def reflexive(e):
return constructors.Forall([constructors.BoundedVariableBinding(x, e)],
Equal(x, x, e))
x = common_vars.x()
y = common_vars.y()
def symmetric(e):
return constructors.Forall([ constructors.BoundedVariableBinding(x, e)
, constructors.BoundedVariableBinding(y, e)],
constructors.Implies(
predicates = [Equal(x, y, e)],
consequent = Equal(y, x, e)))
x = common_vars.x()
y = common_vars.y()
z = common_vars.z()
def transitive(e):
return constructors.Forall([ constructors.BoundedVariableBinding(x, e)
, constructors.BoundedVariableBinding(y, e)
, constructors.BoundedVariableBinding(z, e)],
constructors.Implies(
predicates = [ Equal(x, y, e)
, Equal(y, z, e) ],
consequent = Equal(x, z, e)))
A = common_vars.A()
claim = constructors.Forall([constructors.OrdinaryVariableBinding(A)],
constructors.Iff(
right = constructors.And(
[ reflexive(A)
, symmetric(A)
def ExpandTimes(a, b, c):
base = And([Equal(natural.zero, a), Equal(natural.zero, c)])
z = common_vars.z()
step = ExistsNatural([z], And([Equal(S(z), a), Equal(Plus(b, Times(z, b)), c)]))
return Or([base, step])