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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** * 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 uint_scope with uint.
Bind Scope uint_scope with uint.
Delimit Scope int_scope with int.
Bind 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 *)
Fixpoint nzhead d :=
match d with
| D0 d => nzhead d
| _ => d
end.
(** [unorm] : normalization of unsigned integers *)
Definition unorm d :=
match nzhead d with
| Nil => zero
| d => d
end.
(** [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.
(** 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.
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