Bar induction is a reasoning principle used in intuitionistic mathematics, introduced by L.E.J. Brouwer. It is useful in giving constructive versions of classical results. It is based on an inductive argument.
The goal of the principle is to prove properties of infinite streams of natural numbers, called choice sequences in intuitionistic terminology, by inductively reducing them to decidable properties of finite lists.
Given two predicates R and S on finite lists of natural numbers such that the following conditions hold:
then we can conclude that S holds for the empty list.
In classical reverse mathematics, "bar induction" (BI) denotes the related principle stating that if a relation R is a well-order, then we have the schema of transfinite induction over R for arbitrary formulas.