Abstract
AbstractWe investigate (co-) induction in classical logic under the propositions-as-types paradigm, considering propositional, second-order and (co-) inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-) iterator and show that it is strongly normalizing. We prove this using a reducibility argument.
Highlights
As an example of the kind of analysis that can be done by focusing on this separation of concepts, Curien and Herbelin [6] and Wadler [22] devised LK-based calculi that showed, syntacticly, the duality of the two most common evaluations strategies: call-by-name and call-by-value. While originally such classical calculi included only propositional types—i.e. conjunction, disjunction, negation, implication and subtraction—they were later extended with second-order types [15, 20] and with positive inductive types [15]
The typing constraints on Mendler induction correspond—model-wise—to a monotonization step. This turns out to be what we need to guarantee that an inductive type can be modeled by a least fix-point; without this step, the interpretation of a type scheme would be a function on complete lattices that would not necessarily be monotone
28 Classical logic with Mendler induction case, we extend any such γ with those type variables that appear in A but were not contemplated in γ
Summary
As an example of the kind of analysis that can be done by focusing on this separation of concepts, Curien and Herbelin [6] and Wadler [22] devised LK-based calculi that showed, syntacticly, the duality of the two most common evaluations strategies: call-by-name and call-by-value While originally such classical calculi included only propositional types—i.e. conjunction, disjunction, negation, implication and subtraction (the dual connective of implication)—they were later extended with second-order types [15, 20] and with positive (co-) inductive types [15]. This is comparable to the use of existential types as a basis for modeling ad hoc polymorphism in functional languages [18] as opposed to the more elaborate encoding by means of universal types—and stresses the point that duality brings forth gains in expressiveness at little cost for the designer
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