def test_new_relational():
x = Symbol('x')
assert Eq(x, 0) == Relational(x, 0) # None ==> Equality
assert Eq(x, 0) == Relational(x, 0, '==')
assert Eq(x, 0) == Relational(x, 0, 'eq')
assert Eq(x, 0) == Equality(x, 0)
assert Eq(x, 0) != Relational(x, 1) # None ==> Equality
assert Eq(x, 0) != Relational(x, 1, '==')
assert Eq(x, 0) != Relational(x, 1, 'eq')
assert Eq(x, 0) != Equality(x, 1)
assert Eq(x, -1) == Relational(x, -1) # None ==> Equality
assert Eq(x, -1) == Relational(x, -1, '==')
assert Eq(x, -1) == Relational(x, -1, 'eq')
assert Eq(x, -1) == Equality(x, -1)
assert Eq(x, -1) != Relational(x, 1) # None ==> Equality
assert Eq(x, -1) != Relational(x, 1, '==')
assert Eq(x, -1) != Relational(x, 1, 'eq')
assert Eq(x, -1) != Equality(x, 1)
assert Ne(x, 0) == Relational(x, 0, '!=')
assert Ne(x, 0) == Relational(x, 0, '<>')
assert Ne(x, 0) == Relational(x, 0, 'ne')
assert Ne(x, 0) == Unequality(x, 0)
assert Ne(x, 0) != Relational(x, 1, '!=')
assert Ne(x, 0) != Relational(x, 1, '<>')
assert Ne(x, 0) != Relational(x, 1, 'ne')
assert Ne(x, 0) != Unequality(x, 1)
assert Ge(x, 0) == Relational(x, 0, '>=')
assert Ge(x, 0) == Relational(x, 0, 'ge')
assert Ge(x, 0) == GreaterThan(x, 0)
assert Ge(x, 1) != Relational(x, 0, '>=')
assert Ge(x, 1) != Relational(x, 0, 'ge')
assert Ge(x, 1) != GreaterThan(x, 0)
assert (x >= 1) == Relational(x, 1, '>=')
assert (x >= 1) == Relational(x, 1, 'ge')
assert (x >= 1) == GreaterThan(x, 1)
assert (x >= 0) != Relational(x, 1, '>=')
assert (x >= 0) != Relational(x, 1, 'ge')
assert (x >= 0) != GreaterThan(x, 1)
assert Le(x, 0) == Relational(x, 0, '<=')
assert Le(x, 0) == Relational(x, 0, 'le')
assert Le(x, 0) == LessThan(x, 0)
assert Le(x, 1) != Relational(x, 0, '<=')
assert Le(x, 1) != Relational(x, 0, 'le')
assert Le(x, 1) != LessThan(x, 0)
assert (x <= 1) == Relational(x, 1, '<=')
assert (x <= 1) == Relational(x, 1, 'le')
assert (x <= 1) == LessThan(x, 1)
assert (x <= 0) != Relational(x, 1, '<=')
assert (x <= 0) != Relational(x, 1, 'le')
assert (x <= 0) != LessThan(x, 1)
assert Gt(x, 0) == Relational(x, 0, '>')
assert Gt(x, 0) == Relational(x, 0, 'gt')
assert Gt(x, 0) == StrictGreaterThan(x, 0)
assert Gt(x, 1) != Relational(x, 0, '>')
assert Gt(x, 1) != Relational(x, 0, 'gt')
assert Gt(x, 1) != StrictGreaterThan(x, 0)
assert (x > 1) == Relational(x, 1, '>')
assert (x > 1) == Relational(x, 1, 'gt')
assert (x > 1) == StrictGreaterThan(x, 1)
assert (x > 0) != Relational(x, 1, '>')
assert (x > 0) != Relational(x, 1, 'gt')
assert (x > 0) != StrictGreaterThan(x, 1)
assert Lt(x, 0) == Relational(x, 0, '<')
assert Lt(x, 0) == Relational(x, 0, 'lt')
assert Lt(x, 0) == StrictLessThan(x, 0)
assert Lt(x, 1) != Relational(x, 0, '<')
assert Lt(x, 1) != Relational(x, 0, 'lt')
assert Lt(x, 1) != StrictLessThan(x, 0)
assert (x < 1) == Relational(x, 1, '<')
assert (x < 1) == Relational(x, 1, 'lt')
assert (x < 1) == StrictLessThan(x, 1)
