Proof theory of higher-order equations: conservativity, normal forms and term rewriting

We introduce a necessary and sufficient condition for the ω-extensionality rule of higherorder equational logic to be conservative over first-order many-sorted equational logic for ground first-order equations. This gives a precise condition under which computation in the higher-order initial model by term rewriting is possible. The condition is then generalised to characterise a normal form for higher-order equational proofs in which extensionality inferences occur only as the final proof inferences. The main result is based on a notion of observational equivalence between higher-order elements induced by a topology of finite information on such elements. Applied to extensional higher-order algebras with countable first-order carrier sets, the finite information topology is metric and second countable in every type.