As the following worksheet shows, 
 

Download Why_not_consider_subexpressions?.mw

the underlined part is evidently not the simplest. (For instance, shouldn't  and  be converted into  and ?) 
If I understand correctly, , by default, should try combining every part of an expression with every other to apply a vast range of potential transformations to look at many different forms of it and make progress in picking out the simplest possible one. So, why is simplify unable to touch certain sub-expressions when they are encountered at intermediate stages in a computation? 

In my view, <x || (1 .. 2); y || (1 .. 2); 1 $ 2> should return a Matrix without any error messages; however, 

<x || (1 .. 2); y || (1 .. 2); 1 $ 2>; # Arguments are shielded??? 
Error, (in Matrix) this entry is too wide or too narrow: 1

If I understand right, each argument of a function is evaluated in turn (unless the modifier is used). 
So why is it not equivalent to <x1, x2; y1, y2; 1, 1>? 

I am trying to get Maple to simplify the following trigonometric expressions (for "generic" parameters) as much as possible: 

sineExpr(3) := (
   sin(a[2] - b[1])*sin(a[3] - b[1]))/(
   sin(b[2] - b[1])*sin(b[3] - b[1]))*sin(a[1] - b[1]) + (
   sin(a[3] - b[2])*sin(a[1] - b[2]))/(
   sin(b[3] - b[2])*sin(b[1] - b[2]))*sin(a[2] - b[2]) + (
   sin(a[1] - b[3])*sin(a[2] - b[3]))/(
   sin(b[1] - b[3])*sin(b[2] - b[3]))*sin(a[3] - b[3]);
 = 
   


sineExpr(4) := (
   sin(a[2] - b[1])*sin(a[3] - b[1])*sin(a[4] - b[1]))/(
   sin(b[2] - b[1])*sin(b[3] - b[1])*sin(b[4] - b[1]))*
   sin(a[1] - b[1]) + (
   sin(a[3] - b[2])*sin(a[4] - b[2])*sin(a[1] - b[2]))/(
   sin(b[3] - b[2])*sin(b[4] - b[2])*sin(b[1] - b[2]))*
   sin(a[2] - b[2]) + (
   sin(a[4] - b[3])*sin(a[1] - b[3])*sin(a[2] - b[3]))/(
   sin(b[4] - b[3])*sin(b[1] - b[3])*sin(b[2] - b[3]))*
   sin(a[3] - b[3]) + (
   sin(a[1] - b[4])*sin(a[2] - b[4])*sin(a[3] - b[4]))/(
   sin(b[1] - b[4])*sin(b[2] - b[4])*sin(b[3] - b[4]))*
   sin(a[4] - b[4]);
 = 

So far, all of my attempts have failed: 
 

Download sinIdentity.mw

Note that because zero testing is frequently considerably easier, combine always succeeds in showing that the difference between the simplest possible and the original version is zero. 

combine(sin((a[1]+a[2]+a[3])-(b[1]+b[2]+b[3]))-sineExpr(3));
 = 
                               0

combine(sin((a[1]+a[2]+a[3]+a[4])-(b[1]+b[2]+b[3]+b[4]))-sineExpr(4));
 = 
                               0

However, I wonder if Maple can thoroughly simplify them without knowing those known “simplest possible” form beforehand. 
I also tried some other functions like rationalize, radnormal, and `convert/trig`, yet Maple appears to have not been capable of completely simplifying even the sub-simplest case 𝑚＝2. Is there any workaround? 

Of note, it can be demonstrated inductively that ∀m∈ℕ_₊ 

where none of the denominators is 0. Nevertheless, as mentioned above, is it possible to transform  and  (as well as , if possible) into potentially more elegant forms (Ideally,  is rewritten into ,  is rewritten into , and  is rewritten into .) without any such a priori knowledge? 
In Mma, these may be done using TrigReduce directly (cf. ); unfortunately, I cannot found a Maple equivalent to such functionality. 

