Strong Normalization for HA + EM1 by Non-Deterministic Choice

Abstract

We study the strong normalization of a new Curry-Howard correspondence for HA + EM1, constructive Heyting Arithmetic with the excluded middle on Sigma01-formulas. The proof-term language of HA + EM1 consists in the lambda calculus plus an operator ||_a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We give a strong normalization proof for the system based on a technique of "non-deterministic immersion"

On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

“…To this end, we make a detour into a logically inconsistent, yet computationally sound world: the system LC ⋆ , a type system which extends LC . The idea that extending a system can make easier rather than harder to prove its normalization might not seem very intuitive, but it is well tested and very successful (see [32], [5], [3], [7]). LC ⋆ will be our calculus of realizers.…”

“…Termst, u, v ::= x | tu | tm | λx u | λα u | t, u | uπ 0 | uπ1 | ι 0 (u) | ι 1 (u) | t[x.u, y.v] | (m, t) | t[(α, x).u] | (u a v) | H ∀αP a | W ∃αP a | True | Ruvm | rt 1 . .…”