def apply(n, m, b):
i = Symbol.i(integer=True)
return Equality(
Concatenate(
Concatenate(MatProduct[i:m](Swap(n, i, b[i])), ZeroMatrix(n)).T,
Concatenate(ZeroMatrix(n), 1)).T,
MatProduct[i:m](Swap(n + 1, i, b[i])))
def apply(n):
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
return Equality(
Concatenate(
Concatenate(Swap(n, i, j), ZeroMatrix(n)).T,
Concatenate(ZeroMatrix(n), 1)), Swap(n + 1, i, j))
def apply(x, w=None):
n = x.shape[0]
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
if w is None:
w = Symbol.w(shape=(n, n, n), definition=LAMBDA[j:n](Swap(n, 0, j)))
assert w.shape == (n, n, n)
assert w[j].definition == Swap(n, 0, j)
return Equality(
LAMBDA[i:n](x[w[j][i] @ LAMBDA[i:n](i)]), LAMBDA[i:n](Piecewise(
(x[0], Equality(i, j)), (x[j], Equality(i, 0)), (x[i], True))))
def prove(Eq):
n = Symbol.n(domain=Interval(2, oo, integer=True))
i = Symbol.i(domain=Interval(0, n - 1, integer=True))
j = Symbol.j(domain=Interval(0, n - 1, integer=True))
assert Identity(n).is_integer
w = Symbol.w(integer=True, shape=(n, n, n, n), definition=LAMBDA[j, i](Swap(n, i, j)))
Eq << apply(w)
Eq << w[j, i].equality_defined()
Eq << Eq[0] @ Eq[-1]
Eq << Eq[-1].this.rhs.expand()
Eq << Eq[-1].this.rhs.simplify(deep=True, wrt=Eq[-1].rhs.variable)
Eq << Eq[-1].this.rhs.function.as_KroneckerDelta()
Eq << Eq[0] @ Eq[0]
Eq << Eq[-1].subs(Eq[-2].reversed)
Eq << w[i, j].inverse() @ Eq[-1]
Eq << Eq[-1].forall((i,), (j,))
def apply(given, i=None, j=None, w=None):
assert given.is_Equality
x_set_comprehension, interval = given.args
n = interval.max() + 1
assert interval.min() == 0
assert len(x_set_comprehension.limits) == 1
k, a, b = x_set_comprehension.limits[0]
assert b - a == n - 1
x = LAMBDA(x_set_comprehension.function.arg,
*x_set_comprehension.limits).simplify()
if j is None:
j = Symbol.j(domain=[0, n - 1], integer=True, given=True)
if i is None:
i = Symbol.i(domain=[0, n - 1], integer=True, given=True)
assert j >= 0 and j < n
assert i >= 0 and i < n
index = index_function(n)
if w is None:
_i = Symbol.i(integer=True)
_j = Symbol.j(integer=True)
w = Symbol.w(definition=LAMBDA[_j:n, _i:n](Swap(n, _i, _j)))
return Equality(index[i](w[index[i](x[:n]), index[j](x[:n])] @ x[:n]),
index[j](x[:n]),
given=given)
def apply(given):
assert given.is_ForAll and len(given.limits) == 2
j, a, n_munis_1 = given.limits[0]
assert a == 1
x, S = given.limits[1]
contains = given.function
assert contains.is_Contains
ref, _S = contains.args
assert S == _S and ref.is_LAMBDA and S.is_set
dtype = S.element_type
assert len(ref.limits) == 1
i, a, _n_munis_1 = ref.limits[0]
assert _n_munis_1 == n_munis_1 and a == 0
piecewise = ref.function
assert piecewise.is_Piecewise and len(piecewise.args) == 3
x0, condition0 = piecewise.args[0]
assert condition0.is_Equality and {*condition0.args} == {i, j}
xj, conditionj = piecewise.args[1]
assert conditionj.is_Equality and {*conditionj.args} == {i, 0}
xi, conditioni = piecewise.args[2]
assert conditioni
n = n_munis_1 + 1
assert x[j] == xj and x[i] == xi and x[0] == x0 and dtype == x.dtype