assert (x < 0) != Relational(x, 1, '<')
assert (x < 0) != Relational(x, 1, 'lt')
assert (x < 0) != StrictLessThan(x, 1)
# finally, some fuzz testing
from random import randint
from sympy.core.compatibility import unichr
for i in range(100):
while 1:
strtype, length = (unichr, 65535) if randint(0, 1) else (chr, 255)
relation_type = strtype(randint(0, length))
if randint(0, 1):
relation_type += strtype(randint(0, length))
if relation_type not in ('==', 'eq', '!=', '<>', 'ne', '>=', 'ge',
'<=', 'le', '>', 'gt', '<', 'lt', ':=',
'+=', '-=', '*=', '/=', '%='):
break
raises(ValueError, lambda: Relational(x, 1, relation_type))
assert all(Relational(x, 0, op).rel_op == '==' for op in ('eq', '=='))
assert all(
Relational(x, 0, op).rel_op == '!=' for op in ('ne', '<>', '!='))
assert all(Relational(x, 0, op).rel_op == '>' for op in ('gt', '>'))
assert all(Relational(x, 0, op).rel_op == '<' for op in ('lt', '<'))
assert all(Relational(x, 0, op).rel_op == '>=' for op in ('ge', '>='))
assert all(Relational(x, 0, op).rel_op == '<=' for op in ('le', '<='))
(r"(x!)!", _factorial(_factorial(x))),
(r"x!!!", _factorial(_factorial(_factorial(x)))),
(r"5!7!", _Mul(_factorial(5), _factorial(7))),
(r"\sqrt{x}", sqrt(x)),
(r"\sqrt{x + b}", sqrt(_Add(x, b))),
(r"\sqrt[3]{\sin x}", root(sin(x), 3)),
(r"\sqrt[y]{\sin x}", root(sin(x), y)),
(r"\sqrt[\theta]{\sin x}", root(sin(x), theta)),
(r"\sqrt{\frac{12}{6}}", _Sqrt(_Mul(12, _Pow(6, -1)))),
(r"\overline{z}", _Conjugate(z)),
(r"\overline{\overline{z}}", _Conjugate(_Conjugate(z))),
(r"\overline{x + y}", _Conjugate(_Add(x, y))),
(r"\overline{x} + \overline{y}", _Conjugate(x) + _Conjugate(y)),
(r"x < y", StrictLessThan(x, y)),
(r"x \leq y", LessThan(x, y)),
(r"x > y", StrictGreaterThan(x, y)),
(r"x \geq y", GreaterThan(x, y)),
(r"\mathit{x}", Symbol('x')),
(r"\mathit{test}", Symbol('test')),
(r"\mathit{TEST}", Symbol('TEST')),
(r"\mathit{HELLO world}", Symbol('HELLO world')),
(r"\sum_{k = 1}^{3} c", Sum(c, (k, 1, 3))),
(r"\sum_{k = 1}^3 c", Sum(c, (k, 1, 3))),
(r"\sum^{3}_{k = 1} c", Sum(c, (k, 1, 3))),
(r"\sum^3_{k = 1} c", Sum(c, (k, 1, 3))),
(r"\sum_{k = 1}^{10} k^2", Sum(k**2, (k, 1, 10))),
(r"\sum_{n = 0}^{\infty} \frac{1}{n!}",
Sum(_Pow(_factorial(n), -1), (n, 0, oo))),
(r"\prod_{a = b}^{c} x", Product(x, (a, b, c))),
(r"\prod_{a = b}^c x", Product(x, (a, b, c))),
(r"\prod^{c}_{a = b} x", Product(x, (a, b, c))),
def test_new_relational():
x = Symbol("x")
assert Eq(x, 0) == Relational(x, 0) # None ==> Equality
assert Eq(x, 0) == Relational(x, 0, "==")
assert Eq(x, 0) == Relational(x, 0, "eq")
assert Eq(x, 0) == Equality(x, 0)
assert Eq(x, 0) != Relational(x, 1) # None ==> Equality
assert Eq(x, 0) != Relational(x, 1, "==")
assert Eq(x, 0) != Relational(x, 1, "eq")
assert Eq(x, 0) != Equality(x, 1)
assert Eq(x, -1) == Relational(x, -1) # None ==> Equality
assert Eq(x, -1) == Relational(x, -1, "==")
