from sympy.combinatorics.free_groups import free_group, FreeGroup from sympy.core import Symbol from sympy.testing.pytest import raises from sympy.core.numbers import oo F, x, y, z = free_group("x, y, z") def test_FreeGroup__init__(): x, y, z = map(Symbol, "xyz") assert len(FreeGroup("x, y, z").generators) == 3 assert len(FreeGroup(x).generators) == 1 assert len(FreeGroup(("x", "y", "z"))) == 3 assert len(FreeGroup((x, y, z)).generators) == 3 def test_free_group(): G, a, b, c = free_group("a, b, c") assert F.generators == (x, y, z) assert x*z**2 in F assert x in F assert y*z**-1 in F assert (y*z)**0 in F assert a not in F assert a**0 not in F assert len(F) == 3 assert str(F) == '' assert not F == G assert F.order() is oo assert F.is_abelian == False assert F.center() == {F.identity} (e,) = free_group("") assert e.order() == 1 assert e.generators == () assert e.elements == {e.identity} assert e.is_abelian == True def test_FreeGroup__hash__(): assert hash(F) def test_FreeGroup__eq__(): assert free_group("x, y, z")[0] == free_group("x, y, z")[0] assert free_group("x, y, z")[0] is free_group("x, y, z")[0] assert free_group("x, y, z")[0] != free_group("a, x, y")[0] assert free_group("x, y, z")[0] is not free_group("a, x, y")[0] assert free_group("x, y")[0] != free_group("x, y, z")[0] assert free_group("x, y")[0] is not free_group("x, y, z")[0] assert free_group("x, y, z")[0] != free_group("x, y")[0] assert free_group("x, y, z")[0] is not free_group("x, y")[0] def test_FreeGroup__getitem__(): assert F[0:] == FreeGroup("x, y, z") assert F[1:] == FreeGroup("y, z") assert F[2:] == FreeGroup("z") def test_FreeGroupElm__hash__(): assert hash(x*y*z) def test_FreeGroupElm_copy(): f = x*y*z**3 g = f.copy() h = x*y*z**7 assert f == g assert f != h def test_FreeGroupElm_inverse(): assert x.inverse() == x**-1 assert (x*y).inverse() == y**-1*x**-1 assert (y*x*y**-1).inverse() == y*x**-1*y**-1 assert (y**2*x**-1).inverse() == x*y**-2 def test_FreeGroupElm_type_error(): raises(TypeError, lambda: 2/x) raises(TypeError, lambda: x**2 + y**2) raises(TypeError, lambda: x/2) def test_FreeGroupElm_methods(): assert (x**0).order() == 1 assert (y**2).order() is oo assert (x**-1*y).commutator(x) == y**-1*x**-1*y*x assert len(x**2*y**-1) == 3 assert len(x**-1*y**3*z) == 5 def test_FreeGroupElm_eliminate_word(): w = x**5*y*x**2*y**-4*x assert w.eliminate_word( x, x**2 ) == x**10*y*x**4*y**-4*x**2 w3 = x**2*y**3*x**-1*y assert w3.eliminate_word(x, x**2) == x**4*y**3*x**-2*y assert w3.eliminate_word(x, y) == y**5 assert w3.eliminate_word(x, y**4) == y**8 assert w3.eliminate_word(y, x**-1) == x**-3 assert w3.eliminate_word(x, y*z) == y*z*y*z*y**3*z**-1 assert (y**-3).eliminate_word(y, x**-1*z**-1) == z*x*z*x*z*x #assert w3.eliminate_word(x, y*x) == y*x*y*x**2*y*x*y*x*y*x*z**3 #assert w3.eliminate_word(x, x*y) == x*y*x**2*y*x*y*x*y*x*y*z**3 def test_FreeGroupElm_array_form(): assert (x*z).array_form == ((Symbol('x'), 1), (Symbol('z'), 1)) assert (x**2*z*y*x**-2).array_form == \ ((Symbol('x'), 2), (Symbol('z'), 1), (Symbol('y'), 1), (Symbol('x'), -2)) assert (x**-2*y**-1).array_form == ((Symbol('x'), -2), (Symbol('y'), -1)) def test_FreeGroupElm_letter_form(): assert (x**3).letter_form == (Symbol('x'), Symbol('x'), Symbol('x')) assert (x**2*z**-2*x).letter_form == \ (Symbol('x'), Symbol('x'), -Symbol('z'), -Symbol('z'), Symbol('x')) def test_FreeGroupElm_ext_rep(): assert (x**2*z**-2*x).ext_rep == \ (Symbol('x'), 2, Symbol('z'), -2, Symbol('x'), 1) assert (x**-2*y**-1).ext_rep == (Symbol('x'), -2, Symbol('y'), -1) assert (x*z).ext_rep == (Symbol('x'), 1, Symbol('z'), 1) def test_FreeGroupElm__mul__pow__(): x1 = x.group.dtype(((Symbol('x'), 1),)) assert x**2 == x1*x assert (x**2*y*x**-2)**4 == x**2*y**4*x**-2 assert (x**2)**2 == x**4 assert (x**-1)**-1 == x assert (x**-1)**0 == F.identity assert (y**2)**-2 == y**-4 assert x**2*x**-1 == x assert x**2*y**2*y**-1 == x**2*y assert x*x**-1 == F.identity assert x/x == F.identity assert x/x**2 == x**-1 assert (x**2*y)/(x**2*y**-1) == x**2*y**2*x**-2 assert (x**2*y)/(y**-1*x**2) == x**2*y*x**-2*y assert x*(x**-1*y*z*y**-1) == y*z*y**-1 assert x**2*(x**-2*y**-1*z**2*y) == y**-1*z**2*y a = F.identity for n in range(10): assert a == x**n assert a**-1 == x**-n a *= x def test_FreeGroupElm__len__(): assert len(x**5*y*x**2*y**-4*x) == 13 assert len(x**17) == 17 assert len(y**0) == 0 def test_FreeGroupElm_comparison(): assert not (x*y == y*x) assert x**0 == y**0 assert x**2 < y**3 assert not x**3 < y**2 assert x*y < x**2*y assert x**2*y**2 < y**4 assert not y**4 < y**-4 assert not y**4 < x**-4 assert y**-2 < y**2 assert x**2 <= y**2 assert x**2 <= x**2 assert not y*z > z*y assert x > x**-1 assert not x**2 >= y**2 def test_FreeGroupElm_syllables(): w = x**5*y*x**2*y**-4*x assert w.number_syllables() == 5 assert w.exponent_syllable(2) == 2 assert w.generator_syllable(3) == Symbol('y') assert w.sub_syllables(1, 2) == y assert w.sub_syllables(3, 3) == F.identity def test_FreeGroup_exponents(): w1 = x**2*y**3 assert w1.exponent_sum(x) == 2 assert w1.exponent_sum(x**-1) == -2 assert w1.generator_count(x) == 2 w2 = x**2*y**4*x**-3 assert w2.exponent_sum(x) == -1 assert w2.generator_count(x) == 5 def test_FreeGroup_generators(): assert (x**2*y**4*z**-1).contains_generators() == {x, y, z} assert (x**-1*y**3).contains_generators() == {x, y} def test_FreeGroupElm_words(): w = x**5*y*x**2*y**-4*x assert w.subword(2, 6) == x**3*y assert w.subword(3, 2) == F.identity assert w.subword(6, 10) == x**2*y**-2 assert w.substituted_word(0, 7, y**-1) == y**-1*x*y**-4*x assert w.substituted_word(0, 7, y**2*x) == y**2*x**2*y**-4*x