From 83dcdd358e61654ecdcc2cbe1a256adbc66027b3 Mon Sep 17 00:00:00 2001 From: miwer106 <michellewerth@hotmail.com> Date: Thu, 26 Mar 2020 12:09:27 +0100 Subject: [PATCH] add rush hour puzzle, sudoku puzzle and first try for jobs puzzle, they are not completed MC missing --- ...=> 00_ProB_Jupyter_Notebook_Overview.ipynb | 0 Rush_Hour_Puzzle.ipynb | 316 ++++++++++++++ Sudoku_Solved.ipynb | 391 ++++++++++++++++++ Untitled.ipynb | 347 ++++++++++++++++ 4 files changed, 1054 insertions(+) rename ProB_Jupyter_Notebook_Overview.ipynb => 00_ProB_Jupyter_Notebook_Overview.ipynb (100%) create mode 100644 Rush_Hour_Puzzle.ipynb create mode 100644 Sudoku_Solved.ipynb create mode 100644 Untitled.ipynb diff --git a/ProB_Jupyter_Notebook_Overview.ipynb b/00_ProB_Jupyter_Notebook_Overview.ipynb similarity index 100% rename from ProB_Jupyter_Notebook_Overview.ipynb rename to 00_ProB_Jupyter_Notebook_Overview.ipynb diff --git a/Rush_Hour_Puzzle.ipynb b/Rush_Hour_Puzzle.ipynb new file mode 100644 index 0000000..f3a4d3f --- /dev/null +++ b/Rush_Hour_Puzzle.ipynb @@ -0,0 +1,316 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Rush Hour Puzzle\n", + "\n", + "The [rush hour puzzle](https://en.wikipedia.org/wiki/Rush_Hour_(puzzle)) was invented in 1970 by Nob Yoshigahara. Even though the puzzle is one of the older puzzles, it is very interesting to model. \n", + "In this notebook we are going to look at one encoding for the rush hour board game in which cars are packed on a 6-by-6 grid and can either move horizontally or vertically. The goal is to move the red car to the exit.\n", + "In this particular instance we try to solve the [hardest puzzle of the original game Nr. 40](www.puzzles.com/products/RushHour/RHfromMarkRiedel/Jam.html?40).\n", + "\n", + "Inspired by discussions with Neng-Fa Zhou at ICLP'14 in Vienna, we now have a new version of the B model for this puzzle.\n", + "\n", + "The old version can still be found in the our [modelling examples](https://www3.hhu.de/stups/handbook/prob2/modelling_examples.html#rush-hour-puzzle)\n", + "\n", + "Let us take a look at the B model of the Rush Hour Puzzle. " + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Loaded machine: RushHour" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "::load\n", + "MACHINE RushHour\n", + "/* a more elegant encoding of the Rush Hour puzzle */\n", + "/* Michael Leuschel, July 2014 */\n", + "/* ProB finds solution in about 10.5 secs (turning invariant checking off) */\n", + "/* This version has been slightly adapted for TLC by adding the c:1..red guards:\n", + " it finds a solution in 3 seconds\n", + " (most of this time is spent replaying the counter ; the model checking seems less than a\n", + " second) */\n", + "SETS DIR = {h,v}\n", + "CONSTANTS sze, dir, red, dim, free_initial\n", + "PROPERTIES\n", + " sze = [2,2,2,2,2, 2,2,2,2, 3,3,3, 2] & /* the sizes of the cars */\n", + " dir = [v,v,v,v,v, h,h,h,h, v,v,h, h] & /* indicating whether the cars move vertically or horizontally */\n", + " red = size(sze) & /* the last car is the red one */\n", + " dim = 5 & /* the grid goes from 0..dim */\n", + " free_initial = {(0,3),(1,3), (0,5), (3,4),(4,0),(4,1),(5,5)}\n", + "DEFINITIONS\n", + " GOAL == (col(red) = 4); /* The target : move red car to the right */\n", + " ANIMATION_STR_JUSTIFY_RIGHT == TRUE;\n", + " ANIMATION_FUNCTION_DEFAULT == (0..dim)*(0..dim)*{-1};\n", + " ANIMATION_FUNCTION ==\n", + " {r,c,i| i:1..red & dir(i)=h & row(i)=r & c:col(i)..col(i)+sze(i)-1} \\/\n", + " {r,c,i| i:1..red & dir(i)=v & col(i)=c & r:row(i)..row(i)+sze(i)-1} \\/\n", + " free * {0}\n", + "VARIABLES free, row, col\n", + "INVARIANT\n", + " free <: (0..dim)*(0..dim) & /* the currently free blocks */\n", + " card(free) = card(free_initial) &\n", + " row : 1..red --> 0..dim & /* the row of each car */\n", + " col : 1..red --> 0..dim /* the column for each car */\n", + "INITIALISATION\n", + " free := free_initial\n", + " ||\n", + " col := [(1),(2),(2),(3),(4), /* vertical 2-size cars */\n", + " (0),(1),(3),(4), /* horiz. 2-size cars */\n", + " (0),(5), /* vertical 3-size cars */\n", + " (0), /* horiz. 3-size cars */\n", + " (3)] /* red car */\n", + " ||\n", + " row := [(1),(1),(4),(3),(0),\n", + " (5),(0),(5),(4),\n", + " (0),(1),\n", + " (3),\n", + " (2)] /* red car */\n", + "OPERATIONS\n", + " mv_down(c,F) = PRE c:1..red & c |-> v : dir & F = row(c)+sze(c)|->col(c) &\n", + " F : free THEN\n", + " free := free - {F} \\/ {row(c)|->col(c)} ||\n", + " row(c) := row(c)+1\n", + " END;\n", + " mv_up(c,F) = PRE c:1..red & c |-> v : dir & F = row(c)-1|->col(c) &\n", + " F : free THEN\n", + " free := free - {F} \\/ {row(c)+sze(c)-1|->col(c)} ||\n", + " row(c) := row(c)-1\n", + " END;\n", + " mv_right(c,F) = PRE c:1..red & c |-> h : dir & F = row(c)|->col(c)+sze(c) &\n", + " F : free THEN\n", + " free := free - {F} \\/ {row(c)|->col(c)} ||\n", + " col(c) := col(c)+1\n", + " END;\n", + " mv_left(c,F) = PRE c:1..red & c |-> h : dir & F = row(c)|->col(c)-1 &\n", + " F : free THEN\n", + " free := free - {F} \\/ {row(c)|->col(c)+sze(c)-1} ||\n", + " col(c) := col(c)-1\n", + " END\n", + "END" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Finding a Solution\n", + "\n", + "Since jupyter notebook does not " + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Machine constants set up using operation 0: $setup_constants()" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":constants" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Machine initialised using operation 1: $initialise_machine()" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":init" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "Enter a B expression or predicate to evaluate it. To load a B machine, enter its source code directly, or use `:load` to load an external machine file.\n", + "\n", + "You can also use any of the following commands. For more help on a particular command, run `:help commandname`.\n", + "\n", + "## Evaluation\n", + "\n", + "* `:eval` - Evaluate a formula and display the result.\n", + "* `:solve` - Solve a predicate with the specified solver.\n", + "* `:table` - Display an expression as a table.\n", + "* `:type` - Display the static type of a formula.\n", + "* `:prettyprint` - Pretty-print a predicate.\n", + "* `:let` - Evaluate an expression and store it in a local variable.\n", + "* `:unlet` - Remove a local variable.\n", + "* `:assert` - Ensure that the predicate is true, and show an error otherwise.\n", + "\n", + "## Animation\n", + "\n", + "* `::load` - Load a B machine from the given source code.\n", + "* `:load` - Load a machine from a file.\n", + "* `:constants` - Set up the current machine's constants.\n", + "* `:init` - Initialise the current machine with the specified predicate\n", + "* `:exec` - Execute an operation.\n", + "* `:browse` - Show information about the current state.\n", + "* `:trace` - Display all states and executed operations in the current trace.\n", + "* `:goto` - Go to the state with the specified index in the current trace.\n", + "* `:find` - Try to find a state for which the given predicate is true (in addition to the machine's invariant).\n", + "\n", + "## Visualisation\n", + "\n", + "* `:show` - Show the machine's animation function visualisation for the current state.\n", + "* `:dot` - Execute and show a dot visualisation.\n", + "\n", + "## Verification\n", + "\n", + "* `:check` - Check the machine's properties, invariant, or assertions in the current state.\n", + "* `:modelcheck` - Run the ProB model checker on the current model.\n", + "\n", + "## Other\n", + "\n", + "* `::render` - Render some content with the specified MIME type.\n", + "* `:bsymb` - Load all bsymb.sty command definitions, so that they can be used in $\\LaTeX$ formulas in Markdown cells.\n", + "* `:groovy` - Evaluate the given Groovy expression.\n", + "* `:help` - Display help for a specific command, or general help about the REPL.\n", + "* `:pref` - View or change the value of one or more preferences.\n", + "* `:stats` - Show statistics about the state space.\n", + "* `:time` - Execute the given command and measure how long it takes to execute.\n", + "* `:version` - Display version info about the ProB 2 Jupyter kernel, ProB 2, and the underlying ProB CLI.\n" + ], + "text/plain": [ + "Enter a B expression or predicate to evaluate it. To load a B machine, enter its source code directly, or use :load to load an external machine file.\n", + "You can also use any of the following commands. For more help on a particular command, run :help commandname.\n", + "\n", + "Evaluation:\n", + ":eval - Evaluate a formula and display the result.\n", + ":solve - Solve a predicate with the specified solver.\n", + ":table - Display an expression as a table.\n", + ":type - Display the static type of a formula.\n", + ":prettyprint - Pretty-print a predicate.\n", + ":let - Evaluate an expression and store it in a local variable.\n", + ":unlet - Remove a local variable.\n", + ":assert - Ensure that the predicate is true, and show an error otherwise.\n", + "\n", + "Animation:\n", + "::load - Load a B machine from the given source code.\n", + ":load - Load a machine from a file.\n", + ":constants - Set up the current machine's constants.\n", + ":init - Initialise the current machine with the specified predicate\n", + ":exec - Execute an operation.\n", + ":browse - Show information about the current state.\n", + ":trace - Display all states and executed operations in the current trace.\n", + ":goto - Go to the state with the specified index in the current trace.\n", + ":find - Try to find a state for which the given predicate is true (in addition to the machine's invariant).\n", + "\n", + "Visualisation:\n", + ":show - Show the machine's animation function visualisation for the current state.\n", + ":dot - Execute and show a dot visualisation.\n", + "\n", + "Verification:\n", + ":check - Check the machine's properties, invariant, or assertions in the current state.\n", + ":modelcheck - Run the ProB model checker on the current model.\n", + "\n", + "Other:\n", + "::render - Render some content with the specified MIME type.\n", + ":bsymb - Load all bsymb.sty command definitions, so that they can be used in $\\LaTeX$ formulas in Markdown cells.\n", + ":groovy - Evaluate the given Groovy expression.\n", + ":help - Display help for a specific command, or general help about the REPL.\n", + ":pref - View or change the value of one or more preferences.\n", + ":stats - Show statistics about the state space.\n", + ":time - Execute the given command and measure how long it takes to execute.\n", + ":version - Display version info about the ProB 2 Jupyter kernel, ProB 2, and the underlying ProB CLI.\n" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":help" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "14.355 sec, 4782 of 4782 states processed, 29890 transitions" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "Model Checking complete. No error nodes found." + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":modelcheck" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "ProB 2", + "language": "prob", + "name": "prob2" + }, + "language_info": { + "codemirror_mode": "prob2_jupyter_repl", + "file_extension": ".prob", + "mimetype": "text/x-prob2-jupyter-repl", + "name": "prob" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Sudoku_Solved.ipynb b/Sudoku_Solved.ipynb new file mode 100644 index 0000000..85bd3f1 --- /dev/null +++ b/Sudoku_Solved.ipynb @@ -0,0 +1,391 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Sudoku Solving in Jupyter Notebook\n", + "\n", + "Another very interesting [modelling example](https://www3.hhu.de/stups/handbook/prob2/modelling_examples.html#sudoku-solved-in-the-prob-repl) is the Sudoku Solved in the ProB REPL. Since jupyter notebook can do just as much, I wanted to replicate that example for jupyter notebook. \n", + "\n", + "For this notebook you will need to understand how to interact with a machine, in this case the default empty machine, which is loaded at the beginning of each notebook.\n", + "\n", + "I recommend you check out the ProB Jupyter Notebook Overview before you start with this session. Let us begin by defining the domain `DOM` for the numbers to be put inside the Sudoku:" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{1,2,3,4,5,6,7,8,9\\}$" + ], + "text/plain": [ + "{1,2,3,4,5,6,7,8,9}" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let DOM 1..9" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And the group of indices `SUBSQ` which can be used to costruct 3x3 sub-squares:" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{\\{1,2,3\\},\\{4,5,6\\},\\{7,8,9\\}\\}$" + ], + "text/plain": [ + "{{1,2,3},{4,5,6},{7,8,9}}" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let SUBSQ {{1,2,3},{4,5,6},{7,8,9}}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we need to encode the sets of pairs in `Diff1` and `Diff2` of coordinates at which the values in a sudoku have to be different. (This is the case if they lie on the same column or row respectively.) " + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{(1\\mapsto 2\\mapsto 1\\mapsto 1),(1\\mapsto 2\\mapsto 2\\mapsto 2),(1\\mapsto 2\\mapsto 3\\mapsto 3),(1\\mapsto 2\\mapsto 4\\mapsto 4),(1\\mapsto 2\\mapsto 5\\mapsto 5),(1\\mapsto 2\\mapsto 6\\mapsto 6),(1\\mapsto 2\\mapsto 7\\mapsto 7),(1\\mapsto 2\\mapsto 8\\mapsto 8),(1\\mapsto 2\\mapsto 9\\mapsto 9),(1\\mapsto 3\\mapsto 1\\mapsto 1),(1\\mapsto 3\\mapsto 2\\mapsto 2),(1\\mapsto 3\\mapsto 3\\mapsto 3),(1\\mapsto 3\\mapsto 4\\mapsto 4),(1\\mapsto 3\\mapsto 5\\mapsto 5),(1\\mapsto 3\\mapsto 6\\mapsto 6),(1\\mapsto 3\\mapsto 7\\mapsto 7),(1\\mapsto 3\\mapsto 8\\mapsto 8),(1\\mapsto 3\\mapsto 9\\mapsto 9),(1\\mapsto 4\\mapsto 1\\mapsto 1),(1\\mapsto 4\\mapsto 2\\mapsto 2),(1\\mapsto 4\\mapsto 3\\mapsto 3),(1\\mapsto 4\\mapsto 4\\mapsto 4),(1\\mapsto 4\\mapsto 5\\mapsto 5),(1\\mapsto 4\\mapsto 6\\mapsto 6),(1\\mapsto 4\\mapsto 7\\mapsto 7),(1\\mapsto 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+ ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let Diff1 {x1,x2,y1,y2| x1:DOM & x2:DOM & y1:DOM & y2:DOM & x1<x2 & y1=y2}" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{(1\\mapsto 1\\mapsto 1\\mapsto 2),(1\\mapsto 1\\mapsto 1\\mapsto 3),(1\\mapsto 1\\mapsto 1\\mapsto 4),(1\\mapsto 1\\mapsto 1\\mapsto 5),(1\\mapsto 1\\mapsto 1\\mapsto 6),(1\\mapsto 1\\mapsto 1\\mapsto 7),(1\\mapsto 1\\mapsto 1\\mapsto 8),(1\\mapsto 1\\mapsto 1\\mapsto 9),(1\\mapsto 1\\mapsto 2\\mapsto 3),(1\\mapsto 1\\mapsto 2\\mapsto 4),(1\\mapsto 1\\mapsto 2\\mapsto 5),(1\\mapsto 1\\mapsto 2\\mapsto 6),(1\\mapsto 1\\mapsto 2\\mapsto 7),(1\\mapsto 1\\mapsto 2\\mapsto 8),(1\\mapsto 1\\mapsto 2\\mapsto 9),(1\\mapsto 1\\mapsto 3\\mapsto 4),(1\\mapsto 1\\mapsto 3\\mapsto 5),(1\\mapsto 1\\mapsto 3\\mapsto 6),(1\\mapsto 1\\mapsto 3\\mapsto 7),(1\\mapsto 1\\mapsto 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+ ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let Diff2 {x1,x2,y1,y2| x1:DOM & x2:DOM & y1:DOM & y2:DOM & x1=x2 & y1<y2}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We have not yet encoded that in each sub-square the values must also all be distinct. Nonetheless, let us try and solve the puzzle as it stands, by looking for a full board (of type `DOM → (DOM → DOM)`) which has distinct values on each row and column:" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\mathit{TRUE}$\n", + "\n", + "**Solution:**\n", + "* $\\mathit{Diff2} = \\{(1\\mapsto 1\\mapsto 1\\mapsto 2),(1\\mapsto 1\\mapsto 1\\mapsto 3),(1\\mapsto 1\\mapsto 1\\mapsto 4),(1\\mapsto 1\\mapsto 1\\mapsto 5),(1\\mapsto 1\\mapsto 1\\mapsto 6),(1\\mapsto 1\\mapsto 1\\mapsto 7),(1\\mapsto 1\\mapsto 1\\mapsto 8),(1\\mapsto 1\\mapsto 1\\mapsto 9),(1\\mapsto 1\\mapsto 2\\mapsto 3),(1\\mapsto 1\\mapsto 2\\mapsto 4),(1\\mapsto 1\\mapsto 2\\mapsto 5),(1\\mapsto 1\\mapsto 2\\mapsto 6),(1\\mapsto 1\\mapsto 2\\mapsto 7),(1\\mapsto 1\\mapsto 2\\mapsto 8),(1\\mapsto 1\\mapsto 2\\mapsto 9),(1\\mapsto 1\\mapsto 3\\mapsto 4),(1\\mapsto 1\\mapsto 3\\mapsto 5),(1\\mapsto 1\\mapsto 3\\mapsto 6),(1\\mapsto 1\\mapsto 3\\mapsto 7),(1\\mapsto 1\\mapsto 3\\mapsto 8),(1\\mapsto 1\\mapsto 3\\mapsto 9),(1\\mapsto 1\\mapsto 4\\mapsto 5),(1\\mapsto 1\\mapsto 4\\mapsto 6),(1\\mapsto 1\\mapsto 4\\mapsto 7),(1\\mapsto 1\\mapsto 4\\mapsto 8),(1\\mapsto 1\\mapsto 4\\mapsto 9),(1\\mapsto 1\\mapsto 5\\mapsto 6),(1\\mapsto 1\\mapsto 5\\mapsto 7),(1\\mapsto 1\\mapsto 5\\mapsto 8),(1\\mapsto 1\\mapsto 5\\mapsto 9),(1\\mapsto 1\\mapsto 6\\mapsto 7),(1\\mapsto 1\\mapsto 6\\mapsto 8),(1\\mapsto 1\\mapsto 6\\mapsto 9),(1\\mapsto 1\\mapsto 7\\mapsto 8),(1\\mapsto 1\\mapsto 7\\mapsto 9),(1\\mapsto 1\\mapsto 8\\mapsto 9),(2\\mapsto 2\\mapsto 1\\mapsto 2),(2\\mapsto 2\\mapsto 1\\mapsto 3),(2\\mapsto 2\\mapsto 1\\mapsto 4),(2\\mapsto 2\\mapsto 1\\mapsto 5),(2\\mapsto 2\\mapsto 1\\mapsto 6),(2\\mapsto 2\\mapsto 1\\mapsto 7),(2\\mapsto 2\\mapsto 1\\mapsto 8),(2\\mapsto 2\\mapsto 1\\mapsto 9),(2\\mapsto 2\\mapsto 2\\mapsto 3),(2\\mapsto 2\\mapsto 2\\mapsto 4),(2\\mapsto 2\\mapsto 2\\mapsto 5),(2\\mapsto 2\\mapsto 2\\mapsto 6),(2\\mapsto 2\\mapsto 2\\mapsto 7),(2\\mapsto 2\\mapsto 2\\mapsto 8),(2\\mapsto 2\\mapsto 2\\mapsto 9),(2\\mapsto 2\\mapsto 3\\mapsto 4),(2\\mapsto 2\\mapsto 3\\mapsto 5),(2\\mapsto 2\\mapsto 3\\mapsto 6),(2\\mapsto 2\\mapsto 3\\mapsto 7),(2\\mapsto 2\\mapsto 3\\mapsto 8),(2\\mapsto 2\\mapsto 3\\mapsto 9),(2\\mapsto 2\\mapsto 4\\mapsto 5),(2\\mapsto 2\\mapsto 4\\mapsto 6),(2\\mapsto 2\\mapsto 4\\mapsto 7),(2\\mapsto 2\\mapsto 4\\mapsto 8),(2\\mapsto 2\\mapsto 4\\mapsto 9),(2\\mapsto 2\\mapsto 5\\mapsto 6),(2\\mapsto 2\\mapsto 5\\mapsto 7),(2\\mapsto 2\\mapsto 5\\mapsto 8),(2\\mapsto 2\\mapsto 5\\mapsto 9),(2\\mapsto 2\\mapsto 6\\mapsto 7),(2\\mapsto 2\\mapsto 6\\mapsto 8),(2\\mapsto 2\\mapsto 6\\mapsto 9),(2\\mapsto 2\\mapsto 7\\mapsto 8),(2\\mapsto 2\\mapsto 7\\mapsto 9),(2\\mapsto 2\\mapsto 8\\mapsto 9),(3\\mapsto 3\\mapsto 1\\mapsto 2),(3\\mapsto 3\\mapsto 1\\mapsto 3),(3\\mapsto 3\\mapsto 1\\mapsto 4),(3\\mapsto 3\\mapsto 1\\mapsto 5),(3\\mapsto 3\\mapsto 1\\mapsto 6),(3\\mapsto 3\\mapsto 1\\mapsto 7),(3\\mapsto 3\\mapsto 1\\mapsto 8),(3\\mapsto 3\\mapsto 1\\mapsto 9),(3\\mapsto 3\\mapsto 2\\mapsto 3),(3\\mapsto 3\\mapsto 2\\mapsto 4),(3\\mapsto 3\\mapsto 2\\mapsto 5),(3\\mapsto 3\\mapsto 2\\mapsto 6),(3\\mapsto 3\\mapsto 2\\mapsto 7),(3\\mapsto 3\\mapsto 2\\mapsto 8),(3\\mapsto 3\\mapsto 2\\mapsto 9),(3\\mapsto 3\\mapsto 3\\mapsto 4),(3\\mapsto 3\\mapsto 3\\mapsto 5),(3\\mapsto 3\\mapsto 3\\mapsto 6),(3\\mapsto 3\\mapsto 3\\mapsto 7),(3\\mapsto 3\\mapsto 3\\mapsto 8),(3\\mapsto 3\\mapsto 3\\mapsto 9),(3\\mapsto 3\\mapsto 4\\mapsto 5),(3\\mapsto 3\\mapsto 4\\mapsto 6),(3\\mapsto 3\\mapsto 4\\mapsto 7),(3\\mapsto 3\\mapsto 4\\mapsto 8),(3\\mapsto 3\\mapsto 4\\mapsto 9),(3\\mapsto 3\\mapsto 5\\mapsto 6),(3\\mapsto 3\\mapsto 5\\mapsto 7),(3\\mapsto 3\\mapsto 5\\mapsto 8),(3\\mapsto 3\\mapsto 5\\mapsto 9),(3\\mapsto 3\\mapsto 6\\mapsto 7),(3\\mapsto 3\\mapsto 6\\mapsto 8),(3\\mapsto 3\\mapsto 6\\mapsto 9),(3\\mapsto 3\\mapsto 7\\mapsto 8),(3\\mapsto 3\\mapsto 7\\mapsto 9),(3\\mapsto 3\\mapsto 8\\mapsto 9),(4\\mapsto 4\\mapsto 1\\mapsto 2),(4\\mapsto 4\\mapsto 1\\mapsto 3),(4\\mapsto 4\\mapsto 1\\mapsto 4),(4\\mapsto 4\\mapsto 1\\mapsto 5),(4\\mapsto 4\\mapsto 1\\mapsto 6),(4\\mapsto 4\\mapsto 1\\mapsto 7),(4\\mapsto 4\\mapsto 1\\mapsto 8),(4\\mapsto 4\\mapsto 1\\mapsto 9),(4\\mapsto 4\\mapsto 2\\mapsto 3),(4\\mapsto 4\\mapsto 2\\mapsto 4),(4\\mapsto 4\\mapsto 2\\mapsto 5),(4\\mapsto 4\\mapsto 2\\mapsto 6),(4\\mapsto 4\\mapsto 2\\mapsto 7),(4\\mapsto 4\\mapsto 2\\mapsto 