# Jolverine Turing-completeness proof

Jolverine can be shown Turing complete by a reduction from ternary Reversible Brainfuck. We put the construction on a separate page because of the size of one of its parts.

A ternary Reversible Brainfuck program is converted to Jolverine as a series of diagonal elements concatenated with each other. The first element is the initialization; then follows the elements for the parts of the RB program in order. Loops are considerably more complicated to represent than any of the other RB instructions, and contain their inner instructions embedded within them.

Note that I/O is not needed for theoretical Turing-completeness, and may not have completely matching semantics between all esolangs involved. For best results only use input on a cell containing a zero value.

The following properties are kept for all elements other than initialization (whose job is to set them up in the first place):

• Each element takes a multiple of 7 ticks to execute. Apart from loop elements, this means each of them is a multiple of 7 long.
• Each element starts with the instruction wheel in the state
left <--
input
output
rot
right
with the top instruction (left) the current one, and with the next executed instruction to be re-inserted at the top.
• Instructions which may turn the IP will only be executed as the sixth instruction on the wheel. This ensures that the wheel sixth instructions are always at 7x7 grid positions in the larger program, thus ensuring there can be no position mismatch not fixable simply by adding a multiple of 7 non-* symbols.
• At the start of an element, the current cell of the Jolverine tape corresponds to the current RB cell. RB cells are spaced two cells apart on the Jolverine tape, with the intermediate cell being zero.

Since editing this stuff has been rather awkward, the reduction has only been tested on one program:

[>,.]

## Reduction tables

The following table lists the elements other than loops.

initialization
==============
..*.*
.        +      -      >      <      .      ,
.       =      =      =      =      =      =
*      .      .      *      *      .      .
.      .      .      .      .      .      .
.      .      .      .      .      .      *
.      .      .      .      .      .      .
.      .      .      .      .      .      .
.      *      *      .      .      *      .
.      .      .      *      .      .      *
*      *      *      .      *      *      .
.      .      .      .      .      .      .
.      .      .      .      .      .      *
.      .      .      .      .      .      .
.      .      .      .      .      .      .
.      .      .      .      .      .      .
.      *      .      *      *      *      .
*      *      *      *      .      .      .
*      .      .      .      .      .      .
.      .      .      .      .      .      *
.      .      .      .      .      .      .
.      .      .      .      .      *      .
.      .      *      .      .      *      .
.      .      .      .      *      .      *
*                           .      .
.                           .      .
.                           .      .
.                           .      .
.                           .      .
*                           *      .
.                           .      *
.
.
*
.
.
.
.
.
.
*
.
.
.
*
*
.
.
.
.
.
.
*
*
.
.
.
.
.

The following shows the general structure of the large loop elements. The instructions for the loop contents are embedded at the spot marked "INSIDE LOOP". The surrounding paths must be stretched as necessary by inserting multiples of 7 non-* elements in order to fit the inside content. (The version below has space enough for the loop [>,.] as that was what it was tested with.)

