Library Coq.Init.Decimal
Decimal numbers
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.
For signed integers, we use two constructors Pos and Neg.
For decimal numbers, we use two constructors Decimal and
DecimalExp, depending on whether or not they are given with an
exponent (e.g., 1.02e+01). i is the integral part while f is
the fractional part (beware that leading zeroes do matter).
Variant decimal :=
| Decimal (i:int) (f:uint)
| DecimalExp (i:int) (f:uint) (e:int).
Scheme Equality for uint.
Scheme Equality for int.
Scheme Equality for decimal.
Notation int_eq_dec := signed_int_eq_dec.
Notation int_beq := signed_int_beq.
Notation internal_int_dec_lb := internal_signed_int_dec_lb.
Notation internal_int_dec_bl := internal_signed_int_dec_bl.
Declare Scope dec_uint_scope.
Delimit Scope dec_uint_scope with uint.
Bind Scope dec_uint_scope with uint.
Declare Scope dec_int_scope.
Delimit Scope dec_int_scope with int.
Bind Scope dec_int_scope with int.
Register uint as num.uint.type.
Register int as num.int.type.
Register decimal as num.decimal.type.
Fixpoint nb_digits d :=
match d with
| Nil => O
| D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d =>
S (nb_digits d)
end.
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.
Definition opp (d:int) :=
match d with
| Pos d => Neg d
| Neg d => Pos d
end.
Definition abs (d:int) : uint :=
match d with
| Pos d => d
| Neg d => d
end.
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.
Definition app d d' := revapp (rev d) d'.
Definition app_int d1 d2 :=
match d1 with Pos d1 => Pos (app d1 d2) | Neg d1 => Neg (app d1 d2) end.
nztail removes all trailing zero digits and return both the
result and the number of removed digits.
Definition nztail d :=
let fix aux d_rev :=
match d_rev with
| D0 d_rev => let (r, n) := aux d_rev in pair r (S n)
| _ => pair d_rev O
end in
let (r, n) := aux (rev d) in pair (rev r) n.
Definition nztail_int d :=
match d with
| Pos d => let (r, n) := nztail d in pair (Pos r) n
| Neg d => let (r, n) := nztail d in pair (Neg r) n
end.
del_head n d removes n digits at beginning of d
or returns zero if d has less than n digits.
Fixpoint del_head n d :=
match n with
| O => d
| S n =>
match d with
| Nil => zero
| D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d =>
del_head n d
end
end.
Definition del_head_int n d :=
match d with
| Pos d => del_head n d
| Neg d => del_head n d
end.
del_tail n d removes n digits at end of d
or returns zero if d has less than n digits.
Definition del_tail n d := rev (del_head n (rev d)).
Definition del_tail_int n d :=
match d with
| Pos d => Pos (del_tail n d)
| Neg d => Neg (del_tail n d)
end.
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.
Pseudo-conversion functions used when declaring
Number Notations on uint and int.