Program extraction from normalization proofs

Abstract

This paper describes formalizations of Tait's normalization proof for the simply typed λ-calculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machine-extracted that implement variants of the well-known normalization-by-evaluation algorithm. The case study is used to test and compare the program extraction machineries of the three proof assistants in a non-trivial setting.

Adding Type Constructor Parameterization to Java.

“…We show that types that are well-formed with respect to the conformance of arities (we call such types well-kinded types) can never be infinitely reduced (they are strongly normalizable). The proof of this property is similar to the proof of strong normalization for terms of the simply typed lambda calculus [Tai67,BBLS05]. This similarity is not surprising since well-kinded FGJ ω types are almost isomorphic to well-typed lambda-terms.…”

Program Extraction from Nested Definitions

“…These quantifiers, which were first introduced by Berger [2], can be viewed as a refinement of the Set/Prop distinction in constructive type systems like Coq. Intuitively, a proof of ∀ nc x A(x) (A(x) nonHarrop) represents a procedure that assigns to any x a proof M (x) of A(x) where M (x) does not make "computational use" of x, i.e., the extracted program et(M (x)) does not depend on x. Dually, a proof of ∃ nc x A(x) is a proof of M (x) for some x where the witness x is "hidden", that is, not available for computational use; in fact, ∃ nc can be seen as inductively defined by the clause ∀ nc x (A → ∃ nc x A).…”

Minlog - A Tool for Program Extraction Supporting Algebras and Coalgebras

“…Program extraction from proofs is also implemented in Isabelle [Isa] (for algebras) and in Coq [Coq] (cf [BBLS06] for a joint case study). The implementation in Isabelle has been modelled after Minlog's extraction.…”