8),(4\\mapsto 4\\mapsto 2\\mapsto 9),(4\\mapsto 4\\mapsto 3\\mapsto 4),(4\\mapsto 4\\mapsto 3\\mapsto 5),(4\\mapsto 4\\mapsto 3\\mapsto 6),(4\\mapsto 4\\mapsto 3\\mapsto 7),(4\\mapsto 4\\mapsto 3\\mapsto 8),(4\\mapsto 4\\mapsto 3\\mapsto 9),(4\\mapsto 4\\mapsto 4\\mapsto 5),(4\\mapsto 4\\mapsto 4\\mapsto 6),(4\\mapsto 4\\mapsto 4\\mapsto 7),(4\\mapsto 4\\mapsto 4\\mapsto 8),(4\\mapsto 4\\mapsto 4\\mapsto 9),(4\\mapsto 4\\mapsto 5\\mapsto 6),(4\\mapsto 4\\mapsto 5\\mapsto 7),(4\\mapsto 4\\mapsto 5\\mapsto 8),(4\\mapsto 4\\mapsto 5\\mapsto 9),(4\\mapsto 4\\mapsto 6\\mapsto 7),(4\\mapsto 4\\mapsto 6\\mapsto 8),(4\\mapsto 4\\mapsto 6\\mapsto 9),(4\\mapsto 4\\mapsto 7\\mapsto 8),(4\\mapsto 4\\mapsto 7\\mapsto 9),(4\\mapsto 4\\mapsto 8\\mapsto 9),(5\\mapsto 5\\mapsto 1\\mapsto 2),(5\\mapsto 5\\mapsto 1\\mapsto 3),(5\\mapsto 5\\mapsto 1\\mapsto 4),(5\\mapsto 5\\mapsto 1\\mapsto 5),(5\\mapsto 5\\mapsto 1\\mapsto 6),(5\\mapsto 5\\mapsto 1\\mapsto 7),(5\\mapsto 5\\mapsto 1\\mapsto 8),(5\\mapsto 5\\mapsto 1\\mapsto 9),(5\\mapsto 5\\mapsto 2\\mapsto 3),(5\\mapsto 5\\mapsto 2\\mapsto 4),(5\\mapsto 5\\mapsto 2\\mapsto 5),(5\\mapsto 5\\mapsto 2\\mapsto 6),(5\\mapsto 5\\mapsto 2\\mapsto 7),(5\\mapsto 5\\mapsto 2\\mapsto 8),(5\\mapsto 5\\mapsto 2\\mapsto 9),(5\\mapsto 5\\mapsto 3\\mapsto 4),(5\\mapsto 5\\mapsto 3\\mapsto 5),(5\\mapsto 5\\mapsto 3\\mapsto 6),(5\\mapsto 5\\mapsto 3\\mapsto 7),(5\\mapsto 5\\mapsto 3\\mapsto 8),(5\\mapsto 5\\mapsto 3\\mapsto 9),(5\\mapsto 5\\mapsto 4\\mapsto 5),(5\\mapsto 5\\mapsto 4\\mapsto 6),(5\\mapsto 5\\mapsto 4\\mapsto 7),(5\\mapsto 5\\mapsto 4\\mapsto 8),(5\\mapsto 5\\mapsto 4\\mapsto 9),(5\\mapsto 5\\mapsto 5\\mapsto 6),(5\\mapsto 5\\mapsto 5\\mapsto 7),(5\\mapsto 5\\mapsto 5\\mapsto 8),(5\\mapsto 5\\mapsto 5\\mapsto 9),(5\\mapsto 5\\mapsto 6\\mapsto 7),(5\\mapsto 5\\mapsto 6\\mapsto 8),(5\\mapsto 5\\mapsto 6\\mapsto 9),(5\\mapsto 5\\mapsto 7\\mapsto 8),(5\\mapsto 5\\mapsto 7\\mapsto 9),(5\\mapsto 5\\mapsto 8\\mapsto 9),(6\\mapsto 6\\mapsto 1\\mapsto 2),(6\\mapsto 6\\mapsto 1\\mapsto 3),(6\\mapsto 6\\mapsto 1\\mapsto 4),(6\\mapsto 6\\mapsto 1\\mapsto 5),(6\\mapsto 6\\mapsto 1\\mapsto 6),(6\\mapsto 6\\mapsto 1\\mapsto 7),(6\\mapsto 6\\mapsto 1\\mapsto 8),(6\\mapsto 6\\mapsto 1\\mapsto 9),(6\\mapsto 6\\mapsto 2\\mapsto 3),(6\\mapsto 6\\mapsto 2\\mapsto 4),(6\\mapsto 6\\mapsto 2\\mapsto 5),(6\\mapsto 6\\mapsto 2\\mapsto 6),(6\\mapsto 6\\mapsto 2\\mapsto 7),(6\\mapsto 6\\mapsto 2\\mapsto 8),(6\\mapsto 6\\mapsto 2\\mapsto 9),(6\\mapsto 6\\mapsto 3\\mapsto 4),(6\\mapsto 6\\mapsto 3\\mapsto 5),(6\\mapsto 6\\mapsto 3\\mapsto 6),(6\\mapsto 6\\mapsto 3\\mapsto 7),(6\\mapsto 6\\mapsto 3\\mapsto 8),(6\\mapsto 6\\mapsto 3\\mapsto 9),(6\\mapsto 6\\mapsto 4\\mapsto 5),(6\\mapsto 6\\mapsto 4\\mapsto 6),(6\\mapsto 6\\mapsto 4\\mapsto 7),(6\\mapsto 6\\mapsto 4\\mapsto 8),(6\\mapsto 6\\mapsto 4\\mapsto 9),(6\\mapsto 6\\mapsto 5\\mapsto 6),(6\\mapsto 6\\mapsto 5\\mapsto 7),(6\\mapsto 6\\mapsto 5\\mapsto 8),(6\\mapsto 6\\mapsto 5\\mapsto 9),(6\\mapsto 6\\mapsto 6\\mapsto 7),(6\\mapsto 6\\mapsto 6\\mapsto 8),(6\\mapsto 6\\mapsto 6\\mapsto 9),(6\\mapsto 6\\mapsto 7\\mapsto 8),(6\\mapsto 6\\mapsto 7\\mapsto 9),(6\\mapsto 6\\mapsto 8\\mapsto 9),(7\\mapsto 7\\mapsto 1\\mapsto 2),(7\\mapsto 7\\mapsto 1\\mapsto 3),(7\\mapsto 7\\mapsto 1\\mapsto 4),(7\\mapsto 7\\mapsto 1\\mapsto 5),(7\\mapsto 7\\mapsto 1\\mapsto 6),(7\\mapsto 7\\mapsto 1\\mapsto 7),(7\\mapsto 7\\mapsto 1\\mapsto 8),(7\\mapsto 7\\mapsto 1\\mapsto 9),(7\\mapsto 7\\mapsto 2\\mapsto 3),(7\\mapsto 7\\mapsto 2\\mapsto 4),(7\\mapsto 7\\mapsto 2\\mapsto 5),(7\\mapsto 7\\mapsto 2\\mapsto 6),(7\\mapsto 7\\mapsto 2\\mapsto 7),(7\\mapsto 7\\mapsto 2\\mapsto 8),(7\\mapsto 7\\mapsto 2\\mapsto 9),(7\\mapsto 7\\mapsto 3\\mapsto 4),(7\\mapsto 7\\mapsto 3\\mapsto 5),(7\\mapsto 7\\mapsto 3\\mapsto 6),(7\\mapsto 7\\mapsto 3\\mapsto 7),(7\\mapsto 7\\mapsto 3\\mapsto 8),(7\\mapsto 7\\mapsto 3\\mapsto 9),(7\\mapsto 7\\mapsto 4\\mapsto 5),(7\\mapsto 7\\mapsto 4\\mapsto 6),(7\\mapsto 7\\mapsto 4\\mapsto 7),(7\\mapsto 7\\mapsto 4\\mapsto 8),(7\\mapsto 7\\mapsto 4\\mapsto 9),(7\\mapsto 7\\mapsto 5\\mapsto 6),(7\\mapsto 7\\mapsto 5\\mapsto 7),(7\\mapsto 7\\mapsto 5\\mapsto 8),(7\\mapsto 7\\mapsto 5\\mapsto 9),(7\\mapsto 7\\mapsto 6\\mapsto 7),(7\\mapsto 7\\mapsto 6\\mapsto 8),(7\\mapsto 7\\mapsto 6\\mapsto 9),(7\\mapsto 7\\mapsto 7\\mapsto 8),(7\\mapsto 7\\mapsto 7\\mapsto 9),(7\\mapsto 7\\mapsto 8\\mapsto 9),(8\\mapsto 8\\mapsto 1\\mapsto 2),(8\\mapsto 8\\mapsto 1\\mapsto 3),(8\\mapsto 8\\mapsto 1\\mapsto 4),(8\\mapsto 8\\mapsto 1\\mapsto 5),(8\\mapsto 8\\mapsto 1\\mapsto 6),(8\\mapsto 8\\mapsto 1\\mapsto 7),(8\\mapsto 8\\mapsto 1\\mapsto 8),(8\\mapsto 8\\mapsto 1\\mapsto 9),(8\\mapsto 8\\mapsto 2\\mapsto 3),(8\\mapsto 8\\mapsto 2\\mapsto 4),(8\\mapsto 8\\mapsto 2\\mapsto 5),(8\\mapsto 8\\mapsto 2\\mapsto 6),(8\\mapsto 8\\mapsto 2\\mapsto 7),(8\\mapsto 8\\mapsto 2\\mapsto 8),(8\\mapsto 8\\mapsto 2\\mapsto 9),(8\\mapsto 8\\mapsto 3\\mapsto 4),(8\\mapsto 8\\mapsto 3\\mapsto 5),(8\\mapsto 8\\mapsto 3\\mapsto 6),(8\\mapsto 8\\mapsto 3\\mapsto 7),(8\\mapsto 8\\mapsto 3\\mapsto 8),(8\\mapsto 8\\mapsto 3\\mapsto 9),(8\\mapsto 8\\mapsto 4\\mapsto 5),(8\\mapsto 8\\mapsto 4\\mapsto 6),(8\\mapsto 8\\mapsto 4\\mapsto 7),(8\\mapsto 8\\mapsto 4\\mapsto 8),(8\\mapsto 8\\mapsto 4\\mapsto 9),(8\\mapsto 8\\mapsto 5\\mapsto 6),(8\\mapsto 8\\mapsto 5\\mapsto 7),(8\\mapsto 8\\mapsto 5\\mapsto 8),(8\\mapsto 8\\mapsto 5\\mapsto 9),(8\\mapsto 8\\mapsto 6\\mapsto 7),(8\\mapsto 8\\mapsto 6\\mapsto 8),(8\\mapsto 8\\mapsto 6\\mapsto 9),(8\\mapsto 8\\mapsto 7\\mapsto 8),(8\\mapsto 8\\mapsto 7\\mapsto 9),(8\\mapsto 8\\mapsto 8\\mapsto 9),(9\\mapsto 9\\mapsto 1\\mapsto 2),(9\\mapsto 9\\mapsto 1\\mapsto 3),(9\\mapsto 9\\mapsto 1\\mapsto 4),(9\\mapsto 9\\mapsto 1\\mapsto 5),(9\\mapsto 9\\mapsto 1\\mapsto 6),(9\\mapsto 9\\mapsto 1\\mapsto 7),(9\\mapsto 9\\mapsto 1\\mapsto 8),(9\\mapsto 9\\mapsto 1\\mapsto 9),(9\\mapsto 9\\mapsto 2\\mapsto 3),(9\\mapsto 9\\mapsto 2\\mapsto 4),(9\\mapsto 9\\mapsto 2\\mapsto 5),(9\\mapsto 9\\mapsto 2\\mapsto 6),(9\\mapsto 9\\mapsto 2\\mapsto 7),(9\\mapsto 9\\mapsto 2\\mapsto 8),(9\\mapsto 9\\mapsto 2\\mapsto 9),(9\\mapsto 9\\mapsto 3\\mapsto 4),(9\\mapsto 9\\mapsto 3\\mapsto 5),(9\\mapsto 9\\mapsto 3\\mapsto 6),(9\\mapsto 9\\mapsto 3\\mapsto 7),(9\\mapsto 9\\mapsto 3\\mapsto 8),(9\\mapsto 9\\mapsto 3\\mapsto 9),(9\\mapsto 9\\mapsto 4\\mapsto 5),(9\\mapsto 9\\mapsto 4\\mapsto 6),(9\\mapsto 9\\mapsto 4\\mapsto 7),(9\\mapsto 9\\mapsto 4\\mapsto 8),(9\\mapsto 9\\mapsto 4\\mapsto 9),(9\\mapsto 9\\mapsto 5\\mapsto 6),(9\\mapsto 9\\mapsto 5\\mapsto 7),(9\\mapsto 9\\mapsto 5\\mapsto 8),(9\\mapsto 9\\mapsto 5\\mapsto 9),(9\\mapsto 9\\mapsto 6\\mapsto 7),(9\\mapsto 9\\mapsto 6\\mapsto 8),(9\\mapsto 9\\mapsto 6\\mapsto 9),(9\\mapsto 9\\mapsto 7\\mapsto 8),(9\\mapsto 9\\mapsto 7\\mapsto 9),(9\\mapsto 9\\mapsto 8\\mapsto 9)\\}$\n", + "* $\\mathit{Board} = \\{(1\\mapsto\\{(1\\mapsto 2),(2\\mapsto 5),(3\\mapsto 4),(4\\mapsto 1),(5\\mapsto 3),(6\\mapsto 6),(7\\mapsto 7),(8\\mapsto 8),(9\\mapsto 9)\\}),(2\\mapsto\\{(1\\mapsto 1),(2\\mapsto 4),(3\\mapsto 2),(4\\mapsto 3),(5\\mapsto 6),(6\\mapsto 5),(7\\mapsto 9),(8\\mapsto 7),(9\\mapsto 8)\\}),(3\\mapsto\\{(1\\mapsto 4),(2\\mapsto 9),(3\\mapsto 3),(4\\mapsto 2),(5\\mapsto 1),(6\\mapsto 8),(7\\mapsto 5),(8\\mapsto 6),(9\\mapsto 7)\\}),(4\\mapsto\\{(1\\mapsto 3),(2\\mapsto 1),(3\\mapsto 8),(4\\mapsto 7),(5\\mapsto 9),(6\\mapsto 2),(7\\mapsto 4),(8\\mapsto 5),(9\\mapsto 6)\\}),(5\\mapsto\\{(1\\mapsto 6),(2\\mapsto 3),(3\\mapsto 9),(4\\mapsto 8),(5\\mapsto 7),(6\\mapsto 1),(7\\mapsto 2),(8\\mapsto 4),(9\\mapsto 5)\\}),(6\\mapsto\\{(1\\mapsto 5),(2\\mapsto 2),(3\\mapsto 1),(4\\mapsto 9),(5\\mapsto 8),(6\\mapsto 7),(7\\mapsto 6),(8\\mapsto 3),(9\\mapsto 4)\\}),(7\\mapsto\\{(1\\mapsto 7),(2\\mapsto 6),(3\\mapsto 5),(4\\mapsto 4),(5\\mapsto 2),(6\\mapsto 9),(7\\mapsto 8),(8\\mapsto 1),(9\\mapsto 3)\\}),(8\\mapsto\\{(1\\mapsto 8),(2\\mapsto 7),(3\\mapsto 6),(4\\mapsto 5),(5\\mapsto 4),(6\\mapsto 3),(7\\mapsto 1),(8\\mapsto 9),(9\\mapsto 2)\\}),(9\\mapsto\\{(1\\mapsto 9),(2\\mapsto 8),(3\\mapsto 7),(4\\mapsto 6),(5\\mapsto 5),(6\\mapsto 4),(7\\mapsto 3),(8\\mapsto 2),(9\\mapsto 1)\\})\\}$\n", + "* $\\mathit{DOM} = \\{1,2,3,4,5,6,7,8,9\\}$\n", + "* $\\mathit{Diff1} = \\{(1\\mapsto 2\\mapsto 1\\mapsto 1),(1\\mapsto 2\\mapsto 2\\mapsto 2),(1\\mapsto 2\\mapsto 3\\mapsto 3),(1\\mapsto 2\\mapsto 4\\mapsto 4),(1\\mapsto 2\\mapsto 5\\mapsto 5),(1\\mapsto 2\\mapsto 6\\mapsto 6),(1\\mapsto 2\\mapsto 7\\mapsto 7),(1\\mapsto 2\\mapsto 8\\mapsto 8),(1\\mapsto 2\\mapsto 9\\mapsto 9),(1\\mapsto 3\\mapsto 1\\mapsto 1),(1\\mapsto 3\\mapsto 2\\mapsto 2),(1\\mapsto 3\\mapsto 3\\mapsto 3),(1\\mapsto 3\\mapsto 4\\mapsto 4),(1\\mapsto 3\\mapsto 5\\mapsto 5),(1\\mapsto 3\\mapsto 6\\mapsto 6),(1\\mapsto 3\\mapsto 7\\mapsto 7),(1\\mapsto 3\\mapsto 8\\mapsto 8),(1\\mapsto 3\\mapsto 9\\mapsto 9),(1\\mapsto 4\\mapsto 1\\mapsto 1),(1\\mapsto 4\\mapsto 2\\mapsto 2),(1\\mapsto 4\\mapsto 3\\mapsto 3),(1\\mapsto 4\\mapsto 4\\mapsto 4),(1\\mapsto 4\\mapsto 5\\mapsto 5),(1\\mapsto 4\\mapsto 6\\mapsto 6),(1\\mapsto 4\\mapsto 7\\mapsto 7),(1\\mapsto 4\\mapsto 8\\mapsto 8),(1\\mapsto 4\\mapsto 9\\mapsto 9),(1\\mapsto 5\\mapsto 1\\mapsto 1),(1\\mapsto 5\\mapsto 2\\mapsto 2),(1\\mapsto 5\\mapsto 3\\mapsto 3),(1\\mapsto 5\\mapsto 4\\mapsto 4),(1\\mapsto 5\\mapsto 5\\mapsto 5),(1\\mapsto 5\\mapsto 6\\mapsto 6),(1\\mapsto 5\\mapsto 7\\mapsto 7),(1\\mapsto 5\\mapsto 8\\mapsto 8),(1\\mapsto 5\\mapsto 9\\mapsto 9),(1\\mapsto 6\\mapsto 1\\mapsto 1),(1\\mapsto 6\\mapsto 2\\mapsto 2),(1\\mapsto 6\\mapsto 3\\mapsto 3),(1\\mapsto 6\\mapsto 4\\mapsto 4),(1\\mapsto 6\\mapsto 5\\mapsto 5),(1\\mapsto 6\\mapsto 6\\mapsto 6),(1\\mapsto 6\\mapsto 7\\mapsto 7),(1\\mapsto 6\\mapsto 8\\mapsto 8),(1\\mapsto 6\\mapsto 9\\mapsto 9),(1\\mapsto 7\\mapsto 1\\mapsto 1),(1\\mapsto 7\\mapsto 2\\mapsto 2),(1\\mapsto 7\\mapsto 3\\mapsto 3),(1\\mapsto 7\\mapsto 4\\mapsto 4),(1\\mapsto 7\\mapsto 5\\mapsto 5),(1\\mapsto 7\\mapsto 6\\mapsto 6),(1\\mapsto 7\\mapsto 7\\mapsto 7),(1\\mapsto 7\\mapsto 8\\mapsto 8),(1\\mapsto 7\\mapsto 9\\mapsto 9),(1\\mapsto 8\\mapsto 1\\mapsto 1),(1\\mapsto 8\\mapsto 2\\mapsto 2),(1\\mapsto 8\\mapsto 3\\mapsto 3),(1\\mapsto 8\\mapsto 4\\mapsto 4),(1\\mapsto 8\\mapsto 5\\mapsto 5),(1\\mapsto 8\\mapsto 6\\mapsto 6),(1\\mapsto 8\\mapsto 7\\mapsto 7),(1\\mapsto 8\\mapsto 8\\mapsto 8),(1\\mapsto 8\\mapsto 9\\mapsto 9),(1\\mapsto 9\\mapsto 1\\mapsto 1),(1\\mapsto 9\\mapsto 2\\mapsto 2),(1\\mapsto 9\\mapsto 3\\mapsto 3),(1\\mapsto 9\\mapsto 4\\mapsto 4),(1\\mapsto 9\\mapsto 5\\mapsto 5),(1\\mapsto 9\\mapsto 6\\mapsto 6),(1\\mapsto 9\\mapsto 7\\mapsto 7),(1\\mapsto 9\\mapsto 8\\mapsto 8),(1\\mapsto 9\\mapsto 9\\mapsto 9),(2\\mapsto 3\\mapsto 1\\mapsto 1),(2\\mapsto 3\\mapsto 2\\mapsto 2),(2\\mapsto 3\\mapsto 3\\mapsto 3),(2\\mapsto 3\\mapsto 4\\mapsto 4),(2\\mapsto 3\\mapsto 5\\mapsto 5),(2\\mapsto 3\\mapsto 6\\mapsto 6),(2\\mapsto 3\\mapsto 7\\mapsto 7),(2\\mapsto 3\\mapsto 8\\mapsto 8),(2\\mapsto 3\\mapsto 9\\mapsto 9),(2\\mapsto 4\\mapsto 1\\mapsto 1),(2\\mapsto 4\\mapsto 2\\mapsto 2),(2\\mapsto 4\\mapsto 3\\mapsto 3),(2\\mapsto 4\\mapsto 4\\mapsto 4),(2\\mapsto 4\\mapsto 5\\mapsto 5),(2\\mapsto 4\\mapsto 6\\mapsto 6),(2\\mapsto 4\\mapsto 7\\mapsto 7),(2\\mapsto 4\\mapsto 8\\mapsto 8),(2\\mapsto 4\\mapsto 9\\mapsto 9),(2\\mapsto 5\\mapsto 1\\mapsto 1),(2\\mapsto 5\\mapsto 2\\mapsto 2),(2\\mapsto 5\\mapsto 3\\mapsto 3),(2\\mapsto 5\\mapsto 4\\mapsto 4),(2\\mapsto 5\\mapsto 5\\mapsto 5),(2\\mapsto 5\\mapsto 6\\mapsto 6),(2\\mapsto 5\\mapsto 7\\mapsto 7),(2\\mapsto 5\\mapsto 8\\mapsto 8),(2\\mapsto 5\\mapsto 9\\mapsto 9),(2\\mapsto 6\\mapsto 1\\mapsto 1),(2\\mapsto 6\\mapsto 2\\mapsto 2),(2\\mapsto 6\\mapsto 3\\mapsto 3),(2\\mapsto 6\\mapsto 4\\mapsto 4),(2\\mapsto 6\\mapsto 5\\mapsto 5),(2\\mapsto 6\\mapsto 6\\mapsto 6),(2\\mapsto 6\\mapsto 7\\mapsto 7),(2\\mapsto 6\\mapsto 8\\mapsto 8),(2\\mapsto 6\\mapsto 9\\mapsto 9),(2\\mapsto 7\\mapsto 1\\mapsto 1),(2\\mapsto 7\\mapsto 2\\mapsto 2),(2\\mapsto 7\\mapsto 3\\mapsto 3),(2\\mapsto 7\\mapsto 4\\mapsto 4),(2\\mapsto 7\\mapsto 5\\mapsto 5),(2\\mapsto 7\\mapsto 6\\mapsto 6),(2\\mapsto 7\\mapsto 7\\mapsto 7),(2\\mapsto 7\\mapsto 8\\mapsto 8),(2\\mapsto 7\\mapsto 9\\mapsto 9),(2\\mapsto 8\\mapsto 1\\mapsto 1),(2\\mapsto 8\\mapsto 2\\mapsto 2),(2\\mapsto 8\\mapsto 3\\mapsto 3),(2\\mapsto 8\\mapsto 4\\mapsto 4),(2\\mapsto 8\\mapsto 5\\mapsto 5),(2\\mapsto 8\\mapsto 6\\mapsto 6),(2\\mapsto 8\\mapsto 7\\mapsto 7),(2\\mapsto 8\\mapsto 8\\mapsto 8),(2\\mapsto 8\\mapsto 9\\mapsto 9),(2\\mapsto 9\\mapsto 1\\mapsto 1),(2\\mapsto 9\\mapsto 2\\mapsto 2),(2\\mapsto 9\\mapsto 3\\mapsto 3),(2\\mapsto 9\\mapsto 4\\mapsto 4),(2\\mapsto 9\\mapsto 5\\mapsto 5),(2\\mapsto 9\\mapsto 6\\mapsto 6),(2\\mapsto 9\\mapsto 7\\mapsto 7),(2\\mapsto 9\\mapsto 8\\mapsto 8),(2\\mapsto 9\\mapsto 9\\mapsto 9),(3\\mapsto 4\\mapsto 1\\mapsto 1),(3\\mapsto 4\\mapsto 2\\mapsto 2),(3\\mapsto 4\\mapsto 3\\mapsto 3),(3\\mapsto 4\\mapsto 4\\mapsto 4),(3\\mapsto 4\\mapsto 5\\mapsto 5),(3\\mapsto 4\\mapsto 6\\mapsto 6),(3\\mapsto 4\\mapsto 7\\mapsto 7),(3\\mapsto 4\\mapsto 8\\mapsto 8),(3\\mapsto 4\\mapsto 9\\mapsto 9),(3\\mapsto 5\\mapsto 1\\mapsto 1),(3\\mapsto 5\\mapsto 2\\mapsto 2),(3\\mapsto 5\\mapsto 3\\mapsto 3),(3\\mapsto 5\\mapsto 4\\mapsto 4),(3\\mapsto 5\\mapsto 5\\mapsto 5),(3\\mapsto 5\\mapsto 6\\mapsto 6),(3\\mapsto 5\\mapsto 7\\mapsto 7),(3\\mapsto 5\\mapsto 8\\mapsto 8),(3\\mapsto 5\\mapsto 9\\mapsto 9),(3\\mapsto 6\\mapsto 1\\mapsto 1),(3\\mapsto 6\\mapsto 2\\mapsto 2),(3\\mapsto 6\\mapsto 3\\mapsto 3),(3\\mapsto 6\\mapsto 4\\mapsto 4),(3\\mapsto 6\\mapsto 5\\mapsto 5),(3\\mapsto 6\\mapsto 6\\mapsto 6),(3\\mapsto 6\\mapsto 7\\mapsto 7),(3\\mapsto 6\\mapsto 8\\mapsto 8),(3\\mapsto 6\\mapsto 9\\mapsto 9),(3\\mapsto 7\\mapsto 1\\mapsto 1),(3\\mapsto 7\\mapsto 2\\mapsto 2),(3\\mapsto 7\\mapsto 3\\mapsto 3),(3\\mapsto 7\\mapsto 4\\mapsto 4),(3\\mapsto 7\\mapsto 5\\mapsto 5),(3\\mapsto 7\\mapsto 6\\mapsto 6),(3\\mapsto 7\\mapsto 7\\mapsto 7),(3\\mapsto 7\\mapsto 8\\mapsto 8),(3\\mapsto 7\\mapsto 9\\mapsto 9),(3\\mapsto 8\\mapsto 1\\mapsto 1),(3\\mapsto 8\\mapsto 2\\mapsto 2),(3\\mapsto 8\\mapsto 3\\mapsto 3),(3\\mapsto 8\\mapsto 4\\mapsto 4),(3\\mapsto 8\\mapsto 5\\mapsto 5),(3\\mapsto 8\\mapsto 6\\mapsto 6),(3\\mapsto 8\\mapsto 7\\mapsto 7),(3\\mapsto 8\\mapsto 8\\mapsto 8),(3\\mapsto 8\\mapsto 9\\mapsto 9),(3\\mapsto 9\\mapsto 1\\mapsto 1),(3\\mapsto 9\\mapsto 2\\mapsto 2),(3\\mapsto 9\\mapsto 3\\mapsto 3),(3\\mapsto 9\\mapsto 4\\mapsto 4),(3\\mapsto 9\\mapsto 5\\mapsto 5),(3\\mapsto 9\\mapsto 6\\mapsto 6),(3\\mapsto 9\\mapsto 7\\mapsto 7),(3\\mapsto 9\\mapsto 8\\mapsto 8),(3\\mapsto 9\\mapsto 9\\mapsto 9),(4\\mapsto 5\\mapsto 1\\mapsto 1),(4\\mapsto 5\\mapsto 2\\mapsto 2),(4\\mapsto 5\\mapsto 3\\mapsto 3),(4\\mapsto 5\\mapsto 4\\mapsto 4),(4\\mapsto 5\\mapsto 5\\mapsto 5),(4\\mapsto 5\\mapsto 6\\mapsto 6),(4\\mapsto 5\\mapsto 7\\mapsto 7),(4\\mapsto 5\\mapsto 8\\mapsto 8),(4\\mapsto 5\\mapsto 9\\mapsto 9),(4\\mapsto 6\\mapsto 1\\mapsto 1),(4\\mapsto 6\\mapsto 2\\mapsto 2),(4\\mapsto 6\\mapsto 3\\mapsto 3),(4\\mapsto 6\\mapsto 4\\mapsto 4),(4\\mapsto 6\\mapsto 5\\mapsto 5),(4\\mapsto 6\\mapsto 6\\mapsto 6),(4\\mapsto 6\\mapsto 7\\mapsto 7),(4\\mapsto 6\\mapsto 8\\mapsto 8),(4\\mapsto 6\\mapsto 9\\mapsto 9),(4\\mapsto 7\\mapsto 1\\mapsto 1),(4\\mapsto 7\\mapsto 2\\mapsto 2),(4\\mapsto 7\\mapsto 3\\mapsto 3),(4\\mapsto 7\\mapsto 4\\mapsto 4),(4\\mapsto 7\\mapsto 5\\mapsto 5),(4\\mapsto 7\\mapsto 6\\mapsto 6),(4\\mapsto 7\\mapsto 7\\mapsto 7),(4\\mapsto 7\\mapsto 8\\mapsto 8),(4\\mapsto 7\\mapsto 9\\mapsto 9),(4\\mapsto 8\\mapsto 1\\mapsto 1),(4\\mapsto 8\\mapsto 2\\mapsto 2),(4\\mapsto 8\\mapsto 3\\mapsto 3),(4\\mapsto 8\\mapsto 4\\mapsto 4),(4\\mapsto 8\\mapsto 5\\mapsto 5),(4\\mapsto 8\\mapsto 6\\mapsto 6),(4\\mapsto 8\\mapsto 7\\mapsto 7),(4\\mapsto 8\\mapsto 8\\mapsto 8),(4\\mapsto 8\\mapsto 9\\mapsto 9),(4\\mapsto 9\\mapsto 1\\mapsto 1),(4\\mapsto 9\\mapsto 2\\mapsto 2),(4\\mapsto 9\\mapsto 3\\mapsto 3),(4\\mapsto 9\\mapsto 4\\mapsto 4),(4\\mapsto 9\\mapsto 5\\mapsto 5),(4\\mapsto 9\\mapsto 6\\mapsto 6),(4\\mapsto 9\\mapsto 7\\mapsto 7),(4\\mapsto 9\\mapsto 8\\mapsto 8),(4\\mapsto 9\\mapsto 9\\mapsto 9),(5\\mapsto 6\\mapsto 1\\mapsto 1),(5\\mapsto 6\\mapsto 2\\mapsto 2),(5\\mapsto 6\\mapsto 3\\mapsto 3),(5\\mapsto 6\\mapsto 4\\mapsto 4),(5\\mapsto 6\\mapsto 5\\mapsto 5),(5\\mapsto 6\\mapsto 6\\mapsto 6),(5\\mapsto 6\\mapsto 7\\mapsto 7),(5\\mapsto 6\\mapsto 8\\mapsto 8),(5\\mapsto 6\\mapsto 9\\mapsto 9),(5\\mapsto 7\\mapsto 1\\mapsto 1),(5\\mapsto 7\\mapsto 2\\mapsto 2),(5\\mapsto 7\\mapsto 3\\mapsto 3),(5\\mapsto 7\\mapsto 4\\mapsto 4),(5\\mapsto 7\\mapsto 5\\mapsto 5),(5\\mapsto 7\\mapsto 6\\mapsto 6),(5\\mapsto 7\\mapsto 7\\mapsto 7),(5\\mapsto 7\\mapsto 8\\mapsto 8),(5\\mapsto 7\\mapsto 9\\mapsto 9),(5\\mapsto 8\\mapsto 1\\mapsto 1),(5\\mapsto 8\\mapsto 2\\mapsto 2),(5\\mapsto 8\\mapsto 3\\mapsto 3),(5\\mapsto 8\\mapsto 4\\mapsto 4),(5\\mapsto 8\\mapsto 5\\mapsto 5),(5\\mapsto 8\\mapsto 6\\mapsto 6),(5\\mapsto 8\\mapsto 7\\mapsto 7),(5\\mapsto 8\\mapsto 8\\mapsto 8),(5\\mapsto 8\\mapsto 9\\mapsto 9),(5\\mapsto 9\\mapsto 1\\mapsto 1),(5\\mapsto 9\\mapsto 2\\mapsto 2),(5\\mapsto 9\\mapsto 3\\mapsto 3),(5\\mapsto 9\\mapsto 4\\mapsto 4),(5\\mapsto 9\\mapsto 5\\mapsto 5),(5\\mapsto 9\\mapsto 6\\mapsto 6),(5\\mapsto 9\\mapsto 7\\mapsto 7),(5\\mapsto 9\\mapsto 8\\mapsto 8),(5\\mapsto 9\\mapsto 9\\mapsto 9),(6\\mapsto 7\\mapsto 1\\mapsto 1),(6\\mapsto 7\\mapsto 2\\mapsto 2),(6\\mapsto 7\\mapsto 3\\mapsto 3),(6\\mapsto 7\\mapsto 4\\mapsto 4),(6\\mapsto 7\\mapsto 5\\mapsto 5),(6\\mapsto 7\\mapsto 6\\mapsto 6),(6\\mapsto 7\\mapsto 7\\mapsto 7),(6\\mapsto 7\\mapsto 8\\mapsto 8),(6\\mapsto 7\\mapsto 9\\mapsto 9),(6\\mapsto 8\\mapsto 1\\mapsto 1),(6\\mapsto 8\\mapsto 2\\mapsto 2),(6\\mapsto 8\\mapsto 3\\mapsto 3),(6\\mapsto 8\\mapsto 4\\mapsto 4),(6\\mapsto 8\\mapsto 5\\mapsto 5),(6\\mapsto 8\\mapsto 6\\mapsto 6),(6\\mapsto 8\\mapsto 7\\mapsto 7),(6\\mapsto 8\\mapsto 8\\mapsto 8),(6\\mapsto 8\\mapsto 9\\mapsto 9),(6\\mapsto 9\\mapsto 1\\mapsto 1),(6\\mapsto 9\\mapsto 2\\mapsto 2),(6\\mapsto 9\\mapsto 3\\mapsto 3),(6\\mapsto 9\\mapsto 4\\mapsto 4),(6\\mapsto 9\\mapsto 5\\mapsto 5),(6\\mapsto 9\\mapsto 6\\mapsto 6),(6\\mapsto 9\\mapsto 7\\mapsto 7),(6\\mapsto 9\\mapsto 8\\mapsto 8),(6\\mapsto 9\\mapsto 9\\mapsto 9),(7\\mapsto 8\\mapsto 1\\mapsto 1),(7\\mapsto 8\\mapsto 2\\mapsto 2),(7\\mapsto 8\\mapsto 3\\mapsto 3),(7\\mapsto 8\\mapsto 4\\mapsto 4),(7\\mapsto 8\\mapsto 5\\mapsto 5),(7\\mapsto 8\\mapsto 6\\mapsto 6),(7\\mapsto 8\\mapsto 7\\mapsto 7),(7\\mapsto 8\\mapsto 8\\mapsto 8),(7\\mapsto 8\\mapsto 9\\mapsto 9),(7\\mapsto 9\\mapsto 1\\mapsto 1),(7\\mapsto 9\\mapsto 2\\mapsto 2),(7\\mapsto 9\\mapsto 3\\mapsto 3),(7\\mapsto 9\\mapsto 4\\mapsto 4),(7\\mapsto 9\\mapsto 5\\mapsto 5),(7\\mapsto 9\\mapsto 6\\mapsto 6),(7\\mapsto 9\\mapsto 7\\mapsto 7),(7\\mapsto 9\\mapsto 8\\mapsto 8),(7\\mapsto 9\\mapsto 9\\mapsto 9),(8\\mapsto 9\\mapsto 1\\mapsto 1),(8\\mapsto 9\\mapsto 2\\mapsto 2),(8\\mapsto 9\\mapsto 3\\mapsto 3),(8\\mapsto 9\\mapsto 4\\mapsto 4),(8\\mapsto 9\\mapsto 5\\mapsto 5),(8\\mapsto 9\\mapsto 6\\mapsto 6),(8\\mapsto 9\\mapsto 7\\mapsto 7),(8\\mapsto 9\\mapsto 8\\mapsto 8),(8\\mapsto 9\\mapsto 9\\mapsto 9)\\}$\n", + "* $\\mathit{SUBSQ} = \\{\\{1,2,3\\},\\{4,5,6\\},\\{7,8,9\\}\\}$" + ], + "text/plain": [ + "TRUE\n", + "\n", + "Solution:\n", + "\tDiff2 = {(1↦1↦1↦2),(1↦1↦1↦3),(1↦1↦1↦4),(1↦1↦1↦5),(1↦1↦1↦6),(1↦1↦1↦7),(1↦1↦1↦8),(1↦1↦1↦9),(1↦1↦2↦3),(1↦1↦2↦4),(1↦1↦2↦5),(1↦1↦2↦6),(1↦1↦2↦7),(1↦1↦2↦8),(1↦1↦2↦9),(1↦1↦3↦4),(1↦1↦3↦5),(1↦1↦3↦6),(1↦1↦3↦7),(1↦1↦3↦8),(1↦1↦3↦9),(1↦1↦4↦5),(1↦1↦4↦6),(1↦1↦4↦7),(1↦1↦4↦8),(1↦1↦4↦9),(1↦1↦5↦6),(1↦1↦5↦7),(1↦1↦5↦8),(1↦1↦5↦9),(1↦1↦6↦7),(1↦1↦6↦8),(1↦1↦6↦9),(1↦1↦7↦8),(1↦1↦7↦9),(1↦1↦8↦9),(2↦2↦1↦2),(2↦2↦1↦3),(2↦2↦1↦4),(2↦2↦1↦5),(2↦2↦1↦6),(2↦2↦1↦7),(2↦2↦1↦8),(2↦2↦1↦9),(2↦2↦2↦3),(2↦2↦2↦4),(2↦2↦2↦5),(2↦2↦2↦6),(2↦2↦2↦7),(2↦2↦2↦8),(2↦2↦2↦9),(2↦2↦3↦4),(2↦2↦3↦5),(2↦2↦3↦6),(2↦2↦3↦7),(2↦2↦3↦8),(2↦2↦3↦9),(2↦2↦4↦5),(2↦2↦4↦6),(2↦2↦4↦7),(2↦2↦4↦8),(2↦2↦4↦9),(2↦2↦5↦6),(2↦2↦5↦7),(2↦2↦5↦8),(2↦2↦5↦9),(2↦2↦6↦7),(2↦2↦6↦8),(2↦2↦6↦9),(2↦2↦7↦8),(2↦2↦7↦9),(2↦2↦8↦9),(3↦3↦1↦2),(3↦3↦1↦3),(3↦3↦1↦4),(3↦3↦1↦5),(3↦3↦1↦6),(3↦3↦1↦7),(3↦3↦1↦8),(3↦3↦1↦9),(3↦3↦2↦3),(3↦3↦2↦4),(3↦3↦2↦5),(3↦3↦2↦6),(3↦3↦2↦7),(3↦3↦2↦8),(3↦3↦2↦9),(3↦3↦3↦4),(3↦3↦3↦5),(3↦3↦3↦6),(3↦3↦3↦7),(3↦3↦3↦8),(3↦3↦3↦9),(3↦3↦4↦5),(3↦3↦4↦6),(3↦3↦4↦7),(3↦3↦4↦8),(3↦3↦4↦9),(3↦3↦5↦6),(3↦3↦5↦7),(3↦3↦5↦8),(3↦3↦5↦9),(3↦3↦6↦7),(3↦3↦6↦8),(3↦3↦6↦9),(3↦3↦7↦8),(3↦3↦7↦9),(3↦3↦8↦9),(4↦4↦1↦2),(4↦4↦1↦3),(4↦4↦1↦4),(4↦4↦1↦5),(4↦4↦1↦6),(4↦4↦1↦7),(4↦4↦1↦8),(4↦4↦1↦9),(4↦4↦2↦3),(4↦4↦2↦4),(4↦4↦2↦5),(4↦4↦2↦6),(4↦4↦2↦7),(4↦4↦2↦8),(4↦4↦2↦9),(4↦4↦3↦4),(4↦4↦3↦5),(4↦4↦3↦6),(4↦4↦3↦7),(4↦4↦3↦8),(4↦4↦3↦9),(4↦4↦4↦5),(4↦4↦4↦6),(4↦4↦4↦7),(4↦4↦4↦8),(4↦4↦4↦9),(4↦4↦5↦6),(4↦4↦5↦7),(4↦4↦5↦8),(4↦4↦5↦9),(4↦4↦6↦7),(4↦4↦6↦8),(4↦4↦6↦9),(4↦4↦7↦8),(4↦4↦7↦9),(4↦4↦8↦9),(5↦5↦1↦2),(5↦5↦1↦3),(5↦5↦1↦4),(5↦5↦1↦5),(5↦5↦1↦6),(5↦5↦1↦7),(5↦5↦1↦8),(5↦5↦1↦9),(5↦5↦2↦3),(5↦5↦2↦4),(5↦5↦2↦5),(5↦5↦2↦6),(5↦5↦2↦7),(5↦5↦2↦8),(5↦5↦2↦9),(5↦5↦3↦4),(5↦5↦3↦5),(5↦5↦3↦6),(5↦5↦3↦7),(5↦5↦3↦8),(5↦5↦3↦9),(5↦5↦4↦5),(5↦5↦4↦6),(5↦5↦4↦7),(5↦5↦4↦8),(5↦5↦4↦9),(5↦5↦5↦6),(5↦5↦5↦7),(5↦5↦5↦8),(5↦5↦5↦9),(5↦5↦6↦7),(5↦5↦6↦8),(5↦5↦6↦9),(5↦5↦7↦8),(5↦5↦7↦9),(5↦5↦8↦9),(6↦6↦1↦2),(6↦6↦1↦3),(6↦6↦1↦4),(6↦6↦1↦5),(6↦6↦1↦6),(6↦6↦1↦7),(6↦6↦1↦8),(6↦6↦1↦9),(6↦6↦2↦3),(6↦6↦2↦4),(6↦6↦2↦5),(6↦6↦2↦6),(6↦6↦2↦7),(6↦6↦2↦8),(6↦6↦2↦9),(6↦6↦3↦4),(6↦6↦3↦5),(6↦6↦3↦6),(6↦6↦3↦7),(6↦6↦3↦8),(6↦6↦3↦9),(6↦6↦4↦5),(6↦6↦4↦6),(6↦6↦4↦7),(6↦6↦4↦8),(6↦6↦4↦9),(6↦6↦5↦6),(6↦6↦5↦7),(6↦6↦5↦8),(6↦6↦5↦9),(6↦6↦6↦7),(6↦6↦6↦8),(6↦6↦6↦9),(6↦6↦7↦8),(6↦6↦7↦9),(6↦6↦8↦9),(7↦7↦1↦2),(7↦7↦1↦3),(7↦7↦1↦4),(7↦7↦1↦5),(7↦7↦1↦6),(7↦7↦1↦7),(7↦7↦1↦8),(7↦7↦1↦9),(7↦7↦2↦3),(7↦7↦2↦4),(7↦7↦2↦5),(7↦7↦2↦6),(7↦7↦2↦7),(7↦7↦2↦8),(7↦7↦2↦9),(7↦7↦3↦4),(7↦7↦3↦5),(7↦7↦3↦6),(7↦7↦3↦7),(7↦7↦3↦8),(7↦7↦3↦9),(7↦7↦4↦5),(7↦7↦4↦6),(7↦7↦4↦7),(7↦7↦4↦8),(7↦7↦4↦9),(7↦7↦5↦6),(7↦7↦5↦7),(7↦7↦5↦8),(7↦7↦5↦9),(7↦7↦6↦7),(7↦7↦6↦8),(7↦7↦6↦9),(7↦7↦7↦8),(7↦7↦7↦9),(7↦7↦8↦9),(8↦8↦1↦2),(8↦8↦1↦3),(8↦8↦1↦4),(8↦8↦1↦5),(8↦8↦1↦6),(8↦8↦1↦7),(8↦8↦1↦8),(8↦8↦1↦9),(8↦8↦2↦3),(8↦8↦2↦4),(8↦8↦2↦5),(8↦8↦2↦6),(8↦8↦2↦7),(8↦8↦2↦8),(8↦8↦2↦9),(8↦8↦3↦4),(8↦8↦3↦5),(8↦8↦3↦6),(8↦8↦3↦7),(8↦8↦3↦8),(8↦8↦3↦9),(8↦8↦4↦5),(8↦8↦4↦6),(8↦8↦4↦7),(8↦8↦4↦8),(8↦8↦4↦9),(8↦8↦5↦6),(8↦8↦5↦7),(8↦8↦5↦8),(8↦8↦5↦9),(8↦8↦6↦7),(8↦8↦6↦8),(8↦8↦6↦9),(8↦8↦7↦8),(8↦8↦7↦9),(8↦8↦8↦9),(9↦9↦1↦2),(9↦9↦1↦3),(9↦9↦1↦4),(9↦9↦1↦5),(9↦9↦1↦6),(9↦9↦1↦7),(9↦9↦1↦8),(9↦9↦1↦9),(9↦9↦2↦3),(9↦9↦2↦4),(9↦9↦2↦5),(9↦9↦2↦6),(9↦9↦2↦7),(9↦9↦2↦8),(9↦9↦2↦9),(9↦9↦3↦4),(9↦9↦3↦5),(9↦9↦3↦6),(9↦9↦3↦7),(9↦9↦3↦8),(9↦9↦3↦9),(9↦9↦4↦5),(9↦9↦4↦6),(9↦9↦4↦7),(9↦9↦4↦8),(9↦9↦4↦9),(9↦9↦5↦6),(9↦9↦5↦7),(9↦9↦5↦8),(9↦9↦5↦9),(9↦9↦6↦7),(9↦9↦6↦8),(9↦9↦6↦9),(9↦9↦7↦8),(9↦9↦7↦9),(9↦9↦8↦9)}\n", + "\tBoard = {(1↦{(1↦2),(2↦5),(3↦4),(4↦1),(5↦3),(6↦6),(7↦7),(8↦8),(9↦9)}),(2↦{(1↦1),(2↦4),(3↦2),(4↦3),(5↦6),(6↦5),(7↦9),(8↦7),(9↦8)}),(3↦{(1↦4),(2↦9),(3↦3),(4↦2),(5↦1),(6↦8),(7↦5),(8↦6),(9↦7)}),(4↦{(1↦3),(2↦1),(3↦8),(4↦7),(5↦9),(6↦2),(7↦4),(8↦5),(9↦6)}),(5↦{(1↦6),(2↦3),(3↦9),(4↦8),(5↦7),(6↦1),(7↦2),(8↦4),(9↦5)}),(6↦{(1↦5),(2↦2),(3↦1),(4↦9),(5↦8),(6↦7),(7↦6),(8↦3),(9↦4)}),(7↦{(1↦7),(2↦6),(3↦5),(4↦4),(5↦2),(6↦9),(7↦8),(8↦1),(9↦3)}),(8↦{(1↦8),(2↦7),(3↦6),(4↦5),(5↦4),(6↦3),(7↦1),(8↦9),(9↦2)}),(9↦{(1↦9),(2↦8),(3↦7),(4↦6),(5↦5),(6↦4),(7↦3),(8↦2),(9↦1)})}\n", + "\tDOM = {1,2,3,4,5,6,7,8,9}\n", + "\tDiff1 = {(1↦2↦1↦1),(1↦2↦2↦2),(1↦2↦3↦3),(1↦2↦4↦4),(1↦2↦5↦5),(1↦2↦6↦6),(1↦2↦7↦7),(1↦2↦8↦8),(1↦2↦9↦9),(1↦3↦1↦1),(1↦3↦2↦2),(1↦3↦3↦3),(1↦3↦4↦4),(1↦3↦5↦5),(1↦3↦6↦6),(1↦3↦7↦7),(1↦3↦8↦8),(1↦3↦9↦9),(1↦4↦1↦1),(1↦4↦2↦2),(1↦4↦3↦3),(1↦4↦4↦4),(1↦4↦5↦5),(1↦4↦6↦6),(1↦4↦7↦7),(1↦4↦8↦8),(1↦4↦9↦9),(1↦5↦1↦1),(1↦5↦2↦2),(1↦5↦3↦3),(1↦5↦4↦4),(1↦5↦5↦5),(1↦5↦6↦6),(1↦5↦7↦7),(1↦5↦8↦8),(1↦5↦9↦9),(1↦6↦1↦1),(1↦6↦2↦2),(1↦6↦3↦3),(1↦6↦4↦4),(1↦6↦5↦5),(1↦6↦6↦6),(1↦6↦7↦7),(1↦6↦8↦8),(1↦6↦9↦9),(1↦7↦1↦1),(1↦7↦2↦2),(1↦7↦3↦3),(1↦7↦4↦4),(1↦7↦5↦5),(1↦7↦6↦6),(1↦7↦7↦7),(1↦7↦8↦8),(1↦7↦9↦9),(1↦8↦1↦1),(1↦8↦2↦2),(1↦8↦3↦3),(1↦8↦4↦4),(1↦8↦5↦5),(1↦8↦6↦6),(1↦8↦7↦7),(1↦8↦8↦8),(1↦8↦9↦9),(1↦9↦1↦1),(1↦9↦2↦2),(1↦9↦3↦3),(1↦9↦4↦4),(1↦9↦5↦5),(1↦9↦6↦6),(1↦9↦7↦7),(1↦9↦8↦8),(1↦9↦9↦9),(2↦3↦1↦1),(2↦3↦2↦2),(2↦3↦3↦3),(2↦3↦4↦4),(2↦3↦5↦5),(2↦3↦6↦6),(2↦3↦7↦7),(2↦3↦8↦8),(2↦3↦9↦9),(2↦4↦1↦1),(2↦4↦2↦2),(2↦4↦3↦3),(2↦4↦4↦4),(2↦4↦5↦5),(2↦4↦6↦6),(2↦4↦7↦7),(2↦4↦8↦8),(2↦4↦9↦9),(2↦5↦1↦1),(2↦5↦2↦2),(2↦5↦3↦3),(2↦5↦4↦4),(2↦5↦5↦5),(2↦5↦6↦6),(2↦5↦7↦7),(2↦5↦8↦8),(2↦5↦9↦9),(2↦6↦1↦1),(2↦6↦2↦2),(2↦6↦3↦3),(2↦6↦4↦4),(2↦6↦5↦5),(2↦6↦6↦6),(2↦6↦7↦7),(2↦6↦8↦8),(2↦6↦9↦9),(2↦7↦1↦1),(2↦7↦2↦2),(2↦7↦3↦3),(2↦7↦4↦4),(2↦7↦5↦5),(2↦7↦6↦6),(2↦7↦7↦7),(2↦7↦8↦8),(2↦7↦9↦9),(2↦8↦1↦1),(2↦8↦2↦2),(2↦8↦3↦3),(2↦8↦4↦4),(2↦8↦5↦5),(2↦8↦6↦6),(2↦8↦7↦7),(2↦8↦8↦8),(2↦8↦9↦9),(2↦9↦1↦1),(2↦9↦2↦2),(2↦9↦3↦3),(2↦9↦4↦4),(2↦9↦5↦5),(2↦9↦6↦6),(2↦9↦7↦7),(2↦9↦8↦8),(2↦9↦9↦9),(3↦4↦1↦1),(3↦4↦2↦2),(3↦4↦3↦3),(3↦4↦4↦4),(3↦4↦5↦5),(3↦4↦6↦6),(3↦4↦7↦7),(3↦4↦8↦8),(3↦4↦9↦9),(3↦5↦1↦1),(3↦5↦2↦2),(3↦5↦3↦3),(3↦5↦4↦4),(3↦5↦5↦5),(3↦5↦6↦6),(3↦5↦7↦7),(3↦5↦8↦8),(3↦5↦9↦9),(3↦6↦1↦1),(3↦6↦2↦2),(3↦6↦3↦3),(3↦6↦4↦4),(3↦6↦5↦5),(3↦6↦6↦6),(3↦6↦7↦7),(3↦6↦8↦8),(3↦6↦9↦9),(3↦7↦1↦1),(3↦7↦2↦2),(3↦7↦3↦3),(3↦7↦4↦4),(3↦7↦5↦5),(3↦7↦6↦6),(3↦7↦7↦7),(3↦7↦8↦8),(3↦7↦9↦9),(3↦8↦1↦1),(3↦8↦2↦2),(3↦8↦3↦3),(3↦8↦4↦4),(3↦8↦5↦5),(3↦8↦6↦6),(3↦8↦7↦7),(3↦8↦8↦8),(3↦8↦9↦9),(3↦9↦1↦1),(3↦9↦2↦2),(3↦9↦3↦3),(3↦9↦4↦4),(3↦9↦5↦5),(3↦9↦6↦6),(3↦9↦7↦7),(3↦9↦8↦8),(3↦9↦9↦9),(4↦5↦1↦1),(4↦5↦2↦2),(4↦5↦3↦3),(4↦5↦4↦4),(4↦5↦5↦5),(4↦5↦6↦6),(4↦5↦7↦7),(4↦5↦8↦8),(4↦5↦9↦9),(4↦6↦1↦1),(4↦6↦2↦2),(4↦6↦3↦3),(4↦6↦4↦4),(4↦6↦5↦5),(4↦6↦6↦6),(4↦6↦7↦7),(4↦6↦8↦8),(4↦6↦9↦9),(4↦7↦1↦1),(4↦7↦2↦2),(4↦7↦3↦3),(4↦7↦4↦4),(4↦7↦5↦5),(4↦7↦6↦6),(4↦7↦7↦7),(4↦7↦8↦8),(4↦7↦9↦9),(4↦8↦1↦1),(4↦8↦2↦2),(4↦8↦3↦3),(4↦8↦4↦4),(4↦8↦5↦5),(4↦8↦6↦6),(4↦8↦7↦7),(4↦8↦8↦8),(4↦8↦9↦9),(4↦9↦1↦1),(4↦9↦2↦2),(4↦9↦3↦3),(4↦9↦4↦4),(4↦9↦5↦5),(4↦9↦6↦6),(4↦9↦7↦7),(4↦9↦8↦8),(4↦9↦9↦9),(5↦6↦1↦1),(5↦6↦2↦2),(5↦6↦3↦3),(5↦6↦4↦4),(5↦6↦5↦5),(5↦6↦6↦6),(5↦6↦7↦7),(5↦6↦8↦8),(5↦6↦9↦9),(5↦7↦1↦1),(5↦7↦2↦2),(5↦7↦3↦3),(5↦7↦4↦4),(5↦7↦5↦5),(5↦7↦6↦6),(5↦7↦7↦7),(5↦7↦8↦8),(5↦7↦9↦9),(5↦8↦1↦1),(5↦8↦2↦2),(5↦8↦3↦3),(5↦8↦4↦4),(5↦8↦5↦5),(5↦8↦6↦6),(5↦8↦7↦7),(5↦8↦8↦8),(5↦8↦9↦9),(5↦9↦1↦1),(5↦9↦2↦2),(5↦9↦3↦3),(5↦9↦4↦4),(5↦9↦5↦5),(5↦9↦6↦6),(5↦9↦7↦7),(5↦9↦8↦8),(5↦9↦9↦9),(6↦7↦1↦1),(6↦7↦2↦2),(6↦7↦3↦3),(6↦7↦4↦4),(6↦7↦5↦5),(6↦7↦6↦6),(6↦7↦7↦7),(6↦7↦8↦8),(6↦7↦9↦9),(6↦8↦1↦1),(6↦8↦2↦2),(6↦8↦3↦3),(6↦8↦4↦4),(6↦8↦5↦5),(6↦8↦6↦6),(6↦8↦7↦7),(6↦8↦8↦8),(6↦8↦9↦9),(6↦9↦1↦1),(6↦9↦2↦2),(6↦9↦3↦3),(6↦9↦4↦4),(6↦9↦5↦5),(6↦9↦6↦6),(6↦9↦7↦7),(6↦9↦8↦8),(6↦9↦9↦9),(7↦8↦1↦1),(7↦8↦2↦2),(7↦8↦3↦3),(7↦8↦4↦4),(7↦8↦5↦5),(7↦8↦6↦6),(7↦8↦7↦7),(7↦8↦8↦8),(7↦8↦9↦9),(7↦9↦1↦1),(7↦9↦2↦2),(7↦9↦3↦3),(7↦9↦4↦4),(7↦9↦5↦5),(7↦9↦6↦6),(7↦9↦7↦7),(7↦9↦8↦8),(7↦9↦9↦9),(8↦9↦1↦1),(8↦9↦2↦2),(8↦9↦3↦3),(8↦9↦4↦4),(8↦9↦5↦5),(8↦9↦6↦6),(8↦9↦7↦7),(8↦9↦8↦8),(8↦9↦9↦9)}\n", + "\tSUBSQ = {{1,2,3},{4,5,6},{7,8,9}}" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Board : DOM --> (DOM --> DOM) & !(x1,x2,y1,y2).((x1,x2,y1,y2):Diff1\\/Diff2 => Board(x1)(y1) /= Board(x2)(y2))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now, we will try and complete the constraints and put pairs of co-ordinates within each sub-square into the variable `Diff3`, and computing the union of `Diff1`, `Diff2` and `Diff3`:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{(1\\mapsto 1\\mapsto 2\\mapsto 1),(1\\mapsto 1\\mapsto 3\\mapsto 1),(1\\mapsto 1\\mapsto 3\\mapsto 2),(1\\mapsto 1\\mapsto 5\\mapsto 4),(1\\mapsto 1\\mapsto 6\\mapsto 4),(1\\mapsto 1\\mapsto 6\\mapsto 5),(1\\mapsto 1\\mapsto 8\\mapsto 7),(1\\mapsto 1\\mapsto 9\\mapsto 7),(1\\mapsto 1\\mapsto 9\\mapsto 8),(2\\mapsto 1\\mapsto 1\\mapsto 1),(2\\mapsto 1\\mapsto 1\\mapsto 2),(2\\mapsto 1\\mapsto 1\\mapsto 3),(2\\mapsto 1\\mapsto 2\\mapsto 1),(2\\mapsto 1\\mapsto 2\\mapsto 2),(2\\mapsto 1\\mapsto 2\\mapsto 3),(2\\mapsto 1\\mapsto 3\\mapsto 1),(2\\mapsto 1\\mapsto 3\\mapsto 2),(2\\mapsto 1\\mapsto 3\\mapsto 3),(2\\mapsto 1\\mapsto 4\\mapsto 4),(2\\mapsto 1\\mapsto 4\\mapsto 5),(2\\mapsto 1\\mapsto 4\\mapsto 6),(2\\mapsto 1\\mapsto 5\\mapsto 4),(2\\mapsto 1\\mapsto 5\\mapsto 5),(2\\mapsto 1\\mapsto 5\\mapsto 6),(2\\mapsto 1\\mapsto 6\\mapsto 4),(2\\mapsto 1\\mapsto 6\\mapsto 5),(2\\mapsto 1\\mapsto 6\\mapsto 6),(2\\mapsto 1\\mapsto 7\\mapsto 7),(2\\mapsto 1\\mapsto 7\\mapsto 8),(2\\mapsto 1\\mapsto 7\\mapsto 9),(2\\mapsto 1\\mapsto 8\\mapsto 7),(2\\mapsto 1\\mapsto 8\\mapsto 8),(2\\mapsto 1\\mapsto 8\\mapsto 9),(2\\mapsto 1\\mapsto 9\\mapsto 7),(2\\mapsto 1\\mapsto 9\\mapsto 8),(2\\mapsto 1\\mapsto 9\\mapsto 9),(2\\mapsto 2\\mapsto 2\\mapsto 1),(2\\mapsto 2\\mapsto 3\\mapsto 1),(2\\mapsto 2\\mapsto 3\\mapsto 2),(2\\mapsto 2\\mapsto 5\\mapsto 4),(2\\mapsto 2\\mapsto 6\\mapsto 4),(2\\mapsto 2\\mapsto 6\\mapsto 5),(2\\mapsto 2\\mapsto 8\\mapsto 7),(2\\mapsto 2\\mapsto 9\\mapsto 7),(2\\mapsto 2\\mapsto 9\\mapsto 8),(3\\mapsto 1\\mapsto 1\\mapsto 1),(3\\mapsto 1\\mapsto 1\\mapsto 2),(3\\mapsto 1\\mapsto 1\\mapsto 3),(3\\mapsto 1\\mapsto 2\\mapsto 1),(3\\mapsto 1\\mapsto 2\\mapsto 2),(3\\mapsto 1\\mapsto 2\\mapsto 3),(3\\mapsto 1\\mapsto 3\\mapsto 1),(3\\mapsto 1\\mapsto 3\\mapsto 2),(3\\mapsto 1\\mapsto 3\\mapsto 3),(3\\mapsto 1\\mapsto 4\\mapsto 4),(3\\mapsto 1\\mapsto 4\\mapsto 5),(3\\mapsto 1\\mapsto 4\\mapsto 6),(3\\mapsto 1\\mapsto 5\\mapsto 4),(3\\mapsto 1\\mapsto 5\\mapsto 5),(3\\mapsto 1\\mapsto 5\\mapsto 6),(3\\mapsto 1\\mapsto 6\\mapsto 4),(3\\mapsto 1\\mapsto 6\\mapsto 5),(3\\mapsto 1\\mapsto 6\\mapsto 6),(3\\mapsto 1\\mapsto 7\\mapsto 7),(3\\mapsto 1\\mapsto 7\\mapsto 8),(3\\mapsto 1\\mapsto 7\\mapsto 9),(3\\mapsto 1\\mapsto 8\\mapsto 7),(3\\mapsto 1\\mapsto 8\\mapsto 8),(3\\mapsto 1\\mapsto 8\\mapsto 9),(3\\mapsto 1\\mapsto 9\\mapsto 7),(3\\mapsto 1\\mapsto 9\\mapsto 8),(3\\mapsto 1\\mapsto 9\\mapsto 9),(3\\mapsto 2\\mapsto 1\\mapsto 1),(3\\mapsto 2\\mapsto 1\\mapsto 2),(3\\mapsto 2\\mapsto 1\\mapsto 3),(3\\mapsto 2\\mapsto 2\\mapsto 1),(3\\mapsto 2\\mapsto 2\\mapsto 2),(3\\mapsto 2\\mapsto 2\\mapsto 3),(3\\mapsto 2\\mapsto 3\\mapsto 1),(3\\mapsto 2\\mapsto 3\\mapsto 2),(3\\mapsto 2\\mapsto 3\\mapsto 3),(3\\mapsto 2\\mapsto 4\\mapsto 4),(3\\mapsto 2\\mapsto 4\\mapsto 5),(3\\mapsto 2\\mapsto 4\\mapsto 6),(3\\mapsto 2\\mapsto 5\\mapsto 4),(3\\mapsto 2\\mapsto 5\\mapsto 5),(3\\mapsto 2\\mapsto 5\\mapsto 6),(3\\mapsto 2\\mapsto 6\\mapsto 4),(3\\mapsto 2\\mapsto 6\\mapsto 5),(3\\mapsto 2\\mapsto 6\\mapsto 6),(3\\mapsto 2\\mapsto 7\\mapsto 7),(3\\mapsto 2\\mapsto 7\\mapsto 8),(3\\mapsto 2\\mapsto 7\\mapsto 9),(3\\mapsto 2\\mapsto 8\\mapsto 7),(3\\mapsto 2\\mapsto 8\\mapsto 8),(3\\mapsto 2\\mapsto 8\\mapsto 9),(3\\mapsto 2\\mapsto 9\\mapsto 7),(3\\mapsto 2\\mapsto 9\\mapsto 8),(3\\mapsto 2\\mapsto 9\\mapsto 9),(3\\mapsto 3\\mapsto 2\\mapsto 1),(3\\mapsto 3\\mapsto 3\\mapsto 1),(3\\mapsto 3\\mapsto 3\\mapsto 2),(3\\mapsto 3\\mapsto 5\\mapsto 4),(3\\mapsto 3\\mapsto 6\\mapsto 4),(3\\mapsto 3\\mapsto 6\\mapsto 5),(3\\mapsto 3\\mapsto 8\\mapsto 7),(3\\mapsto 3\\mapsto 9\\mapsto 7),(3\\mapsto 3\\mapsto 9\\mapsto 8),(4\\mapsto 4\\mapsto 2\\mapsto 1),(4\\mapsto 4\\mapsto 3\\mapsto 1),(4\\mapsto 4\\mapsto 3\\mapsto 2),(4\\mapsto 4\\mapsto 5\\mapsto 4),(4\\mapsto 4\\mapsto 6\\mapsto 4),(4\\mapsto 4\\mapsto 6\\mapsto 5),(4\\mapsto 4\\mapsto 8\\mapsto 7),(4\\mapsto 4\\mapsto 9\\mapsto 7),(4\\mapsto 4\\mapsto 9\\mapsto 8),(5\\mapsto 4\\mapsto 1\\mapsto 1),(5\\mapsto 4\\mapsto 1\\mapsto 2),(5\\mapsto 4\\mapsto 1\\mapsto 3),(5\\mapsto 4\\mapsto 2\\mapsto 1),(5\\mapsto 4\\mapsto 2\\mapsto 2),(5\\mapsto 4\\mapsto 2\\mapsto 3),(5\\mapsto 4\\mapsto 3\\mapsto 1),(5\\mapsto 4\\mapsto 3\\mapsto 2),(5\\mapsto 4\\mapsto 3\\mapsto 3),(5\\mapsto 4\\mapsto 4\\mapsto 4),(5\\mapsto 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7\\mapsto 8),(8\\mapsto 7\\mapsto 7\\mapsto 9),(8\\mapsto 7\\mapsto 8\\mapsto 7),(8\\mapsto 7\\mapsto 8\\mapsto 8),(8\\mapsto 7\\mapsto 8\\mapsto 9),(8\\mapsto 7\\mapsto 9\\mapsto 7),(8\\mapsto 7\\mapsto 9\\mapsto 8),(8\\mapsto 7\\mapsto 9\\mapsto 9),(8\\mapsto 8\\mapsto 2\\mapsto 1),(8\\mapsto 8\\mapsto 3\\mapsto 1),(8\\mapsto 8\\mapsto 3\\mapsto 2),(8\\mapsto 8\\mapsto 5\\mapsto 4),(8\\mapsto 8\\mapsto 6\\mapsto 4),(8\\mapsto 8\\mapsto 6\\mapsto 5),(8\\mapsto 8\\mapsto 8\\mapsto 7),(8\\mapsto 8\\mapsto 9\\mapsto 7),(8\\mapsto 8\\mapsto 9\\mapsto 8),(9\\mapsto 7\\mapsto 1\\mapsto 1),(9\\mapsto 7\\mapsto 1\\mapsto 2),(9\\mapsto 7\\mapsto 1\\mapsto 3),(9\\mapsto 7\\mapsto 2\\mapsto 1),(9\\mapsto 7\\mapsto 2\\mapsto 2),(9\\mapsto 7\\mapsto 2\\mapsto 3),(9\\mapsto 7\\mapsto 3\\mapsto 1),(9\\mapsto 7\\mapsto 3\\mapsto 2),(9\\mapsto 7\\mapsto 3\\mapsto 3),(9\\mapsto 7\\mapsto 4\\mapsto 4),(9\\mapsto 7\\mapsto 4\\mapsto 5),(9\\mapsto 7\\mapsto 4\\mapsto 6),(9\\mapsto 7\\mapsto 5\\mapsto 4),(9\\mapsto 7\\mapsto 5\\mapsto 5),(9\\mapsto 7\\mapsto 5\\mapsto 6),(9\\mapsto 7\\mapsto 6\\mapsto 4),(9\\mapsto 7\\mapsto 6\\mapsto 5),(9\\mapsto 7\\mapsto 6\\mapsto 6),(9\\mapsto 7\\mapsto 7\\mapsto 7),(9\\mapsto 7\\mapsto 7\\mapsto 8),(9\\mapsto 7\\mapsto 7\\mapsto 9),(9\\mapsto 7\\mapsto 8\\mapsto 7),(9\\mapsto 7\\mapsto 8\\mapsto 8),(9\\mapsto 7\\mapsto 8\\mapsto 9),(9\\mapsto 7\\mapsto 9\\mapsto 7),(9\\mapsto 7\\mapsto 9\\mapsto 8),(9\\mapsto 7\\mapsto 9\\mapsto 9),(9\\mapsto 8\\mapsto 1\\mapsto 1),(9\\mapsto 8\\mapsto 1\\mapsto 2),(9\\mapsto 8\\mapsto 1\\mapsto 3),(9\\mapsto 8\\mapsto 2\\mapsto 1),(9\\mapsto 8\\mapsto 2\\mapsto 2),(9\\mapsto 8\\mapsto 2\\mapsto 3),(9\\mapsto 8\\mapsto 3\\mapsto 1),(9\\mapsto 8\\mapsto 3\\mapsto 2),(9\\mapsto 8\\mapsto 3\\mapsto 3),(9\\mapsto 8\\mapsto 4\\mapsto 4),(9\\mapsto 8\\mapsto 4\\mapsto 5),(9\\mapsto 8\\mapsto 4\\mapsto 6),(9\\mapsto 8\\mapsto 5\\mapsto 4),(9\\mapsto 8\\mapsto 5\\mapsto 5),(9\\mapsto 8\\mapsto 5\\mapsto 6),(9\\mapsto 8\\mapsto 6\\mapsto 4),(9\\mapsto 8\\mapsto 6\\mapsto 5),(9\\mapsto 8\\mapsto 6\\mapsto 6),(9\\mapsto 8\\mapsto 7\\mapsto 7),(9\\mapsto 8\\mapsto 7\\mapsto 8),(9\\mapsto 8\\mapsto 7\\mapsto 9),(9\\mapsto 8\\mapsto 8\\mapsto 7),(9\\mapsto 8\\mapsto 8\\mapsto 8),(9\\mapsto 8\\mapsto 8\\mapsto 9),(9\\mapsto 8\\mapsto 9\\mapsto 7),(9\\mapsto 8\\mapsto 9\\mapsto 8),(9\\mapsto 8\\mapsto 9\\mapsto 9),(9\\mapsto 9\\mapsto 2\\mapsto 1),(9\\mapsto 9\\mapsto 3\\mapsto 1),(9\\mapsto 9\\mapsto 3\\mapsto 2),(9\\mapsto 9\\mapsto 5\\mapsto 4),(9\\mapsto 9\\mapsto 6\\mapsto 4),(9\\mapsto 9\\mapsto 6\\mapsto 5),(9\\mapsto 9\\mapsto 8\\mapsto 7),(9\\mapsto 9\\mapsto 9\\mapsto 7),(9\\mapsto 9\\mapsto 9\\mapsto 8)\\}$" + ], + "text/plain": [ + 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+ ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let Diff3 {x1,x2,y1,y2|#(s1,s2).(s1:SUBSQ & s2:SUBSQ & x1:s1 & x2:s1 & x1>=x2 & (x1=x2 => y1>y2) & y1:s2 & y2:s2 & (x1,y1) /= (x2,y2))}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "A full Sudoku solution, with distinct values in each row, column and sub-square, can now be found as follows:" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\mathit{TRUE}$\n", + "\n", + "**Solution:**\n", + "* $\\mathit{Diff3} = \\{(1\\mapsto 1\\mapsto 2\\mapsto 1),(1\\mapsto 1\\mapsto 3\\mapsto 1),(1\\mapsto 1\\mapsto 3\\mapsto 2),(1\\mapsto 1\\mapsto 5\\mapsto 4),(1\\mapsto 1\\mapsto 6\\mapsto 4),(1\\mapsto 1\\mapsto 6\\mapsto 5),(1\\mapsto 1\\mapsto 8\\mapsto 7),(1\\mapsto 1\\mapsto 9\\mapsto 7),(1\\mapsto 1\\mapsto 9\\mapsto 8),(2\\mapsto 1\\mapsto 1\\mapsto 1),(2\\mapsto 1\\mapsto 1\\mapsto 2),(2\\mapsto 1\\mapsto 1\\mapsto 3),(2\\mapsto 1\\mapsto 2\\mapsto 1),(2\\mapsto 1\\mapsto 2\\mapsto 2),(2\\mapsto 1\\mapsto 2\\mapsto 3),(2\\mapsto 1\\mapsto 3\\mapsto 1),(2\\mapsto 1\\mapsto 3\\mapsto 2),(2\\mapsto 1\\mapsto 3\\mapsto 3),(2\\mapsto 1\\mapsto 4\\mapsto 4),(2\\mapsto 1\\mapsto 4\\mapsto 5),(2\\mapsto 1\\mapsto 4\\mapsto 6),(2\\mapsto 1\\mapsto 5\\mapsto 4),(2\\mapsto 1\\mapsto 5\\mapsto 5),(2\\mapsto 1\\mapsto 5\\mapsto 6),(2\\mapsto 1\\mapsto 6\\mapsto 4),(2\\mapsto 1\\mapsto 6\\mapsto 5),(2\\mapsto 1\\mapsto 6\\mapsto 6),(2\\mapsto 1\\mapsto 7\\mapsto 7),(2\\mapsto 1\\mapsto 7\\mapsto 8),(2\\mapsto 1\\mapsto 7\\mapsto 9),(2\\mapsto 1\\mapsto 8\\mapsto 7),(2\\mapsto 1\\mapsto 8\\mapsto 8),(2\\mapsto 1\\mapsto 8\\mapsto 9),(2\\mapsto 1\\mapsto 9\\mapsto 7),(2\\mapsto 1\\mapsto 9\\mapsto 8),(2\\mapsto 1\\mapsto 9\\mapsto 9),(2\\mapsto 2\\mapsto 2\\mapsto 1),(2\\mapsto 2\\mapsto 3\\mapsto 1),(2\\mapsto 2\\mapsto 3\\mapsto 2),(2\\mapsto 2\\mapsto 5\\mapsto 4),(2\\mapsto 2\\mapsto 6\\mapsto 4),(2\\mapsto 2\\mapsto 6\\mapsto 5),(2\\mapsto 2\\mapsto 8\\mapsto 7),(2\\mapsto 2\\mapsto 9\\mapsto 7),(2\\mapsto 2\\mapsto 9\\mapsto 8),(3\\mapsto 1\\mapsto 1\\mapsto 1),(3\\mapsto 1\\mapsto 1\\mapsto 2),(3\\mapsto 1\\mapsto 1\\mapsto 3),(3\\mapsto 1\\mapsto 2\\mapsto 1),(3\\mapsto 1\\mapsto 2\\mapsto 2),(3\\mapsto 1\\mapsto 2\\mapsto 3),(3\\mapsto 1\\mapsto 3\\mapsto 1),(3\\mapsto 1\\mapsto 3\\mapsto 2),(3\\mapsto 1\\mapsto 3\\mapsto 3),(3\\mapsto 1\\mapsto 4\\mapsto 4),(3\\mapsto 1\\mapsto 4\\mapsto 5),(3\\mapsto 1\\mapsto 4\\mapsto 6),(3\\mapsto 1\\mapsto 5\\mapsto 4),(3\\mapsto 1\\mapsto 5\\mapsto 5),(3\\mapsto 1\\mapsto 5\\mapsto 6),(3\\mapsto 1\\mapsto 6\\mapsto 4),(3\\mapsto 1\\mapsto 6\\mapsto 5),(3\\mapsto 1\\mapsto 6\\mapsto 6),(3\\mapsto 1\\mapsto 7\\mapsto 7),(3\\mapsto 1\\mapsto 7\\mapsto 8),(3\\mapsto 1\\mapsto 7\\mapsto 9),(3\\mapsto 1\\mapsto 8\\mapsto 7),(3\\mapsto 1\\mapsto 8\\mapsto 8),(3\\mapsto 1\\mapsto 8\\mapsto 9),(3\\mapsto 1\\mapsto 9\\mapsto 7),(3\\mapsto 1\\mapsto 9\\mapsto 8),(3\\mapsto 1\\mapsto 9\\mapsto 9),(3\\mapsto 2\\mapsto 1\\mapsto 1),(3\\mapsto 2\\mapsto 1\\mapsto 2),(3\\mapsto 2\\mapsto 1\\mapsto 3),(3\\mapsto 2\\mapsto 2\\mapsto 1),(3\\mapsto 2\\mapsto 2\\mapsto 2),(3\\mapsto 2\\mapsto 2\\mapsto 3),(3\\mapsto 2\\mapsto 3\\mapsto 1),(3\\mapsto 2\\mapsto 3\\mapsto 2),(3\\mapsto 2\\mapsto 3\\mapsto 3),(3\\mapsto 2\\mapsto 4\\mapsto 4),(3\\mapsto 2\\mapsto 4\\mapsto 5),(3\\mapsto 2\\mapsto 4\\mapsto 6),(3\\mapsto 2\\mapsto 5\\mapsto 4),(3\\mapsto 2\\mapsto 5\\mapsto 5),(3\\mapsto 2\\mapsto 5\\mapsto 6),(3\\mapsto 2\\mapsto 6\\mapsto 4),(3\\mapsto 2\\mapsto 6\\mapsto 5),(3\\mapsto 2\\mapsto 6\\mapsto 6),(3\\mapsto 2\\mapsto 7\\mapsto 7),(3\\mapsto 2\\mapsto 7\\mapsto 8),(3\\mapsto 2\\mapsto 7\\mapsto 9),(3\\mapsto 2\\mapsto 8\\mapsto 7),(3\\mapsto 2\\mapsto 8\\mapsto 8),(3\\mapsto 2\\mapsto 8\\mapsto 9),(3\\mapsto 2\\mapsto 9\\mapsto 7),(3\\mapsto 2\\mapsto 9\\mapsto 8),(3\\mapsto 2\\mapsto 9\\mapsto 9),(3\\mapsto 3\\mapsto 2\\mapsto 1),(3\\mapsto 3\\mapsto 3\\mapsto 1),(3\\mapsto 3\\mapsto 3\\mapsto 2),(3\\mapsto 3\\mapsto 5\\mapsto 4),(3\\mapsto 3\\mapsto 6\\mapsto 4),(3\\mapsto 3\\mapsto 6\\mapsto 5),(3\\mapsto 3\\mapsto 8\\mapsto 7),(3\\mapsto 3\\mapsto 9\\mapsto 7),(3\\mapsto 3\\mapsto 9\\mapsto 8),(4\\mapsto 4\\mapsto 2\\mapsto 1),(4\\mapsto 4\\mapsto 3\\mapsto 1),(4\\mapsto 4\\mapsto 3\\mapsto 2),(4\\mapsto 4\\mapsto 5\\mapsto 4),(4\\mapsto 4\\mapsto 6\\mapsto 4),(4\\mapsto 4\\mapsto 6\\mapsto 5),(4\\mapsto 4\\mapsto 8\\mapsto 7),(4\\mapsto 4\\mapsto 9\\mapsto 7),(4\\mapsto 4\\mapsto 9\\mapsto 8),(5\\mapsto 4\\mapsto 1\\mapsto 1),(5\\mapsto 4\\mapsto 1\\mapsto 2),(5\\mapsto 4\\mapsto 1\\mapsto 3),(5\\mapsto 4\\mapsto 2\\mapsto 1),(5\\mapsto 4\\mapsto 2\\mapsto 2),(5\\mapsto 4\\mapsto 2\\mapsto 3),(5\\mapsto 4\\mapsto 3\\mapsto 1),(5\\mapsto 4\\mapsto 3\\mapsto 2),(5\\mapsto 4\\mapsto 3\\mapsto 3),(5\\mapsto 4\\mapsto 4\\mapsto 4),(5\\mapsto 4\\mapsto 4\\mapsto 5),(5\\mapsto 4\\mapsto 4\\mapsto 6),(5\\mapsto 4\\mapsto 5\\mapsto 4),(5\\mapsto 4\\mapsto 5\\mapsto 5),(5\\mapsto 4\\mapsto 5\\mapsto 6),(5\\mapsto 4\\mapsto 6\\mapsto 4),(5\\mapsto 4\\mapsto 6\\mapsto 5),(5\\mapsto 4\\mapsto 6\\mapsto 6),(5\\mapsto 4\\mapsto 7\\mapsto 7),(5\\mapsto 