assert Eq(x, -1) == Relational(x, -1, "eq")
assert Eq(x, -1) == Equality(x, -1)
assert Eq(x, -1) != Relational(x, 1) # None ==> Equality
assert Eq(x, -1) != Relational(x, 1, "==")
assert Eq(x, -1) != Relational(x, 1, "eq")
assert Eq(x, -1) != Equality(x, 1)
assert Ne(x, 0) == Relational(x, 0, "!=")
assert Ne(x, 0) == Relational(x, 0, "<>")
assert Ne(x, 0) == Relational(x, 0, "ne")
assert Ne(x, 0) == Unequality(x, 0)
assert Ne(x, 0) != Relational(x, 1, "!=")
assert Ne(x, 0) != Relational(x, 1, "<>")
assert Ne(x, 0) != Relational(x, 1, "ne")
assert Ne(x, 0) != Unequality(x, 1)
assert Ge(x, 0) == Relational(x, 0, ">=")
assert Ge(x, 0) == Relational(x, 0, "ge")
assert Ge(x, 0) == GreaterThan(x, 0)
assert Ge(x, 1) != Relational(x, 0, ">=")
assert Ge(x, 1) != Relational(x, 0, "ge")
assert Ge(x, 1) != GreaterThan(x, 0)
assert (x >= 1) == Relational(x, 1, ">=")
assert (x >= 1) == Relational(x, 1, "ge")
assert (x >= 1) == GreaterThan(x, 1)
assert (x >= 0) != Relational(x, 1, ">=")
assert (x >= 0) != Relational(x, 1, "ge")
assert (x >= 0) != GreaterThan(x, 1)
assert Le(x, 0) == Relational(x, 0, "<=")
assert Le(x, 0) == Relational(x, 0, "le")
assert Le(x, 0) == LessThan(x, 0)
assert Le(x, 1) != Relational(x, 0, "<=")
assert Le(x, 1) != Relational(x, 0, "le")
assert Le(x, 1) != LessThan(x, 0)
assert (x <= 1) == Relational(x, 1, "<=")
assert (x <= 1) == Relational(x, 1, "le")
assert (x <= 1) == LessThan(x, 1)
assert (x <= 0) != Relational(x, 1, "<=")
assert (x <= 0) != Relational(x, 1, "le")
assert (x <= 0) != LessThan(x, 1)
assert Gt(x, 0) == Relational(x, 0, ">")
assert Gt(x, 0) == Relational(x, 0, "gt")
assert Gt(x, 0) == StrictGreaterThan(x, 0)
assert Gt(x, 1) != Relational(x, 0, ">")
assert Gt(x, 1) != Relational(x, 0, "gt")
assert Gt(x, 1) != StrictGreaterThan(x, 0)
assert (x > 1) == Relational(x, 1, ">")
assert (x > 1) == Relational(x, 1, "gt")
assert (x > 1) == StrictGreaterThan(x, 1)
assert (x > 0) != Relational(x, 1, ">")
assert (x > 0) != Relational(x, 1, "gt")
assert (x > 0) != StrictGreaterThan(x, 1)
assert Lt(x, 0) == Relational(x, 0, "<")
assert Lt(x, 0) == Relational(x, 0, "lt")
assert Lt(x, 0) == StrictLessThan(x, 0)
assert Lt(x, 1) != Relational(x, 0, "<")
assert Lt(x, 1) != Relational(x, 0, "lt")
assert Lt(x, 1) != StrictLessThan(x, 0)
assert (x < 1) == Relational(x, 1, "<")
assert (x < 1) == Relational(x, 1, "lt")
assert (x < 1) == StrictLessThan(x, 1)
assert (x < 0) != Relational(x, 1, "<")
assert (x < 0) != Relational(x, 1, "lt")
assert (x < 0) != StrictLessThan(x, 1)
# finally, some fuzz testing
from random import randint
from sympy.core.compatibility import unichr
for i in range(100):
while 1:
strtype, length = (unichr, 65535) if randint(0, 1) else (chr, 255)
relation_type = strtype(randint(0, length))
if randint(0, 1):
relation_type += strtype(randint(0, length))
if relation_type not in (
"==",
"eq",
"!=",
"<>",
