The general tool environment is identical to the single-state tool environment, except that actions can change the states of tools in addition to the states of materials. We can solve the general tool case using a solution to the single-state tool case by resetting tools whenever they are dirtied.
The proof is simple, but requires that we formalize the notion of
being a tool. Let E be an environment with a state space of the
form 
. Let a be an action of
E and 
be a component of its state space. We will say
if a
never changes and it has the same result regardless of the value
of . if it
is independent of all other components. is a tool if it has some privileged value
such that:
of E, we can reach
the state 
using only actions focused
on . , focused on , or else only defined on
states whose component is 
.
Proof:
Let 
be a solution (policy) for D. By the definition of a tool,
there must be a policy 
that will bring D' from any state (s,
t) to 
without changing the S component. Let 
be the projection from D' to D given by
For each 
, we define the -implementation of a, 
by
and so D' is reducible to D. The general case of multiple tools
follows from induction.