4\\mapsto 7\\mapsto 8),(5\\mapsto 4\\mapsto 7\\mapsto 9),(5\\mapsto 4\\mapsto 8\\mapsto 7),(5\\mapsto 4\\mapsto 8\\mapsto 8),(5\\mapsto 4\\mapsto 8\\mapsto 9),(5\\mapsto 4\\mapsto 9\\mapsto 7),(5\\mapsto 4\\mapsto 9\\mapsto 8),(5\\mapsto 4\\mapsto 9\\mapsto 9),(5\\mapsto 5\\mapsto 2\\mapsto 1),(5\\mapsto 5\\mapsto 3\\mapsto 1),(5\\mapsto 5\\mapsto 3\\mapsto 2),(5\\mapsto 5\\mapsto 5\\mapsto 4),(5\\mapsto 5\\mapsto 6\\mapsto 4),(5\\mapsto 5\\mapsto 6\\mapsto 5),(5\\mapsto 5\\mapsto 8\\mapsto 7),(5\\mapsto 5\\mapsto 9\\mapsto 7),(5\\mapsto 5\\mapsto 9\\mapsto 8),(6\\mapsto 4\\mapsto 1\\mapsto 1),(6\\mapsto 4\\mapsto 1\\mapsto 2),(6\\mapsto 4\\mapsto 1\\mapsto 3),(6\\mapsto 4\\mapsto 2\\mapsto 1),(6\\mapsto 4\\mapsto 2\\mapsto 2),(6\\mapsto 4\\mapsto 2\\mapsto 3),(6\\mapsto 4\\mapsto 3\\mapsto 1),(6\\mapsto 4\\mapsto 3\\mapsto 2),(6\\mapsto 4\\mapsto 3\\mapsto 3),(6\\mapsto 4\\mapsto 4\\mapsto 4),(6\\mapsto 4\\mapsto 4\\mapsto 5),(6\\mapsto 4\\mapsto 4\\mapsto 6),(6\\mapsto 4\\mapsto 5\\mapsto 4),(6\\mapsto 4\\mapsto 5\\mapsto 5),(6\\mapsto 4\\mapsto 5\\mapsto 6),(6\\mapsto 4\\mapsto 6\\mapsto 4),(6\\mapsto 4\\mapsto 6\\mapsto 5),(6\\mapsto 4\\mapsto 6\\mapsto 6),(6\\mapsto 4\\mapsto 7\\mapsto 7),(6\\mapsto 4\\mapsto 7\\mapsto 8),(6\\mapsto 4\\mapsto 7\\mapsto 9),(6\\mapsto 4\\mapsto 8\\mapsto 7),(6\\mapsto 4\\mapsto 8\\mapsto 8),(6\\mapsto 4\\mapsto 8\\mapsto 9),(6\\mapsto 4\\mapsto 9\\mapsto 7),(6\\mapsto 4\\mapsto 9\\mapsto 8),(6\\mapsto 4\\mapsto 9\\mapsto 9),(6\\mapsto 5\\mapsto 1\\mapsto 1),(6\\mapsto 5\\mapsto 1\\mapsto 2),(6\\mapsto 5\\mapsto 1\\mapsto 3),(6\\mapsto 5\\mapsto 2\\mapsto 1),(6\\mapsto 5\\mapsto 2\\mapsto 2),(6\\mapsto 5\\mapsto 2\\mapsto 3),(6\\mapsto 5\\mapsto 3\\mapsto 1),(6\\mapsto 5\\mapsto 3\\mapsto 2),(6\\mapsto 5\\mapsto 3\\mapsto 3),(6\\mapsto 5\\mapsto 4\\mapsto 4),(6\\mapsto 5\\mapsto 4\\mapsto 5),(6\\mapsto 5\\mapsto 4\\mapsto 6),(6\\mapsto 5\\mapsto 5\\mapsto 4),(6\\mapsto 5\\mapsto 5\\mapsto 5),(6\\mapsto 5\\mapsto 5\\mapsto 6),(6\\mapsto 5\\mapsto 6\\mapsto 4),(6\\mapsto 5\\mapsto 6\\mapsto 5),(6\\mapsto 5\\mapsto 6\\mapsto 6),(6\\mapsto 5\\mapsto 7\\mapsto 7),(6\\mapsto 5\\mapsto 7\\mapsto 8),(6\\mapsto 5\\mapsto 7\\mapsto 9),(6\\mapsto 5\\mapsto 8\\mapsto 7),(6\\mapsto 5\\mapsto 8\\mapsto 8),(6\\mapsto 5\\mapsto 8\\mapsto 9),(6\\mapsto 5\\mapsto 9\\mapsto 7),(6\\mapsto 5\\mapsto 9\\mapsto 8),(6\\mapsto 5\\mapsto 9\\mapsto 9),(6\\mapsto 6\\mapsto 2\\mapsto 1),(6\\mapsto 6\\mapsto 3\\mapsto 1),(6\\mapsto 6\\mapsto 3\\mapsto 2),(6\\mapsto 6\\mapsto 5\\mapsto 4),(6\\mapsto 6\\mapsto 6\\mapsto 4),(6\\mapsto 6\\mapsto 6\\mapsto 5),(6\\mapsto 6\\mapsto 8\\mapsto 7),(6\\mapsto 6\\mapsto 9\\mapsto 7),(6\\mapsto 6\\mapsto 9\\mapsto 8),(7\\mapsto 7\\mapsto 2\\mapsto 1),(7\\mapsto 7\\mapsto 3\\mapsto 1),(7\\mapsto 7\\mapsto 3\\mapsto 2),(7\\mapsto 7\\mapsto 5\\mapsto 4),(7\\mapsto 7\\mapsto 6\\mapsto 4),(7\\mapsto 7\\mapsto 6\\mapsto 5),(7\\mapsto 7\\mapsto 8\\mapsto 7),(7\\mapsto 7\\mapsto 9\\mapsto 7),(7\\mapsto 7\\mapsto 9\\mapsto 8),(8\\mapsto 7\\mapsto 1\\mapsto 1),(8\\mapsto 7\\mapsto 1\\mapsto 2),(8\\mapsto 7\\mapsto 1\\mapsto 3),(8\\mapsto 7\\mapsto 2\\mapsto 1),(8\\mapsto 7\\mapsto 2\\mapsto 2),(8\\mapsto 7\\mapsto 2\\mapsto 3),(8\\mapsto 7\\mapsto 3\\mapsto 1),(8\\mapsto 7\\mapsto 3\\mapsto 2),(8\\mapsto 7\\mapsto 3\\mapsto 3),(8\\mapsto 7\\mapsto 4\\mapsto 4),(8\\mapsto 7\\mapsto 4\\mapsto 5),(8\\mapsto 7\\mapsto 4\\mapsto 6),(8\\mapsto 7\\mapsto 5\\mapsto 4),(8\\mapsto 7\\mapsto 5\\mapsto 5),(8\\mapsto 7\\mapsto 5\\mapsto 6),(8\\mapsto 7\\mapsto 6\\mapsto 4),(8\\mapsto 7\\mapsto 6\\mapsto 5),(8\\mapsto 7\\mapsto 6\\mapsto 6),(8\\mapsto 7\\mapsto 7\\mapsto 7),(8\\mapsto 7\\mapsto 7\\mapsto 8),(8\\mapsto 7\\mapsto 7\\mapsto 9),(8\\mapsto 7\\mapsto 8\\mapsto 7),(8\\mapsto 7\\mapsto 8\\mapsto 8),(8\\mapsto 7\\mapsto 8\\mapsto 9),(8\\mapsto 7\\mapsto 9\\mapsto 7),(8\\mapsto 7\\mapsto 9\\mapsto 8),(8\\mapsto 7\\mapsto 9\\mapsto 9),(8\\mapsto 8\\mapsto 2\\mapsto 1),(8\\mapsto 8\\mapsto 3\\mapsto 1),(8\\mapsto 8\\mapsto 3\\mapsto 2),(8\\mapsto 8\\mapsto 5\\mapsto 4),(8\\mapsto 8\\mapsto 6\\mapsto 4),(8\\mapsto 8\\mapsto 6\\mapsto 5),(8\\mapsto 8\\mapsto 8\\mapsto 7),(8\\mapsto 8\\mapsto 9\\mapsto 7),(8\\mapsto 8\\mapsto 9\\mapsto 8),(9\\mapsto 7\\mapsto 1\\mapsto 1),(9\\mapsto 7\\mapsto 1\\mapsto 2),(9\\mapsto 7\\mapsto 1\\mapsto 3),(9\\mapsto 7\\mapsto 2\\mapsto 1),(9\\mapsto 7\\mapsto 2\\mapsto 2),(9\\mapsto 7\\mapsto 2\\mapsto 3),(9\\mapsto 7\\mapsto 3\\mapsto 1),(9\\mapsto 7\\mapsto 3\\mapsto 2),(9\\mapsto 7\\mapsto 3\\mapsto 3),(9\\mapsto 7\\mapsto 4\\mapsto 4),(9\\mapsto 7\\mapsto 4\\mapsto 5),(9\\mapsto 7\\mapsto 4\\mapsto 6),(9\\mapsto 7\\mapsto 5\\mapsto 4),(9\\mapsto 7\\mapsto 5\\mapsto 5),(9\\mapsto 7\\mapsto 5\\mapsto 6),(9\\mapsto 7\\mapsto 6\\mapsto 4),(9\\mapsto 7\\mapsto 6\\mapsto 5),(9\\mapsto 7\\mapsto 6\\mapsto 6),(9\\mapsto 7\\mapsto 7\\mapsto 7),(9\\mapsto 7\\mapsto 7\\mapsto 8),(9\\mapsto 7\\mapsto 7\\mapsto 9),(9\\mapsto 7\\mapsto 8\\mapsto 7),(9\\mapsto 7\\mapsto 8\\mapsto 8),(9\\mapsto 7\\mapsto 8\\mapsto 9),(9\\mapsto 7\\mapsto 9\\mapsto 7),(9\\mapsto 7\\mapsto 9\\mapsto 8),(9\\mapsto 7\\mapsto 9\\mapsto 9),(9\\mapsto 8\\mapsto 1\\mapsto 1),(9\\mapsto 8\\mapsto 1\\mapsto 2),(9\\mapsto 8\\mapsto 1\\mapsto 3),(9\\mapsto 8\\mapsto 2\\mapsto 1),(9\\mapsto 8\\mapsto 2\\mapsto 2),(9\\mapsto 8\\mapsto 2\\mapsto 3),(9\\mapsto 8\\mapsto 3\\mapsto 1),(9\\mapsto 8\\mapsto 3\\mapsto 2),(9\\mapsto 8\\mapsto 3\\mapsto 3),(9\\mapsto 8\\mapsto 4\\mapsto 4),(9\\mapsto 8\\mapsto 4\\mapsto 5),(9\\mapsto 8\\mapsto 4\\mapsto 6),(9\\mapsto 8\\mapsto 5\\mapsto 4),(9\\mapsto 8\\mapsto 5\\mapsto 5),(9\\mapsto 8\\mapsto 5\\mapsto 6),(9\\mapsto 8\\mapsto 6\\mapsto 4),(9\\mapsto 8\\mapsto 6\\mapsto 5),(9\\mapsto 8\\mapsto 6\\mapsto 6),(9\\mapsto 8\\mapsto 7\\mapsto 7),(9\\mapsto 8\\mapsto 7\\mapsto 8),(9\\mapsto 8\\mapsto 7\\mapsto 9),(9\\mapsto 8\\mapsto 8\\mapsto 7),(9\\mapsto 8\\mapsto 8\\mapsto 8),(9\\mapsto 8\\mapsto 8\\mapsto 9),(9\\mapsto 8\\mapsto 9\\mapsto 7),(9\\mapsto 8\\mapsto 9\\mapsto 8),(9\\mapsto 8\\mapsto 9\\mapsto 9),(9\\mapsto 9\\mapsto 2\\mapsto 1),(9\\mapsto 9\\mapsto 3\\mapsto 1),(9\\mapsto 9\\mapsto 3\\mapsto 2),(9\\mapsto 9\\mapsto 5\\mapsto 4),(9\\mapsto 9\\mapsto 6\\mapsto 4),(9\\mapsto 9\\mapsto 6\\mapsto 5),(9\\mapsto 9\\mapsto 8\\mapsto 7),(9\\mapsto 9\\mapsto 9\\mapsto 7),(9\\mapsto 9\\mapsto 9\\mapsto 8)\\}$\n", + "* $\\mathit{Diff2} = \\{(1\\mapsto 1\\mapsto 1\\mapsto 2),(1\\mapsto 1\\mapsto 1\\mapsto 3),(1\\mapsto 1\\mapsto 1\\mapsto 4),(1\\mapsto 1\\mapsto 1\\mapsto 5),(1\\mapsto 1\\mapsto 1\\mapsto 6),(1\\mapsto 1\\mapsto 1\\mapsto 7),(1\\mapsto 1\\mapsto 1\\mapsto 8),(1\\mapsto 1\\mapsto 1\\mapsto 9),(1\\mapsto 1\\mapsto 2\\mapsto 3),(1\\mapsto 1\\mapsto 2\\mapsto 4),(1\\mapsto 1\\mapsto 2\\mapsto 5),(1\\mapsto 1\\mapsto 2\\mapsto 6),(1\\mapsto 1\\mapsto 2\\mapsto 7),(1\\mapsto 1\\mapsto 2\\mapsto 8),(1\\mapsto 1\\mapsto 2\\mapsto 9),(1\\mapsto 1\\mapsto 3\\mapsto 4),(1\\mapsto 1\\mapsto 3\\mapsto 5),(1\\mapsto 1\\mapsto 3\\mapsto 6),(1\\mapsto 1\\mapsto 3\\mapsto 7),(1\\mapsto 1\\mapsto 3\\mapsto 8),(1\\mapsto 1\\mapsto 3\\mapsto 9),(1\\mapsto 1\\mapsto 4\\mapsto 5),(1\\mapsto 1\\mapsto 4\\mapsto 6),(1\\mapsto 1\\mapsto 4\\mapsto 7),(1\\mapsto 1\\mapsto 4\\mapsto 8),(1\\mapsto 1\\mapsto 4\\mapsto 9),(1\\mapsto 1\\mapsto 5\\mapsto 6),(1\\mapsto 1\\mapsto 5\\mapsto 7),(1\\mapsto 1\\mapsto 5\\mapsto 8),(1\\mapsto 1\\mapsto 5\\mapsto 9),(1\\mapsto 1\\mapsto 6\\mapsto 7),(1\\mapsto 1\\mapsto 6\\mapsto 8),(1\\mapsto 1\\mapsto 6\\mapsto 9),(1\\mapsto 1\\mapsto 7\\mapsto 8),(1\\mapsto 1\\mapsto 7\\mapsto 9),(1\\mapsto 1\\mapsto 8\\mapsto 9),(2\\mapsto 2\\mapsto 1\\mapsto 2),(2\\mapsto 2\\mapsto 1\\mapsto 3),(2\\mapsto 2\\mapsto 1\\mapsto 4),(2\\mapsto 2\\mapsto 1\\mapsto 5),(2\\mapsto 2\\mapsto 1\\mapsto 6),(2\\mapsto 2\\mapsto 1\\mapsto 7),(2\\mapsto 2\\mapsto 1\\mapsto 8),(2\\mapsto 2\\mapsto 1\\mapsto 9),(2\\mapsto 2\\mapsto 2\\mapsto 3),(2\\mapsto 2\\mapsto 2\\mapsto 4),(2\\mapsto 2\\mapsto 2\\mapsto 5),(2\\mapsto 2\\mapsto 2\\mapsto 6),(2\\mapsto 2\\mapsto 2\\mapsto 7),(2\\mapsto 2\\mapsto 2\\mapsto 8),(2\\mapsto 2\\mapsto 2\\mapsto 9),(2\\mapsto 2\\mapsto 3\\mapsto 4),(2\\mapsto 2\\mapsto 3\\mapsto 5),(2\\mapsto 2\\mapsto 3\\mapsto 6),(2\\mapsto 2\\mapsto 3\\mapsto 7),(2\\mapsto 2\\mapsto 3\\mapsto 8),(2\\mapsto 2\\mapsto 3\\mapsto 9),(2\\mapsto 2\\mapsto 4\\mapsto 5),(2\\mapsto 2\\mapsto 4\\mapsto 6),(2\\mapsto 2\\mapsto 4\\mapsto 7),(2\\mapsto 2\\mapsto 4\\mapsto 8),(2\\mapsto 2\\mapsto 4\\mapsto 9),(2\\mapsto 2\\mapsto 5\\mapsto 6),(2\\mapsto 2\\mapsto 5\\mapsto 7),(2\\mapsto 2\\mapsto 5\\mapsto 8),(2\\mapsto 2\\mapsto 5\\mapsto 9),(2\\mapsto 2\\mapsto 6\\mapsto 7),(2\\mapsto 2\\mapsto 6\\mapsto 8),(2\\mapsto 2\\mapsto 6\\mapsto 9),(2\\mapsto 2\\mapsto 7\\mapsto 8),(2\\mapsto 2\\mapsto 7\\mapsto 9),(2\\mapsto 2\\mapsto 8\\mapsto 9),(3\\mapsto 3\\mapsto 1\\mapsto 2),(3\\mapsto 3\\mapsto 1\\mapsto 3),(3\\mapsto 3\\mapsto 1\\mapsto 4),(3\\mapsto 3\\mapsto 1\\mapsto 5),(3\\mapsto 3\\mapsto 1\\mapsto 6),(3\\mapsto 3\\mapsto 1\\mapsto 7),(3\\mapsto 3\\mapsto 1\\mapsto 8),(3\\mapsto 3\\mapsto 1\\mapsto 9),(3\\mapsto 3\\mapsto 2\\mapsto 3),(3\\mapsto 3\\mapsto 2\\mapsto 4),(3\\mapsto 3\\mapsto 2\\mapsto 5),(3\\mapsto 3\\mapsto 2\\mapsto 6),(3\\mapsto 3\\mapsto 2\\mapsto 7),(3\\mapsto 3\\mapsto 2\\mapsto 8),(3\\mapsto 3\\mapsto 2\\mapsto 9),(3\\mapsto 3\\mapsto 3\\mapsto 4),(3\\mapsto 3\\mapsto 3\\mapsto 5),(3\\mapsto 3\\mapsto 3\\mapsto 6),(3\\mapsto 3\\mapsto 3\\mapsto 7),(3\\mapsto 3\\mapsto 3\\mapsto 8),(3\\mapsto 3\\mapsto 3\\mapsto 9),(3\\mapsto 3\\mapsto 4\\mapsto 5),(3\\mapsto 3\\mapsto 4\\mapsto 6),(3\\mapsto 3\\mapsto 4\\mapsto 7),(3\\mapsto 3\\mapsto 4\\mapsto 8),(3\\mapsto 3\\mapsto 4\\mapsto 9),(3\\mapsto 3\\mapsto 5\\mapsto 6),(3\\mapsto 3\\mapsto 5\\mapsto 7),(3\\mapsto 3\\mapsto 5\\mapsto 8),(3\\mapsto 3\\mapsto 5\\mapsto 9),(3\\mapsto 3\\mapsto 6\\mapsto 7),(3\\mapsto 3\\mapsto 6\\mapsto 8),(3\\mapsto 3\\mapsto 6\\mapsto 9),(3\\mapsto 3\\mapsto 7\\mapsto 8),(3\\mapsto 3\\mapsto 7\\mapsto 9),(3\\mapsto 3\\mapsto 8\\mapsto 9),(4\\mapsto 4\\mapsto 1\\mapsto 2),(4\\mapsto 4\\mapsto 1\\mapsto 3),(4\\mapsto 4\\mapsto 1\\mapsto 4),(4\\mapsto 4\\mapsto 1\\mapsto 5),(4\\mapsto 4\\mapsto 1\\mapsto 6),(4\\mapsto 4\\mapsto 1\\mapsto 7),(4\\mapsto 4\\mapsto 1\\mapsto 8),(4\\mapsto 4\\mapsto 1\\mapsto 9),(4\\mapsto 4\\mapsto 2\\mapsto 3),(4\\mapsto 4\\mapsto 2\\mapsto 4),(4\\mapsto 4\\mapsto 2\\mapsto 5),(4\\mapsto 4\\mapsto 2\\mapsto 6),(4\\mapsto 4\\mapsto 2\\mapsto 7),(4\\mapsto 4\\mapsto 2\\mapsto 8),(4\\mapsto 4\\mapsto 2\\mapsto 9),(4\\mapsto 4\\mapsto 3\\mapsto 4),(4\\mapsto 4\\mapsto 3\\mapsto 5),(4\\mapsto 4\\mapsto 3\\mapsto 6),(4\\mapsto 4\\mapsto 3\\mapsto 7),(4\\mapsto 4\\mapsto 3\\mapsto 8),(4\\mapsto 4\\mapsto 3\\mapsto 9),(4\\mapsto 4\\mapsto 4\\mapsto 5),(4\\mapsto 4\\mapsto 4\\mapsto 6),(4\\mapsto 4\\mapsto 4\\mapsto 7),(4\\mapsto 4\\mapsto 4\\mapsto 8),(4\\mapsto 4\\mapsto 4\\mapsto 9),(4\\mapsto 4\\mapsto 5\\mapsto 6),(4\\mapsto 4\\mapsto 5\\mapsto 7),(4\\mapsto 4\\mapsto 5\\mapsto 8),(4\\mapsto 4\\mapsto 5\\mapsto 9),(4\\mapsto 4\\mapsto 6\\mapsto 7),(4\\mapsto 4\\mapsto 6\\mapsto 8),(4\\mapsto 4\\mapsto 6\\mapsto 9),(4\\mapsto 4\\mapsto 7\\mapsto 8),(4\\mapsto 4\\mapsto 7\\mapsto 9),(4\\mapsto 4\\mapsto 8\\mapsto 9),(5\\mapsto 5\\mapsto 1\\mapsto 2),(5\\mapsto 5\\mapsto 1\\mapsto 3),(5\\mapsto 5\\mapsto 1\\mapsto 4),(5\\mapsto 5\\mapsto 1\\mapsto 5),(5\\mapsto 5\\mapsto 1\\mapsto 6),(5\\mapsto 5\\mapsto 1\\mapsto 7),(5\\mapsto 5\\mapsto 1\\mapsto 8),(5\\mapsto 5\\mapsto 1\\mapsto 9),(5\\mapsto 5\\mapsto 2\\mapsto 3),(5\\mapsto 5\\mapsto 2\\mapsto 4),(5\\mapsto 5\\mapsto 2\\mapsto 5),(5\\mapsto 5\\mapsto 2\\mapsto 6),(5\\mapsto 5\\mapsto 2\\mapsto 7),(5\\mapsto 5\\mapsto 2\\mapsto 8),(5\\mapsto 5\\mapsto 2\\mapsto 9),(5\\mapsto 5\\mapsto 3\\mapsto 4),(5\\mapsto 5\\mapsto 3\\mapsto 5),(5\\mapsto 5\\mapsto 3\\mapsto 6),(5\\mapsto 5\\mapsto 3\\mapsto 7),(5\\mapsto 5\\mapsto 3\\mapsto 8),(5\\mapsto 5\\mapsto 3\\mapsto 9),(5\\mapsto 5\\mapsto 4\\mapsto 5),(5\\mapsto 5\\mapsto 4\\mapsto 6),(5\\mapsto 5\\mapsto 4\\mapsto 7),(5\\mapsto 5\\mapsto 4\\mapsto 8),(5\\mapsto 5\\mapsto 4\\mapsto 9),(5\\mapsto 5\\mapsto 5\\mapsto 6),(5\\mapsto 5\\mapsto 5\\mapsto 7),(5\\mapsto 5\\mapsto 5\\mapsto 8),(5\\mapsto 5\\mapsto 5\\mapsto 9),(5\\mapsto 5\\mapsto 6\\mapsto 7),(5\\mapsto 5\\mapsto 6\\mapsto 8),(5\\mapsto 5\\mapsto 6\\mapsto 9),(5\\mapsto 5\\mapsto 7\\mapsto 8),(5\\mapsto 5\\mapsto 7\\mapsto 9),(5\\mapsto 5\\mapsto 8\\mapsto 9),(6\\mapsto 6\\mapsto 1\\mapsto 2),(6\\mapsto 6\\mapsto 1\\mapsto 3),(6\\mapsto 6\\mapsto 1\\mapsto 4),(6\\mapsto 6\\mapsto 1\\mapsto 5),(6\\mapsto 6\\mapsto 1\\mapsto 6),(6\\mapsto 6\\mapsto 1\\mapsto 7),(6\\mapsto 6\\mapsto 1\\mapsto 8),(6\\mapsto 6\\mapsto 1\\mapsto 9),(6\\mapsto 6\\mapsto 2\\mapsto 3),(6\\mapsto 6\\mapsto 2\\mapsto 4),(6\\mapsto 6\\mapsto 2\\mapsto 5),(6\\mapsto 6\\mapsto 2\\mapsto 6),(6\\mapsto 6\\mapsto 2\\mapsto 7),(6\\mapsto 6\\mapsto 2\\mapsto 8),(6\\mapsto 6\\mapsto 2\\mapsto 9),(6\\mapsto 6\\mapsto 3\\mapsto 4),(6\\mapsto 6\\mapsto 3\\mapsto 5),(6\\mapsto 6\\mapsto 3\\mapsto 6),(6\\mapsto 6\\mapsto 3\\mapsto 7),(6\\mapsto 6\\mapsto 3\\mapsto 8),(6\\mapsto 6\\mapsto 3\\mapsto 9),(6\\mapsto 6\\mapsto 4\\mapsto 5),(6\\mapsto 6\\mapsto 4\\mapsto 6),(6\\mapsto 6\\mapsto 4\\mapsto 7),(6\\mapsto 6\\mapsto 4\\mapsto 8),(6\\mapsto 6\\mapsto 4\\mapsto 9),(6\\mapsto 6\\mapsto 5\\mapsto 6),(6\\mapsto 6\\mapsto 5\\mapsto 7),(6\\mapsto 6\\mapsto 5\\mapsto 8),(6\\mapsto 6\\mapsto 5\\mapsto 9),(6\\mapsto 6\\mapsto 6\\mapsto 7),(6\\mapsto 6\\mapsto 6\\mapsto 8),(6\\mapsto 6\\mapsto 6\\mapsto 9),(6\\mapsto 6\\mapsto 7\\mapsto 8),(6\\mapsto 6\\mapsto 7\\mapsto 9),(6\\mapsto 6\\mapsto 8\\mapsto 9),(7\\mapsto 7\\mapsto 1\\mapsto 2),(7\\mapsto 7\\mapsto 1\\mapsto 3),(7\\mapsto 7\\mapsto 1\\mapsto 4),(7\\mapsto 7\\mapsto 1\\mapsto 5),(7\\mapsto 7\\mapsto 1\\mapsto 6),(7\\mapsto 7\\mapsto 1\\mapsto 7),(7\\mapsto 7\\mapsto 1\\mapsto 8),(7\\mapsto 7\\mapsto 1\\mapsto 9),(7\\mapsto 7\\mapsto 2\\mapsto 3),(7\\mapsto 7\\mapsto 2\\mapsto 4),(7\\mapsto 7\\mapsto 2\\mapsto 5),(7\\mapsto 7\\mapsto 2\\mapsto 6),(7\\mapsto 7\\mapsto 2\\mapsto 7),(7\\mapsto 7\\mapsto 2\\mapsto 8),(7\\mapsto 7\\mapsto 2\\mapsto 9),(7\\mapsto 7\\mapsto 3\\mapsto 4),(7\\mapsto 7\\mapsto 3\\mapsto 5),(7\\mapsto 7\\mapsto 3\\mapsto 6),(7\\mapsto 7\\mapsto 3\\mapsto 7),(7\\mapsto 7\\mapsto 3\\mapsto 8),(7\\mapsto 7\\mapsto 3\\mapsto 9),(7\\mapsto 7\\mapsto 4\\mapsto 5),(7\\mapsto 7\\mapsto 4\\mapsto 6),(7\\mapsto 7\\mapsto 4\\mapsto 7),(7\\mapsto 7\\mapsto 4\\mapsto 8),(7\\mapsto 7\\mapsto 4\\mapsto 9),(7\\mapsto 7\\mapsto 5\\mapsto 6),(7\\mapsto 7\\mapsto 5\\mapsto 7),(7\\mapsto 7\\mapsto 5\\mapsto 8),(7\\mapsto 7\\mapsto 5\\mapsto 9),(7\\mapsto 7\\mapsto 6\\mapsto 7),(7\\mapsto 7\\mapsto 6\\mapsto 8),(7\\mapsto 7\\mapsto 6\\mapsto 9),(7\\mapsto 7\\mapsto 7\\mapsto 8),(7\\mapsto 7\\mapsto 7\\mapsto 9),(7\\mapsto 7\\mapsto 8\\mapsto 9),(8\\mapsto 8\\mapsto 1\\mapsto 2),(8\\mapsto 8\\mapsto 1\\mapsto 3),(8\\mapsto 8\\mapsto 1\\mapsto 4),(8\\mapsto 8\\mapsto 1\\mapsto 5),(8\\mapsto 8\\mapsto 1\\mapsto 6),(8\\mapsto 8\\mapsto 1\\mapsto 7),(8\\mapsto 8\\mapsto 1\\mapsto 8),(8\\mapsto 8\\mapsto 1\\mapsto 9),(8\\mapsto 8\\mapsto 2\\mapsto 3),(8\\mapsto 8\\mapsto 2\\mapsto 4),(8\\mapsto 8\\mapsto 2\\mapsto 5),(8\\mapsto 8\\mapsto 2\\mapsto 6),(8\\mapsto 8\\mapsto 2\\mapsto 7),(8\\mapsto 8\\mapsto 2\\mapsto 8),(8\\mapsto 8\\mapsto 2\\mapsto 9),(8\\mapsto 8\\mapsto 3\\mapsto 4),(8\\mapsto 8\\mapsto 3\\mapsto 5),(8\\mapsto 8\\mapsto 3\\mapsto 6),(8\\mapsto 8\\mapsto 3\\mapsto 7),(8\\mapsto 8\\mapsto 3\\mapsto 8),(8\\mapsto 8\\mapsto 3\\mapsto 9),(8\\mapsto 8\\mapsto 4\\mapsto 5),(8\\mapsto 8\\mapsto 4\\mapsto 6),(8\\mapsto 8\\mapsto 4\\mapsto 7),(8\\mapsto 8\\mapsto 4\\mapsto 8),(8\\mapsto 8\\mapsto 4\\mapsto 9),(8\\mapsto 8\\mapsto 5\\mapsto 6),(8\\mapsto 8\\mapsto 5\\mapsto 7),(8\\mapsto 8\\mapsto 5\\mapsto 8),(8\\mapsto 8\\mapsto 5\\mapsto 9),(8\\mapsto 8\\mapsto 6\\mapsto 7),(8\\mapsto 8\\mapsto 6\\mapsto 8),(8\\mapsto 8\\mapsto 6\\mapsto 9),(8\\mapsto 8\\mapsto 7\\mapsto 8),(8\\mapsto 8\\mapsto 7\\mapsto 9),(8\\mapsto 8\\mapsto 8\\mapsto 9),(9\\mapsto 9\\mapsto 1\\mapsto 2),(9\\mapsto 9\\mapsto 1\\mapsto 3),(9\\mapsto 9\\mapsto 1\\mapsto 4),(9\\mapsto 9\\mapsto 1\\mapsto 5),(9\\mapsto 9\\mapsto 1\\mapsto 6),(9\\mapsto 9\\mapsto 1\\mapsto 7),(9\\mapsto 9\\mapsto 1\\mapsto 8),(9\\mapsto 9\\mapsto 1\\mapsto 9),(9\\mapsto 9\\mapsto 2\\mapsto 3),(9\\mapsto 9\\mapsto 2\\mapsto 4),(9\\mapsto 9\\mapsto 2\\mapsto 5),(9\\mapsto 9\\mapsto 2\\mapsto 6),(9\\mapsto 9\\mapsto 2\\mapsto 7),(9\\mapsto 9\\mapsto 2\\mapsto 8),(9\\mapsto 9\\mapsto 2\\mapsto 9),(9\\mapsto 9\\mapsto 3\\mapsto 4),(9\\mapsto 9\\mapsto 3\\mapsto 5),(9\\mapsto 9\\mapsto 3\\mapsto 6),(9\\mapsto 9\\mapsto 3\\mapsto 7),(9\\mapsto 9\\mapsto 3\\mapsto 8),(9\\mapsto 9\\mapsto 3\\mapsto 9),(9\\mapsto 9\\mapsto 4\\mapsto 5),(9\\mapsto 9\\mapsto 4\\mapsto 6),(9\\mapsto 9\\mapsto 4\\mapsto 7),(9\\mapsto 9\\mapsto 4\\mapsto 8),(9\\mapsto 9\\mapsto 4\\mapsto 9),(9\\mapsto 9\\mapsto 5\\mapsto 6),(9\\mapsto 9\\mapsto 5\\mapsto 7),(9\\mapsto 9\\mapsto 5\\mapsto 8),(9\\mapsto 9\\mapsto 5\\mapsto 9),(9\\mapsto 9\\mapsto 6\\mapsto 7),(9\\mapsto 9\\mapsto 6\\mapsto 8),(9\\mapsto 9\\mapsto 6\\mapsto 9),(9\\mapsto 9\\mapsto 7\\mapsto 8),(9\\mapsto 9\\mapsto 7\\mapsto 9),(9\\mapsto 9\\mapsto 8\\mapsto 9)\\}$\n", + "* $\\mathit{Board} = \\{(1\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(2\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(3\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(4\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(5\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(6\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(7\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(8\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(9\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\})\\}$\n", + "* $\\mathit{DOM} = \\{1,2,3,4,5,6,7,8,9\\}$\n", + "* $\\mathit{Diff1} = \\{(1\\mapsto 2\\mapsto 1\\mapsto 1),(1\\mapsto 2\\mapsto 2\\mapsto 2),(1\\mapsto 2\\mapsto 3\\mapsto 3),(1\\mapsto 2\\mapsto 4\\mapsto 4),(1\\mapsto 2\\mapsto 5\\mapsto 5),(1\\mapsto 2\\mapsto 6\\mapsto 6),(1\\mapsto 2\\mapsto 7\\mapsto 7),(1\\mapsto 2\\mapsto 8\\mapsto 8),(1\\mapsto 2\\mapsto 9\\mapsto 9),(1\\mapsto 3\\mapsto 1\\mapsto 1),(1\\mapsto 3\\mapsto 2\\mapsto 2),(1\\mapsto 3\\mapsto 3\\mapsto 3),(1\\mapsto 3\\mapsto 4\\mapsto 4),(1\\mapsto 3\\mapsto 5\\mapsto 5),(1\\mapsto 3\\mapsto 6\\mapsto 6),(1\\mapsto 3\\mapsto 7\\mapsto 7),(1\\mapsto 3\\mapsto 8\\mapsto 8),(1\\mapsto 3\\mapsto 9\\mapsto 9),(1\\mapsto 4\\mapsto 1\\mapsto 1),(1\\mapsto 4\\mapsto 2\\mapsto 2),(1\\mapsto 4\\mapsto 3\\mapsto 3),(1\\mapsto 4\\mapsto 4\\mapsto 4),(1\\mapsto 4\\mapsto 5\\mapsto 5),(1\\mapsto 4\\mapsto 6\\mapsto 6),(1\\mapsto 4\\mapsto 7\\mapsto 7),(1\\mapsto 4\\mapsto 8\\mapsto 8),(1\\mapsto 4\\mapsto 9\\mapsto 9),(1\\mapsto 5\\mapsto 1\\mapsto 1),(1\\mapsto 5\\mapsto 2\\mapsto 2),(1\\mapsto 5\\mapsto 3\\mapsto 3),(1\\mapsto 5\\mapsto 4\\mapsto 4),(1\\mapsto 5\\mapsto 5\\mapsto 5),(1\\mapsto 5\\mapsto 6\\mapsto 6),(1\\mapsto 5\\mapsto 7\\mapsto 7),(1\\mapsto 5\\mapsto 8\\mapsto 8),(1\\mapsto 5\\mapsto 9\\mapsto 9),(1\\mapsto 6\\mapsto 1\\mapsto 1),(1\\mapsto 6\\mapsto 2\\mapsto 2),(1\\mapsto 6\\mapsto 3\\mapsto 3),(1\\mapsto 6\\mapsto 4\\mapsto 4),(1\\mapsto 6\\mapsto 5\\mapsto 5),(1\\mapsto 6\\mapsto 6\\mapsto 6),(1\\mapsto 6\\mapsto 7\\mapsto 7),(1\\mapsto 6\\mapsto 8\\mapsto 8),(1\\mapsto 6\\mapsto 9\\mapsto 9),(1\\mapsto 7\\mapsto 1\\mapsto 1),(1\\mapsto 7\\mapsto 2\\mapsto 