Editing Lambda cube (section)

===λω===

In λω, the following operator<math display="block">AND := \lambda \alpha : * . \lambda \beta : * . \Pi \gamma : * . (\alpha \to \beta \to \gamma) \to \gamma</math>is definable, that is <math>\vdash AND : * \to * \to *</math>. The derivation<math display="block">\alpha : *, \beta : * \vdash \Pi \gamma : * . (\alpha \to \beta \to \gamma) \to \gamma : *</math>can be obtained already in λ2, however the polymorphic <math display="inline">AND</math> can only be defined if the rule <math display="inline">(\square, *)</math> is also present.

From a computing point of view, λω is extremely strong, and has been considered as a basis for programming languages.<ref>{{Cite book|title=Programming in higher-order typed lambda-calculi |last1=Pierce |first1=Benjamin |first2=Scott |last2=Dietzen |first3=Spiro |last3=Michaylov |date=1989 |publisher=Computer Science Department, Carnegie Mellon University |id=CMU-CS-89-111 ERGO-89-075 |oclc=20442222}}</ref>