On Extensibility of Proof Checkers

My suggestion is little different from LCF, just replacing one computational meta language (ML) with another (ECC, FS0,...). The philosophical point is that it is then possible to accept non canonical proof notations as object level proofs, removing the need to actually normalize them. There are problems to be worked out in practice, such as extraction of programs from constructive proof, and efficient execution of pure, total programs. Although this approach doesn't address the difficulty of proving correctness of tactics in the meta level, it is immediatly useful for tactics with structural justification (e.g. weakening) which are not even representable in LCF, and are infeasible in the Nuprl variant of LCF. Since it can be used for any object system without adding new principles such as reflection, and is compatible with other approaches to extensibility (especially partial reflection), it should be considered as part of the answer to extensibility in proof checkers.