@ARTICLE{Blass95seventrees, author = {Andreas Blass}, title = {Seven Trees in One}, journal = {J. Pure Appl. Algebra}, year = {1995}, volume = {103}, pages = {1--21} }

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Abstract

. Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T 7 of seventuples of such trees. "Particularly elementary" means that the application of the bijection to a seven-tuple of trees involves case distinctions only down to a fixed depth (namely four) in the given seven-tuple. We clarify how this and similar bijections are related to the free commutative semiring on one generator X subject to X = 1 + X 2 . Finally, our main theorem is that the existence of particularly elementary bijections can be deduced from the provable existence, in intuitionistic type theory, of any bijections at all. Introduction This paper was motivated by a remark of Lawvere [8], which implies that there is a particularly elementary coding of seven-tuples of binary trees as single binary trees. In Section 1, we explicitly exhibit such a coding and discuss the sense in which it is particularly elementary. In S...