2),(1\\mapsto 7\\mapsto 3\\mapsto 3),(1\\mapsto 7\\mapsto 4\\mapsto 4),(1\\mapsto 7\\mapsto 5\\mapsto 5),(1\\mapsto 7\\mapsto 6\\mapsto 6),(1\\mapsto 7\\mapsto 7\\mapsto 7),(1\\mapsto 7\\mapsto 8\\mapsto 8),(1\\mapsto 7\\mapsto 9\\mapsto 9),(1\\mapsto 8\\mapsto 1\\mapsto 1),(1\\mapsto 8\\mapsto 2\\mapsto 2),(1\\mapsto 8\\mapsto 3\\mapsto 3),(1\\mapsto 8\\mapsto 4\\mapsto 4),(1\\mapsto 8\\mapsto 5\\mapsto 5),(1\\mapsto 8\\mapsto 6\\mapsto 6),(1\\mapsto 8\\mapsto 7\\mapsto 7),(1\\mapsto 8\\mapsto 8\\mapsto 8),(1\\mapsto 8\\mapsto 9\\mapsto 9),(1\\mapsto 9\\mapsto 1\\mapsto 1),(1\\mapsto 9\\mapsto 2\\mapsto 2),(1\\mapsto 9\\mapsto 3\\mapsto 3),(1\\mapsto 9\\mapsto 4\\mapsto 4),(1\\mapsto 9\\mapsto 5\\mapsto 5),(1\\mapsto 9\\mapsto 6\\mapsto 6),(1\\mapsto 9\\mapsto 7\\mapsto 7),(1\\mapsto 9\\mapsto 8\\mapsto 8),(1\\mapsto 9\\mapsto 9\\mapsto 9),(2\\mapsto 3\\mapsto 1\\mapsto 1),(2\\mapsto 3\\mapsto 2\\mapsto 2),(2\\mapsto 3\\mapsto 3\\mapsto 3),(2\\mapsto 3\\mapsto 4\\mapsto 4),(2\\mapsto 3\\mapsto 5\\mapsto 5),(2\\mapsto 3\\mapsto 6\\mapsto 6),(2\\mapsto 3\\mapsto 7\\mapsto 7),(2\\mapsto 3\\mapsto 8\\mapsto 8),(2\\mapsto 3\\mapsto 9\\mapsto 9),(2\\mapsto 4\\mapsto 1\\mapsto 1),(2\\mapsto 4\\mapsto 2\\mapsto 2),(2\\mapsto 4\\mapsto 3\\mapsto 3),(2\\mapsto 4\\mapsto 4\\mapsto 4),(2\\mapsto 4\\mapsto 5\\mapsto 5),(2\\mapsto 4\\mapsto 6\\mapsto 6),(2\\mapsto 4\\mapsto 7\\mapsto 7),(2\\mapsto 4\\mapsto 8\\mapsto 8),(2\\mapsto 4\\mapsto 9\\mapsto 9),(2\\mapsto 5\\mapsto 1\\mapsto 1),(2\\mapsto 5\\mapsto 2\\mapsto 2),(2\\mapsto 5\\mapsto 3\\mapsto 3),(2\\mapsto 5\\mapsto 4\\mapsto 4),(2\\mapsto 5\\mapsto 5\\mapsto 5),(2\\mapsto 5\\mapsto 6\\mapsto 6),(2\\mapsto 5\\mapsto 7\\mapsto 7),(2\\mapsto 5\\mapsto 8\\mapsto 8),(2\\mapsto 5\\mapsto 9\\mapsto 9),(2\\mapsto 6\\mapsto 1\\mapsto 1),(2\\mapsto 6\\mapsto 2\\mapsto 2),(2\\mapsto 6\\mapsto 3\\mapsto 3),(2\\mapsto 6\\mapsto 4\\mapsto 4),(2\\mapsto 6\\mapsto 5\\mapsto 5),(2\\mapsto 6\\mapsto 6\\mapsto 6),(2\\mapsto 6\\mapsto 7\\mapsto 7),(2\\mapsto 6\\mapsto 8\\mapsto 8),(2\\mapsto 6\\mapsto 9\\mapsto 9),(2\\mapsto 7\\mapsto 1\\mapsto 1),(2\\mapsto 7\\mapsto 2\\mapsto 2),(2\\mapsto 7\\mapsto 3\\mapsto 3),(2\\mapsto 7\\mapsto 4\\mapsto 4),(2\\mapsto 7\\mapsto 5\\mapsto 5),(2\\mapsto 7\\mapsto 6\\mapsto 6),(2\\mapsto 7\\mapsto 7\\mapsto 7),(2\\mapsto 7\\mapsto 8\\mapsto 8),(2\\mapsto 7\\mapsto 9\\mapsto 9),(2\\mapsto 8\\mapsto 1\\mapsto 1),(2\\mapsto 8\\mapsto 2\\mapsto 2),(2\\mapsto 8\\mapsto 3\\mapsto 3),(2\\mapsto 8\\mapsto 4\\mapsto 4),(2\\mapsto 8\\mapsto 5\\mapsto 5),(2\\mapsto 8\\mapsto 6\\mapsto 6),(2\\mapsto 8\\mapsto 7\\mapsto 7),(2\\mapsto 8\\mapsto 8\\mapsto 8),(2\\mapsto 8\\mapsto 9\\mapsto 9),(2\\mapsto 9\\mapsto 1\\mapsto 1),(2\\mapsto 9\\mapsto 2\\mapsto 2),(2\\mapsto 9\\mapsto 3\\mapsto 3),(2\\mapsto 9\\mapsto 4\\mapsto 4),(2\\mapsto 9\\mapsto 5\\mapsto 5),(2\\mapsto 9\\mapsto 6\\mapsto 6),(2\\mapsto 9\\mapsto 7\\mapsto 7),(2\\mapsto 9\\mapsto 8\\mapsto 8),(2\\mapsto 9\\mapsto 9\\mapsto 9),(3\\mapsto 4\\mapsto 1\\mapsto 1),(3\\mapsto 4\\mapsto 2\\mapsto 2),(3\\mapsto 4\\mapsto 3\\mapsto 3),(3\\mapsto 4\\mapsto 4\\mapsto 4),(3\\mapsto 4\\mapsto 5\\mapsto 5),(3\\mapsto 4\\mapsto 6\\mapsto 6),(3\\mapsto 4\\mapsto 7\\mapsto 7),(3\\mapsto 4\\mapsto 8\\mapsto 8),(3\\mapsto 4\\mapsto 9\\mapsto 9),(3\\mapsto 5\\mapsto 1\\mapsto 1),(3\\mapsto 5\\mapsto 2\\mapsto 2),(3\\mapsto 5\\mapsto 3\\mapsto 3),(3\\mapsto 5\\mapsto 4\\mapsto 4),(3\\mapsto 5\\mapsto 5\\mapsto 5),(3\\mapsto 5\\mapsto 6\\mapsto 6),(3\\mapsto 5\\mapsto 7\\mapsto 7),(3\\mapsto 5\\mapsto 8\\mapsto 8),(3\\mapsto 5\\mapsto 9\\mapsto 9),(3\\mapsto 6\\mapsto 1\\mapsto 1),(3\\mapsto 6\\mapsto 2\\mapsto 2),(3\\mapsto 6\\mapsto 3\\mapsto 3),(3\\mapsto 6\\mapsto 4\\mapsto 4),(3\\mapsto 6\\mapsto 5\\mapsto 5),(3\\mapsto 6\\mapsto 6\\mapsto 6),(3\\mapsto 6\\mapsto 7\\mapsto 7),(3\\mapsto 6\\mapsto 8\\mapsto 8),(3\\mapsto 6\\mapsto 9\\mapsto 9),(3\\mapsto 7\\mapsto 1\\mapsto 1),(3\\mapsto 7\\mapsto 2\\mapsto 2),(3\\mapsto 7\\mapsto 3\\mapsto 3),(3\\mapsto 7\\mapsto 4\\mapsto 4),(3\\mapsto 7\\mapsto 5\\mapsto 5),(3\\mapsto 7\\mapsto 6\\mapsto 6),(3\\mapsto 7\\mapsto 7\\mapsto 7),(3\\mapsto 7\\mapsto 8\\mapsto 8),(3\\mapsto 7\\mapsto 9\\mapsto 9),(3\\mapsto 8\\mapsto 1\\mapsto 1),(3\\mapsto 8\\mapsto 2\\mapsto 2),(3\\mapsto 8\\mapsto 3\\mapsto 3),(3\\mapsto 8\\mapsto 4\\mapsto 4),(3\\mapsto 8\\mapsto 5\\mapsto 5),(3\\mapsto 8\\mapsto 6\\mapsto 6),(3\\mapsto 8\\mapsto 7\\mapsto 7),(3\\mapsto 8\\mapsto 8\\mapsto 8),(3\\mapsto 8\\mapsto 9\\mapsto 9),(3\\mapsto 9\\mapsto 1\\mapsto 1),(3\\mapsto 9\\mapsto 2\\mapsto 2),(3\\mapsto 9\\mapsto 3\\mapsto 3),(3\\mapsto 9\\mapsto 4\\mapsto 4),(3\\mapsto 9\\mapsto 5\\mapsto 5),(3\\mapsto 9\\mapsto 6\\mapsto 6),(3\\mapsto 9\\mapsto 7\\mapsto 7),(3\\mapsto 9\\mapsto 8\\mapsto 8),(3\\mapsto 9\\mapsto 9\\mapsto 9),(4\\mapsto 5\\mapsto 1\\mapsto 1),(4\\mapsto 5\\mapsto 2\\mapsto 2),(4\\mapsto 5\\mapsto 3\\mapsto 3),(4\\mapsto 5\\mapsto 4\\mapsto 4),(4\\mapsto 5\\mapsto 5\\mapsto 5),(4\\mapsto 5\\mapsto 6\\mapsto 6),(4\\mapsto 5\\mapsto 7\\mapsto 7),(4\\mapsto 5\\mapsto 8\\mapsto 8),(4\\mapsto 5\\mapsto 9\\mapsto 9),(4\\mapsto 6\\mapsto 1\\mapsto 1),(4\\mapsto 6\\mapsto 2\\mapsto 2),(4\\mapsto 6\\mapsto 3\\mapsto 3),(4\\mapsto 6\\mapsto 4\\mapsto 4),(4\\mapsto 6\\mapsto 5\\mapsto 5),(4\\mapsto 6\\mapsto 6\\mapsto 6),(4\\mapsto 6\\mapsto 7\\mapsto 7),(4\\mapsto 6\\mapsto 8\\mapsto 8),(4\\mapsto 6\\mapsto 9\\mapsto 9),(4\\mapsto 7\\mapsto 1\\mapsto 1),(4\\mapsto 7\\mapsto 2\\mapsto 2),(4\\mapsto 7\\mapsto 3\\mapsto 3),(4\\mapsto 7\\mapsto 4\\mapsto 4),(4\\mapsto 7\\mapsto 5\\mapsto 5),(4\\mapsto 7\\mapsto 6\\mapsto 6),(4\\mapsto 7\\mapsto 7\\mapsto 7),(4\\mapsto 7\\mapsto 8\\mapsto 8),(4\\mapsto 7\\mapsto 9\\mapsto 9),(4\\mapsto 8\\mapsto 1\\mapsto 1),(4\\mapsto 8\\mapsto 2\\mapsto 2),(4\\mapsto 8\\mapsto 3\\mapsto 3),(4\\mapsto 8\\mapsto 4\\mapsto 4),(4\\mapsto 8\\mapsto 5\\mapsto 5),(4\\mapsto 8\\mapsto 6\\mapsto 6),(4\\mapsto 8\\mapsto 7\\mapsto 7),(4\\mapsto 8\\mapsto 8\\mapsto 8),(4\\mapsto 8\\mapsto 9\\mapsto 9),(4\\mapsto 9\\mapsto 1\\mapsto 1),(4\\mapsto 9\\mapsto 2\\mapsto 2),(4\\mapsto 9\\mapsto 3\\mapsto 3),(4\\mapsto 9\\mapsto 4\\mapsto 4),(4\\mapsto 9\\mapsto 5\\mapsto 5),(4\\mapsto 9\\mapsto 6\\mapsto 6),(4\\mapsto 9\\mapsto 7\\mapsto 7),(4\\mapsto 9\\mapsto 8\\mapsto 8),(4\\mapsto 9\\mapsto 9\\mapsto 9),(5\\mapsto 6\\mapsto 1\\mapsto 1),(5\\mapsto 6\\mapsto 2\\mapsto 2),(5\\mapsto 6\\mapsto 3\\mapsto 3),(5\\mapsto 6\\mapsto 4\\mapsto 4),(5\\mapsto 6\\mapsto 5\\mapsto 5),(5\\mapsto 6\\mapsto 6\\mapsto 6),(5\\mapsto 6\\mapsto 7\\mapsto 7),(5\\mapsto 6\\mapsto 8\\mapsto 8),(5\\mapsto 6\\mapsto 9\\mapsto 9),(5\\mapsto 7\\mapsto 1\\mapsto 1),(5\\mapsto 7\\mapsto 2\\mapsto 2),(5\\mapsto 7\\mapsto 3\\mapsto 3),(5\\mapsto 7\\mapsto 4\\mapsto 4),(5\\mapsto 7\\mapsto 5\\mapsto 5),(5\\mapsto 7\\mapsto 6\\mapsto 6),(5\\mapsto 7\\mapsto 7\\mapsto 7),(5\\mapsto 7\\mapsto 8\\mapsto 8),(5\\mapsto 7\\mapsto 9\\mapsto 9),(5\\mapsto 8\\mapsto 1\\mapsto 1),(5\\mapsto 8\\mapsto 2\\mapsto 2),(5\\mapsto 8\\mapsto 3\\mapsto 3),(5\\mapsto 8\\mapsto 4\\mapsto 4),(5\\mapsto 8\\mapsto 5\\mapsto 5),(5\\mapsto 8\\mapsto 6\\mapsto 6),(5\\mapsto 8\\mapsto 7\\mapsto 7),(5\\mapsto 8\\mapsto 8\\mapsto 8),(5\\mapsto 8\\mapsto 9\\mapsto 9),(5\\mapsto 9\\mapsto 1\\mapsto 1),(5\\mapsto 9\\mapsto 2\\mapsto 2),(5\\mapsto 9\\mapsto 3\\mapsto 3),(5\\mapsto 9\\mapsto 4\\mapsto 4),(5\\mapsto 9\\mapsto 5\\mapsto 5),(5\\mapsto 9\\mapsto 6\\mapsto 6),(5\\mapsto 9\\mapsto 7\\mapsto 7),(5\\mapsto 9\\mapsto 8\\mapsto 8),(5\\mapsto 9\\mapsto 9\\mapsto 9),(6\\mapsto 7\\mapsto 1\\mapsto 1),(6\\mapsto 7\\mapsto 2\\mapsto 2),(6\\mapsto 7\\mapsto 3\\mapsto 3),(6\\mapsto 7\\mapsto 4\\mapsto 4),(6\\mapsto 7\\mapsto 5\\mapsto 5),(6\\mapsto 7\\mapsto 6\\mapsto 6),(6\\mapsto 7\\mapsto 7\\mapsto 7),(6\\mapsto 7\\mapsto 8\\mapsto 8),(6\\mapsto 7\\mapsto 9\\mapsto 9),(6\\mapsto 8\\mapsto 1\\mapsto 1),(6\\mapsto 8\\mapsto 2\\mapsto 2),(6\\mapsto 8\\mapsto 3\\mapsto 3),(6\\mapsto 8\\mapsto 4\\mapsto 4),(6\\mapsto 8\\mapsto 5\\mapsto 5),(6\\mapsto 8\\mapsto 6\\mapsto 6),(6\\mapsto 8\\mapsto 7\\mapsto 7),(6\\mapsto 8\\mapsto 8\\mapsto 8),(6\\mapsto 8\\mapsto 9\\mapsto 9),(6\\mapsto 9\\mapsto 1\\mapsto 1),(6\\mapsto 9\\mapsto 2\\mapsto 2),(6\\mapsto 9\\mapsto 3\\mapsto 3),(6\\mapsto 9\\mapsto 4\\mapsto 4),(6\\mapsto 9\\mapsto 5\\mapsto 5),(6\\mapsto 9\\mapsto 6\\mapsto 6),(6\\mapsto 9\\mapsto 7\\mapsto 7),(6\\mapsto 9\\mapsto 8\\mapsto 8),(6\\mapsto 9\\mapsto 9\\mapsto 9),(7\\mapsto 8\\mapsto 1\\mapsto 1),(7\\mapsto 8\\mapsto 2\\mapsto 2),(7\\mapsto 8\\mapsto 3\\mapsto 3),(7\\mapsto 8\\mapsto 4\\mapsto 4),(7\\mapsto 8\\mapsto 5\\mapsto 5),(7\\mapsto 8\\mapsto 6\\mapsto 6),(7\\mapsto 8\\mapsto 7\\mapsto 7),(7\\mapsto 8\\mapsto 8\\mapsto 8),(7\\mapsto 8\\mapsto 9\\mapsto 9),(7\\mapsto 9\\mapsto 1\\mapsto 1),(7\\mapsto 9\\mapsto 2\\mapsto 2),(7\\mapsto 9\\mapsto 3\\mapsto 3),(7\\mapsto 9\\mapsto 4\\mapsto 4),(7\\mapsto 9\\mapsto 5\\mapsto 5),(7\\mapsto 9\\mapsto 6\\mapsto 6),(7\\mapsto 9\\mapsto 7\\mapsto 7),(7\\mapsto 9\\mapsto 8\\mapsto 8),(7\\mapsto 9\\mapsto 9\\mapsto 9),(8\\mapsto 9\\mapsto 1\\mapsto 1),(8\\mapsto 9\\mapsto 2\\mapsto 2),(8\\mapsto 9\\mapsto 3\\mapsto 3),(8\\mapsto 9\\mapsto 4\\mapsto 4),(8\\mapsto 9\\mapsto 5\\mapsto 5),(8\\mapsto 9\\mapsto 6\\mapsto 6),(8\\mapsto 9\\mapsto 7\\mapsto 7),(8\\mapsto 9\\mapsto 8\\mapsto 8),(8\\mapsto 9\\mapsto 9\\mapsto 9)\\}$\n", + "* $\\mathit{Diff} = \\emptyset$\n", + "* $\\mathit{SUBSQ} = \\{\\{1,2,3\\},\\{4,5,6\\},\\{7,8,9\\}\\}$" + ], + "text/plain": [ + "TRUE\n", + "\n", + "Solution:\n", + "\tDiff3 = {(1↦1↦2↦1),(1↦1↦3↦1),(1↦1↦3↦2),(1↦1↦5↦4),(1↦1↦6↦4),(1↦1↦6↦5),(1↦1↦8↦7),(1↦1↦9↦7),(1↦1↦9↦8),(2↦1↦1↦1),(2↦1↦1↦2),(2↦1↦1↦3),(2↦1↦2↦1),(2↦1↦2↦2),(2↦1↦2↦3),(2↦1↦3↦1),(2↦1↦3↦2),(2↦1↦3↦3),(2↦1↦4↦4),(2↦1↦4↦5),(2↦1↦4↦6),(2↦1↦5↦4),(2↦1↦5↦5),(2↦1↦5↦6),(2↦1↦6↦4),(2↦1↦6↦5),(2↦1↦6↦6),(2↦1↦7↦7),(2↦1↦7↦8),(2↦1↦7↦9),(2↦1↦8↦7),(2↦1↦8↦8),(2↦1↦8↦9),(2↦1↦9↦7),(2↦1↦9↦8),(2↦1↦9↦9),(2↦2↦2↦1),(2↦2↦3↦1),(2↦2↦3↦2),(2↦2↦5↦4),(2↦2↦6↦4),(2↦2↦6↦5),(2↦2↦8↦7),(2↦2↦9↦7),(2↦2↦9↦8),(3↦1↦1↦1),(3↦1↦1↦2),(3↦1↦1↦3),(3↦1↦2↦1),(3↦1↦2↦2),(3↦1↦2↦3),(3↦1↦3↦1),(3↦1↦3↦2),(3↦1↦3↦3),(3↦1↦4↦4),(3↦1↦4↦5),(3↦1↦4↦6),(3↦1↦5↦4),(3↦1↦5↦5),(3↦1↦5↦6),(3↦1↦6↦4),(3↦1↦6↦5),(3↦1↦6↦6),(3↦1↦7↦7),(3↦1↦7↦8),(3↦1↦7↦9),(3↦1↦8↦7),(3↦1↦8↦8),(3↦1↦8↦9),(3↦1↦9↦7),(3↦1↦9↦8),(3↦1↦9↦9),(3↦2↦1↦1),(3↦2↦1↦2),(3↦2↦1↦3),(3↦2↦2↦1),(3↦2↦2↦2),(3↦2↦2↦3),(3↦2↦3↦1),(3↦2↦3↦2),(3↦2↦3↦3),(3↦2↦4↦4),(3↦2↦4↦5),(3↦2↦4↦6),(3↦2↦5↦4),(3↦2↦5↦5),(3↦2↦5↦6),(3↦2↦6↦4),(3↦2↦6↦5),(3↦2↦6↦6),(3↦2↦7↦7),(3↦2↦7↦8),(3↦2↦7↦9),(3↦2↦8↦7),(3↦2↦8↦8),(3↦2↦8↦9),(3↦2↦9↦7),(3↦2↦9↦8),(3↦2↦9↦9),(3↦3↦2↦1),(3↦3↦3↦1),(3↦3↦3↦2),(3↦3↦5↦4),(3↦3↦6↦4),(3↦3↦6↦5),(3↦3↦8↦7),(3↦3↦9↦7),(3↦3↦9↦8),(4↦4↦2↦1),(4↦4↦3↦1),(4↦4↦3↦2),(4↦4↦5↦4),(4↦4↦6↦4),(4↦4↦6↦5),(4↦4↦8↦7),(4↦4↦9↦7),(4↦4↦9↦8),(5↦4↦1↦1),(5↦4↦1↦2),(5↦4↦1↦3),(5↦4↦2↦1),(5↦4↦2↦2),(5↦4↦2↦3),(5↦4↦3↦1),(5↦4↦3↦2),(5↦4↦3↦3),(5↦4↦4↦4),(5↦4↦4↦5),(5↦4↦4↦6),(5↦4↦5↦4),(5↦4↦5↦5),(5↦4↦5↦6),(5↦4↦6↦4),(5↦4↦6↦5),(5↦4↦6↦6),(5↦4↦7↦7),(5↦4↦7↦8),(5↦4↦7↦9),(5↦4↦8↦7),(5↦4↦8↦8),(5↦4↦8↦9),(5↦4↦9↦7),(5↦4↦9↦8),(5↦4↦9↦9),(5↦5↦2↦1),(5↦5↦3↦1),(5↦5↦3↦2),(5↦5↦5↦4),(5↦5↦6↦4),(5↦5↦6↦5),(5↦5↦8↦7),(5↦5↦9↦7),(5↦5↦9↦8),(6↦4↦1↦1),(6↦4↦1↦2),(6↦4↦1↦3),(6↦4↦2↦1),(6↦4↦2↦2),(6↦4↦2↦3),(6↦4↦3↦1),(6↦4↦3↦2),(6↦4↦3↦3),(6↦4↦4↦4),(6↦4↦4↦5),(6↦4↦4↦6),(6↦4↦5↦4),(6↦4↦5↦5),(6↦4↦5↦6),(6↦4↦6↦4),(6↦4↦6↦5),(6↦4↦6↦6),(6↦4↦7↦7),(6↦4↦7↦8),(6↦4↦7↦9),(6↦4↦8↦7),(6↦4↦8↦8),(6↦4↦8↦9),(6↦4↦9↦7),(6↦4↦9↦8),(6↦4↦9↦9),(6↦5↦1↦1),(6↦5↦1↦2),(6↦5↦1↦3),(6↦5↦2↦1),(6↦5↦2↦2),(6↦5↦2↦3),(6↦5↦3↦1),(6↦5↦3↦2),(6↦5↦3↦3),(6↦5↦4↦4),(6↦5↦4↦5),(6↦5↦4↦6),(6↦5↦5↦4),(6↦5↦5↦5),(6↦5↦5↦6),(6↦5↦6↦4),(6↦5↦6↦5),(6↦5↦6↦6),(6↦5↦7↦7),(6↦5↦7↦8),(6↦5↦7↦9),(6↦5↦8↦7),(6↦5↦8↦8),(6↦5↦8↦9),(6↦5↦9↦7),(6↦5↦9↦8),(6↦5↦9↦9),(6↦6↦2↦1),(6↦6↦3↦1),(6↦6↦3↦2),(6↦6↦5↦4),(6↦6↦6↦4),(6↦6↦6↦5),(6↦6↦8↦7),(6↦6↦9↦7),(6↦6↦9↦8),(7↦7↦2↦1),(7↦7↦3↦1),(7↦7↦3↦2),(7↦7↦5↦4),(7↦7↦6↦4),(7↦7↦6↦5),(7↦7↦8↦7),(7↦7↦9↦7),(7↦7↦9↦8),(8↦7↦1↦1),(8↦7↦1↦2),(8↦7↦1↦3),(8↦7↦2↦1),(8↦7↦2↦2),(8↦7↦2↦3),(8↦7↦3↦1),(8↦7↦3↦2),(8↦7↦3↦3),(8↦7↦4↦4),(8↦7↦4↦5),(8↦7↦4↦6),(8↦7↦5↦4),(8↦7↦5↦5),(8↦7↦5↦6),(8↦7↦6↦4),(8↦7↦6↦5),(8↦7↦6↦6),(8↦7↦7↦7),(8↦7↦7↦8),(8↦7↦7↦9),(8↦7↦8↦7),(8↦7↦8↦8),(8↦7↦8↦9),(8↦7↦9↦7),(8↦7↦9↦8),(8↦7↦9↦9),(8↦8↦2↦1),(8↦8↦3↦1),(8↦8↦3↦2),(8↦8↦5↦4),(8↦8↦6↦4),(8↦8↦6↦5),(8↦8↦8↦7),(8↦8↦9↦7),(8↦8↦9↦8),(9↦7↦1↦1),(9↦7↦1↦2),(9↦7↦1↦3),(9↦7↦2↦1),(9↦7↦2↦2),(9↦7↦2↦3),(9↦7↦3↦1),(9↦7↦3↦2),(9↦7↦3↦3),(9↦7↦4↦4),(9↦7↦4↦5),(9↦7↦4↦6),(9↦7↦5↦4),(9↦7↦5↦5),(9↦7↦5↦6),(9↦7↦6↦4),(9↦7↦6↦5),(9↦7↦6↦6),(9↦7↦7↦7),(9↦7↦7↦8),(9↦7↦7↦9),(9↦7↦8↦7),(9↦7↦8↦8),(9↦7↦8↦9),(9↦7↦9↦7),(9↦7↦9↦8),(9↦7↦9↦9),(9↦8↦1↦1),(9↦8↦1↦2),(9↦8↦1↦3),(9↦8↦2↦1),(9↦8↦2↦2),(9↦8↦2↦3),(9↦8↦3↦1),(9↦8↦3↦2),(9↦8↦3↦3),(9↦8↦4↦4),(9↦8↦4↦5),(9↦8↦4↦6),(9↦8↦5↦4),(9↦8↦5↦5),(9↦8↦5↦6),(9↦8↦6↦4),(9↦8↦6↦5),(9↦8↦6↦6),(9↦8↦7↦7),(9↦8↦7↦8),(9↦8↦7↦9),(9↦8↦8↦7),(9↦8↦8↦8),(9↦8↦8↦9),(9↦8↦9↦7),(9↦8↦9↦8),(9↦8↦9↦9),(9↦9↦2↦1),(9↦9↦3↦1),(9↦9↦3↦2),(9↦9↦5↦4),(9↦9↦6↦4),(9↦9↦6↦5),(9↦9↦8↦7),(9↦9↦9↦7),(9↦9↦9↦8)}\n", + "\tDiff2 = {(1↦1↦1↦2),(1↦1↦1↦3),(1↦1↦1↦4),(1↦1↦1↦5),(1↦1↦1↦6),(1↦1↦1↦7),(1↦1↦1↦8),(1↦1↦1↦9),(1↦1↦2↦3),(1↦1↦2↦4),(1↦1↦2↦5),(1↦1↦2↦6),(1↦1↦2↦7),(1↦1↦2↦8),(1↦1↦2↦9),(1↦1↦3↦4),(1↦1↦3↦5),(1↦1↦3↦6),(1↦1↦3↦7),(1↦1↦3↦8),(1↦1↦3↦9),(1↦1↦4↦5),(1↦1↦4↦6),(1↦1↦4↦7),(1↦1↦4↦8),(1↦1↦4↦9),(1↦1↦5↦6),(1↦1↦5↦7),(1↦1↦5↦8),(1↦1↦5↦9),(1↦1↦6↦7),(1↦1↦6↦8),(1↦1↦6↦9),(1↦1↦7↦8),(1↦1↦7↦9),(1↦1↦8↦9),(2↦2↦1↦2),(2↦2↦1↦3),(2↦2↦1↦4),(2↦2↦1↦5),(2↦2↦1↦6),(2↦2↦1↦7),(2↦2↦1↦8),(2↦2↦1↦9),(2↦2↦2↦3),(2↦2↦2↦4),(2↦2↦2↦5),(2↦2↦2↦6),(2↦2↦2↦7),(2↦2↦2↦8),(2↦2↦2↦9),(2↦2↦3↦4),(2↦2↦3↦5),(2↦2↦3↦6),(2↦2↦3↦7),(2↦2↦3↦8),(2↦2↦3↦9),(2↦2↦4↦5),(2↦2↦4↦6),(2↦2↦4↦7),(2↦2↦4↦8),(2↦2↦4↦9),(2↦2↦5↦6),(2↦2↦5↦7),(2↦2↦5↦8),(2↦2↦5↦9),(2↦2↦6↦7),(2↦2↦6↦8),(2↦2↦6↦9),(2↦2↦7↦8),(2↦2↦7↦9),(2↦2↦8↦9),(3↦3↦1↦2),(3↦3↦1↦3),(3↦3↦1↦4),(3↦3↦1↦5),(3↦3↦1↦6),(3↦3↦1↦7),(3↦3↦1↦8),(3↦3↦1↦9),(3↦3↦2↦3),(3↦3↦2↦4),(3↦3↦2↦5),(3↦3↦2↦6),(3↦3↦2↦7),(3↦3↦2↦8),(3↦3↦2↦9),(3↦3↦3↦4),(3↦3↦3↦5),(3↦3↦3↦6),(3↦3↦3↦7),(3↦3↦3↦8),(3↦3↦3↦9),(3↦3↦4↦5),(3↦3↦4↦6),(3↦3↦4↦7),(3↦3↦4↦8),(3↦3↦4↦9),(3↦3↦5↦6),(3↦3↦5↦7),(3↦3↦5↦8),(3↦3↦5↦9),(3↦3↦6↦7),(3↦3↦6↦8),(3↦3↦6↦9),(3↦3↦7↦8),(3↦3↦7↦9),(3↦3↦8↦9),(4↦4↦1↦2),(4↦4↦1↦3),(4↦4↦1↦4),(4↦4↦1↦5),(4↦4↦1↦6),(4↦4↦1↦7),(4↦4↦1↦8),(4↦4↦1↦9),(4↦4↦2↦3),(4↦4↦2↦4),(4↦4↦2↦5),(4↦4↦2↦6),(4↦4↦2↦7),(4↦4↦2↦8),(4↦4↦2↦9),(4↦4↦3↦4),(4↦4↦3↦5),(4↦4↦3↦6),(4↦4↦3↦7),(4↦4↦3↦8),(4↦4↦3↦9),(4↦4↦4↦5),(4↦4↦4↦6),(4↦4↦4↦7),(4↦4↦4↦8),(4↦4↦4↦9),(4↦4↦5↦6),(4↦4↦5↦7),(4↦4↦5↦8),(4↦4↦5↦9),(4↦4↦6↦7),(4↦4↦6↦8),(4↦4↦6↦9),(4↦4↦7↦8),(4↦4↦7↦9),(4↦4↦8↦9),(5↦5↦1↦2),(5↦5↦1↦3),(5↦5↦1↦4),(5↦5↦1↦5),(5↦5↦1↦6),(5↦5↦1↦7),(5↦5↦1↦8),(5↦5↦1↦9),(5↦5↦2↦3),(5↦5↦2↦4),(5↦5↦2↦5),(5↦5↦2↦6),(5↦5↦2↦7),(5↦5↦2↦8),(5↦5↦2↦9),(5↦5↦3↦4),(5↦5↦3↦5),(5↦5↦3↦6),(5↦5↦3↦7),(5↦5↦3↦8),(5↦5↦3↦9),(5↦5↦4↦5),(5↦5↦4↦6),(5↦5↦4↦7),(5↦5↦4↦8),(5↦5↦4↦9),(5↦5↦5↦6),(5↦5↦5↦7),(5↦5↦5↦8),(5↦5↦5↦9),(5↦5↦6↦7),(5↦5↦6↦8),(5↦5↦6↦9),(5↦5↦7↦8),(5↦5↦7↦9),(5↦5↦8↦9),(6↦6↦1↦2),(6↦6↦1↦3),(6↦6↦1↦4),(6↦6↦1↦5),(6↦6↦1↦6),(6↦6↦1↦7),(6↦6↦1↦8),(6↦6↦1↦9),(6↦6↦2↦3),(6↦6↦2↦4),(6↦6↦2↦5),(6↦6↦2↦6),(6↦6↦2↦7),(6↦6↦2↦8),(6↦6↦2↦9),(6↦6↦3↦4),(6↦6↦3↦5),(6↦6↦3↦6),(6↦6↦3↦7),(6↦6↦3↦8),(6↦6↦3↦9),(6↦6↦4↦5),(6↦6↦4↦6),(6↦6↦4↦7),(6↦6↦4↦8),(6↦6↦4↦9),(6↦6↦5↦6),(6↦6↦5↦7),(6↦6↦5↦8),(6↦6↦5↦9),(6↦6↦6↦7),(6↦6↦6↦8),(6↦6↦6↦9),(6↦6↦7↦8),(6↦6↦7↦9),(6↦6↦8↦9),(7↦7↦1↦2),(7↦7↦1↦3),(7↦7↦1↦4),(7↦7↦1↦5),(7↦7↦1↦6),(7↦7↦1↦7),(7↦7↦1↦8),(7↦7↦1↦9),(7↦7↦2↦3),(7↦7↦2↦4),(7↦7↦2↦5),(7↦7↦2↦6),(7↦7↦2↦7),(7↦7↦2↦8),(7↦7↦2↦9),(7↦7↦3↦4),(7↦7↦3↦5),(7↦7↦3↦6),(7↦7↦3↦7),(7↦7↦3↦8),(7↦7↦3↦9),(7↦7↦4↦5),(7↦7↦4↦6),(7↦7↦4↦7),(7↦7↦4↦8),(7↦7↦4↦9),(7↦7↦5↦6),(7↦7↦5↦7),(7↦7↦5↦8),(7↦7↦5↦9),(7↦7↦6↦7),(7↦7↦6↦8),(7↦7↦6↦9),(7↦7↦7↦8),(7↦7↦7↦9),(7↦7↦8↦9),(8↦8↦1↦2),(8↦8↦1↦3),(8↦8↦1↦4),(8↦8↦1↦5),(8↦8↦1↦6),(8↦8↦1↦7),(8↦8↦1↦8),(8↦8↦1↦9),(8↦8↦2↦3),(8↦8↦2↦4),(8↦8↦2↦5),(8↦8↦2↦6),(8↦8↦2↦7),(8↦8↦2↦8),(8↦8↦2↦9),(8↦8↦3↦4),(8↦8↦3↦5),(8↦8↦3↦6),(8↦8↦3↦7),(8↦8↦3↦8),(8↦8↦3↦9),(8↦8↦4↦5),(8↦8↦4↦6),(8↦8↦4↦7),(8↦8↦4↦8),(8↦8↦4↦9),(8↦8↦5↦6),(8↦8↦5↦7),(8↦8↦5↦8),(8↦8↦5↦9),(8↦8↦6↦7),(8↦8↦6↦8),(8↦8↦6↦9),(8↦8↦7↦8),(8↦8↦7↦9),(8↦8↦8↦9),(9↦9↦1↦2),(9↦9↦1↦3),(9↦9↦1↦4),(9↦9↦1↦5),(9↦9↦1↦6),(9↦9↦1↦7),(9↦9↦1↦8),(9↦9↦1↦9),(9↦9↦2↦3),(9↦9↦2↦4),(9↦9↦2↦5),(9↦9↦2↦6),(9↦9↦2↦7),(9↦9↦2↦8),(9↦9↦2↦9),(9↦9↦3↦4),(9↦9↦3↦5),(9↦9↦3↦6),(9↦9↦3↦7),(9↦9↦3↦8),(9↦9↦3↦9),(9↦9↦4↦5),(9↦9↦4↦6),(9↦9↦4↦7),(9↦9↦4↦8),(9↦9↦4↦9),(9↦9↦5↦6),(9↦9↦5↦7),(9↦9↦5↦8),(9↦9↦5↦9),(9↦9↦6↦7),(9↦9↦6↦8),(9↦9↦6↦9),(9↦9↦7↦8),(9↦9↦7↦9),(9↦9↦8↦9)}\n", + "\tBoard = {(1↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(2↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(3↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(4↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(5↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(6↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(7↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(8↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(9↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)})}\n", + "\tDOM = {1,2,3,4,5,6,7,8,9}\n", + "\tDiff1 = {(1↦2↦1↦1),(1↦2↦2↦2),(1↦2↦3↦3),(1↦2↦4↦4),(1↦2↦5↦5),(1↦2↦6↦6),(1↦2↦7↦7),(1↦2↦8↦8),(1↦2↦9↦9),(1↦3↦1↦1),(1↦3↦2↦2),(1↦3↦3↦3),(1↦3↦4↦4),(1↦3↦5↦5),(1↦3↦6↦6),(1↦3↦7↦7),(1↦3↦8↦8),(1↦3↦9↦9),(1↦4↦1↦1),(1↦4↦2↦2),(1↦4↦3↦3),(1↦4↦4↦4),(1↦4↦5↦5),(1↦4↦6↦6),(1↦4↦7↦7),(1↦4↦8↦8),(1↦4↦9↦9),(1↦5↦1↦1),(1↦5↦2↦2),(1↦5↦3↦3),(1↦5↦4↦4),(1↦5↦5↦5),(1↦5↦6↦6),(1↦5↦7↦7),(1↦5↦8↦8),(1↦5↦9↦9),(1↦6↦1↦1),(1↦6↦2↦2),(1↦6↦3↦3),(1↦6↦4↦4),(1↦6↦5↦5),(1↦6↦6↦6),(1↦6↦7↦7),(1↦6↦8↦8),(1↦6↦9↦9),(1↦7↦1↦1),(1↦7↦2↦2),(1↦7↦3↦3),(1↦7↦4↦4),(1↦7↦5↦5),(1↦7↦6↦6),(1↦7↦7↦7),(1↦7↦8↦8),(1↦7↦9↦9),(1↦8↦1↦1),(1↦8↦2↦2),(1↦8↦3↦3),(1↦8↦4↦4),(1↦8↦5↦5),(1↦8↦6↦6),(1↦8↦7↦7),(1↦8↦8↦8),(1↦8↦9↦9),(1↦9↦1↦1),(1↦9↦2↦2),(1↦9↦3↦3),(1↦9↦4↦4),(1↦9↦5↦5),(1↦9↦6↦6),(1↦9↦7↦7),(1↦9↦8↦8),(1↦9↦9↦9),(2↦3↦1↦1),(2↦3↦2↦2),(2↦3↦3↦3),(2↦3↦4↦4),(2↦3↦5↦5),(2↦3↦6↦6),(2↦3↦7↦7),(2↦3↦8↦8),(2↦3↦9↦9),(2↦4↦1↦1),(2↦4↦2↦2),(2↦4↦3↦3),(2↦4↦4↦4),(2↦4↦5↦5),(2↦4↦6↦6),(2↦4↦7↦7),(2↦4↦8↦8),(2↦4↦9↦9),(2↦5↦1↦1),(2↦5↦2↦2),(2↦5↦3↦3),(2↦5↦4↦4),(2↦5↦5↦5),(2↦5↦6↦6),(2↦5↦7↦7),(2↦5↦8↦8),(2↦5↦9↦9),(2↦6↦1↦1),(2↦6↦2↦2),(2↦6↦3↦3),(2↦6↦4↦4),(2↦6↦5↦5),(2↦6↦6↦6),(2↦6↦7↦7),(2↦6↦8↦8),(2↦6↦9↦9),(2↦7↦1↦1),(2↦7↦2↦2),(2↦7↦3↦3),(2↦7↦4↦4),(2↦7↦5↦5),(2↦7↦6↦6),(2↦7↦7↦7),(2↦7↦8↦8),(2↦7↦9↦9),(2↦8↦1↦1),(2↦8↦2↦2),(2↦8↦3↦3),(2↦8↦4↦4),(2↦8↦5↦5),(2↦8↦6↦6),(2↦8↦7↦7),(2↦8↦8↦8),(2↦8↦9↦9),(2↦9↦1↦1),(2↦9↦2↦2),(2↦9↦3↦3),(2↦9↦4↦4),(2↦9↦5↦5),(2↦9↦6↦6),(2↦9↦7↦7),(2↦9↦8↦8),(2↦9↦9↦9),(3↦4↦1↦1),(3↦4↦2↦2),(3↦4↦3↦3),(3↦4↦4↦4),(3↦4↦5↦5),(3↦4↦6↦6),(3↦4↦7↦7),(3↦4↦8↦8),(3↦4↦9↦9),(3↦5↦1↦1),(3↦5↦2↦2),(3↦5↦3↦3),(3↦5↦4↦4),(3↦5↦5↦5),(3↦5↦6↦6),(3↦5↦7↦7),(3↦5↦8↦8),(3↦5↦9↦9),(3↦6↦1↦1),(3↦6↦2↦2),(3↦6↦3↦3),(3↦6↦4↦4),(3↦6↦5↦5),(3↦6↦6↦6),(3↦6↦7↦7),(3↦6↦8↦8),(3↦6↦9↦9),(3↦7↦1↦1),(3↦7↦2↦2),(3↦7↦3↦3),(3↦7↦4↦4),(3↦7↦5↦5),(3↦7↦6↦6),(3↦7↦7↦7),(3↦7↦8↦8),(3↦7↦9↦9),(3↦8↦1↦1),(3↦8↦2↦2),(3↦8↦3↦3),(3↦8↦4↦4),(3↦8↦5↦5),(3↦8↦6↦6),(3↦8↦7↦7),(3↦8↦8↦8),(3↦8↦9↦9),(3↦9↦1↦1),(3↦9↦2↦2),(3↦9↦3↦3),(3↦9↦4↦4),(3↦9↦5↦5),(3↦9↦6↦6),(3↦9↦7↦7),(3↦9↦8↦8),(3↦9↦9↦9),(4↦5↦1↦1),(4↦5↦2↦2),(4↦5↦3↦3),(4↦5↦4↦4),(4↦5↦5↦5),(4↦5↦6↦6),(4↦5↦7↦7),(4↦5↦8↦8),(4↦5↦9↦9),(4↦6↦1↦1),(4↦6↦2↦2),(4↦6↦3↦3),(4↦6↦4↦4),(4↦6↦5↦5),(4↦6↦6↦6),(4↦6↦7↦7),(4↦6↦8↦8),(4↦6↦9↦9),(4↦7↦1↦1),(4↦7↦2↦2),(4↦7↦3↦3),(4↦7↦4↦4),(4↦7↦5↦5),(4↦7↦6↦6),(4↦7↦7↦7),(4↦7↦8↦8),(4↦7↦9↦9),(4↦8↦1↦1),(4↦8↦2↦2),(4↦8↦3↦3),(4↦8↦4↦4),(4↦8↦5↦5),(4↦8↦6↦6),(4↦8↦7↦7),(4↦8↦8↦8),(4↦8↦9↦9),(4↦9↦1↦1),(4↦9↦2↦2),(4↦9↦3↦3),(4↦9↦4↦4),(4↦9↦5↦5),(4↦9↦6↦6),(4↦9↦7↦7),(4↦9↦8↦8),(4↦9↦9↦9),(5↦6↦1↦1),(5↦6↦2↦2),(5↦6↦3↦3),(5↦6↦4↦4),(5↦6↦5↦5),(5↦6↦6↦6),(5↦6↦7↦7),(5↦6↦8↦8),(5↦6↦9↦9),(5↦7↦1↦1),(5↦7↦2↦2),(5↦7↦3↦3),(5↦7↦4↦4),(5↦7↦5↦5),(5↦7↦6↦6),(5↦7↦7↦7),(5↦7↦8↦8),(5↦7↦9↦9),(5↦8↦1↦1),(5↦8↦2↦2),(5↦8↦3↦3),(5↦8↦4↦4),(5↦8↦5↦5),(5↦8↦6↦6),(5↦8↦7↦7),(5↦8↦8↦8),(5↦8↦9↦9),(5↦9↦1↦1),(5↦9↦2↦2),(5↦9↦3↦3),(5↦9↦4↦4),(5↦9↦5↦5),(5↦9↦6↦6),(5↦9↦7↦7),(5↦9↦8↦8),(5↦9↦9↦9),(6↦7↦1↦1),(6↦7↦2↦2),(6↦7↦3↦3),(6↦7↦4↦4),(6↦7↦5↦5),(6↦7↦6↦6),(6↦7↦7↦7),(6↦7↦8↦8),(6↦7↦9↦9),(6↦8↦1↦1),(6↦8↦2↦2),(6↦8↦3↦3),(6↦8↦4↦4),(6↦8↦5↦5),(6↦8↦6↦6),(6↦8↦7↦7),(6↦8↦8↦8),(6↦8↦9↦9),(6↦9↦1↦1),(6↦9↦2↦2),(6↦9↦3↦3),(6↦9↦4↦4),(6↦9↦5↦5),(6↦9↦6↦6),(6↦9↦7↦7),(6↦9↦8↦8),(6↦9↦9↦9),(7↦8↦1↦1),(7↦8↦2↦2),(7↦8↦3↦3),(7↦8↦4↦4),(7↦8↦5↦5),(7↦8↦6↦6),(7↦8↦7↦7),(7↦8↦8↦8),(7↦8↦9↦9),(7↦9↦1↦1),(7↦9↦2↦2),(7↦9↦3↦3),(7↦9↦4↦4),(7↦9↦5↦5),(7↦9↦6↦6),(7↦9↦7↦7),(7↦9↦8↦8),(7↦9↦9↦9),(8↦9↦1↦1),(8↦9↦2↦2),(8↦9↦3↦3),(8↦9↦4↦4),(8↦9↦5↦5),(8↦9↦6↦6),(8↦9↦7↦7),(8↦9↦8↦8),(8↦9↦9↦9)}\n", + "\tDiff = ∅\n", + "\tSUBSQ = {{1,2,3},{4,5,6},{7,8,9}}" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Board : DOM --> (DOM --> DOM) & !