Does Sympy's error during solve "[...] undecidable set membership is found in every candidates" mean that no solution exists?

Question

I'm trying to solve the following equation for variable x (in dependence on other symbols b, c, d):

from sympy import symbols, sin, cos, solve, solveset, S
  
b, c, d, x = symbols('b c d x', real=True)

expr = sin(x)*sin(c - 2*d - x) - sin(2*b - 2*d - x)*sin(c - x)  # find `x` such that `expr == 0`

There exist additional constraints which apply to the various symbols, namely:

0 < d < b < d+x < c

So I tried to use solve for incorporating these inequality constraints:

result = solve(
    [
        sin(x)*sin(c - 2*d - x) - sin(2*b - 2*d - x)*sin(c - x),
        0 < d,
        d < b,
        b < d+x,
        d+x < c,
    ],
    x,
)

However, I get the following error:

Traceback (most recent call last):
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polyutils.py", line 211, in _parallel_dict_from_expr_if_gens
    monom[indices[base]] = exp
KeyError: sin(c - x)

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.py", line 818, in _solve_inequality
    p = Poly(expr, s)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polytools.py", line 181, in __new__
    return cls._from_expr(rep, opt)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polytools.py", line 310, in _from_expr
    rep, opt = _dict_from_expr(rep, opt)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polyutils.py", line 368, in _dict_from_expr
    rep, gens = _dict_from_expr_if_gens(expr, opt)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polyutils.py", line 307, in _dict_from_expr_if_gens
    (poly,), gens = _parallel_dict_from_expr_if_gens((expr,), opt)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polyutils.py", line 216, in _parallel_dict_from_expr_if_gens
    raise PolynomialError("%s contains an element of "
sympy.polys.polyerrors.PolynomialError: sin(c - x) contains an element of the set of generators.

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polyutils.py", line 211, in _parallel_dict_from_expr_if_gens
    monom[indices[base]] = exp
KeyError: sin(c - x)

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.py", line 246, in reduce_rational_inequalities
    (numer, denom), opt = parallel_poly_from_expr(
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polytools.py", line 4414, in parallel_poly_from_expr
    return _parallel_poly_from_expr(exprs, opt)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polytools.py", line 4467, in _parallel_poly_from_expr
    reps, opt = _parallel_dict_from_expr(exprs, opt)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polyutils.py", line 332, in _parallel_dict_from_expr
    reps, gens = _parallel_dict_from_expr_if_gens(exprs, opt)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/polys/polyutils.py", line 216, in _parallel_dict_from_expr_if_gens
    raise PolynomialError("%s contains an element of "
sympy.polys.polyerrors.PolynomialError: sin(c - x) contains an element of the set of generators.

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.py", line 827, in _solve_inequality
    rv = reduce_rational_inequalities([[ie]], s)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.py", line 249, in reduce_rational_inequalities
    raise PolynomialError(filldedent('''
sympy.polys.polyerrors.PolynomialError: 
only polynomials and rational functions are supported in this context.

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/tmp/test2.py", line 5, in <module>
    result = solve(
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/solvers.py", line 920, in solve
    return reduce_inequalities(f, symbols=symbols)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.py", line 996, in reduce_inequalities
    rv = _reduce_inequalities(inequalities, symbols)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.py", line 912, in _reduce_inequalities
    other.append(_solve_inequality(Relational(expr, 0, rel), gen))
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.py", line 829, in _solve_inequality
    rv = solve_univariate_inequality(ie, s)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.py", line 500, in solve_univariate_inequality
    frange = function_range(e, gen, domain)
  File "/path/to/venv/lib/python3.9/site-packages/sympy/calculus/util.py", line 208, in function_range
    for critical_point in critical_points:
  File "/path/to/venv/lib/python3.9/site-packages/sympy/utilities/iterables.py", line 2881, in roundrobin
    yield nxt()
  File "/path/to/venv/lib/python3.9/site-packages/sympy/sets/sets.py", line 1439, in __iter__
    raise TypeError(
TypeError: The computation had not completed because of the undecidable set membership is found in every candidates.

I'm not sure what that error means. Does it mean that there exists no solution for the given inequality constraints?

(sympy version: 1.10.1)

---

solveset

I also tried using solveset (however, it doesn't seem to be possible to include the inequality constraints here):

result = solveset(
    sin(x)*sin(c - 2*d - x) - sin(2*b - 2*d - x)*sin(c - x),
    x,
    domain=S.Reals,
)

However, after a long computation, it just returns the original expression:

ConditionSet(x, Eq(-sin(x)*sin(-c + 2*d + x) + sin(c - x)*sin(-2*b + 2*d + x), 0), Reals)

---

matlab

I also tried using Matlab for solving the equation:

syms b c d x

eq1 = sin(x)*sin(c - 2*d - x) - sin(2*b - 2*d - x)*sin(c - x) == 0;
eq2 = 0 < d;
eq3 = d < b;
eq4 = b < d+x;
eq5 = d+x < c;
result = solve([eq1 eq2 eq3 eq4 eq5], x, "Real",true, "ReturnConditions",true);

However, the result is something entirely different, involving a fourth order polynomial of which I can't make any sense:

