### Abstract

Alpha-beta pruning is an efficient search strategy for two-player game trees.
It was invented in the late 1950s and is at the heart of most implementations
of combinatorial game playing programs. These theories formalize and verify a number of
variations of alpha-beta pruning, in particular fail-hard and fail-soft,
and valuations into linear orders, distributive lattices and domains with negative values.
A detailed presentation of these theories can be found in the chapter

*Alpha-Beta Pruning*in the (forthcoming) book Functional Data Structures and Algorithms --- A Proof Assistant Approach.