(x1,x2,y1,y2).((x1,x2,y1,y2):Diff => Board(x1)(y1) /= Board(x2)(y2))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let us now try and add some additional constraints for certain pre-established positions on the board, and put those into the variable P and require that the solution Board contains those values:" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{(1\\mapsto 1\\mapsto 7),(1\\mapsto 2\\mapsto 8),(1\\mapsto 3\\mapsto 1),(2\\mapsto 1\\mapsto 9)\\}$" + ], + "text/plain": [ + "{(1↦1↦7),(1↦2↦8),(1↦3↦1),(2↦1↦9)}" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let P {(1,1,7), (1,2,8), (1,3,1), (2,1,9)}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "You can visualise the solution using the show command of the REPL:" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\mathit{TRUE}$\n", + "\n", + "**Solution:**\n", + "* $\\mathit{P} = \\{(1\\mapsto 1\\mapsto 7),(1\\mapsto 2\\mapsto 8),(1\\mapsto 3\\mapsto 1),(2\\mapsto 1\\mapsto 9)\\}$\n", + "* $\\mathit{Diff3} = \\{(1\\mapsto 1\\mapsto 2\\mapsto 1),(1\\mapsto 1\\mapsto 3\\mapsto 1),(1\\mapsto 1\\mapsto 3\\mapsto 2),(1\\mapsto 1\\mapsto 5\\mapsto 4),(1\\mapsto 1\\mapsto 6\\mapsto 4),(1\\mapsto 1\\mapsto 6\\mapsto 5),(1\\mapsto 1\\mapsto 8\\mapsto 7),(1\\mapsto 1\\mapsto 9\\mapsto 7),(1\\mapsto 1\\mapsto 9\\mapsto 8),(2\\mapsto 1\\mapsto 1\\mapsto 1),(2\\mapsto 1\\mapsto 1\\mapsto 2),(2\\mapsto 1\\mapsto 1\\mapsto 3),(2\\mapsto 1\\mapsto 2\\mapsto 1),(2\\mapsto 1\\mapsto 2\\mapsto 2),(2\\mapsto 1\\mapsto 2\\mapsto 3),(2\\mapsto 1\\mapsto 3\\mapsto 1),(2\\mapsto 1\\mapsto 3\\mapsto 2),(2\\mapsto 1\\mapsto 3\\mapsto 3),(2\\mapsto 1\\mapsto 4\\mapsto 4),(2\\mapsto 1\\mapsto 4\\mapsto 5),(2\\mapsto 1\\mapsto 4\\mapsto 6),(2\\mapsto 1\\mapsto 5\\mapsto 4),(2\\mapsto 1\\mapsto 5\\mapsto 5),(2\\mapsto 1\\mapsto 5\\mapsto 6),(2\\mapsto 1\\mapsto 6\\mapsto 4),(2\\mapsto 1\\mapsto 6\\mapsto 5),(2\\mapsto 1\\mapsto 6\\mapsto 6),(2\\mapsto 1\\mapsto 7\\mapsto 7),(2\\mapsto 1\\mapsto 7\\mapsto 8),(2\\mapsto 1\\mapsto 7\\mapsto 9),(2\\mapsto 1\\mapsto 8\\mapsto 7),(2\\mapsto 1\\mapsto 8\\mapsto 8),(2\\mapsto 1\\mapsto 8\\mapsto 9),(2\\mapsto 1\\mapsto 9\\mapsto 7),(2\\mapsto 1\\mapsto 9\\mapsto 8),(2\\mapsto 1\\mapsto 9\\mapsto 9),(2\\mapsto 2\\mapsto 2\\mapsto 1),(2\\mapsto 2\\mapsto 3\\mapsto 1),(2\\mapsto 2\\mapsto 3\\mapsto 2),(2\\mapsto 2\\mapsto 5\\mapsto 4),(2\\mapsto 2\\mapsto 6\\mapsto 4),(2\\mapsto 2\\mapsto 6\\mapsto 5),(2\\mapsto 2\\mapsto 8\\mapsto 7),(2\\mapsto 2\\mapsto 9\\mapsto 7),(2\\mapsto 2\\mapsto 9\\mapsto 8),(3\\mapsto 1\\mapsto 1\\mapsto 1),(3\\mapsto 1\\mapsto 1\\mapsto 2),(3\\mapsto 1\\mapsto 1\\mapsto 3),(3\\mapsto 1\\mapsto 2\\mapsto 1),(3\\mapsto 1\\mapsto 2\\mapsto 2),(3\\mapsto 1\\mapsto 2\\mapsto 3),(3\\mapsto 1\\mapsto 3\\mapsto 1),(3\\mapsto 1\\mapsto 3\\mapsto 2),(3\\mapsto 1\\mapsto 3\\mapsto 3),(3\\mapsto 1\\mapsto 4\\mapsto 4),(3\\mapsto 1\\mapsto 4\\mapsto 5),(3\\mapsto 1\\mapsto 4\\mapsto 6),(3\\mapsto 1\\mapsto 5\\mapsto 4),(3\\mapsto 1\\mapsto 5\\mapsto 5),(3\\mapsto 1\\mapsto 5\\mapsto 6),(3\\mapsto 1\\mapsto 6\\mapsto 4),(3\\mapsto 1\\mapsto 6\\mapsto 5),(3\\mapsto 1\\mapsto 6\\mapsto 6),(3\\mapsto 1\\mapsto 7\\mapsto 7),(3\\mapsto 1\\mapsto 7\\mapsto 8),(3\\mapsto 1\\mapsto 7\\mapsto 9),(3\\mapsto 1\\mapsto 8\\mapsto 7),(3\\mapsto 1\\mapsto 8\\mapsto 8),(3\\mapsto 1\\mapsto 8\\mapsto 9),(3\\mapsto 1\\mapsto 9\\mapsto 7),(3\\mapsto 1\\mapsto 9\\mapsto 8),(3\\mapsto 1\\mapsto 9\\mapsto 9),(3\\mapsto 2\\mapsto 1\\mapsto 1),(3\\mapsto 2\\mapsto 1\\mapsto 2),(3\\mapsto 2\\mapsto 1\\mapsto 3),(3\\mapsto 2\\mapsto 2\\mapsto 1),(3\\mapsto 2\\mapsto 2\\mapsto 2),(3\\mapsto 2\\mapsto 2\\mapsto 3),(3\\mapsto 2\\mapsto 3\\mapsto 1),(3\\mapsto 2\\mapsto 3\\mapsto 2),(3\\mapsto 2\\mapsto 3\\mapsto 3),(3\\mapsto 2\\mapsto 4\\mapsto 4),(3\\mapsto 2\\mapsto 4\\mapsto 5),(3\\mapsto 2\\mapsto 4\\mapsto 6),(3\\mapsto 2\\mapsto 5\\mapsto 4),(3\\mapsto 2\\mapsto 5\\mapsto 5),(3\\mapsto 2\\mapsto 5\\mapsto 6),(3\\mapsto 2\\mapsto 6\\mapsto 4),(3\\mapsto 2\\mapsto 6\\mapsto 5),(3\\mapsto 2\\mapsto 6\\mapsto 6),(3\\mapsto 2\\mapsto 7\\mapsto 7),(3\\mapsto 2\\mapsto 7\\mapsto 8),(3\\mapsto 2\\mapsto 7\\mapsto 9),(3\\mapsto 2\\mapsto 8\\mapsto 7),(3\\mapsto 2\\mapsto 8\\mapsto 8),(3\\mapsto 2\\mapsto 8\\mapsto 9),(3\\mapsto 2\\mapsto 9\\mapsto 7),(3\\mapsto 2\\mapsto 9\\mapsto 8),(3\\mapsto 2\\mapsto 9\\mapsto 9),(3\\mapsto 3\\mapsto 2\\mapsto 1),(3\\mapsto 3\\mapsto 3\\mapsto 1),(3\\mapsto 3\\mapsto 3\\mapsto 2),(3\\mapsto 3\\mapsto 5\\mapsto 4),(3\\mapsto 3\\mapsto 6\\mapsto 4),(3\\mapsto 3\\mapsto 6\\mapsto 5),(3\\mapsto 3\\mapsto 8\\mapsto 7),(3\\mapsto 3\\mapsto 9\\mapsto 7),(3\\mapsto 3\\mapsto 9\\mapsto 8),(4\\mapsto 4\\mapsto 2\\mapsto 1),(4\\mapsto 4\\mapsto 3\\mapsto 1),(4\\mapsto 4\\mapsto 3\\mapsto 2),(4\\mapsto 4\\mapsto 5\\mapsto 4),(4\\mapsto 4\\mapsto 6\\mapsto 4),(4\\mapsto 4\\mapsto 6\\mapsto 5),(4\\mapsto 4\\mapsto 8\\mapsto 7),(4\\mapsto 4\\mapsto 9\\mapsto 7),(4\\mapsto 4\\mapsto 9\\mapsto 8),(5\\mapsto 4\\mapsto 1\\mapsto 1),(5\\mapsto 4\\mapsto 1\\mapsto 2),(5\\mapsto 4\\mapsto 1\\mapsto 3),(5\\mapsto 4\\mapsto 2\\mapsto 1),(5\\mapsto 4\\mapsto 2\\mapsto 2),(5\\mapsto 4\\mapsto 2\\mapsto 3),(5\\mapsto 4\\mapsto 3\\mapsto 1),(5\\mapsto 4\\mapsto 3\\mapsto 2),(5\\mapsto 4\\mapsto 3\\mapsto 3),(5\\mapsto 4\\mapsto 4\\mapsto 4),(5\\mapsto 4\\mapsto 4\\mapsto 5),(5\\mapsto 4\\mapsto 4\\mapsto 6),(5\\mapsto 4\\mapsto 5\\mapsto 4),(5\\mapsto 4\\mapsto 5\\mapsto 5),(5\\mapsto 4\\mapsto 5\\mapsto 6),(5\\mapsto 4\\mapsto 6\\mapsto 4),(5\\mapsto 4\\mapsto 6\\mapsto 5),(5\\mapsto 4\\mapsto 6\\mapsto 6),(5\\mapsto 4\\mapsto 7\\mapsto 7),(5\\mapsto 4\\mapsto 7\\mapsto 8),(5\\mapsto 4\\mapsto 7\\mapsto 9),(5\\mapsto 4\\mapsto 8\\mapsto 7),(5\\mapsto 4\\mapsto 8\\mapsto 8),(5\\mapsto 4\\mapsto 8\\mapsto 9),(5\\mapsto 4\\mapsto 9\\mapsto 7),(5\\mapsto 4\\mapsto 9\\mapsto 8),(5\\mapsto 4\\mapsto 9\\mapsto 9),(5\\mapsto 5\\mapsto 2\\mapsto 1),(5\\mapsto 5\\mapsto 3\\mapsto 1),(5\\mapsto 5\\mapsto 3\\mapsto 2),(5\\mapsto 5\\mapsto 5\\mapsto 4),(5\\mapsto 5\\mapsto 6\\mapsto 4),(5\\mapsto 5\\mapsto 6\\mapsto 5),(5\\mapsto 5\\mapsto 8\\mapsto 7),(5\\mapsto 5\\mapsto 9\\mapsto 7),(5\\mapsto 5\\mapsto 9\\mapsto 8),(6\\mapsto 4\\mapsto 1\\mapsto 1),(6\\mapsto 4\\mapsto 1\\mapsto 2),(6\\mapsto 4\\mapsto 1\\mapsto 3),(6\\mapsto 4\\mapsto 2\\mapsto 1),(6\\mapsto 4\\mapsto 2\\mapsto 2),(6\\mapsto 4\\mapsto 2\\mapsto 3),(6\\mapsto 4\\mapsto 3\\mapsto 1),(6\\mapsto 4\\mapsto 3\\mapsto 2),(6\\mapsto 4\\mapsto 3\\mapsto 3),(6\\mapsto 4\\mapsto 4\\mapsto 4),(6\\mapsto 4\\mapsto 4\\mapsto 5),(6\\mapsto 4\\mapsto 4\\mapsto 6),(6\\mapsto 4\\mapsto 5\\mapsto 4),(6\\mapsto 4\\mapsto 5\\mapsto 5),(6\\mapsto 4\\mapsto 5\\mapsto 6),(6\\mapsto 4\\mapsto 6\\mapsto 4),(6\\mapsto 4\\mapsto 6\\mapsto 5),(6\\mapsto 4\\mapsto 6\\mapsto 6),(6\\mapsto 4\\mapsto 7\\mapsto 7),(6\\mapsto 4\\mapsto 7\\mapsto 8),(6\\mapsto 4\\mapsto 7\\mapsto 9),(6\\mapsto 4\\mapsto 8\\mapsto 7),(6\\mapsto 4\\mapsto 8\\mapsto 8),(6\\mapsto 4\\mapsto 8\\mapsto 9),(6\\mapsto 4\\mapsto 9\\mapsto 7),(6\\mapsto 4\\mapsto 9\\mapsto 8),(6\\mapsto 4\\mapsto 9\\mapsto 9),(6\\mapsto 5\\mapsto 1\\mapsto 1),(6\\mapsto 5\\mapsto 1\\mapsto 2),(6\\mapsto 5\\mapsto 1\\mapsto 3),(6\\mapsto 5\\mapsto 2\\mapsto 1),(6\\mapsto 5\\mapsto 2\\mapsto 2),(6\\mapsto 5\\mapsto 2\\mapsto 3),(6\\mapsto 5\\mapsto 3\\mapsto 1),(6\\mapsto 5\\mapsto 3\\mapsto 2),(6\\mapsto 5\\mapsto 3\\mapsto 3),(6\\mapsto 5\\mapsto 4\\mapsto 4),(6\\mapsto 5\\mapsto 4\\mapsto 5),(6\\mapsto 5\\mapsto 4\\mapsto 6),(6\\mapsto 5\\mapsto 5\\mapsto 4),(6\\mapsto 5\\mapsto 5\\mapsto 5),(6\\mapsto 5\\mapsto 5\\mapsto 6),(6\\mapsto 5\\mapsto 6\\mapsto 4),(6\\mapsto 5\\mapsto 6\\mapsto 5),(6\\mapsto 5\\mapsto 6\\mapsto 6),(6\\mapsto 5\\mapsto 7\\mapsto 7),(6\\mapsto 5\\mapsto 7\\mapsto 8),(6\\mapsto 5\\mapsto 7\\mapsto 9),(6\\mapsto 5\\mapsto 8\\mapsto 7),(6\\mapsto 5\\mapsto 8\\mapsto 8),(6\\mapsto 5\\mapsto 8\\mapsto 9),(6\\mapsto 5\\mapsto 9\\mapsto 7),(6\\mapsto 5\\mapsto 9\\mapsto 8),(6\\mapsto 5\\mapsto 9\\mapsto 9),(6\\mapsto 6\\mapsto 2\\mapsto 1),(6\\mapsto 6\\mapsto 3\\mapsto 1),(6\\mapsto 6\\mapsto 3\\mapsto 2),(6\\mapsto 6\\mapsto 5\\mapsto 4),(6\\mapsto 6\\mapsto 6\\mapsto 4),(6\\mapsto 6\\mapsto 6\\mapsto 5),(6\\mapsto 6\\mapsto 8\\mapsto 7),(6\\mapsto 6\\mapsto 9\\mapsto 7),(6\\mapsto 6\\mapsto 9\\mapsto 8),(7\\mapsto 7\\mapsto 2\\mapsto 1),(7\\mapsto 7\\mapsto 3\\mapsto 1),(7\\mapsto 7\\mapsto 3\\mapsto 2),(7\\mapsto 7\\mapsto 5\\mapsto 4),(7\\mapsto 7\\mapsto 6\\mapsto 4),(7\\mapsto 7\\mapsto 6\\mapsto 5),(7\\mapsto 7\\mapsto 8\\mapsto 7),(7\\mapsto 7\\mapsto 9\\mapsto 7),(7\\mapsto 7\\mapsto 9\\mapsto 8),(8\\mapsto 7\\mapsto 1\\mapsto 1),(8\\mapsto 7\\mapsto 1\\mapsto 2),(8\\mapsto 7\\mapsto 1\\mapsto 3),(8\\mapsto 7\\mapsto 2\\mapsto 1),(8\\mapsto 7\\mapsto 2\\mapsto 2),(8\\mapsto 7\\mapsto 2\\mapsto 3),(8\\mapsto 7\\mapsto 3\\mapsto 1),(8\\mapsto 7\\mapsto 3\\mapsto 2),(8\\mapsto 7\\mapsto 3\\mapsto 3),(8\\mapsto 7\\mapsto 4\\mapsto 4),(8\\mapsto 7\\mapsto 4\\mapsto 5),(8\\mapsto 7\\mapsto 4\\mapsto 6),(8\\mapsto 7\\mapsto 5\\mapsto 4),(8\\mapsto 7\\mapsto 5\\mapsto 5),(8\\mapsto 7\\mapsto 5\\mapsto 6),(8\\mapsto 7\\mapsto 6\\mapsto 4),(8\\mapsto 7\\mapsto 6\\mapsto 5),(8\\mapsto 7\\mapsto 6\\mapsto 6),(8\\mapsto 7\\mapsto 7\\mapsto 7),(8\\mapsto 7\\mapsto 7\\mapsto 8),(8\\mapsto 7\\mapsto 7\\mapsto 9),(8\\mapsto 7\\mapsto 8\\mapsto 7),(8\\mapsto 7\\mapsto 8\\mapsto 8),(8\\mapsto 7\\mapsto 8\\mapsto 9),(8\\mapsto 7\\mapsto 9\\mapsto 7),(8\\mapsto 7\\mapsto 9\\mapsto 8),(8\\mapsto 7\\mapsto 9\\mapsto 9),(8\\mapsto 8\\mapsto 2\\mapsto 1),(8\\mapsto 8\\mapsto 3\\mapsto 1),(8\\mapsto 8\\mapsto 3\\mapsto 2),(8\\mapsto 8\\mapsto 5\\mapsto 4),(8\\mapsto 8\\mapsto 6\\mapsto 4),(8\\mapsto 8\\mapsto 6\\mapsto 5),(8\\mapsto 8\\mapsto 8\\mapsto 7),(8\\mapsto 8\\mapsto 9\\mapsto 7),(8\\mapsto 8\\mapsto 9\\mapsto 8),(9\\mapsto 7\\mapsto 1\\mapsto 1),(9\\mapsto 7\\mapsto 1\\mapsto 2),(9\\mapsto 7\\mapsto 1\\mapsto 3),(9\\mapsto 7\\mapsto 2\\mapsto 1),(9\\mapsto 7\\mapsto 2\\mapsto 2),(9\\mapsto 7\\mapsto 2\\mapsto 3),(9\\mapsto 7\\mapsto 3\\mapsto 1),(9\\mapsto 7\\mapsto 3\\mapsto 2),(9\\mapsto 7\\mapsto 3\\mapsto 3),(9\\mapsto 7\\mapsto 4\\mapsto 4),(9\\mapsto 7\\mapsto 4\\mapsto 5),(9\\mapsto 7\\mapsto 4\\mapsto 6),(9\\mapsto 7\\mapsto 5\\mapsto 4),(9\\mapsto 7\\mapsto 5\\mapsto 5),(9\\mapsto 7\\mapsto 5\\mapsto 6),(9\\mapsto 7\\mapsto 6\\mapsto 4),(9\\mapsto 7\\mapsto 6\\mapsto 5),(9\\mapsto 7\\mapsto 6\\mapsto 6),(9\\mapsto 7\\mapsto 7\\mapsto 7),(9\\mapsto 7\\mapsto 7\\mapsto 8),(9\\mapsto 7\\mapsto 7\\mapsto 9),(9\\mapsto 7\\mapsto 8\\mapsto 7),(9\\mapsto 7\\mapsto 8\\mapsto 8),(9\\mapsto 7\\mapsto 8\\mapsto 9),(9\\mapsto 7\\mapsto 9\\mapsto 7),(9\\mapsto 7\\mapsto 9\\mapsto 8),(9\\mapsto 7\\mapsto 9\\mapsto 9),(9\\mapsto 8\\mapsto 1\\mapsto 1),(9\\mapsto 8\\mapsto 1\\mapsto 2),(9\\mapsto 8\\mapsto 1\\mapsto 3),(9\\mapsto 8\\mapsto 2\\mapsto 1),(9\\mapsto 8\\mapsto 2\\mapsto 2),(9\\mapsto 8\\mapsto 2\\mapsto 3),(9\\mapsto 8\\mapsto 3\\mapsto 1),(9\\mapsto 8\\mapsto 3\\mapsto 2),(9\\mapsto 8\\mapsto 3\\mapsto 3),(9\\mapsto 8\\mapsto 4\\mapsto 4),(9\\mapsto 8\\mapsto 4\\mapsto 5),(9\\mapsto 8\\mapsto 4\\mapsto 6),(9\\mapsto 8\\mapsto 5\\mapsto 4),(9\\mapsto 8\\mapsto 5\\mapsto 5),(9\\mapsto 8\\mapsto 5\\mapsto 6),(9\\mapsto 8\\mapsto 6\\mapsto 4),(9\\mapsto 8\\mapsto 6\\mapsto 5),(9\\mapsto 8\\mapsto 6\\mapsto 6),(9\\mapsto 8\\mapsto 7\\mapsto 7),(9\\mapsto 8\\mapsto 7\\mapsto 8),(9\\mapsto 8\\mapsto 7\\mapsto 9),(9\\mapsto 8\\mapsto 8\\mapsto 7),(9\\mapsto 8\\mapsto 8\\mapsto 8),(9\\mapsto 8\\mapsto 8\\mapsto 9),(9\\mapsto 8\\mapsto 9\\mapsto 7),(9\\mapsto 8\\mapsto 9\\mapsto 8),(9\\mapsto 8\\mapsto 9\\mapsto 9),(9\\mapsto 9\\mapsto 2\\mapsto 1),(9\\mapsto 9\\mapsto 3\\mapsto 1),(9\\mapsto 9\\mapsto 3\\mapsto 2),(9\\mapsto 9\\mapsto 5\\mapsto 4),(9\\mapsto 9\\mapsto 6\\mapsto 4),(9\\mapsto 9\\mapsto 6\\mapsto 5),(9\\mapsto 9\\mapsto 8\\mapsto 7),(9\\mapsto 9\\mapsto 9\\mapsto 7),(9\\mapsto 9\\mapsto 9\\mapsto 8)\\}$\n", + "* $\\mathit{Diff2} = \\{(1\\mapsto 1\\mapsto 1\\mapsto 2),(1\\mapsto 1\\mapsto 1\\mapsto 3),(1\\mapsto 1\\mapsto 1\\mapsto 4),(1\\mapsto 1\\mapsto 1\\mapsto 5),(1\\mapsto 1\\mapsto 1\\mapsto 6),(1\\mapsto 1\\mapsto 1\\mapsto 7),(1\\mapsto 1\\mapsto 1\\mapsto 8),(1\\mapsto 1\\mapsto 1\\mapsto 9),(1\\mapsto 1\\mapsto 2\\mapsto 3),(1\\mapsto 1\\mapsto 2\\mapsto 4),(1\\mapsto 1\\mapsto 2\\mapsto 5),(1\\mapsto 1\\mapsto 2\\mapsto 6),(1\\mapsto 1\\mapsto 2\\mapsto 7),(1\\mapsto 1\\mapsto 2\\mapsto 8),(1\\mapsto 1\\mapsto 2\\mapsto 9),(1\\mapsto 1\\mapsto 3\\mapsto 4),(1\\mapsto 1\\mapsto 3\\mapsto 5),(1\\mapsto 1\\mapsto 3\\mapsto 6),(1\\mapsto 1\\mapsto 3\\mapsto 7),(1\\mapsto 1\\mapsto 3\\mapsto 8),(1\\mapsto 1\\mapsto 3\\mapsto 9),(1\\mapsto 1\\mapsto 4\\mapsto 5),(1\\mapsto 1\\mapsto 4\\mapsto 6),(1\\mapsto 1\\mapsto 4\\mapsto 7),(1\\mapsto 1\\mapsto 4\\mapsto 8),(1\\mapsto 1\\mapsto 4\\mapsto 9),(1\\mapsto 1\\mapsto 5\\mapsto 6),(1\\mapsto 1\\mapsto 5\\mapsto 7),(1\\mapsto 1\\mapsto 5\\mapsto 8),(1\\mapsto 1\\mapsto 5\\mapsto 9),(1\\mapsto 1\\mapsto 6\\mapsto 7),(1\\mapsto 1\\mapsto 6\\mapsto 8),(1\\mapsto 1\\mapsto 6\\mapsto 9),(1\\mapsto 1\\mapsto 7\\mapsto 8),(1\\mapsto 1\\mapsto 7\\mapsto 9),(1\\mapsto 1\\mapsto 8\\mapsto 9),(2\\mapsto 2\\mapsto 1\\mapsto 2),(2\\mapsto 2\\mapsto 1\\mapsto 3),(2\\mapsto 2\\mapsto 1\\mapsto 4),(2\\mapsto 2\\mapsto 1\\mapsto 5),(2\\mapsto 2\\mapsto 1\\mapsto 6),(2\\mapsto 2\\mapsto 1\\mapsto 7),(2\\mapsto 2\\mapsto 1\\mapsto 8),(2\\mapsto 2\\mapsto 1\\mapsto 9),(2\\mapsto 2\\mapsto 2\\mapsto 3),(2\\mapsto 2\\mapsto 2\\mapsto 4),(2\\mapsto 2\\mapsto 2\\mapsto 5),(2\\mapsto 2\\mapsto 2\\mapsto 6),(2\\mapsto 2\\mapsto 2\\mapsto 7),(2\\mapsto 2\\mapsto 2\\mapsto 8),(2\\mapsto 2\\mapsto 2\\mapsto 9),(2\\mapsto 2\\mapsto 3\\mapsto 4),(2\\mapsto 2\\mapsto 3\\mapsto 5),(2\\mapsto 2\\mapsto 3\\mapsto 6),(2\\mapsto 2\\mapsto 3\\mapsto 7),(2\\mapsto 2\\mapsto 3\\mapsto 8),(2\\mapsto 2\\mapsto 3\\mapsto 9),(2\\mapsto 2\\mapsto 4\\mapsto 5),(2\\mapsto 2\\mapsto 4\\mapsto 6),(2\\mapsto 2\\mapsto 4\\mapsto 7),(2\\mapsto 2\\mapsto 4\\mapsto 8),(2\\mapsto 2\\mapsto 4\\mapsto 9),(2\\mapsto 2\\mapsto 5\\mapsto 6),(2\\mapsto 2\\mapsto 5\\mapsto 7),(2\\mapsto 2\\mapsto 5\\mapsto 8),(2\\mapsto 2\\mapsto 5\\mapsto 9),(2\\mapsto 2\\mapsto 6\\mapsto 7),(2\\mapsto 2\\mapsto 6\\mapsto 8),(2\\mapsto 2\\mapsto 6\\mapsto 9),(2\\mapsto 2\\mapsto 7\\mapsto 8),(2\\mapsto 2\\mapsto 7\\mapsto 9),(2\\mapsto 2\\mapsto 8\\mapsto 9),(3\\mapsto 3\\mapsto 1\\mapsto 2),(3\\mapsto 3\\mapsto 1\\mapsto 3),(3\\mapsto 3\\mapsto 1\\mapsto 4),(3\\mapsto 3\\mapsto 1\\mapsto 5),(3\\mapsto 3\\mapsto 1\\mapsto 6),(3\\mapsto 3\\mapsto 1\\mapsto 7),(3\\mapsto 3\\mapsto 1\\mapsto 8),(3\\mapsto 3\\mapsto 1\\mapsto 9),(3\\mapsto 3\\mapsto 2\\mapsto 3),(3\\mapsto 3\\mapsto 2\\mapsto 4),(3\\mapsto 3\\mapsto 2\\mapsto 5),(3\\mapsto 3\\mapsto 2\\mapsto 6),(3\\mapsto 3\\mapsto 2\\mapsto 7),(3\\mapsto 3\\mapsto 2\\mapsto 8),(3\\mapsto 3\\mapsto 2\\mapsto 9),(3\\mapsto 3\\mapsto 3\\mapsto 4),(3\\mapsto 3\\mapsto 3\\mapsto 5),(3\\mapsto 3\\mapsto 3\\mapsto 6),(3\\mapsto 3\\mapsto 3\\mapsto 7),(3\\mapsto 3\\mapsto 3\\mapsto 8),(3\\mapsto 3\\mapsto 3\\mapsto 9),(3\\mapsto 3\\mapsto 4\\mapsto 5),(3\\mapsto 3\\mapsto 4\\mapsto 6),(3\\mapsto 3\\mapsto 4\\mapsto 7),(3\\mapsto 3\\mapsto 4\\mapsto 8),(3\\mapsto 3\\mapsto 4\\mapsto 9),(3\\mapsto 3\\mapsto 5\\mapsto 6),(3\\mapsto 3\\mapsto 5\\mapsto 7),(3\\mapsto 3\\mapsto 5\\mapsto 8),(3\\mapsto 3\\mapsto 5\\mapsto 9),(3\\mapsto 3\\mapsto 6\\mapsto 7),(3\\mapsto 3\\mapsto 6\\mapsto 8),(3\\mapsto 3\\mapsto 6\\mapsto 9),(3\\mapsto 3\\mapsto 7\\mapsto 8),(3\\mapsto 3\\mapsto 7\\mapsto 9),(3\\mapsto 3\\mapsto 8\\mapsto 9),(4\\mapsto 4\\mapsto 1\\mapsto 2),(4\\mapsto 4\\mapsto 1\\mapsto 3),(4\\mapsto 4\\mapsto 1\\mapsto 4),(4\\mapsto 4\\mapsto 1\\mapsto 5),(4\\mapsto 4\\mapsto 1\\mapsto 6),(4\\mapsto 4\\mapsto 1\\mapsto 7),(4\\mapsto 4\\mapsto 1\\mapsto 8),(4\\mapsto 4\\mapsto 1\\mapsto 9),(4\\mapsto 4\\mapsto 2\\mapsto 3),(4\\mapsto 4\\mapsto 2\\mapsto 4),(4\\mapsto 4\\mapsto 2\\mapsto 5),(4\\mapsto 4\\mapsto 2\\mapsto 6),(4\\mapsto 4\\mapsto 2\\mapsto 7),(4\\mapsto 4\\mapsto 2\\mapsto 8),(4\\mapsto 4\\mapsto 2\\mapsto 9),(4\\mapsto 4\\mapsto 3\\mapsto 4),(4\\mapsto 4\\mapsto 3\\mapsto 5),(4\\mapsto 4\\mapsto 3\\mapsto 6),(4\\mapsto 4\\mapsto 3\\mapsto 7),(4\\mapsto 4\\mapsto 3\\mapsto 8),(4\\mapsto 4\\mapsto 3\\mapsto 9),(4\\mapsto 4\\mapsto 4\\mapsto 5),(4\\mapsto 4\\mapsto 4\\mapsto 6),(4\\mapsto 4\\mapsto 4\\mapsto 7),(4\\mapsto 4\\mapsto 4\\mapsto 8),(4\\mapsto 4\\mapsto 4\\mapsto 9),(4\\mapsto 4\\mapsto 5\\mapsto 6),(4\\mapsto 4\\mapsto 5\\mapsto 7),(4\\mapsto 4\\mapsto 5\\mapsto 8),(4\\mapsto 4\\mapsto 5\\mapsto 9),(4\\mapsto 4\\mapsto 6\\mapsto 7),(4\\mapsto 4\\mapsto 6\\mapsto 8),(4\\mapsto 4\\mapsto 6\\mapsto 9),(4\\mapsto 4\\mapsto 7\\mapsto 8),(4\\mapsto 4\\mapsto 7\\mapsto 9),(4\\mapsto 4\\mapsto 8\\mapsto 9),(5\\mapsto 5\\mapsto 1\\mapsto 2),(5\\mapsto 5\\mapsto 1\\mapsto 3),(5\\mapsto 5\\mapsto 1\\mapsto 4),(5\\mapsto 5\\mapsto 1\\mapsto 5),(5\\mapsto 5\\mapsto 1\\mapsto 6),(5\\mapsto 5\\mapsto 1\\mapsto 7),(5\\mapsto 5\\mapsto 1\\mapsto 8),(5\\mapsto 5\\mapsto 1\\mapsto 9),(5\\mapsto 5\\mapsto 2\\mapsto 3),(5\\mapsto 5\\mapsto 2\\mapsto 4),(5\\mapsto 5\\mapsto 2\\mapsto 5),(5\\mapsto 5\\mapsto 2\\mapsto 6),(5\\mapsto 5\\mapsto 2\\mapsto 7),(5\\mapsto 5\\mapsto 2\\mapsto 8),(5\\mapsto 5\\mapsto 2\\mapsto 9),(5\\mapsto 5\\mapsto 3\\mapsto 4),(5\\mapsto 5\\mapsto 3\\mapsto 5),(5\\mapsto 5\\mapsto 3\\mapsto 6),(5\\mapsto 5\\mapsto 3\\mapsto 7),(5\\mapsto 5\\mapsto 3\\mapsto 8),(5\\mapsto 5\\mapsto 3\\mapsto 9),(5\\mapsto 5\\mapsto 4\\mapsto 5),(5\\mapsto 5\\mapsto 4\\mapsto 6),(5\\mapsto 5\\mapsto 4\\mapsto 7),(5\\mapsto 5\\mapsto 4\\mapsto 8),(5\\mapsto 5\\mapsto 4\\mapsto 9),(5\\mapsto 5\\mapsto 5\\mapsto 6),(5\\mapsto 5\\mapsto 5\\mapsto 7),(5\\mapsto 5\\mapsto 5\\mapsto 8),(5\\mapsto 5\\mapsto 5\\mapsto 9),(5\\mapsto 5\\mapsto 6\\mapsto 7),(5\\mapsto 5\\mapsto 6\\mapsto 8),(5\\mapsto 5\\mapsto 6\\mapsto 9),(5\\mapsto 5\\mapsto 7\\mapsto 8),(5\\mapsto 5\\mapsto 7\\mapsto 9),(5\\mapsto 5\\mapsto 8\\mapsto 9),(6\\mapsto 6\\mapsto 1\\mapsto 2),(6\\mapsto 6\\mapsto 1\\mapsto 3),(6\\mapsto 6\\mapsto 1\\mapsto 4),(6\\mapsto 6\\mapsto 1\\mapsto 5),(6\\mapsto 6\\mapsto 1\\mapsto 6),(6\\mapsto 6\\mapsto 1\\mapsto 7),(6\\mapsto 6\\mapsto 1\\mapsto 8),(6\\mapsto 6\\mapsto 1\\mapsto 9),(6\\mapsto 6\\mapsto 2\\mapsto 3),(6\\mapsto 6\\mapsto 2\\mapsto 4),(6\\mapsto 6\\mapsto 2\\mapsto 5),(6\\mapsto 6\\mapsto 2\\mapsto 6),(6\\mapsto 6\\mapsto 2\\mapsto 7),(6\\mapsto 6\\mapsto 2\\mapsto 8),(6\\mapsto 6\\mapsto 2\\mapsto 9),(6\\mapsto 6\\mapsto 3\\mapsto 4),(6\\mapsto 6\\mapsto 3\\mapsto 5),(6\\mapsto 6\\mapsto 3\\mapsto 6),(6\\mapsto 6\\mapsto 3\\mapsto 7),(6\\mapsto 6\\mapsto 3\\mapsto 8),(6\\mapsto 6\\mapsto 3\\mapsto 9),(6\\mapsto 6\\mapsto 4\\mapsto 5),(6\\mapsto 6\\mapsto 4\\mapsto 6),(6\\mapsto 6\\mapsto 4\\mapsto 7),(6\\mapsto 6\\mapsto 4\\mapsto 8),(6\\mapsto 6\\mapsto 4\\mapsto 9),(6\\mapsto 6\\mapsto 5\\mapsto 6),(6\\mapsto 6\\mapsto 5\\mapsto 7),(6\\mapsto 6\\mapsto 5\\mapsto 8),(6\\mapsto 6\\mapsto 5\\mapsto 9),(6\\mapsto 6\\mapsto 6\\mapsto 7),(6\\mapsto 