Polarised intermediate representation of lambda calculus with sums

The theory of the λ-calculus with extensional sums is more complex than with only pairs and functions. We propose an untyped representation - an intermediate calculus - for the λ-calculus with sums, based on the following principles: 1) Computation is described as the reduction of pairs of an expression and a context, the context must be represented inside-out, 2) Operations are represented abstractly by their transition rule, 3) Positive and negative expressions are respectively eager and lazy, this polarity is an approximation of the type. We offer an introduction from the ground up to our approach, and we review the benefits. A structure of alternating phases naturally emerges through the study of normal forms, offering a reconstruction of focusing. Considering further purity assumption, we obtain maximal multi-focusing. As an application, we can deduce a syntax-directed algorithm to decide the equivalence of normal forms in the simply-typed λ-calculus with sums, and justify it with our intermediate calculus.