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(************************************************************************)
(* * The Rocq Prover / The Rocq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \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) *)
(************************************************************************)
From Stdlib Require Import Ascii String.
From Stdlib Require Import BinNums.
Import BinNatDef.
Import BinIntDef.
Import BinPosDef.
Local Open Scope positive_scope.
Local Open Scope string_scope.
Local Notation "a || b"
:= (if a then true else if b then true else false).
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 if ascii_dec ch "8" then Some 8
else if ascii_dec ch "9" then Some 9
else if ascii_dec ch "a" || ascii_dec ch "A" then Some 10
else if ascii_dec ch "b" || ascii_dec ch "B" then Some 11
else if ascii_dec ch "c" || ascii_dec ch "C" then Some 12
else if ascii_dec ch "d" || ascii_dec ch "D" then Some 13
else if ascii_dec ch "e" || ascii_dec ch "E" then Some 14
else if ascii_dec ch "f" || ascii_dec ch "F" then Some 15
else None)%N.
Fixpoint pos_hex_app (p q:positive) : positive :=
match q with
| 1 => p~0~0~0~1
| 2 => p~0~0~1~0
| 3 => p~0~0~1~1
| 4 => p~0~1~0~0
| 5 => p~0~1~0~1
| 6 => p~0~1~1~0
| 7 => p~0~1~1~1
| 8 => p~1~0~0~0
| 9 => p~1~0~0~1
| 10 => p~1~0~1~0
| 11 => p~1~0~1~1
| 12 => p~1~1~0~0
| 13 => p~1~1~0~1
| 14 => p~1~1~1~0
| 15 => p~1~1~1~1
| q~0~0~0~0 => (pos_hex_app p q)~0~0~0~0
| q~0~0~0~1 => (pos_hex_app p q)~0~0~0~1
| q~0~0~1~0 => (pos_hex_app p q)~0~0~1~0
| q~0~0~1~1 => (pos_hex_app p q)~0~0~1~1
| q~0~1~0~0 => (pos_hex_app p q)~0~1~0~0
| q~0~1~0~1 => (pos_hex_app p q)~0~1~0~1
| q~0~1~1~0 => (pos_hex_app p q)~0~1~1~0
| q~0~1~1~1 => (pos_hex_app p q)~0~1~1~1
| q~1~0~0~0 => (pos_hex_app p q)~1~0~0~0
| q~1~0~0~1 => (pos_hex_app p q)~1~0~0~1
| q~1~0~1~0 => (pos_hex_app p q)~1~0~1~0
| q~1~0~1~1 => (pos_hex_app p q)~1~0~1~1
| q~1~1~0~0 => (pos_hex_app p q)~1~1~0~0
| q~1~1~0~1 => (pos_hex_app p q)~1~1~0~1
| q~1~1~1~0 => (pos_hex_app p q)~1~1~1~0
| q~1~1~1~1 => (pos_hex_app p q)~1~1~1~1
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
| 8 => String "8" rest
| 9 => String "9" rest
| 10 => String "a" rest
| 11 => String "b" rest
| 12 => String "c" rest
| 13 => String "d" rest
| 14 => String "e" rest
| 15 => String "f" rest
| p'~0~0~0~0 => of_pos p' (String "0" rest)
| p'~0~0~0~1 => of_pos p' (String "1" rest)
| p'~0~0~1~0 => of_pos p' (String "2" rest)
| p'~0~0~1~1 => of_pos p' (String "3" rest)
| p'~0~1~0~0 => of_pos p' (String "4" rest)
| p'~0~1~0~1 => of_pos p' (String "5" rest)
| p'~0~1~1~0 => of_pos p' (String "6" rest)
| p'~0~1~1~1 => of_pos p' (String "7" rest)
| p'~1~0~0~0 => of_pos p' (String "8" rest)
| p'~1~0~0~1 => of_pos p' (String "9" rest)
| p'~1~0~1~0 => of_pos p' (String "a" rest)
| p'~1~0~1~1 => of_pos p' (String "b" rest)
| p'~1~1~0~0 => of_pos p' (String "c" rest)
| p'~1~1~0~1 => of_pos p' (String "d" rest)
| p'~1~1~1~0 => of_pos p' (String "e" rest)
| p'~1~1~1~1 => of_pos p' (String "f" 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 16 rest)
| None => N0
end
end.
Fixpoint to_N_of_pos (p : positive) (rest : string) (base : N) {struct p}
: to_N (of_pos p rest) base
= to_N rest match base with
| N0 => N.pos p
| Npos v => Npos (pos_hex_app v p)
end.
Proof.
do 4 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 "x" (Raw.of_pos p "")).
Definition of_N (n : N) : string
:= match n with
| N0 => "0x0"
| Npos p => of_pos p
end.
Definition of_Z (z : Z) : string
:= match z with
| Zneg p => String "-" (of_pos p)
| Z0 => "0x0"
| 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 "x"
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 = "0x1" := eq_refl.
Example of_pos_2 : of_pos 2 = "0x2" := eq_refl.
Example of_pos_3 : of_pos 3 = "0x3" := eq_refl.
Example of_pos_7 : of_pos 7 = "0x7" := eq_refl.
Example of_pos_8 : of_pos 8 = "0x8" := eq_refl.
Example of_pos_9 : of_pos 9 = "0x9" := eq_refl.
Example of_pos_10 : of_pos 10 = "0xa" := eq_refl.
Example of_pos_11 : of_pos 11 = "0xb" := eq_refl.
Example of_pos_12 : of_pos 12 = "0xc" := eq_refl.
Example of_pos_13 : of_pos 13 = "0xd" := eq_refl.
Example of_pos_14 : of_pos 14 = "0xe" := eq_refl.
Example of_pos_15 : of_pos 15 = "0xf" := eq_refl.
Example of_pos_16 : of_pos 16 = "0x10" := eq_refl.
Example of_N_0 : of_N 0 = "0x0" := eq_refl.
Example of_Z_0 : of_Z 0 = "0x0" := eq_refl.
Example of_Z_m1 : of_Z (-1) = "-0x1" := eq_refl.
Example of_nat_0 : of_nat 0 = "0x0" := eq_refl.
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