w = Symbol.w(definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
return ForAll(Contains(w[i, j] @ x, S), (x, S), given=given)
def apply(x, w=None, left=True, reference=True):
n = x.shape[0]
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
k = Symbol.k(integer=True)
if w is None:
w = Symbol.w(integer=True,
shape=(n, n, n, n),
definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
else:
assert len(w.shape) == 4 and all(s == n for s in w.shape)
if left:
if reference:
return Equality(LAMBDA[k:n](x[w[i, j, k] @ LAMBDA[k:n](k)]),
w[i, j] @ x)
else:
return Equality(x[w[i, j, k] @ LAMBDA[k:n](k)], w[i, j, k] @ x)
else:
if reference:
return Equality(LAMBDA[k:n](x[LAMBDA[k:n](k) @ w[i, j, k]]),
x @ w[i, j])
else:
return Equality(x[LAMBDA[k:n](k) @ w[i, j, k]], x @ w[i, j, k])
def prove(Eq):
n = Symbol.n(domain=Interval(2, oo, integer=True))
S = Symbol.S(dtype=dtype.integer * n)
x = Symbol.x(**S.element_symbol().dtype.dict)
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
w = Symbol.w(integer=True,
shape=(n, n, n, n),
definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
k = Symbol.k(integer=True)
given = ForAll[x:S](Contains(LAMBDA[k:n](x[(w[i, j] @ LAMBDA[k:n](k))[k]]),
S))
Eq.P_definition, Eq.w_definition, Eq.swap, Eq.axiom = apply(given)
Eq << algebre.matrix.elementary.swap.identity.apply(x, w)
Eq << Eq.swap.subs(Eq[-1])
Eq << swapn.permutation.apply(Eq[-1])
def apply(x):
n = x.shape[0]
i = Symbol.i(domain=Interval(0, n - 1, integer=True))
j = Symbol.j(domain=Interval(0, n - 1, integer=True))
w = Symbol.w(integer=True, shape=(n, n, n, n), definition=LAMBDA[j, i](Swap(n, i, j)))
return Equality(x @ w[i, j] @ w[i, j], x)
def apply(n, w=None):
i = Symbol.i(domain=Interval(0, n - 1, integer=True))
j = Symbol.j(domain=Interval(0, n - 1, integer=True))
assert n >= 2
if w is None:
w = Symbol.w(definition=LAMBDA[j, i](Swap(n, i, j)))
else:
assert len(w.shape) == 4 and all(s == n for s in w.shape)
return ForAll(Equality(w[0, i] @ w[0, j] @ w[0, i], w[i, j]), (j, Interval(1, n - 1, integer=True) - i.set))
def apply(n, w=None):
domain = Interval(0, n - 1, integer=True)
t = Symbol.t(domain=domain)
i = Symbol.i(domain=domain)
j = Symbol.j(domain=domain)
assert n >= 2
if w is None:
w = Symbol.w(definition=LAMBDA[j, i](Swap(n, i, j)))
return ForAll(Equality(w[t, i] @ w[t, j] @ w[t, i], w[i, j]),
(j, domain - {i, t}))
def predefined_symbols(n):
x = Symbol.x(shape=(oo, ), integer=True, nonnegative=True)
t = Symbol.t(integer=True)
Q = Symbol.Q(definition=LAMBDA[t:n + 1](conditionset(
x[:n + 1],
Equality(x[:n + 1].set_comprehension(), Interval(0, n, integer=True))
& Equality(x[n], t))))
j = Symbol.j(integer=True)
i = Symbol.i(integer=True)
w = Symbol.w(definition=LAMBDA[j:n + 1, i:n + 1](Swap(n + 1, i, j)))
return Q, w, x
def apply(x, i=None, j=None, w=None):
n = x.shape[0]
if i is None:
i = Symbol.i(domain=Interval(0, n - 1, integer=True))
if j is None:
j = Symbol.j(domain=Interval(0, n - 1, integer=True))
if w is None:
w = Symbol.w(definition=LAMBDA[j, i](Swap(n, i, j)))
