Efficient algorithms for isomorphisms of simple types
POPL '03 Proceedings of the 30th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Memoization in Type-Directed Partial Evaluation
GPCE '02 Proceedings of the 1st ACM SIGPLAN/SIGSOFT conference on Generative Programming and Component Engineering
Extensional normalisation and type-directed partial evaluation for typed lambda calculus with sums
Proceedings of the 31st ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Classical isomorphisms of types
Mathematical Structures in Computer Science
Efficient algorithms for isomorphisms of simple types
Mathematical Structures in Computer Science
Isomorphisms of simple inductive types through extensional rewriting
Mathematical Structures in Computer Science
On the building of affine retractions
Mathematical Structures in Computer Science
A Characterisation of Lambda Definability with Sums Via T T-Closure Operators
CSL '08 Proceedings of the 22nd international workshop on Computer Science Logic
Mathematical models of computational and combinatorial structures
FOSSACS'05 Proceedings of the 8th international conference on Foundations of Software Science and Computation Structures
Subtyping recursive types modulo associative commutative products
TLCA'05 Proceedings of the 7th international conference on Typed Lambda Calculi and Applications
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Tarski asked whether the arithmetic identities taught in high school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question for the language of arithmetic expressions using a constant for the number one and the operations of product and exponentiation is affirmative, and the complete equational theory also characterises isomorphism in the typed lambda calculus, where the constant for one and the operations of product and exponentiation respectively correspond to the unit type and the product and arrow type constructors. This paper studies isomorphisms in typed lambda calculiwith empty and sum types from this viewpoint. We close an open problem by establishing that the theory of type isomorphisms in the presence of product, arrow, and sum types (with or without the unit type) is not finitely axiomatisable. Further, we observe that for type theories with arrow, empty and sum types the correspondence between isomorphism and arithmetic equality generally breaks down, but that it still holds in some particular cases including that of type isomorphism with the empty type and equality with zero.