Altenkirch, Thorsten
[Verfasser:in];
Kaposi, Ambrus
[Verfasser:in];
Šinkarovs, Artjoms
[Verfasser:in];
Végh, Tamás
[Verfasser:in]
;
Thorsten Altenkirch and Ambrus Kaposi and Artjoms Šinkarovs and Tamás Végh
[Mitwirkende:r]
Combinatory Logic and Lambda Calculus Are Equal, Algebraically
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Beschreibung:
It is well-known that extensional lambda calculus is equivalent to extensional combinatory logic. In this paper we describe a formalisation of this fact in Cubical Agda. The distinguishing features of our formalisation are the following: (i) Both languages are defined as generalised algebraic theories, the syntaxes are intrinsically typed and quotiented by conversion; we never mention preterms or break the quotients in our construction. (ii) Typing is a parameter, thus the un(i)typed and simply typed variants are special cases of the same proof. (iii) We define syntaxes as quotient inductive-inductive types (QIITs) in Cubical Agda; we prove the equivalence and (via univalence) the equality of these QIITs; we do not rely on any axioms, the conversion functions all compute and can be experimented with.