## 15-816 Linear Logic |

Inversion principles provide us with one way to remove unnecessary
disjunctive non-determinism in proof search. Inversion principles for
right rules in the sequent calculus lead to the notion of a *weakly
goal-directed derivation*.

In this lecture we extend this further by taking advantage of
inversion principles for left rules and one additional observation which
will lead us to the notion of a *weakly focussed derivation*. A
derivation which is both weakly goal-directed and focussed is called
*weakly uniform*, a class of derivations which is sound and
complete for proof search in linear logic. When combined with
unification and occurrence constraints, much of the non-determinism has
been either eliminated or isolated into a few crucial choices.

We show how this can be turned into flexible basis for a theorem
proving procedure combining interactive and automatic deduction using
*tactics* and *tacticals*. In general, a tactic is a
function which may reduce a goal to some subgoals which are sufficient
for proving the original goal, or fail. The idea of possibly failing
functions gives rise to a rich set of combinators for combining tactics
called *tacticals*. Tactics and tacticals are the basis for some
of the most successful theorem proving environments in use today, such
as Coq, Isabelle, or NuPrl.

- Next: Lecture 12: Proof Terms
- Previous: Lecture 10: Unrestricted Resources
- Schedule

Frank Pfenning fp@cs