using a SMT solver like Z3 to define and test invariants is a really interesting application of an extremely powerful tool.<p>from z3 import *<p>def check_invariant(precondition, loop_body, invariant):
"""
Check if a loop invariant holds.<p><pre><code> Args:
precondition (z3.ExprRef): The precondition of the loop.
loop_body (Callable[[z3.ExprRef], z3.ExprRef]): A function representing the loop body.
invariant (z3.ExprRef): The loop invariant to check.
"""
s = Solver()
s.add(precondition)
# Check if the invariant holds initially
s.push()
s.add(Not(invariant))
if s.check() == sat:
print("Invariant does not hold initially.")
return
s.pop()
# Check if the invariant is preserved by the loop body
s.push()
s.add(invariant)
s.add(loop_body(invariant))
s.add(Not(invariant))
if s.check() == sat:
print("Invariant is not preserved by the loop body.")
return
s.pop()
print("Invariant holds.")
</code></pre>
# Example: Proving that the sum of the first n integers is n<i>(n+1)/2
def example():
n = Int('n')
i = Int('i')
sum = Int('sum')<p><pre><code> precondition = And(n >= 0, i == 0, sum == 0)
loop_body = lambda inv: And(i < n, sum == i * (i + 1) / 2, i == i + 1, sum == sum + i)
invariant = sum == i * (i + 1) / 2
check_invariant(precondition, loop_body, invariant)
</code></pre>
if __name__ == "__main__":
example()</i>