"ne",
">=",
"ge",
"<=",
"le",
">",
"gt",
"<",
"lt",
":=",
"+=",
"-=",
"*=",
"/=",
"%=",
):
break
raises(ValueError, lambda: Relational(x, 1, relation_type))
assert all(Relational(x, 0, op).rel_op == "==" for op in ("eq", "=="))
assert all(Relational(x, 0, op).rel_op == "!=" for op in ("ne", "<>", "!="))
assert all(Relational(x, 0, op).rel_op == ">" for op in ("gt", ">"))
assert all(Relational(x, 0, op).rel_op == "<" for op in ("lt", "<"))
assert all(Relational(x, 0, op).rel_op == ">=" for op in ("ge", ">="))
assert all(Relational(x, 0, op).rel_op == "<=" for op in ("le", "<="))
def test_new_relational():
x = Symbol('x')
assert Eq(x) == Relational(x, 0) # None ==> Equality
assert Eq(x) == Relational(x, 0, '==')
assert Eq(x) == Relational(x, 0, 'eq')
assert Eq(x) == Equality(x, 0)
assert Eq(x, -1) == Relational(x, -1) # None ==> Equality
assert Eq(x, -1) == Relational(x, -1, '==')
assert Eq(x, -1) == Relational(x, -1, 'eq')
assert Eq(x, -1) == Equality(x, -1)
assert Eq(x) != Relational(x, 1) # None ==> Equality
assert Eq(x) != Relational(x, 1, '==')
assert Eq(x) != Relational(x, 1, 'eq')
assert Eq(x) != Equality(x, 1)
assert Eq(x, -1) != Relational(x, 1) # None ==> Equality
assert Eq(x, -1) != Relational(x, 1, '==')
assert Eq(x, -1) != Relational(x, 1, 'eq')
assert Eq(x, -1) != Equality(x, 1)
assert Ne(x, 0) == Relational(x, 0, '!=')
assert Ne(x, 0) == Relational(x, 0, '<>')
assert Ne(x, 0) == Relational(x, 0, 'ne')
assert Ne(x, 0) == Unequality(x, 0)
assert Ne(x, 0) != Relational(x, 1, '!=')
assert Ne(x, 0) != Relational(x, 1, '<>')
assert Ne(x, 0) != Relational(x, 1, 'ne')
assert Ne(x, 0) != Unequality(x, 1)
assert Ge(x, 0) == Relational(x, 0, '>=')
assert Ge(x, 0) == Relational(x, 0, 'ge')
assert Ge(x, 0) == GreaterThan(x, 0)
assert Ge(x, 1) != Relational(x, 0, '>=')
assert Ge(x, 1) != Relational(x, 0, 'ge')
assert Ge(x, 1) != GreaterThan(x, 0)
assert (x >= 1) == Relational(x, 1, '>=')
assert (x >= 1) == Relational(x, 1, 'ge')
assert (x >= 1) == GreaterThan(x, 1)
assert (x >= 0) != Relational(x, 1, '>=')
assert (x >= 0) != Relational(x, 1, 'ge')
assert (x >= 0) != GreaterThan(x, 1)
assert Le(x, 0) == Relational(x, 0, '<=')
assert Le(x, 0) == Relational(x, 0, 'le')
assert Le(x, 0) == LessThan(x, 0)
assert Le(x, 1) != Relational(x, 0, '<=')
assert Le(x, 1) != Relational(x, 0, 'le')
assert Le(x, 1) != LessThan(x, 0)
assert (x <= 1) == Relational(x, 1, '<=')
assert (x <= 1) == Relational(x, 1, 'le')
assert (x <= 1) == LessThan(x, 1)
assert (x <= 0) != Relational(x, 1, '<=')
assert (x <= 0) != Relational(x, 1, 'le')
assert (x <= 0) != LessThan(x, 1)
assert Gt(x, 0) == Relational(x, 0, '>')
assert Gt(x, 0) == Relational(x, 0, 'gt')
assert Gt(x, 0) == StrictGreaterThan(x, 0)
assert Gt(x, 1) != Relational(x, 0, '>')
assert Gt(x, 1) != Relational(x, 0, 'gt')
assert Gt(x, 1) != StrictGreaterThan(x, 0)
assert (x > 1) == Relational(x, 1, '>')
