Library Coq.Init.Decimal

Decimal numbers

These numbers coded in base 10 will be used for parsing and printing other Coq numeral datatypes in an human-readable way. See the Numeral Notation command. We represent numbers in base 10 as lists of decimal digits, in big-endian order (most significant digit comes first).
Unsigned integers are just lists of digits. For instance, ten is (D1 (D0 Nil))

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.

Notation zero := (D0 Nil).

For signed integers, we use two constructors Pos and Neg.

Inductive int := Pos (d:uint) | Neg (d:uint).

Delimit Scope dec_uint_scope with uint.
Delimit Scope dec_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

Fixpoint nzhead d :=
  match d with
  | D0 d => nzhead d
  | _ => d

unorm : normalization of unsigned integers

Definition unorm d :=
  match nzhead d with
  | Nil => zero
  | d => d

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

A few easy operations. For more advanced computations, use the conversions with other Coq numeral datatypes (e.g. Z) and the operations on them.

Definition opp (d:int) :=
  match d with
  | Pos d => Neg d
  | Neg d => Pos d

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')

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)

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)

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 Little.

Pseudo-conversion functions used when declaring Numeral Notations on uint and int.

Definition uint_of_uint (i:uint) := i.
Definition int_of_int (i:int) := i.