def prove(Eq):
A = Symbol.A(dtype=dtype.integer)
B = Symbol.B(dtype=dtype.integer)
Eq << apply(Equality(Intersection(A, B), S.EmptySet))
C = Symbol.C(dtype=dtype.integer, definition=A | B)
D = Symbol.D(dtype=dtype.integer, definition=A - B)
Eq << C.this.definition
Eq << D.this.definition
Eq << Eq[-1].union(A)
Eq << Eq[-2].union(B)
Eq << sets.equality.imply.equality.given.emptyset.complement.apply(Eq[0])
Eq << Eq[-1].abs()
Eq << Eq[1].subs(Eq[-1].reversed)
Eq << Eq[-1] - Eq[-1].rhs.args[0]
Eq << (A - B).assertion()
def prove(Eq):
A = Symbol.A(dtype=dtype.integer)
B = Symbol.B(dtype=dtype.integer)
Eq << apply(A, B)
Eq << Equality(abs(A | B), abs(A) + abs(B - A), plausible=True)
Eq << Eq[-1].subs(Eq[-2])
Eq << Eq[-1] - Eq[-1].lhs.args[1]
C = Symbol.C(dtype=dtype.integer, definition=A & B)
D = Symbol.D(dtype=dtype.integer, definition=B - A)
Eq.C_definition = C.this.definition
Eq.D_definition = D.this.definition
Eq << Equality(D & C, S.EmptySet, plausible=True)
Eq << Eq[-1].subs(Eq.C_definition, Eq.D_definition)
Eq << sets.equality.imply.equality.given.intersection.addition_principle.apply(
Eq[-1])
Eq << Eq[-1].subs(Eq.C_definition, Eq.D_definition)
Eq << Equality(D & A, S.EmptySet, plausible=True)
Eq << Eq[-1].subs(Eq.D_definition)
Eq << sets.equality.imply.equality.given.intersection.addition_principle.apply(
Eq[-1])
Eq << Eq[-1].subs(Eq.D_definition)