assert (x > 1) == Relational(x, 1, 'gt')
assert (x > 1) == StrictGreaterThan(x, 1)
assert (x > 0) != Relational(x, 1, '>')
assert (x > 0) != Relational(x, 1, 'gt')
assert (x > 0) != StrictGreaterThan(x, 1)
assert Lt(x, 0) == Relational(x, 0, '<')
assert Lt(x, 0) == Relational(x, 0, 'lt')
assert Lt(x, 0) == StrictLessThan(x, 0)
assert Lt(x, 1) != Relational(x, 0, '<')
assert Lt(x, 1) != Relational(x, 0, 'lt')
assert Lt(x, 1) != StrictLessThan(x, 0)
assert (x < 1) == Relational(x, 1, '<')
assert (x < 1) == Relational(x, 1, 'lt')
assert (x < 1) == StrictLessThan(x, 1)
assert (x < 0) != Relational(x, 1, '<')
assert (x < 0) != Relational(x, 1, 'lt')
assert (x < 0) != StrictLessThan(x, 1)
# finally, some fuzz testing
from random import randint
for i in range(100):
while 1:
strtype, length = (unichr, 65535) if randint(0, 1) else (chr, 255)
relation_type = strtype(randint(0, length))
if randint(0, 1):
relation_type += strtype(randint(0, length))
if relation_type not in ('==', 'eq', '!=', '<>', 'ne', '>=', 'ge',
'<=', 'le', '>', 'gt', '<', 'lt'):
break
raises(ValueError, "Relational(x, 1, relation_type)")
def prove(Eq):
k = Symbol.k(integer=True, positive=True)
n = Symbol.n(integer=True, positive=True)
Eq << apply(n, k)
s2 = Eq[0].lhs
s2_quote = Symbol.s_quote_2(definition=Eq[0].rhs.limits[0][1])
Eq << s2_quote.this.definition
Eq.s2_definition = Eq[0].subs(Eq[-1].reversed)
s0 = Eq[1].lhs
s0_quote = Symbol.s_quote_0(definition=Eq[1].rhs.limits[0][1])
Eq << s0_quote.this.definition
Eq << Eq[1].subs(Eq[-1].reversed)
s0_definition = Eq[-1]
e = Symbol.e(dtype=dtype.integer.set)
s0_ = image_set(Union(e, {n.set}), e, s0)
plausible0 = Subset(s0_, s2, plausible=True)
Eq << plausible0
Eq << Eq[-1].definition
Eq << Eq[-1].this.limits[0][1].subs(s0_definition)
Eq << Eq[-1].subs(Eq.s2_definition)
s0_plausible = Eq[-1]
Eq.s2_quote_definition = s2_quote.assertion()
Eq << s0_quote.assertion()
Eq << Eq[-1].split()
x_abs_positive = Eq[-3]
x_abs_sum = Eq[-2]
x_union_s0 = Eq[-1]
i = Eq[-1].lhs.limits[0][0]
x = Eq[-1].variable.base
Eq << Equality.define(x[k], {n})
x_k_definition = Eq[-1]
Eq << Eq[-1].union(Eq[-2])
x_union = Eq[-1]
Eq << x_k_definition.set
Eq << Eq[-1].union(x[:k].set_comprehension())
Eq << s0_plausible.subs(Eq[-1].reversed)
Eq << Eq[-1].definition.definition
Eq << x_k_definition.abs()
Eq << Eq[-1].subs(StrictGreaterThan(1, 0, plausible=True))
Eq << x_abs_sum + Eq[-2]
Eq << (x_abs_positive & Eq[-2])
Eq << (x_union & Eq[-1] & Eq[-2])
j = Symbol.j(domain=Interval(0, k, integer=True))
B = Eq[2].lhs
Eq << plausible0.subs(Eq[2].reversed)
Eq << s2.this.bisect(conditionset(e, Contains({n}, e), s2))
Eq.subset_B = Subset(Eq[-1].rhs.args[0], Eq[-2].lhs,
plausible=True) # unproven
Eq.supset_B = Supset(Eq[-1].rhs.args[0], Eq[-2].lhs,
plausible=True) # unproven
Eq << Eq.supset_B.subs(Eq[2])
Eq << Eq[-1].definition.definition
Eq << Eq.subset_B.subs(Eq[2])