return Equality(w[i, j] @ w[i, j] @ x, x)
def prove(Eq):
n = Symbol.n(domain=Interval(2, oo, integer=True))
S = Symbol.S(dtype=dtype.integer * n)
x = Symbol.x(**S.element_symbol().dtype.dict)
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
w = Symbol.w(integer=True, shape=(n, n, n, n), definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
given = ForAll[x:S](Contains(w[0, j] @ x, S))
Eq << apply(given)
Eq.given_i = given.subs(j, i)
Eq << given.subs(x, Eq.given_i.function.lhs)
Eq << (Eq.given_i & Eq[-1]).split()[-1]
Eq << Eq.given_i.subs(x, Eq[-1].function.lhs)
Eq.final_statement = (Eq[-2] & Eq[-1]).split()[0]
Eq << swap2.equality.apply(n, w)
Eq << Eq[-1] @ x
Eq << Eq[-1].forall((Eq[-1].limits[0].args[1].args[1].arg,))
Eq.i_complement = Eq.final_statement.subs(Eq[-1])
Eq.plausible = ForAll(Contains(w[i, j] @ x, S), (x, S), (j, Interval(1, n - 1, integer=True)), plausible=True)
Eq << Eq.plausible.bisect(i.set, wrt=j)
Eq.i_complement, Eq.i_intersection = Eq[-1].split()
Eq << sets.imply.equality.intersection.apply(i, Interval(1, n - 1, integer=True))
Eq << Eq.i_intersection.this.limits[1].subs(Eq[-1])
Eq << Eq[-1].subs(w[i, i].equality_defined())
Eq << (Eq.i_complement & Eq.i_intersection)
Eq << elementary.swap.transpose.apply(w).subs(j, 0)
Eq << Eq.given_i.subs(Eq[-1].reversed)
Eq << (Eq[-1] & Eq.plausible)
def apply(x, w=None):
n = x.shape[0]
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
if w is None:
w = Symbol.w(integer=True,
shape=(n, n, n, n),
definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
else:
assert len(w.shape) == 4 and all(s == n for s in w.shape)
assert w[i, j].is_Swap or w[i, j].definition.is_Swap
lhs = (w[i, j] @ x).set_comprehension()
return Equality(lhs, x.set_comprehension(free_symbol=lhs.variable))
def apply(n):
i = Symbol.i(integer=True)
p = Symbol.p(shape=(oo, ), integer=True, nonnegative=True)
P = Symbol.P(dtype=dtype.integer * n,
definition=conditionset(
p[:n],
Equality(p[:n].set_comprehension(),
Interval(0, n - 1, integer=True))))
b = Symbol.b(integer=True, shape=(oo, ), nonnegative=True)
return ForAll[p[:n]:P](Exists[b[:n]](Equality(
p[:n], LAMBDA[i:n](i) @ MatProduct[i:n](Swap(n, i, b[i])))))
def apply(x, d, w=None):
n = x.shape[0]
assert d.shape == (n, )
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
k = Symbol.k(integer=True)
if w is None:
w = Symbol.w(integer=True,
shape=(n, n, n, n),
definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
else:
assert len(w.shape) == 4 and all(s == n for s in w.shape)
assert w[i, j].is_Swap or w[i, j].definition.is_Swap
return Equality(x[d @ w[i, j, k]], LAMBDA[k:n](x[d[k]]) @ w[i, j, k])
def apply(x, d, w=None):
n = x.shape[0]
m = d.shape[0]
assert m.is_integer and m.is_finite
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
k = Symbol.k(integer=True)
if w is None:
w = Symbol.w(integer=True,
shape=(n, n, n, n),
definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
else:
assert len(w.shape) == 4 and all(s == n for s in w.shape)
assert w[i, j].is_Swap or w[i, j].definition.is_Swap
