Clear();
Print ("Beginning demo program: test_2_5.txt\n");
Print ("This demonstrates the use of the following procedures:\n");
Print ("   PartitionRep( partition )\n");
Print ("   IndexToRep( partition )\n");
Print ("   QuotientGroupConvolution( CL, partition )\n");
Print ("   MapRefinement( partition, refinement )\n\n");
Sleep(5);


# Let's create a set [1 .. 10], with partition being
# equivalence under modulus by 5. (note partition 3
# and partition 4 are switched)

partition := [ [1,6], [2,7], [4,9], [3,8], [5,10] ];
rep       := PartitionRep( partition );

Print ("   Okay, let's think about the set [1 .. 10] with a partition\n");
Print ("being equivalence under modulus by 5.  However, for future\n");
Print ("demonstration, we will switch partition 3 and 4.\n\n");
Print ("   partition := [ [1,6], [2,7], [4,9], [3,8], [5,10] ];\n\n\n");
Sleep(5);

Print ("   rep := PartitionRep( partition );\n");
Print (rep ,"\n\n");
Print ("   Here we see that in the sixth location we have a 1.  This\n");
Print ("is because 1 is the first element of the partition containing\n");
Print ("6.  Thus PartitionRep gives us a representative for each\n");
Print ("element of the group with respect to the partition.\n\n\n");
Sleep(8);


# Now let's see what IndexToRep does

rep := IndexToRep( partition );


Print ("   rep := IndexToRep( partition );\n");
Print (rep, "\n\n");
Print ("   We can see that IndexToRep returns a set with the same size\n");
Print ("as partition.  In the i-th location is a representative of the\n");
Print ("i-th subset of the partition.\n\n\n");
Sleep(8);


# Let's consider C8, from test_2_1 we can generate a Cayley table

group    := SmallGroup(8,1);
elements := Elements(group);
func     := x->x;
CL       := ConvolutionTable_f(group, elements, func);

Print ("   Let's consider C8, with a replacement of x^i -> i+1\n");
Print ("So e = 1, x = 2, x^2 = 3, ... \n");
Print ("The Cayley table is:\n");
Print ("   group    := SmallGroup(8,1);;\n");
Print ("   elements := Elements(group);;\n");
Print ("   func     := x->x;;\n");
Print ("   CL       := ConvolutionTable_f(group, elements, func);\n\n");
PrintArray (CL);
Print ("\n\n");
Sleep(5);


# And a partition of this structure is:

partition := [ [1,3,5,7], [2,4,6,8] ];

Print ("A partition of this structure (generated by x^2) is:\n");
Print ("   partition := [ [1,3,5,7], [2,4,6,8] ];;\n\n\n");
Sleep(3);


# thus the convolution table on the quotient group
# C8/<x^2> is:

qCL := QuotientGroupConvolution( CL, partition );

Print ("   The Cayley table on the quotient group C8/<x^2> is:\n");
Print ("   qCL := QuotientGroupConvolution( CL, partition );\n\n");
PrintArray (qCL);
Print ("\n\n");
Sleep(8);


# Let's try again with another...

refinement := [ [1,5], [2,6], [3,7], [4,8] ];
qCL        := QuotientGroupConvolution( CL, refinement );

Print ("   Let's try with a different partition.\n");
Print ("   refinement := [ [1,5], [2,6], [3,7], [4,8] ];;\n");
Print ("   qCL        := QuotientGroupConvolution( CL, refinement );\n\n");
PrintArray (qCL);
Print ("\n");
Sleep(8);
Print ("   Hmmm.... looks an awful lot like C4.\n\n\n");
Sleep(3);


# now let's work on refining partitions

list := MapRefinement( partition, refinement );

Print ("   Now we can refine the partitions.\n");
Print ("   list := MapRefinement( partition, refinement );\n\n");
Print (list, "\n\n");