A Formulae-as-Types Interpretation of Subtractive Logic

T. Crolard
2004 Journal of Logic and Computation  
We present a formulae-as-types interpretation of Subtractive Logic (i.e. bi-intuitionistic logic). This presentation is two-fold: we first define a very natural restriction of the λµ-calculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for first-class coroutines (a
more » ... icted form of first-class continuations). Lemma 6.11. Given two λµ →+×− -terms t, u safe w.r.t. first-class coroutines and such that S [] (u) ⊆ S [] (t) and S δ (u) ⊆ S δ (t) and an arbitrary context C[ ], if C[t ] is safe then C[u] is safe w.r.t. first-class coroutines. Proof. By induction on the context C[ ]. Lemma 6.12. Given an instance r s of a rule of the λµ →+×− -calculus, if r is safe w.r.t. first-class coroutines then s is safe w.r.t. first-class coroutines.
doi:10.1093/logcom/14.4.529 fatcat:tjyiakbzlvblxnggrksoeciszi