We establish a course-of-values induction principle for K-finite sets in intuitionistic type theory. Using this principle, we prove a pigeonhole principle conjectured by Bénabou and Loiseau. We also comment on some variants of this pigeonhole principle.

The pigeonhole principle states that if n items are contained in m boxes, then at least one box has no more than n/m items. It is utilized to solve many data management problems, especially for thresholded similarity searches. Despite many pigeonhole principle-based solutions proposed in the last few decades, the condition stated by the principle is weak. It only constrains the number of items ...

This paper considers the relational versions of the surjective and multifunction weak pigeonhole principles for PV , Σ1 and Θ b 2-formulas. We show that the relational surjective pigeonhole principle for Θ2 formulas in S 1 2 implies a circuit block-recognition principle which in turn implies the surjective weak pigeonhole principle for Σ1 formulas. We introduce a class of predicates correspondi...

This paper considers the relational versions of the surjective, partial surjective, and multifunction weak pigeonhole principles for PV , Σ1, Π b 1, and B(Σ b 1)-formulas as well as relativizations of these formulas to higher levels of the bounded arithmetic hierarchy. We show that the partial surjective weak pigeonhole principle for Π1 formulas implies that for each k there is a string of leng...

Theorem 1.1. If n + 1 objects are put into n boxes, then at least one box contains two or more objects. Proof. Trivial. Example 1.1. Among 13 people there are two who have their birthdays in the same month. Example 1.2. There are n married couples. How many of the 2n people must be selected in order to guarantee that one has selected a married couple? Other principles related to the pigeonhole ...

The combinatorial matching principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a xed bi-partition of the vertices, there is no perfect matching between them. Therefore , it follows from recent lower bounds for the pigeonhole principle that the matching principle requires exponential-size boun...

The combinatorial parity principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a xed bi-partition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the parity principle requires exponential-size bounded-d...