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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Ascii String.
Require Import BinNums.
Import BinNatDef.
Import BinIntDef.
Import BinPosDef.
Local Open Scope positive_scope.
Local Open Scope string_scope.
Definition ascii_to_digit (ch : ascii) : option N
:= (if ascii_dec ch "0" then Some 0
else if ascii_dec ch "1" then Some 1
else if ascii_dec ch "2" then Some 2
else if ascii_dec ch "3" then Some 3
else if ascii_dec ch "4" then Some 4
else if ascii_dec ch "5" then Some 5
else if ascii_dec ch "6" then Some 6
else if ascii_dec ch "7" then Some 7
else None)%N.
Fixpoint pos_oct_app (p q:positive) : positive :=
match q with
| 1 => p~0~0~1
| 2 => p~0~1~0
| 3 => p~0~1~1
| 4 => p~1~0~0
| 5 => p~1~0~1
| 6 => p~1~1~0
| 7 => p~1~1~1
| q~0~0~0 => (pos_oct_app p q)~0~0~0
| q~0~0~1 => (pos_oct_app p q)~0~0~1
| q~0~1~0 => (pos_oct_app p q)~0~1~0
| q~0~1~1 => (pos_oct_app p q)~0~1~1
| q~1~0~0 => (pos_oct_app p q)~1~0~0
| q~1~0~1 => (pos_oct_app p q)~1~0~1
| q~1~1~0 => (pos_oct_app p q)~1~1~0
| q~1~1~1 => (pos_oct_app p q)~1~1~1
end.
Module Raw.
Fixpoint of_pos (p : positive) (rest : string) : string
:= match p with
| 1 => String "1" rest
| 2 => String "2" rest
| 3 => String "3" rest
| 4 => String "4" rest
| 5 => String "5" rest
| 6 => String "6" rest
| 7 => String "7" rest
| p'~0~0~0 => of_pos p' (String "0" rest)
| p'~0~0~1 => of_pos p' (String "1" rest)
| p'~0~1~0 => of_pos p' (String "2" rest)
| p'~0~1~1 => of_pos p' (String "3" rest)
| p'~1~0~0 => of_pos p' (String "4" rest)
| p'~1~0~1 => of_pos p' (String "5" rest)
| p'~1~1~0 => of_pos p' (String "6" rest)
| p'~1~1~1 => of_pos p' (String "7" rest)
end.
Fixpoint to_N (s : string) (rest : N)
: N
:= match s with
| "" => rest
| String ch s'
=> to_N
s'
match ascii_to_digit ch with
| Some v => N.add v (N.mul 8 rest)
| None => N0
end
end.
Fixpoint to_N_of_pos (p : positive) (rest : string) (base : N)
: to_N (of_pos p rest) base
= to_N rest match base with
| N0 => N.pos p
| Npos v => Npos (pos_oct_app v p)
end.
Proof.
do 3 try destruct p as [p|p|]; destruct base; try reflexivity;
cbn; rewrite to_N_of_pos; reflexivity.
Qed.
End Raw.
Definition of_pos (p : positive) : string
:= String "0" (String "o" (Raw.of_pos p "")).
Definition of_N (n : N) : string
:= match n with
| N0 => "0o0"
| Npos p => of_pos p
end.
Definition of_Z (z : Z) : string
:= match z with
| Zneg p => String "-" (of_pos p)
| Z0 => "0o0"
| Zpos p => of_pos p
end.
Definition of_nat (n : nat) : string
:= of_N (N.of_nat n).
Definition to_N (s : string) : N
:= match s with
| String s0 (String so s)
=> if ascii_dec s0 "0"
then if ascii_dec so "o"
then Raw.to_N s N0
else N0
else N0
| _ => N0
end.
Definition to_pos (s : string) : positive
:= match to_N s with
| N0 => 1
| Npos p => p
end.
Definition to_Z (s : string) : Z
:= let '(is_neg, n) := match s with
| String s0 s'
=> if ascii_dec s0 "-"
then (true, to_N s')
else (false, to_N s)
| EmptyString => (false, to_N s)
end in
match n with
| N0 => Z0
| Npos p => if is_neg then Zneg p else Zpos p
end.
Definition to_nat (s : string) : nat
:= N.to_nat (to_N s).
Lemma to_N_of_N (n : N)
: to_N (of_N n)
= n.
Proof.
destruct n; [ reflexivity | apply Raw.to_N_of_pos ].
Qed.
Lemma to_Z_of_Z (z : Z)
: to_Z (of_Z z)
= z.
Proof.
cbv [of_Z to_Z]; destruct z as [|z|z]; cbn;
try reflexivity;
rewrite Raw.to_N_of_pos; cbn; reflexivity.
Qed.
Lemma to_nat_of_nat (n : nat)
: to_nat (of_nat n)
= n.
Proof.
cbv [to_nat of_nat];
rewrite to_N_of_N, Nnat.Nat2N.id; reflexivity.
Qed.
Lemma to_pos_of_pos (p : positive)
: to_pos (of_pos p)
= p.
Proof.
cbv [of_pos to_pos to_N]; cbn;
rewrite Raw.to_N_of_pos; cbn; reflexivity.
Qed.
Example of_pos_1 : of_pos 1 = "0o1" := eq_refl.
Example of_pos_2 : of_pos 2 = "0o2" := eq_refl.
Example of_pos_3 : of_pos 3 = "0o3" := eq_refl.
Example of_pos_7 : of_pos 7 = "0o7" := eq_refl.
Example of_pos_8 : of_pos 8 = "0o10" := eq_refl.
Example of_N_0 : of_N 0 = "0o0" := eq_refl.
Example of_Z_0 : of_Z 0 = "0o0" := eq_refl.
Example of_Z_m1 : of_Z (-1) = "-0o1" := eq_refl.
Example of_nat_0 : of_nat 0 = "0o0" := eq_refl.
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