Now suppose the world contains a material and a set of tools, but those tools always clean or otherwise reset themselves after use. Self-resetting tools have only one state, and so they are a trivial kind of environment. We define the ``singleton'' environment as the environment with exactly one state:
All single-state environments are isomorphic to , so we model an environment consisting of a material M=(S,A) and a self-resetting tool as . Its state space is simply and its actions are just the set
Each such action performs some action from M on the M-component of the product's state and leaves the component unchanged. By induction, we have that:
And so single-state-tool worlds are trivially reducible to tool-free worlds.