Abstract
Publisher Summary This chapter discusses cut-free rules for classical and intuitionistic simple type theory with and without extensionality. The proof uses an extension of the method of computability previously applied to the intuitionistic theory of iterated inductive definitions and the theory of species. Function constants denote functions whose arguments and values are individuals. Free and bound occurrences of a variable in a term or formula are defined as usual. Terms and formulae which only differ in the naming of their bound variables are identified. Free and bound occurrences of a variable in a deduction are defined as in MARTIN-LOF. Deductions that only differ in the naming of their bound variables are identified.
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