from itertools import product
import numpy as np
import sympy as sp


def test_nonnegative_word_expansion_has_no_cancellation():
    # Positive states cannot exhibit exact cancellation. Differences still can.
    A=sp.Matrix([[0,1],[1,0]])
    B=sp.Matrix([[1,0],[0,0]])
    C=sp.Matrix([[1,1]])
    x=sp.Matrix([1,0])
    for u,v in product(range(3),repeat=2):
        lhs=(C*(u*A+v*B)**2*x)[0]
        rhs=u*u*(C*A*A*x)[0]+u*v*(C*(A*B+B*A)*x)[0]+v*v*(C*B*B*x)[0]
        assert lhs==rhs and lhs>=0


def test_dual_rail_lift_preserves_every_short_word():
    matrices=[sp.Matrix([[0,-2],[1,0]]),sp.Matrix([[3,1],[-1,2]])]
    lifts=[]
    for A in matrices:
        plus=A.applyfunc(lambda z:max(z,0)); minus=A.applyfunc(lambda z:max(-z,0))
        lifts.append(plus.row_join(minus).col_join(minus.row_join(plus)))
    D=sp.eye(2).row_join(-sp.eye(2))
    for A,L in zip(matrices,lifts):
        assert D*L==A*D
        assert all(v>=0 for v in L)
    z=sp.Matrix([2,0,0,3])
    for n in range(5):
        for word in product(range(2),repeat=n):
            state=z; direct=D*z
            for index in word:
                state=lifts[index]*state; direct=matrices[index]*direct
            assert D*state==direct


def test_bounded_nonanalytic_cross_recovery_on_valid_data():
    for a,b,x in product([-2.,-.1,.2,1.],[-3.,-.3,.4,2.],[-2.,0.,3.]):
        y1,y2=a*a*x,b*b*x
        weights=np.array([a*b*a*a,a*b*b*b])/(a**4+b**4)
        assert np.isclose(weights@np.array([y1,y2]),a*b*x)
        assert np.linalg.norm(weights)<=1/np.sqrt(2)+1e-12


def test_positive_substochastic_lift_separates_two_uniformly_indistinguishable_states():
    from noulith import TwistedAlgebra
    algebra=TwistedAlgebra(2,{(0,1):sp.Integer(-1)})
    lifts=[]
    for i in range(2):
        A=algebra.right_matrix(i)
        plus=A.applyfunc(lambda x:max(x,0)); minus=A.applyfunc(lambda x:max(-x,0))
        lift=plus.row_join(minus).col_join(minus.row_join(plus))
        assert all(x>=0 for x in lift)
        assert all(sum(lift[:,j])<=1 for j in range(8))
        lifts.append(lift)
    zplus=sp.zeros(8,1); zplus[0]=1
    zminus=sp.zeros(8,1); zminus[4]=1
    C=sp.zeros(1,8); C[0,3]=1
    u,v=sp.symbols('u v')
    mixture=u*lifts[0]+v*lifts[1]
    for d in range(5):
        assert sp.expand((C*mixture**d*(zplus-zminus))[0])==0
    assert sp.expand((C*mixture**2*zplus)[0])==u*v
    assert (C*lifts[1]*lifts[0]*zplus)[0]==1
    assert (C*lifts[1]*lifts[0]*zminus)[0]==0