*                                                                                                 *
. .                                                                                               . .
.   *                                                                                             .   *
.     .                                                                                           .     .
.       .                                                                                         .       .
.         .                                                                                       .         .
*           .                                                                                     *           .
*             .                                                                                   *             .
.               *                                                                                 .               *
.                 .                                                                               .                 .
.                   .                                                                             .                   .
.                     .                                                                           .                     .
*                       .                                                                         *                       .
.                         .                                                                       .                         .
.                           *                                                                     .                           *
*                             .                                                                   *                             .
B                                                                 .                               .                                                                 .                               .
E                                                               .                                 *                                                               .                                 *
F                                                             .                                   .                                                             .                                   .
O                                                           *                                     .                                                           *                                     .
R                                                         .                                       .                                                         .                                       .
E                                                       .                                         .                                                       .                                         .
.                                           *                                                     .                                           *
L                                                   .                                             *                                                   .                                             *
O                                                 .                                               *                                                 .                                               *
O                                               .                                                 .                                               .                                                 .
P                                             *                                                   .                                             *                                                   .
,                                           *                                                     .                                           *                                                     .
,                                         .                                                       .                                         .                                                       .
,                                       .                                                         .                                       .                                                         .
.                                     .                                                           .                                     .                                                           .
.                                   .                                                             *                                   .                                                             *
.                                 .                                                               .                                 .                                                               .
.                               *                                                                 .                               *                                                                 .
.                             *                                                                   .                             *                                                                   .
*                           .                                                                     *                           .                                                                     .
.                         .                                                                       .                         .                                                                       .
*                       .                                                                         *                       .                                                                         ,
.                     .                                                                           .                     .                                                                           ,
.                   .                                                                             .                   .                                                                             ,
.                 *                                                                               .                 *                                                                               ,
.               .                                                                                 .               .                                                                                 ,
.             .                                                                                   .             .                                                                                   ,
*           .                                                                                     *           .                                                                                     ,
.         .                                                                                       .         .                                                                                       ,
.       .                                                                                         .       .                                                                                         ,
.     .                                                                                           .     .                                                                                           ,
.   *                                                                                             .   *                                                                                             ,
. .                                                                                               . .                                                                                               ,
*.*......*.....**.....**......*...*..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....*.*.....*.....*                                                                                                 ,
.                                                                                               .                                                                                                   ,
*                                                                                             .                                                                                                     ,
.                                                                                           .                                                                                                       ,
.                                                                                         .                                                                                                         ,
.                                                                                       .                                                                                                           ,
.                                                                                     *                                                                                                             ,
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.                                                                                 .                                                                                                                 ,
*                                                                               .                                                                                                                   ,
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*                                                                                             .                                                                                                                                                                                                   ,
.                                                                                               .                                                                                                                                                                                                   ,
*.*......*.....**.....**......*...*..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....*.*.....*.....*                                                                                                                                                                                                   ,
. .                                                                                               . .                                                                                                                                                                                                   ,
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.                 *                                                                               .                 *                                                                                                                                                                                                   ,
.                   .                                                                             .                   .                                                                                                                                                                                                   ,
.                     .                                                                           .                     .                                                                                                                                                                                                   ,
*                       .                                                                         *                       .                                                                                                                                                                                                   ,
.                         .                                                                       .                         .                                                                                                                                                                                                   ,
*                           .                                                                     *                           .                                                                                                                                                                                                   ,
.                             *                                                                   .                             *                                                                                                                                                                                                   ,
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*                                 .                                                               .                                 .                                                                                                                                                                                                   ,
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.                                       .                                                         .                                       .                                                                                                                                                                                                   ,
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.                                           *                                                     .                                           *                                                                                                                                                                                                   ,
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.                                                           *                                     .                                                           *                                                                                                                                                                                                   ,
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.                                                               .                                 *                                                               .                                                                                                                                                                                                   ,
.                                                                 .                               .                                                                 .                                                                                                                                                                                                   ,
*                                                                   *                             .                                                                   *                                                                                                                                                                                                   ,
.                                                                     .                           .                                                                     .                                                                                                                                                                                                   ,
.                                                                       *                         .                                                                       .                                                                                                                                                                                                   ,
.                                                                         .                       .                                                                         I                                                                                                                                                                                                   ,
*                                                                           .                     *                                                                           N                                                                                                                                                                                                   ,
.                                                                             .                   *                                                                             S                                                                                                                                                                                                   ,
.                                                                               .                 .                                                                               I                                                                                                                                                                                                   ,
*                                                                                 .               *                                                                                 D                                                                                                                                                                                                   ,
*                                                                                   .             .                                                                                   E                                                                                                                                                                                                   ,
.                                                                                   *           .                                                                                                                                                                                                                                                                                         ,
.                                                                                   .         .                                                                                       L                                                                                                                                                                                                   ,
.                                                                                   .       .                                                                                         O                                                                                                                                                                                                   ,
.                                                                                   .     .                                                                                           O                                                                                                                                                                                                   ,
*                                                                                   .   .                                                                                             P                                                                                                                                                                                                   ,
.                                                                                   . *                                                                                               ,                                                                                                                                                                                                   ,
*                                                                                   *                                                                                                 ,                                                                                                                                                                                                   ,
.                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
.                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
*                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
.                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
.                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
*                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
.                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
.                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
.                                                                                                                                                                                     ,                                                                                                                                                                                                   *
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*.....**....*,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....*......***....*..*.....*.....*.*                                                                                                 ,
,