def prove(Eq):
n = Symbol.n(domain=[2, oo], integer=True)
Eq << apply(n)
_, i, j = Eq[0].rhs.args
_i = i.copy(domain=[0, n - 1])
_j = j.copy(domain=[0, n - 1])
W = Symbol.W(definition=Eq[0].lhs._subs(i, _i)._subs(j, _j))
V = Symbol.V(definition=Eq[0].rhs._subs(i, _i)._subs(j, _j))
Eq << W.this.definition
Eq << V.this.definition
h = Symbol.h(integer=True, domain=[0, n])
k = Symbol.k(integer=True, domain=[0, n])
Eq << (V[h, k].this.definition, W[h, k].this.definition)
Eq << (Eq[-1].this.rhs.as_KroneckerDelta(),
Eq[-2].this.rhs.as_KroneckerDelta())
Eq << Eq[-2] - Eq[-1]
Eq << Eq[-1].reference((k, ), (h, ))
Eq << Eq[-1].subs(Eq[1]).subs(Eq[2])
Eq << Eq[-1].forall((_i, ), (_j, ))
def prove(Eq):
n = Symbol.n(integer=True)
x = Symbol.x(shape=(n, ), real=True)
y = Symbol.y(shape=(n, ), real=True)
W = Symbol.W(shape=(n, n), real=True)
Eq << apply(x, y, W)
i = Symbol.i(domain=Interval(0, n - 1, integer=True))
j = Symbol.j(domain=Interval(0, n - 1, integer=True))
Eq << (x @ W).this.expand(free_symbol={i, j})
Eq << Eq[-1] @ y
Eq << Eq[-1].this.rhs.expand()
Eq.expansion = Eq[-1].this.rhs.function.distribute()
Eq << Eq.expansion.subs(W, W.T)
Eq << Eq[-1].swap(x, y)
Eq << Eq[-1].this.rhs.limits_subs(i, j)
Eq << Eq[-1].this.rhs.limits_swap()
Eq << Eq.expansion.subs(Eq[-1].reversed)
def prove(Eq):
n = Symbol.n(domain=[2, oo], integer=True)
W = Symbol.W(shape=(n, n), complex=True)
Eq << apply(W)
U = Symbol.U(definition=Eq[0].lhs)
V = Symbol.V(definition=Eq[0].rhs)
Eq << U.this.definition
Eq << V.this.definition
i = Symbol.i(integer=True, domain=[0, n])
j = Symbol.j(integer=True, domain=[0, n])
Eq << (V[i, j].this.definition, U[i, j].this.definition)
Eq << (Eq[-1].this.rhs.as_KroneckerDelta(),
Eq[-2].this.rhs.as_KroneckerDelta())
Eq << Eq[-2] - Eq[-1]
Eq << Eq[-1].reference((j, ), (i, ))
Eq << Eq[-1].subs(Eq[1]).subs(Eq[2])
Eq << Eq[-1].forall((j, ), (i, ))
def prove(Eq):
n = Symbol.n(integer=True)
x = Symbol.x(shape=(n, ), real=True)
y = Symbol.y(shape=(n, ), real=True)
W = Symbol.W(shape=(n, n), real=True)
Eq << apply(x, y, Equality(W.T, W))
Eq << bilinear.transpose.apply(x, y, W)
Eq << Eq[-1].subs(Eq[0])