multiplier = MatProduct[i:m](w[i, d[i]])
return Equality(LAMBDA[k:n](x[(LAMBDA[k:n](k) @ multiplier)[k]]),
x @ multiplier)
def apply(n, w=None, left=True, P=None):
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
if w is None:
w = Symbol.w(definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
else:
assert len(w.shape) == 4 and all(s == n for s in w.shape)
assert w[i, j].is_Swap or w[i, j].definition.is_Swap
x = Symbol.x(shape=(oo,), integer=True, nonnegative=True)
x = x[:n]
if P is None:
P = Symbol.P(dtype=dtype.integer * n, definition=conditionset(x, Equality(x.set_comprehension(), Interval(0, n - 1, integer=True))))
if left:
return ForAll[x:P](Contains(w[i, j] @ x, P))
else:
return ForAll[x:P](Contains(x @ w[i, j], P))
def apply(m, d, w=None):
n = d.shape[0]
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
assert m >= 0
if w is None:
w = Symbol.w(definition=LAMBDA[j, i](Swap(n, i, j)))
else:
assert len(w.shape) == 4 and all(s == n for s in w.shape)
assert w[i, j].is_Swap or w[i, j].definition.is_Swap
x = Symbol.x(shape=(oo, ), integer=True, nonnegative=True)
x = x[:n]
P = Symbol.P(dtype=dtype.integer * n,
definition=conditionset(
x,
Equality(x.set_comprehension(),
Interval(0, n - 1, integer=True))))
return ForAll[x:P](Contains(x @ MatProduct[i:m](w[i, d[i]]), P))
def prove(Eq):
n = Symbol.n(domain=Interval(2, oo, integer=True))
S = Symbol.S(dtype=dtype.integer * n, given=True)
x = Symbol.x(shape=(oo, ), integer=True)
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
w = Symbol.w(definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
given = ForAll[x[:n]:S](Contains(w[i, j] @ x[:n], S))
Eq.P_definition, Eq.w_definition, Eq.swap, Eq.axiom = apply(given)
Eq << factorization.apply(n)
*_, b_i = Eq[-1].rhs.args[1].function.args
b, _i = b_i.args
Eq << Eq.w_definition.subs(j, b[_i]).subs(i, _i)
Eq << Eq[-2].subs(Eq[-1].reversed)
k = Eq.axiom.lhs.variable
Eq << Eq[-1][k]
Eq << Eq[-1].this.function.function.rhs.args[0].limits_subs(_i, k)
Eq << swapn.utility.apply(x[:n], b[:n], w)
Eq << Eq[-1].subs(Eq[-2].reversed)
Eq.plausible = Eq.axiom.subs(Eq[-1])
Eq << swapn.mat_product.apply(Eq.swap.T, n, b)
Eq << Eq.plausible.this.function.as_ForAll()
def prove(Eq):
n = Symbol.n(domain=Interval(2, oo, integer=True))
S = Symbol.S(dtype=dtype.integer * n, given=True)
x = Symbol.x(shape=(n,), integer=True)
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
w = Symbol.w(definition=LAMBDA[j:n, i:n](Swap(n, i, j)))
given = ForAll[x[:n]:S](Contains(x[:n] @ w[i, j], S))
Eq.w_definition, Eq.swap, Eq.mat_product = apply(given)
i, _, m_1 = Eq.mat_product.function.lhs.args[1].limits[0]
m = m_1 + 1
b = Eq.mat_product.function.lhs.args[1].function.indices[1].base
Eq << Eq.mat_product.subs(m, 0)
Eq << Eq.mat_product.subs(m, m + 1)
Eq << Eq[-1].function.lhs.args[1].this.bisect(Slice[-1:])
Eq << x @ Eq[-1]
Eq << Eq.swap.subs(i, m).subs(j, b[m])
Eq << Eq[-1].subs(x, Eq[2].rhs.func(*Eq[2].rhs.args[:2]))
Eq << Eq[-1].forall((x, S))
Eq << (Eq[-1] & Eq.mat_product).split()
Eq << Eq[-1].subs(Eq[2].reversed)