In many special cases, generalized hypergeometric functions can be transformed into other (possibly less general) functions, but sometimes Maple fails to convert some of them to other functions when possible.
For instance, 

4*hypergeom([1, 1], [2, 3, 7/2], -(x/2)**2) - 
 35*hypergeom([1, 1], [2, 5/2, 3], -(x/2)**2) - 
 2*hypergeom([1, 1], [3, 3, 7/2], -(x/2)**2) + 
 10*hypergeom([1, 1], [5/2, 3, 3], -(x/2)**2): # which is, in fact, “360*((Ci(x) - ln(x) + 19/18 - gamma)/x**2 - sin(x)/x**3 + (cos(x) + 5/3)/x**4 - 8/3*sin(x)/x**5)” 
expand(%);
 = 



simplify(%);

Is it possible to obtain a non-hypergeometric form of it? E.g., 

. 

Here are more examples that `simplify/hypergeom` (as well as expand) is unable to deal with: 

1. hypergeom([1, 1, 3/2], [5/2, 5/2, 3, 3], -(x/2)**2):
2. hypergeom([1, 1], [5/2, 3, 3], -(x/2)**2):
3. hypergeom([1, 1], [3/2, 2, 2], -(x/2)**2):
4. hypergeom([3/2], [5/2, 5/2], -(x/2)**2):
5. … (to be continued) 

I have read something like How to simplify this hypergeometric function? - MaplePrimes, yet those tricks do not appear to work here. Is there any workaround? 

Edit. There seems to be a bug in `simplify/hypergeom`: 

'hypergeom([-1/3, 1/3, -1/2 - m, -m], [1/2, 1/3 - m, 2/3 - m], 1)':
simplify(`%`, hypergeom) assuming m::nonnegint;
 = 
                               -1
                               --
                               2 

simplify(eval(`%%`, m = 0), 'constant');
 = 
                               1

The expected result should be a piecewise function. 

Here is a "toy" instance. 
As the title suggests, I would like to simplify the following boolean combinations: (Note that the  should be taken into account.) 

formula__0 := 
 convert(And(
   Non(`xor`(Or(0 >- 0, y ** 3=x), 
     And(((y*x - 0^0)^2 + (y*y - x*x)^2)*(y^2 + x^2) > 
       0, 0 <- (y**3 - x)*(y - x**3)**3, 
      `implies`(y + x*2 >= y ** 3*2 + x**3, 
       Or(y + x < y*y*y + x*x*x)), Not((y**3 - x)*(y - x**3) = 0), 
      0 >= 0), y <> x*x*x))), 'boolean_function'):

Regretfully, the built-in command simplify is unable to simplify : 

simplify(formula__0, assume = real); # only rudimentary simplifications 
 = 

The library function SMTLIB:-Simplify seems to simplify it, but the result is incorrect: 

(SMTLIB:-Simplify(formula__0) assuming real); # check {y = 1, x = 0} 
 = 
                    /   /   / 3           3\\\
                 And\Not\And\y  = x, y = x ///

The only procedure I can find that is capable of rewriting it appears to be RealDomain:-solve: 

RealDomain:-solve(formula__0, 'allsolutions'); # see below 
 = 
Warning, solutions may have been lost
    /            3\    /     3       \    /     3        \   
   { x = x, y = x  }, { x = y , y = y }, { y = x , x < -1 }, 
    \             /    \             /    \              /   

      /     3              \    /     3        \   
     { y = x , 0 < x, x < 1 }, { x = y , y < -1 }, 
      \                    /    \              /   

      /     3              \ 
     { x = y , 0 < y, y < 1 }
      \                    / 

Nevertheless, this is more or less overkill, since a complete solution set is somewhat unnecessary, and in practice, an simplified and compact but presumably unsolved form is more applicable to further manipulations. (For example, the simplest form of  (over ℝ²) should at least be x²＋y²＞1, yet  simply returns , which is definitely unsuitable here.) (A more real example can be found in the Ex３ of this compressed file.)

I believe that this is a common problem; curiously, I cannot find any related questions in this forum. So, are such simplifications (similar to SLFQ) available in Maple?