6\\mapsto 6\\mapsto 8),(6\\mapsto 6\\mapsto 6\\mapsto 9),(6\\mapsto 6\\mapsto 7\\mapsto 8),(6\\mapsto 6\\mapsto 7\\mapsto 9),(6\\mapsto 6\\mapsto 8\\mapsto 9),(7\\mapsto 7\\mapsto 1\\mapsto 2),(7\\mapsto 7\\mapsto 1\\mapsto 3),(7\\mapsto 7\\mapsto 1\\mapsto 4),(7\\mapsto 7\\mapsto 1\\mapsto 5),(7\\mapsto 7\\mapsto 1\\mapsto 6),(7\\mapsto 7\\mapsto 1\\mapsto 7),(7\\mapsto 7\\mapsto 1\\mapsto 8),(7\\mapsto 7\\mapsto 1\\mapsto 9),(7\\mapsto 7\\mapsto 2\\mapsto 3),(7\\mapsto 7\\mapsto 2\\mapsto 4),(7\\mapsto 7\\mapsto 2\\mapsto 5),(7\\mapsto 7\\mapsto 2\\mapsto 6),(7\\mapsto 7\\mapsto 2\\mapsto 7),(7\\mapsto 7\\mapsto 2\\mapsto 8),(7\\mapsto 7\\mapsto 2\\mapsto 9),(7\\mapsto 7\\mapsto 3\\mapsto 4),(7\\mapsto 7\\mapsto 3\\mapsto 5),(7\\mapsto 7\\mapsto 3\\mapsto 6),(7\\mapsto 7\\mapsto 3\\mapsto 7),(7\\mapsto 7\\mapsto 3\\mapsto 8),(7\\mapsto 7\\mapsto 3\\mapsto 9),(7\\mapsto 7\\mapsto 4\\mapsto 5),(7\\mapsto 7\\mapsto 4\\mapsto 6),(7\\mapsto 7\\mapsto 4\\mapsto 7),(7\\mapsto 7\\mapsto 4\\mapsto 8),(7\\mapsto 7\\mapsto 4\\mapsto 9),(7\\mapsto 7\\mapsto 5\\mapsto 6),(7\\mapsto 7\\mapsto 5\\mapsto 7),(7\\mapsto 7\\mapsto 5\\mapsto 8),(7\\mapsto 7\\mapsto 5\\mapsto 9),(7\\mapsto 7\\mapsto 6\\mapsto 7),(7\\mapsto 7\\mapsto 6\\mapsto 8),(7\\mapsto 7\\mapsto 6\\mapsto 9),(7\\mapsto 7\\mapsto 7\\mapsto 8),(7\\mapsto 7\\mapsto 7\\mapsto 9),(7\\mapsto 7\\mapsto 8\\mapsto 9),(8\\mapsto 8\\mapsto 1\\mapsto 2),(8\\mapsto 8\\mapsto 1\\mapsto 3),(8\\mapsto 8\\mapsto 1\\mapsto 4),(8\\mapsto 8\\mapsto 1\\mapsto 5),(8\\mapsto 8\\mapsto 1\\mapsto 6),(8\\mapsto 8\\mapsto 1\\mapsto 7),(8\\mapsto 8\\mapsto 1\\mapsto 8),(8\\mapsto 8\\mapsto 1\\mapsto 9),(8\\mapsto 8\\mapsto 2\\mapsto 3),(8\\mapsto 8\\mapsto 2\\mapsto 4),(8\\mapsto 8\\mapsto 2\\mapsto 5),(8\\mapsto 8\\mapsto 2\\mapsto 6),(8\\mapsto 8\\mapsto 2\\mapsto 7),(8\\mapsto 8\\mapsto 2\\mapsto 8),(8\\mapsto 8\\mapsto 2\\mapsto 9),(8\\mapsto 8\\mapsto 3\\mapsto 4),(8\\mapsto 8\\mapsto 3\\mapsto 5),(8\\mapsto 8\\mapsto 3\\mapsto 6),(8\\mapsto 8\\mapsto 3\\mapsto 7),(8\\mapsto 8\\mapsto 3\\mapsto 8),(8\\mapsto 8\\mapsto 3\\mapsto 9),(8\\mapsto 8\\mapsto 4\\mapsto 5),(8\\mapsto 8\\mapsto 4\\mapsto 6),(8\\mapsto 8\\mapsto 4\\mapsto 7),(8\\mapsto 8\\mapsto 4\\mapsto 8),(8\\mapsto 8\\mapsto 4\\mapsto 9),(8\\mapsto 8\\mapsto 5\\mapsto 6),(8\\mapsto 8\\mapsto 5\\mapsto 7),(8\\mapsto 8\\mapsto 5\\mapsto 8),(8\\mapsto 8\\mapsto 5\\mapsto 9),(8\\mapsto 8\\mapsto 6\\mapsto 7),(8\\mapsto 8\\mapsto 6\\mapsto 8),(8\\mapsto 8\\mapsto 6\\mapsto 9),(8\\mapsto 8\\mapsto 7\\mapsto 8),(8\\mapsto 8\\mapsto 7\\mapsto 9),(8\\mapsto 8\\mapsto 8\\mapsto 9),(9\\mapsto 9\\mapsto 1\\mapsto 2),(9\\mapsto 9\\mapsto 1\\mapsto 3),(9\\mapsto 9\\mapsto 1\\mapsto 4),(9\\mapsto 9\\mapsto 1\\mapsto 5),(9\\mapsto 9\\mapsto 1\\mapsto 6),(9\\mapsto 9\\mapsto 1\\mapsto 7),(9\\mapsto 9\\mapsto 1\\mapsto 8),(9\\mapsto 9\\mapsto 1\\mapsto 9),(9\\mapsto 9\\mapsto 2\\mapsto 3),(9\\mapsto 9\\mapsto 2\\mapsto 4),(9\\mapsto 9\\mapsto 2\\mapsto 5),(9\\mapsto 9\\mapsto 2\\mapsto 6),(9\\mapsto 9\\mapsto 2\\mapsto 7),(9\\mapsto 9\\mapsto 2\\mapsto 8),(9\\mapsto 9\\mapsto 2\\mapsto 9),(9\\mapsto 9\\mapsto 3\\mapsto 4),(9\\mapsto 9\\mapsto 3\\mapsto 5),(9\\mapsto 9\\mapsto 3\\mapsto 6),(9\\mapsto 9\\mapsto 3\\mapsto 7),(9\\mapsto 9\\mapsto 3\\mapsto 8),(9\\mapsto 9\\mapsto 3\\mapsto 9),(9\\mapsto 9\\mapsto 4\\mapsto 5),(9\\mapsto 9\\mapsto 4\\mapsto 6),(9\\mapsto 9\\mapsto 4\\mapsto 7),(9\\mapsto 9\\mapsto 4\\mapsto 8),(9\\mapsto 9\\mapsto 4\\mapsto 9),(9\\mapsto 9\\mapsto 5\\mapsto 6),(9\\mapsto 9\\mapsto 5\\mapsto 7),(9\\mapsto 9\\mapsto 5\\mapsto 8),(9\\mapsto 9\\mapsto 5\\mapsto 9),(9\\mapsto 9\\mapsto 6\\mapsto 7),(9\\mapsto 9\\mapsto 6\\mapsto 8),(9\\mapsto 9\\mapsto 6\\mapsto 9),(9\\mapsto 9\\mapsto 7\\mapsto 8),(9\\mapsto 9\\mapsto 7\\mapsto 9),(9\\mapsto 9\\mapsto 8\\mapsto 9)\\}$\n", + "* $\\mathit{Board} = \\{(1\\mapsto\\{(1\\mapsto 7),(2\\mapsto 8),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(2\\mapsto\\{(1\\mapsto 9),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(3\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(4\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(5\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(6\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(7\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(8\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\}),(9\\mapsto\\{(1\\mapsto 1),(2\\mapsto 1),(3\\mapsto 1),(4\\mapsto 1),(5\\mapsto 1),(6\\mapsto 1),(7\\mapsto 1),(8\\mapsto 1),(9\\mapsto 1)\\})\\}$\n", + "* $\\mathit{DOM} = \\{1,2,3,4,5,6,7,8,9\\}$\n", + "* $\\mathit{Diff1} = \\{(1\\mapsto 2\\mapsto 1\\mapsto 1),(1\\mapsto 2\\mapsto 2\\mapsto 2),(1\\mapsto 2\\mapsto 3\\mapsto 3),(1\\mapsto 2\\mapsto 4\\mapsto 4),(1\\mapsto 2\\mapsto 5\\mapsto 5),(1\\mapsto 2\\mapsto 6\\mapsto 6),(1\\mapsto 2\\mapsto 7\\mapsto 7),(1\\mapsto 2\\mapsto 8\\mapsto 8),(1\\mapsto 2\\mapsto 9\\mapsto 9),(1\\mapsto 3\\mapsto 1\\mapsto 1),(1\\mapsto 3\\mapsto 2\\mapsto 2),(1\\mapsto 3\\mapsto 3\\mapsto 3),(1\\mapsto 3\\mapsto 4\\mapsto 4),(1\\mapsto 3\\mapsto 5\\mapsto 5),(1\\mapsto 3\\mapsto 6\\mapsto 6),(1\\mapsto 3\\mapsto 7\\mapsto 7),(1\\mapsto 3\\mapsto 8\\mapsto 8),(1\\mapsto 3\\mapsto 9\\mapsto 9),(1\\mapsto 4\\mapsto 1\\mapsto 1),(1\\mapsto 4\\mapsto 2\\mapsto 2),(1\\mapsto 4\\mapsto 3\\mapsto 3),(1\\mapsto 4\\mapsto 4\\mapsto 4),(1\\mapsto 4\\mapsto 5\\mapsto 5),(1\\mapsto 4\\mapsto 6\\mapsto 6),(1\\mapsto 4\\mapsto 7\\mapsto 7),(1\\mapsto 4\\mapsto 8\\mapsto 8),(1\\mapsto 4\\mapsto 9\\mapsto 9),(1\\mapsto 5\\mapsto 1\\mapsto 1),(1\\mapsto 5\\mapsto 2\\mapsto 2),(1\\mapsto 5\\mapsto 3\\mapsto 3),(1\\mapsto 5\\mapsto 4\\mapsto 4),(1\\mapsto 5\\mapsto 5\\mapsto 5),(1\\mapsto 5\\mapsto 6\\mapsto 6),(1\\mapsto 5\\mapsto 7\\mapsto 7),(1\\mapsto 5\\mapsto 8\\mapsto 8),(1\\mapsto 5\\mapsto 9\\mapsto 9),(1\\mapsto 6\\mapsto 1\\mapsto 1),(1\\mapsto 6\\mapsto 2\\mapsto 2),(1\\mapsto 6\\mapsto 3\\mapsto 3),(1\\mapsto 6\\mapsto 4\\mapsto 4),(1\\mapsto 6\\mapsto 5\\mapsto 5),(1\\mapsto 6\\mapsto 6\\mapsto 6),(1\\mapsto 6\\mapsto 7\\mapsto 7),(1\\mapsto 6\\mapsto 8\\mapsto 8),(1\\mapsto 6\\mapsto 9\\mapsto 9),(1\\mapsto 7\\mapsto 1\\mapsto 1),(1\\mapsto 7\\mapsto 2\\mapsto 2),(1\\mapsto 7\\mapsto 3\\mapsto 3),(1\\mapsto 7\\mapsto 4\\mapsto 4),(1\\mapsto 7\\mapsto 5\\mapsto 5),(1\\mapsto 7\\mapsto 6\\mapsto 6),(1\\mapsto 7\\mapsto 7\\mapsto 7),(1\\mapsto 7\\mapsto 8\\mapsto 8),(1\\mapsto 7\\mapsto 9\\mapsto 9),(1\\mapsto 8\\mapsto 1\\mapsto 1),(1\\mapsto 8\\mapsto 2\\mapsto 2),(1\\mapsto 8\\mapsto 3\\mapsto 3),(1\\mapsto 8\\mapsto 4\\mapsto 4),(1\\mapsto 8\\mapsto 5\\mapsto 5),(1\\mapsto 8\\mapsto 6\\mapsto 6),(1\\mapsto 8\\mapsto 7\\mapsto 7),(1\\mapsto 8\\mapsto 8\\mapsto 8),(1\\mapsto 8\\mapsto 9\\mapsto 9),(1\\mapsto 9\\mapsto 1\\mapsto 1),(1\\mapsto 9\\mapsto 2\\mapsto 2),(1\\mapsto 9\\mapsto 3\\mapsto 3),(1\\mapsto 9\\mapsto 4\\mapsto 4),(1\\mapsto 9\\mapsto 5\\mapsto 5),(1\\mapsto 9\\mapsto 6\\mapsto 6),(1\\mapsto 9\\mapsto 7\\mapsto 7),(1\\mapsto 9\\mapsto 8\\mapsto 8),(1\\mapsto 9\\mapsto 9\\mapsto 9),(2\\mapsto 3\\mapsto 1\\mapsto 1),(2\\mapsto 3\\mapsto 2\\mapsto 2),(2\\mapsto 3\\mapsto 3\\mapsto 3),(2\\mapsto 3\\mapsto 4\\mapsto 4),(2\\mapsto 3\\mapsto 5\\mapsto 5),(2\\mapsto 3\\mapsto 6\\mapsto 6),(2\\mapsto 3\\mapsto 7\\mapsto 7),(2\\mapsto 3\\mapsto 8\\mapsto 8),(2\\mapsto 3\\mapsto 9\\mapsto 9),(2\\mapsto 4\\mapsto 1\\mapsto 1),(2\\mapsto 4\\mapsto 2\\mapsto 2),(2\\mapsto 4\\mapsto 3\\mapsto 3),(2\\mapsto 4\\mapsto 4\\mapsto 4),(2\\mapsto 4\\mapsto 5\\mapsto 5),(2\\mapsto 4\\mapsto 6\\mapsto 6),(2\\mapsto 4\\mapsto 7\\mapsto 7),(2\\mapsto 4\\mapsto 8\\mapsto 8),(2\\mapsto 4\\mapsto 9\\mapsto 9),(2\\mapsto 5\\mapsto 1\\mapsto 1),(2\\mapsto 5\\mapsto 2\\mapsto 2),(2\\mapsto 5\\mapsto 3\\mapsto 3),(2\\mapsto 5\\mapsto 4\\mapsto 4),(2\\mapsto 5\\mapsto 5\\mapsto 5),(2\\mapsto 5\\mapsto 6\\mapsto 6),(2\\mapsto 5\\mapsto 7\\mapsto 7),(2\\mapsto 5\\mapsto 8\\mapsto 8),(2\\mapsto 5\\mapsto 9\\mapsto 9),(2\\mapsto 6\\mapsto 1\\mapsto 1),(2\\mapsto 6\\mapsto 2\\mapsto 2),(2\\mapsto 6\\mapsto 3\\mapsto 3),(2\\mapsto 6\\mapsto 4\\mapsto 4),(2\\mapsto 6\\mapsto 5\\mapsto 5),(2\\mapsto 6\\mapsto 6\\mapsto 6),(2\\mapsto 6\\mapsto 7\\mapsto 7),(2\\mapsto 6\\mapsto 8\\mapsto 8),(2\\mapsto 6\\mapsto 9\\mapsto 9),(2\\mapsto 7\\mapsto 1\\mapsto 1),(2\\mapsto 7\\mapsto 2\\mapsto 2),(2\\mapsto 7\\mapsto 3\\mapsto 3),(2\\mapsto 7\\mapsto 4\\mapsto 4),(2\\mapsto 7\\mapsto 5\\mapsto 5),(2\\mapsto 7\\mapsto 6\\mapsto 6),(2\\mapsto 7\\mapsto 7\\mapsto 7),(2\\mapsto 7\\mapsto 8\\mapsto 8),(2\\mapsto 7\\mapsto 9\\mapsto 9),(2\\mapsto 8\\mapsto 1\\mapsto 1),(2\\mapsto 8\\mapsto 2\\mapsto 2),(2\\mapsto 8\\mapsto 3\\mapsto 3),(2\\mapsto 8\\mapsto 4\\mapsto 4),(2\\mapsto 8\\mapsto 5\\mapsto 5),(2\\mapsto 8\\mapsto 6\\mapsto 6),(2\\mapsto 8\\mapsto 7\\mapsto 7),(2\\mapsto 8\\mapsto 8\\mapsto 8),(2\\mapsto 8\\mapsto 9\\mapsto 9),(2\\mapsto 9\\mapsto 1\\mapsto 1),(2\\mapsto 9\\mapsto 2\\mapsto 2),(2\\mapsto 9\\mapsto 3\\mapsto 3),(2\\mapsto 9\\mapsto 4\\mapsto 4),(2\\mapsto 9\\mapsto 5\\mapsto 5),(2\\mapsto 9\\mapsto 6\\mapsto 6),(2\\mapsto 9\\mapsto 7\\mapsto 7),(2\\mapsto 9\\mapsto 8\\mapsto 8),(2\\mapsto 9\\mapsto 9\\mapsto 9),(3\\mapsto 4\\mapsto 1\\mapsto 1),(3\\mapsto 4\\mapsto 2\\mapsto 2),(3\\mapsto 4\\mapsto 3\\mapsto 3),(3\\mapsto 4\\mapsto 4\\mapsto 4),(3\\mapsto 4\\mapsto 5\\mapsto 5),(3\\mapsto 4\\mapsto 6\\mapsto 6),(3\\mapsto 4\\mapsto 7\\mapsto 7),(3\\mapsto 4\\mapsto 8\\mapsto 8),(3\\mapsto 4\\mapsto 9\\mapsto 9),(3\\mapsto 5\\mapsto 1\\mapsto 1),(3\\mapsto 5\\mapsto 2\\mapsto 2),(3\\mapsto 5\\mapsto 3\\mapsto 3),(3\\mapsto 5\\mapsto 4\\mapsto 4),(3\\mapsto 5\\mapsto 5\\mapsto 5),(3\\mapsto 5\\mapsto 6\\mapsto 6),(3\\mapsto 5\\mapsto 7\\mapsto 7),(3\\mapsto 5\\mapsto 8\\mapsto 8),(3\\mapsto 5\\mapsto 9\\mapsto 9),(3\\mapsto 6\\mapsto 1\\mapsto 1),(3\\mapsto 6\\mapsto 2\\mapsto 2),(3\\mapsto 6\\mapsto 3\\mapsto 3),(3\\mapsto 6\\mapsto 4\\mapsto 4),(3\\mapsto 6\\mapsto 5\\mapsto 5),(3\\mapsto 6\\mapsto 6\\mapsto 6),(3\\mapsto 6\\mapsto 7\\mapsto 7),(3\\mapsto 6\\mapsto 8\\mapsto 8),(3\\mapsto 6\\mapsto 9\\mapsto 9),(3\\mapsto 7\\mapsto 1\\mapsto 1),(3\\mapsto 7\\mapsto 2\\mapsto 2),(3\\mapsto 7\\mapsto 3\\mapsto 3),(3\\mapsto 7\\mapsto 4\\mapsto 4),(3\\mapsto 7\\mapsto 5\\mapsto 5),(3\\mapsto 7\\mapsto 6\\mapsto 6),(3\\mapsto 7\\mapsto 7\\mapsto 7),(3\\mapsto 7\\mapsto 8\\mapsto 8),(3\\mapsto 7\\mapsto 9\\mapsto 9),(3\\mapsto 8\\mapsto 1\\mapsto 1),(3\\mapsto 8\\mapsto 2\\mapsto 2),(3\\mapsto 8\\mapsto 3\\mapsto 3),(3\\mapsto 8\\mapsto 4\\mapsto 4),(3\\mapsto 8\\mapsto 5\\mapsto 5),(3\\mapsto 8\\mapsto 6\\mapsto 6),(3\\mapsto 8\\mapsto 7\\mapsto 7),(3\\mapsto 8\\mapsto 8\\mapsto 8),(3\\mapsto 8\\mapsto 9\\mapsto 9),(3\\mapsto 9\\mapsto 1\\mapsto 1),(3\\mapsto 9\\mapsto 2\\mapsto 2),(3\\mapsto 9\\mapsto 3\\mapsto 3),(3\\mapsto 9\\mapsto 4\\mapsto 4),(3\\mapsto 9\\mapsto 5\\mapsto 5),(3\\mapsto 9\\mapsto 6\\mapsto 6),(3\\mapsto 9\\mapsto 7\\mapsto 7),(3\\mapsto 9\\mapsto 8\\mapsto 8),(3\\mapsto 9\\mapsto 9\\mapsto 9),(4\\mapsto 5\\mapsto 1\\mapsto 1),(4\\mapsto 5\\mapsto 2\\mapsto 2),(4\\mapsto 5\\mapsto 3\\mapsto 3),(4\\mapsto 5\\mapsto 4\\mapsto 4),(4\\mapsto 5\\mapsto 5\\mapsto 5),(4\\mapsto 5\\mapsto 6\\mapsto 6),(4\\mapsto 5\\mapsto 7\\mapsto 7),(4\\mapsto 5\\mapsto 8\\mapsto 8),(4\\mapsto 5\\mapsto 9\\mapsto 9),(4\\mapsto 6\\mapsto 1\\mapsto 1),(4\\mapsto 6\\mapsto 2\\mapsto 2),(4\\mapsto 6\\mapsto 3\\mapsto 3),(4\\mapsto 6\\mapsto 4\\mapsto 4),(4\\mapsto 6\\mapsto 5\\mapsto 5),(4\\mapsto 6\\mapsto 6\\mapsto 6),(4\\mapsto 6\\mapsto 7\\mapsto 7),(4\\mapsto 6\\mapsto 8\\mapsto 8),(4\\mapsto 6\\mapsto 9\\mapsto 9),(4\\mapsto 7\\mapsto 1\\mapsto 1),(4\\mapsto 7\\mapsto 2\\mapsto 2),(4\\mapsto 7\\mapsto 3\\mapsto 3),(4\\mapsto 7\\mapsto 4\\mapsto 4),(4\\mapsto 7\\mapsto 5\\mapsto 5),(4\\mapsto 7\\mapsto 6\\mapsto 6),(4\\mapsto 7\\mapsto 7\\mapsto 7),(4\\mapsto 7\\mapsto 8\\mapsto 8),(4\\mapsto 7\\mapsto 9\\mapsto 9),(4\\mapsto 8\\mapsto 1\\mapsto 1),(4\\mapsto 8\\mapsto 2\\mapsto 2),(4\\mapsto 8\\mapsto 3\\mapsto 3),(4\\mapsto 8\\mapsto 4\\mapsto 4),(4\\mapsto 8\\mapsto 5\\mapsto 5),(4\\mapsto 8\\mapsto 6\\mapsto 6),(4\\mapsto 8\\mapsto 7\\mapsto 7),(4\\mapsto 8\\mapsto 8\\mapsto 8),(4\\mapsto 8\\mapsto 9\\mapsto 9),(4\\mapsto 9\\mapsto 1\\mapsto 1),(4\\mapsto 9\\mapsto 2\\mapsto 2),(4\\mapsto 9\\mapsto 3\\mapsto 3),(4\\mapsto 9\\mapsto 4\\mapsto 4),(4\\mapsto 9\\mapsto 5\\mapsto 5),(4\\mapsto 9\\mapsto 6\\mapsto 6),(4\\mapsto 9\\mapsto 7\\mapsto 7),(4\\mapsto 9\\mapsto 8\\mapsto 8),(4\\mapsto 9\\mapsto 9\\mapsto 9),(5\\mapsto 6\\mapsto 1\\mapsto 1),(5\\mapsto 6\\mapsto 2\\mapsto 2),(5\\mapsto 6\\mapsto 3\\mapsto 3),(5\\mapsto 6\\mapsto 4\\mapsto 4),(5\\mapsto 6\\mapsto 5\\mapsto 5),(5\\mapsto 6\\mapsto 6\\mapsto 6),(5\\mapsto 6\\mapsto 7\\mapsto 7),(5\\mapsto 6\\mapsto 8\\mapsto 8),(5\\mapsto 6\\mapsto 9\\mapsto 9),(5\\mapsto 7\\mapsto 1\\mapsto 1),(5\\mapsto 7\\mapsto 2\\mapsto 2),(5\\mapsto 7\\mapsto 3\\mapsto 3),(5\\mapsto 7\\mapsto 4\\mapsto 4),(5\\mapsto 7\\mapsto 5\\mapsto 5),(5\\mapsto 7\\mapsto 6\\mapsto 6),(5\\mapsto 7\\mapsto 7\\mapsto 7),(5\\mapsto 7\\mapsto 8\\mapsto 8),(5\\mapsto 7\\mapsto 9\\mapsto 9),(5\\mapsto 8\\mapsto 1\\mapsto 1),(5\\mapsto 8\\mapsto 2\\mapsto 2),(5\\mapsto 8\\mapsto 3\\mapsto 3),(5\\mapsto 8\\mapsto 4\\mapsto 4),(5\\mapsto 8\\mapsto 5\\mapsto 5),(5\\mapsto 8\\mapsto 6\\mapsto 6),(5\\mapsto 8\\mapsto 7\\mapsto 7),(5\\mapsto 8\\mapsto 8\\mapsto 8),(5\\mapsto 8\\mapsto 9\\mapsto 9),(5\\mapsto 9\\mapsto 1\\mapsto 1),(5\\mapsto 9\\mapsto 2\\mapsto 2),(5\\mapsto 9\\mapsto 3\\mapsto 3),(5\\mapsto 9\\mapsto 4\\mapsto 4),(5\\mapsto 9\\mapsto 5\\mapsto 5),(5\\mapsto 9\\mapsto 6\\mapsto 6),(5\\mapsto 9\\mapsto 7\\mapsto 7),(5\\mapsto 9\\mapsto 8\\mapsto 8),(5\\mapsto 9\\mapsto 9\\mapsto 9),(6\\mapsto 7\\mapsto 1\\mapsto 1),(6\\mapsto 7\\mapsto 2\\mapsto 2),(6\\mapsto 7\\mapsto 3\\mapsto 3),(6\\mapsto 7\\mapsto 4\\mapsto 4),(6\\mapsto 7\\mapsto 5\\mapsto 5),(6\\mapsto 7\\mapsto 6\\mapsto 6),(6\\mapsto 7\\mapsto 7\\mapsto 7),(6\\mapsto 7\\mapsto 8\\mapsto 8),(6\\mapsto 7\\mapsto 9\\mapsto 9),(6\\mapsto 8\\mapsto 1\\mapsto 1),(6\\mapsto 8\\mapsto 2\\mapsto 2),(6\\mapsto 8\\mapsto 3\\mapsto 3),(6\\mapsto 8\\mapsto 4\\mapsto 4),(6\\mapsto 8\\mapsto 5\\mapsto 5),(6\\mapsto 8\\mapsto 6\\mapsto 6),(6\\mapsto 8\\mapsto 7\\mapsto 7),(6\\mapsto 8\\mapsto 8\\mapsto 8),(6\\mapsto 8\\mapsto 9\\mapsto 9),(6\\mapsto 9\\mapsto 1\\mapsto 1),(6\\mapsto 9\\mapsto 2\\mapsto 2),(6\\mapsto 9\\mapsto 3\\mapsto 3),(6\\mapsto 9\\mapsto 4\\mapsto 4),(6\\mapsto 9\\mapsto 5\\mapsto 5),(6\\mapsto 9\\mapsto 6\\mapsto 6),(6\\mapsto 9\\mapsto 7\\mapsto 7),(6\\mapsto 9\\mapsto 8\\mapsto 8),(6\\mapsto 9\\mapsto 9\\mapsto 9),(7\\mapsto 8\\mapsto 1\\mapsto 1),(7\\mapsto 8\\mapsto 2\\mapsto 2),(7\\mapsto 8\\mapsto 3\\mapsto 3),(7\\mapsto 8\\mapsto 4\\mapsto 4),(7\\mapsto 8\\mapsto 5\\mapsto 5),(7\\mapsto 8\\mapsto 6\\mapsto 6),(7\\mapsto 8\\mapsto 7\\mapsto 7),(7\\mapsto 8\\mapsto 8\\mapsto 8),(7\\mapsto 8\\mapsto 9\\mapsto 9),(7\\mapsto 9\\mapsto 1\\mapsto 1),(7\\mapsto 9\\mapsto 2\\mapsto 2),(7\\mapsto 9\\mapsto 3\\mapsto 3),(7\\mapsto 9\\mapsto 4\\mapsto 4),(7\\mapsto 9\\mapsto 5\\mapsto 5),(7\\mapsto 9\\mapsto 6\\mapsto 6),(7\\mapsto 9\\mapsto 7\\mapsto 7),(7\\mapsto 9\\mapsto 8\\mapsto 8),(7\\mapsto 9\\mapsto 9\\mapsto 9),(8\\mapsto 9\\mapsto 1\\mapsto 1),(8\\mapsto 9\\mapsto 2\\mapsto 2),(8\\mapsto 9\\mapsto 3\\mapsto 3),(8\\mapsto 9\\mapsto 4\\mapsto 4),(8\\mapsto 9\\mapsto 5\\mapsto 5),(8\\mapsto 9\\mapsto 6\\mapsto 6),(8\\mapsto 9\\mapsto 7\\mapsto 7),(8\\mapsto 9\\mapsto 8\\mapsto 8),(8\\mapsto 9\\mapsto 9\\mapsto 9)\\}$\n", + "* $\\mathit{Diff} = \\emptyset$\n", + "* $\\mathit{SUBSQ} = \\{\\{1,2,3\\},\\{4,5,6\\},\\{7,8,9\\}\\}$" + ], + "text/plain": [ + "TRUE\n", + "\n", + "Solution:\n", + "\tP = {(1↦1↦7),(1↦2↦8),(1↦3↦1),(2↦1↦9)}\n", + "\tDiff3 = {(1↦1↦2↦1),(1↦1↦3↦1),(1↦1↦3↦2),(1↦1↦5↦4),(1↦1↦6↦4),(1↦1↦6↦5),(1↦1↦8↦7),(1↦1↦9↦7),(1↦1↦9↦8),(2↦1↦1↦1),(2↦1↦1↦2),(2↦1↦1↦3),(2↦1↦2↦1),(2↦1↦2↦2),(2↦1↦2↦3),(2↦1↦3↦1),(2↦1↦3↦2),(2↦1↦3↦3),(2↦1↦4↦4),(2↦1↦4↦5),(2↦1↦4↦6),(2↦1↦5↦4),(2↦1↦5↦5),(2↦1↦5↦6),(2↦1↦6↦4),(2↦1↦6↦5),(2↦1↦6↦6),(2↦1↦7↦7),(2↦1↦7↦8),(2↦1↦7↦9),(2↦1↦8↦7),(2↦1↦8↦8),(2↦1↦8↦9),(2↦1↦9↦7),(2↦1↦9↦8),(2↦1↦9↦9),(2↦2↦2↦1),(2↦2↦3↦1),(2↦2↦3↦2),(2↦2↦5↦4),(2↦2↦6↦4),(2↦2↦6↦5),(2↦2↦8↦7),(2↦2↦9↦7),(2↦2↦9↦8),(3↦1↦1↦1),(3↦1↦1↦2),(3↦1↦1↦3),(3↦1↦2↦1),(3↦1↦2↦2),(3↦1↦2↦3),(3↦1↦3↦1),(3↦1↦3↦2),(3↦1↦3↦3),(3↦1↦4↦4),(3↦1↦4↦5),(3↦1↦4↦6),(3↦1↦5↦4),(3↦1↦5↦5),(3↦1↦5↦6),(3↦1↦6↦4),(3↦1↦6↦5),(3↦1↦6↦6),(3↦1↦7↦7),(3↦1↦7↦8),(3↦1↦7↦9),(3↦1↦8↦7),(3↦1↦8↦8),(3↦1↦8↦9),(3↦1↦9↦7),(3↦1↦9↦8),(3↦1↦9↦9),(3↦2↦1↦1),(3↦2↦1↦2),(3↦2↦1↦3),(3↦2↦2↦1),(3↦2↦2↦2),(3↦2↦2↦3),(3↦2↦3↦1),(3↦2↦3↦2),(3↦2↦3↦3),(3↦2↦4↦4),(3↦2↦4↦5),(3↦2↦4↦6),(3↦2↦5↦4),(3↦2↦5↦5),(3↦2↦5↦6),(3↦2↦6↦4),(3↦2↦6↦5),(3↦2↦6↦6),(3↦2↦7↦7),(3↦2↦7↦8),(3↦2↦7↦9),(3↦2↦8↦7),(3↦2↦8↦8),(3↦2↦8↦9),(3↦2↦9↦7),(3↦2↦9↦8),(3↦2↦9↦9),(3↦3↦2↦1),(3↦3↦3↦1),(3↦3↦3↦2),(3↦3↦5↦4),(3↦3↦6↦4),(3↦3↦6↦5),(3↦3↦8↦7),(3↦3↦9↦7),(3↦3↦9↦8),(4↦4↦2↦1),(4↦4↦3↦1),(4↦4↦3↦2),(4↦4↦5↦4),(4↦4↦6↦4),(4↦4↦6↦5),(4↦4↦8↦7),(4↦4↦9↦7),(4↦4↦9↦8),(5↦4↦1↦1),(5↦4↦1↦2),(5↦4↦1↦3),(5↦4↦2↦1),(5↦4↦2↦2),(5↦4↦2↦3),(5↦4↦3↦1),(5↦4↦3↦2),(5↦4↦3↦3),(5↦4↦4↦4),(5↦4↦4↦5),(5↦4↦4↦6),(5↦4↦5↦4),(5↦4↦5↦5),(5↦4↦5↦6),(5↦4↦6↦4),(5↦4↦6↦5),(5↦4↦6↦6),(5↦4↦7↦7),(5↦4↦7↦8),(5↦4↦7↦9),(5↦4↦8↦7),(5↦4↦8↦8),(5↦4↦8↦9),(5↦4↦9↦7),(5↦4↦9↦8),(5↦4↦9↦9),(5↦5↦2↦1),(5↦5↦3↦1),(5↦5↦3↦2),(5↦5↦5↦4),(5↦5↦6↦4),(5↦5↦6↦5),(5↦5↦8↦7),(5↦5↦9↦7),(5↦5↦9↦8),(6↦4↦1↦1),(6↦4↦1↦2),(6↦4↦1↦3),(6↦4↦2↦1),(6↦4↦2↦2),(6↦4↦2↦3),(6↦4↦3↦1),(6↦4↦3↦2),(6↦4↦3↦3),(6↦4↦4↦4),(6↦4↦4↦5),(6↦4↦4↦6),(6↦4↦5↦4),(6↦4↦5↦5),(6↦4↦5↦6),(6↦4↦6↦4),(6↦4↦6↦5),(6↦4↦6↦6),(6↦4↦7↦7),(6↦4↦7↦8),(6↦4↦7↦9),(6↦4↦8↦7),(6↦4↦8↦8),(6↦4↦8↦9),(6↦4↦9↦7),(6↦4↦9↦8),(6↦4↦9↦9),(6↦5↦1↦1),(6↦5↦1↦2),(6↦5↦1↦3),(6↦5↦2↦1),(6↦5↦2↦2),(6↦5↦2↦3),(6↦5↦3↦1),(6↦5↦3↦2),(6↦5↦3↦3),(6↦5↦4↦4),(6↦5↦4↦5),(6↦5↦4↦6),(6↦5↦5↦4),(6↦5↦5↦5),(6↦5↦5↦6),(6↦5↦6↦4),(6↦5↦6↦5),(6↦5↦6↦6),(6↦5↦7↦7),(6↦5↦7↦8),(6↦5↦7↦9),(6↦5↦8↦7),(6↦5↦8↦8),(6↦5↦8↦9),(6↦5↦9↦7),(6↦5↦9↦8),(6↦5↦9↦9),(6↦6↦2↦1),(6↦6↦3↦1),(6↦6↦3↦2),(6↦6↦5↦4),(6↦6↦6↦4),(6↦6↦6↦5),(6↦6↦8↦7),(6↦6↦9↦7),(6↦6↦9↦8),(7↦7↦2↦1),(7↦7↦3↦1),(7↦7↦3↦2),(7↦7↦5↦4),(7↦7↦6↦4),(7↦7↦6↦5),(7↦7↦8↦7),(7↦7↦9↦7),(7↦7↦9↦8),(8↦7↦1↦1),(8↦7↦1↦2),(8↦7↦1↦3),(8↦7↦2↦1),(8↦7↦2↦2),(8↦7↦2↦3),(8↦7↦3↦1),(8↦7↦3↦2),(8↦7↦3↦3),(8↦7↦4↦4),(8↦7↦4↦5),(8↦7↦4↦6),(8↦7↦5↦4),(8↦7↦5↦5),(8↦7↦5↦6),(8↦7↦6↦4),(8↦7↦6↦5),(8↦7↦6↦6),(8↦7↦7↦7),(8↦7↦7↦8),(8↦7↦7↦9),(8↦7↦8↦7),(8↦7↦8↦8),(8↦7↦8↦9),(8↦7↦9↦7),(8↦7↦9↦8),(8↦7↦9↦9),(8↦8↦2↦1),(8↦8↦3↦1),(8↦8↦3↦2),(8↦8↦5↦4),(8↦8↦6↦4),(8↦8↦6↦5),(8↦8↦8↦7),(8↦8↦9↦7),(8↦8↦9↦8),(9↦7↦1↦1),(9↦7↦1↦2),(9↦7↦1↦3),(9↦7↦2↦1),(9↦7↦2↦2),(9↦7↦2↦3),(9↦7↦3↦1),(9↦7↦3↦2),(9↦7↦3↦3),(9↦7↦4↦4),(9↦7↦4↦5),(9↦7↦4↦6),(9↦7↦5↦4),(9↦7↦5↦5),(9↦7↦5↦6),(9↦7↦6↦4),(9↦7↦6↦5),(9↦7↦6↦6),(9↦7↦7↦7),(9↦7↦7↦8),(9↦7↦7↦9),(9↦7↦8↦7),(9↦7↦8↦8),(9↦7↦8↦9),(9↦7↦9↦7),(9↦7↦9↦8),(9↦7↦9↦9),(9↦8↦1↦1),(9↦8↦1↦2),(9↦8↦1↦3),(9↦8↦2↦1),(9↦8↦2↦2),(9↦8↦2↦3),(9↦8↦3↦1),(9↦8↦3↦2),(9↦8↦3↦3),(9↦8↦4↦4),(9↦8↦4↦5),(9↦8↦4↦6),(9↦8↦5↦4),(9↦8↦5↦5),(9↦8↦5↦6),(9↦8↦6↦4),(9↦8↦6↦5),(9↦8↦6↦6),(9↦8↦7↦7),(9↦8↦7↦8),(9↦8↦7↦9),(9↦8↦8↦7),(9↦8↦8↦8),(9↦8↦8↦9),(9↦8↦9↦7),(9↦8↦9↦8),(9↦8↦9↦9),(9↦9↦2↦1),(9↦9↦3↦1),(9↦9↦3↦2),(9↦9↦5↦4),(9↦9↦6↦4),(9↦9↦6↦5),(9↦9↦8↦7),(9↦9↦9↦7),(9↦9↦9↦8)}\n", + "\tDiff2 = {(1↦1↦1↦2),(1↦1↦1↦3),(1↦1↦1↦4),(1↦1↦1↦5),(1↦1↦1↦6),(1↦1↦1↦7),(1↦1↦1↦8),(1↦1↦1↦9),(1↦1↦2↦3),(1↦1↦2↦4),(1↦1↦2↦5),(1↦1↦2↦6),(1↦1↦2↦7),(1↦1↦2↦8),(1↦1↦2↦9),(1↦1↦3↦4),(1↦1↦3↦5),(1↦1↦3↦6),(1↦1↦3↦7),(1↦1↦3↦8),(1↦1↦3↦9),(1↦1↦4↦5),(1↦1↦4↦6),(1↦1↦4↦7),(1↦1↦4↦8),(1↦1↦4↦9),(1↦1↦5↦6),(1↦1↦5↦7),(1↦1↦5↦8),(1↦1↦5↦9),(1↦1↦6↦7),(1↦1↦6↦8),(1↦1↦6↦9),(1↦1↦7↦8),(1↦1↦7↦9),(1↦1↦8↦9),(2↦2↦1↦2),(2↦2↦1↦3),(2↦2↦1↦4),(2↦2↦1↦5),(2↦2↦1↦6),(2↦2↦1↦7),(2↦2↦1↦8),(2↦2↦1↦9),(2↦2↦2↦3),(2↦2↦2↦4),(2↦2↦2↦5),(2↦2↦2↦6),(2↦2↦2↦7),(2↦2↦2↦8),(2↦2↦2↦9),(2↦2↦3↦4),(2↦2↦3↦5),(2↦2↦3↦6),(2↦2↦3↦7),(2↦2↦3↦8),(2↦2↦3↦9),(2↦2↦4↦5),(2↦2↦4↦6),(2↦2↦4↦7),(2↦2↦4↦8),(2↦2↦4↦9),(2↦2↦5↦6),(2↦2↦5↦7),(2↦2↦5↦8),(2↦2↦5↦9),(2↦2↦6↦7),(2↦2↦6↦8),(2↦2↦6↦9),(2↦2↦7↦8),(2↦2↦7↦9),(2↦2↦8↦9),(3↦3↦1↦2),(3↦3↦1↦3),(3↦3↦1↦4),(3↦3↦1↦5),(3↦3↦1↦6),(3↦3↦1↦7),(3↦3↦1↦8),(3↦3↦1↦9),(3↦3↦2↦3),(3↦3↦2↦4),(3↦3↦2↦5),(3↦3↦2↦6),(3↦3↦2↦7),(3↦3↦2↦8),(3↦3↦2↦9),(3↦3↦3↦4),(3↦3↦3↦5),(3↦3↦3↦6),(3↦3↦3↦7),(3↦3↦3↦8),(3↦3↦3↦9),(3↦3↦4↦5),(3↦3↦4↦6),(3↦3↦4↦7),(3↦3↦4↦8),(3↦3↦4↦9),(3↦3↦5↦6),(3↦3↦5↦7),(3↦3↦5↦8),(3↦3↦5↦9),(3↦3↦6↦7),(3↦3↦6↦8),(3↦3↦6↦9),(3↦3↦7↦8),(3↦3↦7↦9),(3↦3↦8↦9),(4↦4↦1↦2),(4↦4↦1↦3),(4↦4↦1↦4),(4↦4↦1↦5),(4↦4↦1↦6),(4↦4↦1↦7),(4↦4↦1↦8),(4↦4↦1↦9),(4↦4↦2↦3),(4↦4↦2↦4),(4↦4↦2↦5),(4↦4↦2↦6),(4↦4↦2↦7),(4↦4↦2↦8),(4↦4↦2↦9),(4↦4↦3↦4),(4↦4↦3↦5),(4↦4↦3↦6),(4↦4↦3↦7),(4↦4↦3↦8),(4↦4↦3↦9),(4↦4↦4↦5),(4↦4↦4↦6),(4↦4↦4↦7),(4↦4↦4↦8),(4↦4↦4↦9),(4↦4↦5↦6),(4↦4↦5↦7),(4↦4↦5↦8),(4↦4↦5↦9),(4↦4↦6↦7),(4↦4↦6↦8),(4↦4↦6↦9),(4↦4↦7↦8),(4↦4↦7↦9),(4↦4↦8↦9),(5↦5↦1↦2),(5↦5↦1↦3),(5↦5↦1↦4),(5↦5↦1↦5),(5↦5↦1↦6),(5↦5↦1↦7),(5↦5↦1↦8),(5↦5↦1↦9),(5↦5↦2↦3),(5↦5↦2↦4),(5↦5↦2↦5),(5↦5↦2↦6),(5↦5↦2↦7),(5↦5↦2↦8),(5↦5↦2↦9),(5↦5↦3↦4),(5↦5↦3↦5),(5↦5↦3↦6),(5↦5↦3↦7),(5↦5↦3↦8),(5↦5↦3↦9),(5↦5↦4↦5),(5↦5↦4↦6),(5↦5↦4↦7),(5↦5↦4↦8),(5↦5↦4↦9),(5↦5↦5↦6),(5↦5↦5↦7),(5↦5↦5↦8),(5↦5↦5↦9),(5↦5↦6↦7),(5↦5↦6↦8),(5↦5↦6↦9),(5↦5↦7↦8),(5↦5↦7↦9),(5↦5↦8↦9),(6↦6↦1↦2),(6↦6↦1↦3),(6↦6↦1↦4),(6↦6↦1↦5),(6↦6↦1↦6),(6↦6↦1↦7),(6↦6↦1↦8),(6↦6↦1↦9),(6↦6↦2↦3),(6↦6↦2↦4),(6↦6↦2↦5),(6↦6↦2↦6),(6↦6↦2↦7),(6↦6↦2↦8),(6↦6↦2↦9),(6↦6↦3↦4),(6↦6↦3↦5),(6↦6↦3↦6),(6↦6↦3↦7),(6↦6↦3↦8),(6↦6↦3↦9),(6↦6↦4↦5),(6↦6↦4↦6),(6↦6↦4↦7),(6↦6↦4↦8),(6↦6↦4↦9),(6↦6↦5↦6),(6↦6↦5↦7),(6↦6↦5↦8),(6↦6↦5↦9),(6↦6↦6↦7),(6↦6↦6↦8),(6↦6↦6↦9),(6↦6↦7↦8),(6↦6↦7↦9),(6↦6↦8↦9),(7↦7↦1↦2),(7↦7↦1↦3),(7↦7↦1↦4),(7↦7↦1↦5),(7↦7↦1↦6),(7↦7↦1↦7),(7↦7↦1↦8),(7↦7↦1↦9),(7↦7↦2↦3),(7↦7↦2↦4),(7↦7↦2↦5),(7↦7↦2↦6),(7↦7↦2↦7),(7↦7↦2↦8),(7↦7↦2↦9),(7↦7↦3↦4),(7↦7↦3↦5),(7↦7↦3↦6),(7↦7↦3↦7),(7↦7↦3↦8),(7↦7↦3↦9),(7↦7↦4↦5),(7↦7↦4↦6),(7↦7↦4↦7),(7↦7↦4↦8),(7↦7↦4↦9),(7↦7↦5↦6),(7↦7↦5↦7),(7↦7↦5↦8),(7↦7↦5↦9),(7↦7↦6↦7),(7↦7↦6↦8),(7↦7↦6↦9),(7↦7↦7↦8),(7↦7↦7↦9),(7↦7↦8↦9),(8↦8↦1↦2),(8↦8↦1↦3),(8↦8↦1↦4),(8↦8↦1↦5),(8↦8↦1↦6),(8↦8↦1↦7),(8↦8↦1↦8),(8↦8↦1↦9),(8↦8↦2↦3),(8↦8↦2↦4),(8↦8↦2↦5),(8↦8↦2↦6),(8↦8↦2↦7),(8↦8↦2↦8),(8↦8↦2↦9),(8↦8↦3↦4),(8↦8↦3↦5),(8↦8↦3↦6),(8↦8↦3↦7),(8↦8↦3↦8),(8↦8↦3↦9),(8↦8↦4↦5),(8↦8↦4↦6),(8↦8↦4↦7),(8↦8↦4↦8),(8↦8↦4↦9),(8↦8↦5↦6),(8↦8↦5↦7),(8↦8↦5↦8),(8↦8↦5↦9),(8↦8↦6↦7),(8↦8↦6↦8),(8↦8↦6↦9),(8↦8↦7↦8),(8↦8↦7↦9),(8↦8↦8↦9),(9↦9↦1↦2),(9↦9↦1↦3),(9↦9↦1↦4),(9↦9↦1↦5),(9↦9↦1↦6),(9↦9↦1↦7),(9↦9↦1↦8),(9↦9↦1↦9),(9↦9↦2↦3),(9↦9↦2↦4),(9↦9↦2↦5),(9↦9↦2↦6),(9↦9↦2↦7),(9↦9↦2↦8),(9↦9↦2↦9),(9↦9↦3↦4),(9↦9↦3↦5),(9↦9↦3↦6),(9↦9↦3↦7),(9↦9↦3↦8),(9↦9↦3↦9),(9↦9↦4↦5),(9↦9↦4↦6),(9↦9↦4↦7),(9↦9↦4↦8),(9↦9↦4↦9),(9↦9↦5↦6),(9↦9↦5↦7),(9↦9↦5↦8),(9↦9↦5↦9),(9↦9↦6↦7),(9↦9↦6↦8),(9↦9↦6↦9),(9↦9↦7↦8),(9↦9↦7↦9),(9↦9↦8↦9)}\n", + "\tBoard = {(1↦{(1↦7),(2↦8),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(2↦{(1↦9),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(3↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(4↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(5↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(6↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(7↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(8↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)}),(9↦{(1↦1),(2↦1),(3↦1),(4↦1),(5↦1),(6↦1),(7↦1),(8↦1),(9↦1)})}\n", + "\tDOM = {1,2,3,4,5,6,7,8,9}\n", + "\tDiff1 = {(1↦2↦1↦1),(1↦2↦2↦2),(1↦2↦3↦3),(1↦2↦4↦4),(1↦2↦5↦5),(1↦2↦6↦6),(1↦2↦7↦7),(1↦2↦8↦8),(1↦2↦9↦9),(1↦3↦1↦1),(1↦3↦2↦2),(1↦3↦3↦3),(1↦3↦4↦4),(1↦3↦5↦5),(1↦3↦6↦6),(1↦3↦7↦7),(1↦3↦8↦8),(1↦3↦9↦9),(1↦4↦1↦1),(1↦4↦2↦2),(1↦4↦3↦3),(1↦4↦4↦4),(1↦4↦5↦5),(1↦4↦6↦6),(1↦4↦7↦7),(1↦4↦8↦8),(1↦4↦9↦9),(1↦5↦1↦1),(1↦5↦2↦2),(1↦5↦3↦3),(1↦5↦4↦4),(1↦5↦5↦5),(1↦5↦6↦6),(1↦5↦7↦7),(1↦5↦8↦8),(1↦5↦9↦9),(1↦6↦1↦1),(1↦6↦2↦2),(1↦6↦3↦3),(1↦6↦4↦4),(1↦6↦5↦5),(1↦6↦6↦6),(1↦6↦7↦7),(1↦6↦8↦8),(1↦6↦9↦9),(1↦7↦1↦1),(1↦7↦2↦2),(1↦7↦3↦3),(1↦7↦4↦4),(1↦7↦5↦5),(1↦7↦6↦6),(1↦7↦7↦7),(1↦7↦8↦8),(1↦7↦9↦9),(1↦8↦1↦1),(1↦8↦2↦2),(1↦8↦3↦3),(1↦8↦4↦4),(1↦8↦5↦5),(1↦8↦6↦6),(1↦8↦7↦7),(1↦8↦8↦8),(1↦8↦9↦9),(1↦9↦1↦1),(1↦9↦2↦2),(1↦9↦3↦3),(1↦9↦4↦4),(1↦9↦5↦5),(1↦9↦6↦6),(1↦9↦7↦7),(1↦9↦8↦8),(1↦9↦9↦9),(2↦3↦1↦1),(2↦3↦2↦2),(2↦3↦3↦3),(2↦3↦4↦4),(2↦3↦5↦5),(2↦3↦6↦6),(2↦3↦7↦7),(2↦3↦8↦8),(2↦3↦9↦9),(2↦4↦1↦1),(2↦4↦2↦2),(2↦4↦3↦3),(2↦4↦4↦4),(2↦4↦5↦5),(2↦4↦6↦6),(2↦4↦7↦7),(2↦4↦8↦8),(2↦4↦9↦9),(2↦5↦1↦1),(2↦5↦2↦2),(2↦5↦3↦3),(2↦5↦4↦4),(2↦5↦5↦5),(2↦5↦6↦6),(2↦5↦7↦7),(2↦5↦8↦8),(2↦5↦9↦9),(2↦6↦1↦1),(2↦6↦2↦2),(2↦6↦3↦3),(2↦6↦4↦4),(2↦6↦5↦5),(2↦6↦6↦6),(2↦6↦7↦7),(2↦6↦8↦8),(2↦6↦9↦9),(2↦7↦1↦1),(2↦7↦2↦2),(2↦7↦3↦3),(2↦7↦4↦4),(2↦7↦5↦5),(2↦7↦6↦6),(2↦7↦7↦7),(2↦7↦8↦8),(2↦7↦9↦9),(2↦8↦1↦1),(2↦8↦2↦2),(2↦8↦3↦3),(2↦8↦4↦4),(2↦8↦5↦5),(2↦8↦6↦6),(2↦8↦7↦7),(2↦8↦8↦8),(2↦8↦9↦9),(2↦9↦1↦1),(2↦9↦2↦2),(2↦9↦3↦3),(2↦9↦4↦4),(2↦9↦5↦5),(2↦9↦6↦6),(2↦9↦7↦7),(2↦9↦8↦8),(2↦9↦9↦9),(3↦4↦1↦1),(3↦4↦2↦2),(3↦4↦3↦3),(3↦4↦4↦4),(3↦4↦5↦5),(3↦4↦6↦6),(3↦4↦7↦7),(3↦4↦8↦8),(3↦4↦9↦9),(3↦5↦1↦1),(3↦5↦2↦2),(3↦5↦3↦3),(3↦5↦4↦4),(3↦5↦5↦5),(3↦5↦6↦6),(3↦5↦7↦7),(3↦5↦8↦8),(3↦5↦9↦9),(3↦6↦1↦1),(3↦6↦2↦2),(3↦6↦3↦3),(3↦6↦4↦4),(3↦6↦5↦5),(3↦6↦6↦6),(3↦6↦7↦7),(3↦6↦8↦8),(3↦6↦9↦9),(3↦7↦1↦1),(3↦7↦2↦2),(3↦7↦3↦3),(3↦7↦4↦4),(3↦7↦5↦5),(3↦7↦6↦6),(3↦7↦7↦7),(3↦7↦8↦8),(3↦7↦9↦9),(3↦8↦1↦1),(3↦8↦2↦2),(3↦8↦3↦3),(3↦8↦4↦4),(3↦8↦5↦5),(3↦8↦6↦6),(3↦8↦7↦7),(3↦8↦8↦8),(3↦8↦9↦9),(3↦9↦1↦1),(3↦9↦2↦2),(3↦9↦3↦3),(3↦9↦4↦4),(3↦9↦5↦5),(3↦9↦6↦6),(3↦9↦7↦7),(3↦9↦8↦8),(3↦9↦9↦9),(4↦5↦1↦1),(4↦5↦2↦2),(4↦5↦3↦3),(4↦5↦4↦4),(4↦5↦5↦5),(4↦5↦6↦6),(4↦5↦7↦7),(4↦5↦8↦8),(4↦5↦9↦9),(4↦6↦1↦1),(4↦6↦2↦2),(4↦6↦3↦3),(4↦6↦4↦4),(4↦6↦5↦5),(4↦6↦6↦6),(4↦6↦7↦7),(4↦6↦8↦8),(4↦6↦9↦9),(4↦7↦1↦1),(4↦7↦2↦2),(4↦7↦3↦3),(4↦7↦4↦4),(4↦7↦5↦5),(4↦7↦6↦6),(4↦7↦7↦7),(4↦7↦8↦8),(4↦7↦9↦9),(4↦8↦1↦1),(4↦8↦2↦2),(4↦8↦3↦3),(4↦8↦4↦4),(4↦8↦5↦5),(4↦8↦6↦6),(4↦8↦7↦7),(4↦8↦8↦8),(4↦8↦9↦9),(4↦9↦1↦1),(4↦9↦2↦2),(4↦9↦3↦3),(4↦9↦4↦4),(4↦9↦5↦5),(4↦9↦6↦6),(4↦9↦7↦7),(4↦9↦8↦8),(4↦9↦9↦9),(5↦6↦1↦1),(5↦6↦2↦2),(5↦6↦3↦3),(5↦6↦4↦4),(5↦6↦5↦5),(5↦6↦6↦6),(5↦6↦7↦7),(5↦6↦8↦8),(5↦6↦9↦9),(5↦7↦1↦1),(5↦7↦2↦2),(5↦7↦3↦3),(5↦7↦4↦4),(5↦7↦5↦5),(5↦7↦6↦6),(5↦7↦7↦7),(5↦7↦8↦8),(5↦7↦9↦9),(5↦8↦1↦1),(5↦8↦2↦2),(5↦8↦3↦3),(5↦8↦4↦4),(5↦8↦5↦5),(5↦8↦6↦6),(5↦8↦7↦7),(5↦8↦8↦8),(5↦8↦9↦9),(5↦9↦1↦1),(5↦9↦2↦2),(5↦9↦3↦3),(5↦9↦4↦4),(5↦9↦5↦5),(5↦9↦6↦6),(5↦9↦7↦7),(5↦9↦8↦8),(5↦9↦9↦9),(6↦7↦1↦1),(6↦7↦2↦2),(6↦7↦3↦3),(6↦7↦4↦4),(6↦7↦5↦5),(6↦7↦6↦6),(6↦7↦7↦7),(6↦7↦8↦8),(6↦7↦9↦9),(6↦8↦1↦1),(6↦8↦2↦2),(6↦8↦3↦3),(6↦8↦4↦4),(6↦8↦5↦5),(6↦8↦6↦6),(6↦8↦7↦7),(6↦8↦8↦8),(6↦8↦9↦9),(6↦9↦1↦1),(6↦9↦2↦2),(6↦9↦3↦3),(6↦9↦4↦4),(6↦9↦5↦5),(6↦9↦6↦6),(6↦9↦7↦7),(6↦9↦8↦8),(6↦9↦9↦9),(7↦8↦1↦1),(7↦8↦2↦2),(7↦8↦3↦3),(7↦8↦4↦4),(7↦8↦5↦5),(7↦8↦6↦6),(7↦8↦7↦7),(7↦8↦8↦8),(7↦8↦9↦9),(7↦9↦1↦1),(7↦9↦2↦2),(7↦9↦3↦3),(7↦9↦4↦4),(7↦9↦5↦5),(7↦9↦6↦6),(7↦9↦7↦7),(7↦9↦8↦8),(7↦9↦9↦9),(8↦9↦1↦1),(8↦9↦2↦2),(8↦9↦3↦3),(8↦9↦4↦4),(8↦9↦5↦5),(8↦9↦6↦6),(8↦9↦7↦7),(8↦9↦8↦8),(8↦9↦9↦9)}\n", + "\tDiff = ∅\n", + "\tSUBSQ = {{1,2,3},{4,5,6},{7,8,9}}" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Board : DOM --> (DOM --> DOM) & !(x1,x2,y1,y2).((x1,x2,y1,y2):Diff => Board(x1)(y1) /= Board(x2)(y2)) & !(x,y,z).((x,y,z):P => Board(x)(y)=z)" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Machine initialised using operation 0: $initialise_machine()" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":init" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "<table style=\"font-family:monospace\"><tbody>\n", + "</tbody></table>" + ], + "text/plain": [ + "<Animation function visualisation>" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":show" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "ProB 2", + "language": "prob", + "name": "prob2" + }, + "language_info": { + "codemirror_mode": "prob2_jupyter_repl", + "file_extension": ".prob", + "mimetype": "text/x-prob2-jupyter-repl", + "name": "prob" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Untitled.ipynb b/Untitled.ipynb new file mode 100644 index 0000000..5426bf1 --- /dev/null +++ b/Untitled.ipynb @@ -0,0 +1,347 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The Jobs Puzzle\n", + "\n", + "Based on *Michael Leuschel, David Schneider. Towards B as a High-Level Constraint Modeling Language. In Yamine Ait Amer, Klaus-Dieter Schewe (ed.): Abstract State Machines, Alloy, B, TLA, VDM, and Z, Springer Berlin Heidelberg, 8477: 101-116, 2014.*\n", + "\n", + "This puzzle was originally published in 1984 by [Wos et al., 1984](https://www.mcs.anl.gov/research/projects/AR/book1.html) as part of a collection of puzzles for automatic reasoners. A reference implementation of the puzzle, by one of the authors of the book, using [OTTER, 2003](https://arxiv.org/abs/cs/0310056).\n", + "\n", + "The puzzle consists of eight statements that describe the problem domain and provide some constraints on the elements of the domain. The problem is about a set of people and a set of jobs; the question posed by the puzzle is: who holds which job? The text of the puzzle as presented in \"The jobs puzzle: A challenge for logical expressibility and automated reasoning.\"[S. C. Shapiro, 2011](https://cse.buffalo.edu/~shapiro/Papers/SS11-06-017.pdf) is as follows:\n", + "\n", + "* There are four people: Roberta, Thelma, Steve, and Pete.\n", + "* Among them, they hold eight different jobs.\n", + "* Each holds exactly two jobs.\n", + "* The jobs are: chef, guard, nurse, clerk, police officer (gender not implied), teacher, actor, and boxer.\n", + "* The job of nurse is held by a male.\n", + "* The husband of the chef is the clerk.\n", + "* Roberta is not a boxer.\n", + "* Pete has no education past the ninth grade.\n", + "* Roberta, the chef, and the police officer went golfing together.\n", + "\n", + "What makes this puzzle interesting for automatic reasoners, is that not all the information required to solve the puzzle is provided explicitly in the text.\n", + "\n", + "The puzzle can only be solved if certain implicit assumptions about the world are taken into account, such as: the names in the puzzle denote gender or that some of the job names imply the gender of the person that holds it.\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Shapiro's Challenge\n", + "\n", + "Shapiro, following the original authors' remarks, that formalizing the puzzle was at times hard and tendious, identified three challenges posed by the puat times hard and tedious, identified three challenges posed by the puzzle with regard to automatic reasoners. According to Shapiro, the challenges posed by the jobs puzzle are to:\n", + "\n", + "1. formalize it in a non-difficult, non-tedious way\n", + "2. formalize it in a way that adheres closely to the English statement of the puzzle\n", + "3. have an automated general-purpose commonsense reasoner that can accept that formalization and solve the puzzle quickly.\n", + "\n", + "Any formalization also needs to encode the implicit knowledge used to solve the puzzle for the automatic reasoners while still trying to satisfy the aspects mentioned above. Addressing this challenge makes this puzzle a good case-study for the expressiveness of B to formalize such a problem." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A Solution to the Jobs Puzzle using B\n", + "\n", + "The B encoding of the puzzle uses plain predicate logic, \n", + "combined with set theory and arithmetic. We will show how this \n", + "enables a very concise encoding of the problem, staying very close to the natural language requirements. Moreover, the puzzle can be quickly solved using the constraint solving capabilities of ProB. Following the order of the sentences in the puzzle we will discuss one or more possibilities to formalize them using B.\n", + "\n", + "To express \"*There are four people: Roberta, Thelma, Steve, and Pete*\" we define a set of people, that holds the list of names:" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + ":let PEOPLE {\"Roberta\", \"Thelma\", \"Steve\", \"Pete\"}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We are using strings here to describe the elements of the set. This has the advantage, that the elements of the set are implicitly different.\n", + "\n", + "Alternatively, we could use enumerated or deferred sets defined in the SETS section of a B machine. As stated above we need some additional information that is not included in the puzzle to solve it. \n", + "\n", + "The first bit of information is that the names used in the puzzle imply the gender. In order to express this information we create two sets, MALE and FEMALE which are subsets of PEOPLE and contain the corresponding names." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{\\text{\"Roberta\"},\\text{\"Thelma\"}\\}$" + ], + "text/plain": [ + "{\"Roberta\",\"Thelma\"}" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let FEMALE {\"Roberta\", \"Thelma\"}" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{\\text{\"Pete\"},\\text{\"Steve\"}\\}$" + ], + "text/plain": [ + "{\"Pete\",\"Steve\"}" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let MALE {\"Steve\", \"Pete\"}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The next statement of the puzzle is: \"*among them, they hold eight different jobs*\". This can be formalized in B using a function that maps from a job to the corresponding person that holds this job using a total surjection from JOBS to PEOPLE. \n", + "\n", + "To use that statement, however we have to define JOBS, or the fourth statement." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\{\\text{\"actor\"},\\text{\"boxer\"},\\text{\"chef\"},\\text{\"clerk\"},\\text{\"guard\"},\\text{\"nurse\"},\\text{\"police\"},\\text{\"teacher\"}\\}$" + ], + "text/plain": [ + "{\"actor\",\"boxer\",\"chef\",\"clerk\",\"guard\",\"nurse\",\"police\",\"teacher\"}" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let JOBS {\"chef\", \"guard\", \"nurse\", \"clerk\", \"police\", \"teacher\", \"actor\", \"boxer\"}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we can see what Holds Job will do. " + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\mathit{TRUE}$\n", + "\n", + "**Solution:**\n", + "* $\\mathit{HoldsJob} = \\{(\\text{\"actor\"}\\mapsto\\text{\"Steve\"}),(\\text{\"boxer\"}\\mapsto\\text{\"Thelma\"}),(\\text{\"chef\"}\\mapsto\\text{\"Pete\"}),(\\text{\"clerk\"}\\mapsto\\text{\"Pete\"}),(\\text{\"guard\"}\\mapsto\\text{\"Pete\"}),(\\text{\"nurse\"}\\mapsto\\text{\"Pete\"}),(\\text{\"police\"}\\mapsto\\text{\"Pete\"}),(\\text{\"teacher\"}\\mapsto\\text{\"Roberta\"})\\}$\n", + "* $\\mathit{JOBS} = \\{\\text{\"actor\"},\\text{\"boxer\"},\\text{\"chef\"},\\text{\"clerk\"},\\text{\"guard\"},\\text{\"nurse\"},\\text{\"police\"},\\text{\"teacher\"}\\}$\n", + "* $\\mathit{PEOPLE} = \\{\\text{\"Pete\"},\\text{\"Roberta\"},\\text{\"Steve\"},\\text{\"Thelma\"}\\}$\n", + "* $\\mathit{MALE} = \\{\\text{\"Pete\"},\\text{\"Steve\"}\\}$\n", + "* $\\mathit{FEMALE} = \\{\\text{\"Roberta\"},\\text{\"Thelma\"}\\}$" + ], + "text/plain": [ + "TRUE\n", + "\n", + "Solution:\n", + "\tHoldsJob = {(\"actor\"↦\"Steve\"),(\"boxer\"↦\"Thelma\"),(\"chef\"↦\"Pete\"),(\"clerk\"↦\"Pete\"),(\"guard\"↦\"Pete\"),(\"nurse\"↦\"Pete\"),(\"police\"↦\"Pete\"),(\"teacher\"↦\"Roberta\")}\n", + "\tJOBS = {\"actor\",\"boxer\",\"chef\",\"clerk\",\"guard\",\"nurse\",\"police\",\"teacher\"}\n", + "\tPEOPLE = {\"Pete\",\"Roberta\",\"Steve\",\"Thelma\"}\n", + "\tMALE = {\"Pete\",\"Steve\"}\n", + "\tFEMALE = {\"Roberta\",\"Thelma\"}" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "HoldsJob : JOBS -->> PEOPLE" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Since we have not yet added any additional information, the Jobs are just randomly assigned to each Person. \n", + "\n", + "Although redundant, as we will see below, to express “*Among them, they hold eight different jobs*” we can add the assertion that the cardinality of HoldsJob is 8. \n", + "This is possible, because in B functions and relations can be treated as sets of pairs, where each pair consists of an element of the domain and the corresponding element from the range of the relation." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$\\mathit{TRUE}$\n", + "\n", + "**Solution:**\n", + "* $\\mathit{HoldsJob} = \\{(\\text{\"actor\"}\\mapsto\\text{\"Steve\"}),(\\text{\"boxer\"}\\mapsto\\text{\"Thelma\"}),(\\text{\"chef\"}\\mapsto\\text{\"Pete\"}),(\\text{\"clerk\"}\\mapsto\\text{\"Pete\"}),(\\text{\"guard\"}\\mapsto\\text{\"Pete\"}),(\\text{\"nurse\"}\\mapsto\\text{\"Pete\"}),(\\text{\"police\"}\\mapsto\\text{\"Pete\"}),(\\text{\"teacher\"}\\mapsto\\text{\"Roberta\"})\\}$\n", + "* $\\mathit{JOBS} = \\{\\text{\"actor\"},\\text{\"boxer\"},\\text{\"chef\"},\\text{\"clerk\"},\\text{\"guard\"},\\text{\"nurse\"},\\text{\"police\"},\\text{\"teacher\"}\\}$\n", + "* $\\mathit{PEOPLE} = \\{\\text{\"Pete\"},\\text{\"Roberta\"},\\text{\"Steve\"},\\text{\"Thelma\"}\\}$\n", + "* $\\mathit{MALE} = \\{\\text{\"Pete\"},\\text{\"Steve\"}\\}$\n", + "* $\\mathit{FEMALE} = \\{\\text{\"Roberta\"},\\text{\"Thelma\"}\\}$" + ], + "text/plain": [ + "TRUE\n", + "\n", + "Solution:\n", + "\tHoldsJob = {(\"actor\"↦\"Steve\"),(\"boxer\"↦\"Thelma\"),(\"chef\"↦\"Pete\"),(\"clerk\"↦\"Pete\"),(\"guard\"↦\"Pete\"),(\"nurse\"↦\"Pete\"),(\"police\"↦\"Pete\"),(\"teacher\"↦\"Roberta\")}\n", + "\tJOBS = {\"actor\",\"boxer\",\"chef\",\"clerk\",\"guard\",\"nurse\",\"police\",\"teacher\"}\n", + "\tPEOPLE = {\"Pete\",\"Roberta\",\"Steve\",\"Thelma\"}\n", + "\tMALE = {\"Pete\",\"Steve\"}\n", + "\tFEMALE = {\"Roberta\",\"Thelma\"}" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "HoldsJob : JOBS -->> PEOPLE & card(HoldsJob) = 8" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Constraining the jobs each person holds, the puzzle states: “*Each holds exactly two jobs*”. To express this we use the inverse relation of HoldsJob, it maps a PERSON to the JOBS associated to her. The inverse function or relation is expressed in B using the ~ operator. \n", + "\n", + "For readability we assign the inverse of HoldsJob to a variable called JobsOf. JobsOf is in this case is a relation, because, as stated above, each person holds two jobs.\n", + "\n", + "First of all, we have to add HoldsJob to our globals, however." + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "ename": "EvaluationException", + "evalue": "de.be4.classicalb.core.parser.exceptions.BCompoundException: [2,26] expecting: identifier literal", + "output_type": "error", + "traceback": [ + "\u001b[1m\u001b[31mde.prob.animator.domainobjects.EvaluationException: de.be4.classicalb.core.parser.exceptions.BCompoundException: [2,26] expecting: identifier literal\u001b[0m" + ] + } + ], + "source": [ + ":let HoldsJob JOBS -->> PEOPLE" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Because JobsOf is a relation and not a function, in order to read the values, we need to use B’s relational image operator. This operator maps a subset of the domain to a subset of the range, instead of a single value. To read the jobs Steve holds, the relational image of JobsOf is used as shown below:" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "$8$" + ], + "text/plain": [ + "8" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":let card(HoldsJob) 8" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [ + { + "ename": "EvaluationException", + "evalue": "de.be4.classicalb.core.parser.exceptions.BCompoundException: [2,26] expecting: identifier literal", + "output_type": "error", + "traceback": [ + "\u001b[1m\u001b[31mde.prob.animator.domainobjects.EvaluationException: de.be4.classicalb.core.parser.exceptions.BCompoundException: [2,26] expecting: identifier literal\u001b[0m" + ] + } + ], + "source": [ + ":let JobsOf HoldsJob~" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "ProB 2", + "language": "prob", + "name": "prob2" + }, + "language_info": { + "codemirror_mode": "prob2_jupyter_repl", + "file_extension": ".prob", + "mimetype": "text/x-prob2-jupyter-repl", + "name": "prob" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} -- GitLab