Eq << Eq[-1].definition.definition
Eq.subset_B_definition = Eq[-1] - {n.set}
num_plausibles = len(Eq.plausibles_dict)
Eq.plausible_notcontains = ForAll(NotContains({n}, e), (e, s0),
plausible=True)
Eq << Eq.plausible_notcontains.this.limits[0][1].subs(s0_definition)
Eq << ~Eq[-1]
Eq << Eq[-1].definition
Eq << x_union_s0.union(Eq[-1].reversed).this().function.lhs.simplify()
Eq << Eq[-1].subs(x_union_s0)
assert num_plausibles == len(Eq.plausibles_dict)
Eq << Eq.plausible_notcontains.apply(
sets.notcontains.imply.equality.emptyset)
Eq.s0_complement_n = Eq[-1].apply(
sets.equality.imply.equality.given.emptyset.complement)
Eq << Eq.subset_B_definition.subs(Eq.s0_complement_n)
s2_n = Symbol('s_{2, n}', definition=Eq[-1].limits[0][1])
Eq.s2_n_definition = s2_n.this.definition
Eq << s2_n.assertion()
Eq << Eq[-1].subs(Eq.s2_definition).split()
Eq.s2_n_assertion = Eq[-2].definition
Eq << Eq[-1].subs(Eq.s2_n_assertion)
Eq << Eq[-1].definition
Eq.x_j_definition = Eq[-1].limits_subs(Eq[-1].variable, j).reversed
Eq.x_abs_positive_s2, Eq.x_abs_sum_s2, Eq.x_union_s2 = Eq.s2_quote_definition.split(
)
Eq << Eq.x_union_s2 - Eq.x_j_definition
Eq << Eq[-1].this.function.lhs.args[0].bisect({j})
x_tilde = Symbol(r"\tilde{x}",
shape=(k, ),
dtype=dtype.integer,
definition=LAMBDA[i:k](Piecewise((x[i], i < j),
(x[i + 1], True))))
Eq.x_tilde_definition = x_tilde.equality_defined()
Eq << Eq.x_tilde_definition.union_comprehension((i, 0, k - 1))
Eq << Eq[-1].this.rhs.args[1].limits_subs(i, i - 1)
Eq.x_tilde_union = Eq[-1].subs(Eq[-3])
Eq.x_tilde_abs = Eq.x_tilde_definition.abs()
Eq << Eq.x_tilde_abs.sum((i, 0, k - 1))
Eq << Eq[-1].this.rhs.args[0].limits_subs(i, i - 1)
Eq.x_tilde_abs_sum = Eq[-1].subs(Eq.x_abs_sum_s2, Eq.x_j_definition.abs())
Eq << Eq.x_tilde_abs.as_Or()
Eq << Eq[-1].forall((i, i < j))
Eq << Eq[-2].forall((i, i >= j))
Eq << Eq[-2].subs(Eq.x_abs_positive_s2)
Eq << Eq[-2].subs(Eq.x_abs_positive_s2.limits_subs(i, i + 1))
Eq << (Eq[-1] & Eq[-2])
Eq << (Eq[-1] & Eq.x_tilde_abs_sum & Eq.x_tilde_union)
Eq << Eq[-1].func(
Contains(x_tilde, s0_quote), *Eq[-1].limits, plausible=True)
Eq << Eq[-1].definition
Eq << Eq[-1].this.function.args[0].simplify()
Eq.x_tilde_set_in_s0 = Eq[-3].func(Contains(
UNION.construct_finite_set(x_tilde), s0),
*Eq[-3].limits,
plausible=True)
Eq << Eq.x_tilde_set_in_s0.subs(s0_definition)
Eq << Eq[-1].definition
Eq << Eq.x_tilde_definition.set.union_comprehension((i, 0, k - 1))
Eq << Eq[-1].subs(Eq.x_j_definition)
Eq << Eq[-1].subs(Eq.s2_n_assertion.reversed)
Eq << Eq.x_tilde_set_in_s0.subs(Eq[-1])
Eq << Eq[-1].this.limits[0].subs(Eq.s2_n_definition)
Eq.subset_B_plausible = Eq.subset_B_definition.union({n.set})
Eq << Eq.subset_B_plausible.limits_assertion()
Eq << Eq[-1].definition.split()[1]
Eq << Eq[-1].apply(sets.contains.imply.equality.union)
Eq << Eq.subset_B_plausible.subs(Eq[-1])
Eq << Eq.supset_B.subs(Eq.subset_B)
