Clear();
Print ("Beginning demo program: test_2_6.txt\n\n");
Print ("This demonstrates the use of the following procedures:\n");
Print ("   AutImages( group, elements ) \n");
Print ("   MaxValueOnSets( right_reg, aut_img, subset ) \n");
Print ("   MaxValueOnTuples( right_reg, aut_img, subset ) \n");
Print ("   ReduceSets( right_reg, aut_img, soln , printflag) \n");
Print ("   ReduceTuples( right_reg, aut_img, partition, soln, printflag ) \n\n");
Sleep(5);


# Let's test some automorphisms, now
# Let's begin with C8, and the subset x + x^3

group     := SmallGroup(8,1);
subset    := [2,5];
list      := AutImages( group, Elements(group) );
right_reg := list[1];
aut_grp   := list[2];
aut_img   := list[3];

Print ("   Okay, it's important to first capture the AutImages of\n");
Print ("groups that we use.  Let's look at C8, with the subset\n");
Print ("x + x^3.\n\n");
Print ("   group     := SmallGroup(8,1);;\n");
Print ("   subset    := [2,5];;\n");
Print ("   list      := AutImages( group, Elements(group) );;\n");
Print ("   right_reg := list[1];;\n");
Print ("   aut_grp   := list[2];;\n");
Print ("   aut_img   := list[3];;\n\n");
Sleep(8);


max := MaxValueOnSets(right_reg, aut_img, subset);

Print ("   Now we can look at a maximal representative of 'subset'\n");
Print ("in its equivalence class.\n");
Print ("   max := MaxValueOnSets(right_reg, aut_img, subset);\n\n");
Print (max, "\n\n");
Sleep(8);


# Now let's compare some things that are shift equivalent
# and some that are automorphically equivalent, and some
# that are neither...

shift_subset := [3,4];     # x^2 + x^4
aut_subset   := [6,8];     # x^5 + x^7
ineq_subset  := [3,7];     # x^2 + x^6

max_s := MaxValueOnSets(right_reg, aut_img, shift_subset);
max_a := MaxValueOnSets(right_reg, aut_img, aut_subset);
max_i := MaxValueOnSets(right_reg, aut_img, ineq_subset);

Print ("   Now let's compare some things that are shift equivalent\n");
Print ("and some that are automorphically equivalent, and some that\n");
Print ("are neither...\n\n");
Print ("   shift_subset := [3,4];;     # x^2 + x^4\n");
Print ("   aut_subset   := [6,8];;     # x^5 + x^7\n");
Print ("   ineq_subset  := [3,7];;     # x^2 + x^6\n\n");
Sleep(8);

Print ("   max_s := MaxValueOnSets(right_reg, aut_img, shift_subset);\n");
Print (max_s, "\n\n");
Sleep(3);

Print ("   max_a := MaxValueOnSets(right_reg, aut_img, aut_subset);\n");
Print (max_a, "\n\n");
Sleep(3);

Print ("   max_i := MaxValueOnSets(right_reg, aut_img, ineq_subset);\n");
Print (max_i, "\n\n\n");
Sleep(3);


# Okay, same thing with tuples...

subset := [2, 5, 5];
max    := MaxValueOnTuples(right_reg, aut_img, subset);

Print ("Let's try the same things with tuples:\n");
Print ("   subset := [2, 5, 5];;     # x + 2*x^3\n");
Print ("   max    := MaxValueOnTuples(right_reg, aut_img, subset)\n\n");
Print (max, "\n\n");
Sleep(5);


# Now let's see the entire equivalence class

ec := EquivalenceClass(right_reg, aut_img, subset);

Print ("Now let's see the entire equivalence class:\n");
Print ("    ec := EquivalenceClass(right_reg, aut_img, subset);\n\n");
Print (ec, "\n\n");
Sleep(5);


# Okay, time to try to reduce some sets

list        := [ [2,5], [3,4], [6,8], [3,7] ];
reducedlist := ReduceSets( right_reg, aut_img, list, 0);

Print ("Okay, time to try to reduce some sets.\n");
Print ("   list        := [ [2,5], [3,4], [6,8], [3,7] ];;\n");
Print ("   reducedlist := ReduceSets( right_reg, aut_img, list, 0);\n\n");
Print (reducedlist, "\n\n");
Sleep(8);


# Maybe now we can reduce some tuples?

partition := [ [5,1], [6,2], [7,3], [8,4] ];
soln      := [ [1,1,2], [2,2,3], [1,3,4] ];
rt        := ReduceTuples(right_reg, aut_img, partition, soln, 0);

Print ("Suppose now that I have a partition of C8:\n");
Print ("   partition := [ [5,1], [6,2], [7,3], [8,4] ];;\n\n");
Sleep(3);

Print ("   These are actually cosets of the normal subgroup\n");
Print ("generated by x^2.\n");
Print ("   So now my tuples are actually representative of cosets.\n");
Print ("However, the elements represent the index of the coset.\n");
Print ("That is, the element [1,1,2] means that we have two copies\n");
Print ("of the first coset, [5,1], and one copy of the second, [6,2].\n\n");
Sleep(8);

Print ("   soln := [ [1,1,2], [2,2,3], [1,3,4] ];;\n");
Print ("   rt   := ReduceTuples(right_reg, aut_img, partition, soln, 0);\n");
Print (rt, "\n\n");