## Section: New Results

### Exp-log normal form of types and the axioms for $\eta $-equality of the $\lambda $-calculus with sums

Participant : Danko Ilik.

In the presence of sum types, the $\lambda $-calculus has but one implemented (and incomplete) heuristic for deciding $\beta \eta $-equality of terms, in spite of a dozen of meta-theoretic works showing that the equality is decidable.

In the work discussed here, we first used the exp-log decomposition of
the arrow type—inspired from the analytic transformation
${a}^{b}=exp(b\times loga)$—to obtain a type normal form for
the type languages $\{\to ,\times ,+\}$. We then made a quotient of
the $\beta \eta $-equality of terms modulo the terms coerced into
their representation at the exp-log normal form of their
type. This allows to obtain a *simplification* of the so far
standard axioms for $\beta \eta $-equality.

Moreover, we provided a Coq implementation of a heuristic decision procedure for this equality. Although a heuristic, this implementation manages to tackle examples of equal terms that need a complex program analysis in the only previously implemented heuristic of Vincent Balat.

This work is described in a paper accepted for presentation at POPL 2017, [27].