Proofs and types
Foundations of programming languages
Foundations of programming languages
Phase semantic cut-elimination and normalization proofs of first- and higher-order linear logic
Theoretical Computer Science - Special issue on linear logic, 1
Theoretical Computer Science
Journal of Automated Reasoning
Completeness and Cut-elimination in the Intuitionistic Theory of Types
Journal of Logic and Computation
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We give a fully constructive semantic proof of cut elimination for intuitionistic type theory with axioms. The problem here, as with the original Takeuti conjecture, is that the impredicativity of the formal system involved makes it impossible to define a semantics along conventional lines, in the absence, a priori, of cut, or to prove completeness by induction on subformula structure. In addition, unlike semantic proofs by Tait, Takahashi, and Andrews of variants of the Takeuti conjecture, our arguments are constructive.Our techniques offer also an easier approach than Girard's strong normalization techniques to the problem of extending the cut-elimination result in the presence of axioms. We need only to relativize the Heyting algebras involved in a straightforward way.