Nice problem and solution. Unfortunately the worksheet cannot be executed: you forgot to initialize L.

Remarks.
- It would be interesting to have a unique solution. It seems that a 90 degree rotation invariance of the carpet is enough for this. [A central symmetry gives 2 solutions it seems].
- The isometry can be tested easier because there are only 3 possible rotations.

@Ramakrishnan 

1. Q.E.D. (quod erat demonstrandum) means "what was to be demonstrated" (and marks the end of the proof).
2. In Maple assume(s>0) means that s is assumed to be real and positive. You may check this via about(s);
3. The question asked by coulditbe is correct. It is the answer which has "problems".

@Carl Love 

Indeed, xxx:=eval(varsx,Satisfy(ex))  would have been enough, but it was only a few lines of simple code.

@roubeur 

Probably you did not execute the initialization part; it is better to upload the worksheet in such situations.
Here is a slight modification of the worksheet; just change ee and execute it.

boolean-n.mw

@roubeur 

To go back to the initial variables do:

eval(simp, varsx =~ vars);

The admisible solution xxx was obtained using random numbers. If you re-execute the computation (without restart) xxx will change, so it is normal that you obtained another result (simp).

Note: for some admisible xxx, it could be possible that very few variables must be fixed.
You should experiment.

@acer 

Indeed, in Qgr (in 2d)

Representative(a) . Representative(b)

was parsed as:

Typesetting:-delayDotProduct(Representative(a), Representative(b));

Works without errors in Maple 2017.3.

You should try to present the mathematical problem.
Otherwise, for suggestions or alternate solutions a "reverse engineering" of the Maple code will be necessary.

@Rouben Rostamian  

For an "lndirect documentation" see showstat(log).

@Carl Love 

I was also curious to understand on what basis Maple returns true for is(exp(2),irrational);
This information should (probably) be found in `property/exp` but it's not there.
It would be very disappointing if the answer is obtained using approximations because Digits cannot be set to infinity and the answer is simply a guess in this case.

is(x,irrational) seems indeed to be a pure guess because of:

is(sum(10^(-k^3),k=1..infinity), irrational);
        false

@Rouben Rostamian  

If f is the name of a procedure then  f[u](x)  actually calls  f(x), but in the body of f the index/indices
u can be retrived (and used) via  op(procname).
To check whether f is invoked with an index one may use  type(procname,indexed).

(I am not sure whether this is documented, but I found about it long time ago in a Maple book which I do not remember). 

@Carl Love 

But ?type,property clearly says:

Note: Types are properties, but not all properties are types.

Do you know a type which is not property? (I don't mean "proper").

@Jason Lee 

Download linsyst.mw

@Jason Lee 

What exactly is not clear? I have used your notations.
But I just noticed that you are using Maple T.A. and I am not sure whether this works here.

@Carl Love 

Why dou you think that is is using list(rational) as a type? Of course, any type is a property.

is(list(rational), property);
   true

type(list(rational), property);
    true

Probably your notion of "proper property" is useful.