d => revapp d (D8 d')
| D9 d => revapp d (D9 d')
end.
Definition rev d := revapp d Nil.
Module Little.
Successor of little-endian numbers 
Fixpoint succ d :=
match d with
| Nil => D1 Nil
| D0 d => D1 d
| D1 d => D2 d
| D2 d => D3 d
| D3 d => D4 d
| D4 d => D5 d
| D5 d => D6 d
| D6 d => D7 d
| D7 d => D8 d
| D8 d => D9 d
| D9 d => D0 (succ d)
end.
Doubling little-endian numbers 
Fixpoint double d :=
match d with
| Nil => Nil
| D0 d => D0 (double d)
| D1 d => D2 (double d)
| D2 d => D4 (double d)
| D3 d => D6 (double d)
| D4 d => D8 (double d)
| D5 d => D0 (succ_double d)
| D6 d => D2 (succ_double d)
| D7 d => D4 (succ_double d)
| D8 d => D6 (succ_double d)
| D9 d => D8 (succ_double d)
end
with succ_double d :=
match d with
| Nil => D1 Nil
| D0 d => D1 (double d)
| D1 d => D3 (double d)
| D2 d => D5 (double d)
| D3 d => D7 (double d)
| D4 d => D9 (double d)
| D5 d => D1 (succ_double d)
| D6 d => D3 (succ_double d)
| D7 d => D5 (succ_double d)
| D8 d => D7 (succ_double d)
| D9 d => D9 (succ_double d)
end.
End Little.
