import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import ExprTuple, U, eps, n, s, t
from proveit.linear_algebra import MatrixMult, ScalarMult, Unitary
from proveit.logic import Equals, InSet
from proveit.numbers import Add, Ceil, Exp, IntervalCO, IntervalOC, Log, Mult, NaturalPos, Real, e, frac, greater_eq, i, one, pi, two, zero
from proveit.physics.quantum import normalized_var_ket_u, var_ket_u
from proveit.physics.quantum.QPE import phase, s_ket_domain, two_pow_s
# build up the expression from sub-expressions
expr = ExprTuple(InSet(eps, IntervalOC(zero, one)), InSet(s, NaturalPos), InSet(n, NaturalPos), greater_eq(n, two), InSet(U, Unitary(two_pow_s)), InSet(var_ket_u, s_ket_domain), InSet(phase, Real), InSet(phase, IntervalCO(zero, one)), normalized_var_ket_u, Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u)), InSet(t, NaturalPos), greater_eq(t, Add(n, Ceil(Log(two, Add(two, frac(one, Mult(two, eps))))))))
# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())
stored_expr.style_options()
# display the expression information
stored_expr.expr_info()