>> result.x
 
ans =
 
2*pi*k + atan2(y, 1) - atan2(-y, 1)

>> result.conditions
 
ans =
 
d < b & b + atan2(-y, 1) < d + atan2(y, 1) + 2*pi*k & d + atan2(y, 1) + 2*pi*k < c + atan2(-y, 1) & cos(b)*sin(b) + 2*cos(b)^2*cos(d)*sin(d) ~= cos(d)*sin(d) + 2*cos(b)*cos(d)^2*sin(b) & in(k, 'integer') & ~in(c/pi, 'integer') & in(y, 'real') & 4*y^2*cos(c - 2*d) + 2*y^3*sin(c - 2*d) + sin(2*b - 2*d)*sin(c) + y^2*cos(c - 2*b + 2*d) + 3*y^2*cos(2*b + c - 2*d) + 2*y^3*sin(2*b + c - 2*d) + y^4*sin(2*b - 2*d)*sin(c) == 2*y*sin(2*b + c - 2*d) + 2*y*sin(c - 2*d) & 0 < d

Answer 1

Here is how you can obtain something similar to the Matlab solution using SymPy (note that I redefined the symbols to make the output simpler):

In [19]: a, b, c, d, e, x, y = symbols('a, b, c, d, e, x, y', real=True)

In [20]: expr = sin(x)*sin(a - x) - sin(e - x)*sin(c - x)

In [21]: expr.expand(trig=True)
Out[21]: 
                                        2                                                          
sin(a)?sin(x)?cos(x) - sin(c)?sin(e)?cos (x) + sin(c)?sin(x)?cos(e)?cos(x) + sin(e)?sin(x)?cos(c)?c

           2                2                 
os(x) - sin (x)?cos(a) - sin (x)?cos(c)?cos(e)

In [22]: expr2 = resultant(expr.expand(trig=True), cos(x)**2 + sin(x)**2 - 1, cos(x))

In [23]: expr2.collect(sin(x))
Out[23]: 
?     2                                                             2       2         2       2    
?- sin (a) - 2?sin(a)?sin(c)?cos(e) - 2?sin(a)?sin(e)?cos(c) - 2?sin (c)?sin (e) - sin (c)?cos (e) 

                              2       2   ?    2      ?   2                                        
+ 2?sin(c)?sin(e)?cos(a) - sin (e)?cos (c)??sin (x) + ?sin (a) + 2?sin(a)?sin(c)?cos(e) + 2?sin(a)?

                   2       2         2       2                                  2       2         2
sin(e)?cos(c) + sin (c)?sin (e) + sin (c)?cos (e) - 2?sin(c)?sin(e)?cos(a) + sin (e)?cos (c) + cos 

                                  2       2   ?    4         2       2   
(a) + 2?cos(a)?cos(c)?cos(e) + cos (c)?cos (e)??sin (x) + sin (c)?sin (e)

So we see that what we have is a polynomial of degree 4 in sin(x). This one is a bit nicer to solve than the one that Matlab found because it only has even powers of sin(x). The explicit solutions for sin(x) can be found using roots e.g. the first is:

In [26]: r1, r2, r3, r4 = roots(expr2.subs(sin(x), y), y)

In [27]: r1
Out[27]: 
          _________________________________________________________________________________________
         ?                         ________________________________________________________________
        ?                         ?                                                                
       ?                         ?                               2       2      (cos(2?a) + cos(2?c
      ?                         ?   - 8?(cos(-a + c + e) + 1)?sin (c)?sin (e) + ???????????????????
     ?                        ??                                                                   
-   ?      - ??????????????????????????????????????????????????????????????????????????????????????
   ?           ?   2                                                           2       2         2 
 ??          2??sin (a) + 2?sin(a)?sin(c)?cos(e) + 2?sin(a)?sin(e)?cos(c) + sin (c)?sin (e) + sin (

___________________________________________________________________________________________________
__________________________________________________________________________________________         
                                                                                        2          
) + cos(2?e) - 3?cos(-a + c + e) + cos(a - c + e) + cos(a + c - e) + cos(a + c + e) - 3)           
?????????????????????????????????????????????????????????????????????????????????????????          
                                  4                                                                
???????????????????????????????????????????????????????????????????????????????????????????????????
      2                                  2       2         2                                  2    
c)?cos (e) - 2?sin(c)?sin(e)?cos(a) + sin (e)?cos (c) + cos (a) + 2?cos(a)?cos(c)?cos(e) + cos (c)?

___________________________________________________________________________________________________
                                                                                                   
                                                                                                   
                                                                                                   
                                                                                                   
           -cos(2?a) - cos(2?c) - cos(2?e) + 3?cos(-a + c + e) - cos(a - c + e) - cos(a + c - e) - 
???????? + ????????????????????????????????????????????????????????????????????????????????????????
   2   ?                                            8?(cos(-a + c + e) + 1)                        
cos (e)?                                                                                           

___________________
                   
                   
                   
                   
cos(a + c + e) + 3 
?????????????????? 
                

Whether or not this is real or satisfies the inequalities will depend on the values of the symbolic parameters so I doubt that the inequalities can be used to choose amongst the 4 expressions for the roots. It is perhaps possible to answer different questions using the polynomial expr2 though.