Abstract

Martin-Löf's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and relations formed by inverse images, addition, multiplication, and exponentiation of other relations. The constructions are given in full detail to allow their use in theorem provers for Type Theory, such as Nuprl. The theory is compared with work in the field of ordinal recursion over higher types.

Citations

713	Theory of Recursive Functions and Effective Computability - Rogers - 1967
505	A Computational Logic - Boyer, Moore - 1979
235	Constructive mathematics and computer programming - Martin-Lof - 1982
172	Proving termination with multiset orderings - Dershowitz, Manna - 1979
123	Intuitionistic type theory. Bibliopolis - Martin-Löf - 1984
121	Edinburgh LCF: A Mechanised Logic of Computation - Gordon, Milner, et al. - 1979
33	Waldinger: Deductive Synthesis of the Unification Algorithm - Manna, R - 1981
14	Proving Termination of Normalization Functions for Conditional Expressions - Paulson - 1986
13	A Programming System for Type Theory - Petersson - 1982
12	Constructive mathematics as a programming logic I: Some principles of theory - Constable - 1985
11	New version of the consistency proof for elementary number theory - Gentzen - 1938
11	An interpretation of Martin-Löf’s Type Theory in a type-free theory of propositions - Smith - 1984
10	The constructive second number class - Church - 1938
10	Natural Deduction Proof as Higher-Order Resolution - Paulson - 1985
7	Programming in constructive set theory: Some examples - Nordstrom - 1981
6	Lessons learned from LCF: a survey of natural deduction proofs - Paulson - 1985
6	The identification of propositions and types in Martin-Lof's type theory: A programming example - Smith - 1983
5	Derivation of a Parsing Algorithm in Martin-Lof's theory of types - Chisholm - 1987
5	Program verification in a logical theory of constructions - Dybjer - 1985
4	The Logical Basis for Computer Programming: Volume II: Deductive Systems - Manna, Waldinger - 1990
4	Propositions and specifications of programs - Nordström, Smith - 1984
4	The subset type former and the type of small types in Martin-Löf’s theory of types - Petersson - 1984
3	Khamiss: The while-rule in Martin-Löf's theory of types - Backhouse - 1984
3	On definition trees of ordinal recursive functionals: reduction of the recursion orders by means of type level raising - Terlouw - 1982
2	Functionals defined by transfinite recursion - Tait - 1965
1	Constable and the PRL staff - L - 1986