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## 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 ${a}^{b}=exp\left(b×loga\right)$—to obtain a type normal form for the type languages $\left\{\to ,×,+\right\}$. 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].