Information and Computation - Semantics of Data Types
Proofs and types
Theoretical Computer Science
Encoding of data types in pure construction calculus: a semantic justification
Papers presented at the second annual Workshop on Logical environments
Modified Realizability Toposes and Strong Normalization Proofs
TLCA '93 Proceedings of the International Conference on Typed Lambda Calculi and Applications
Extending Models of Second Order Predicate Logic to Models of Second Dependent Type Theory
CSL '96 Selected Papers from the10th International Workshop on Computer Science Logic
A Simple Model Construction for the Calculus of Constructions
TYPES '95 Selected papers from the International Workshop on Types for Proofs and Programs
CPS translating inductive and coinductive types
PEPM '02 Proceedings of the 2002 ACM SIGPLAN workshop on Partial evaluation and semantics-based program manipulation
An Introduction to Dependent Type Theory
Applied Semantics, International Summer School, APPSEM 2000, Caminha, Portugal, September 9-15, 2000, Advanced Lectures
Proceedings of the 15th ACM SIGPLAN international conference on Functional programming
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This paper proves the non-derivability of induction in second order dependent type theory (λP2). This is done by providing a model construction for λP2, based on a saturated sets like interpretation of types as sets of terms of a weakly extensional combinatory algebra. We give counter-models in which the induction principle over natural numbers is not valid. The proof does not depend on the specific encoding for natural numbers that has been chosen (like e.g. polymorphic Church numerals), so in fact we prove that there can not be an encoding of natural numbers in λP2 such that the induction principle is satisfied. The method extends immediately to other data types, like booleans, lists, trees, etc. In the process of the proof we establish some general properties of the models, which we think are of independent interest. Moreover, we show that the Axiom of Choice is not derivable in λP2.