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This problem has been solved – see our TYPES 2018 abstract (attached).
Basically the idea is to define a relation between “pre-terms” and the semantics and then show that this is contractible for “well-typed objects”, this way you avoid the mutual dependency. This was an idea by Andras Kovacs. In this year’s TYPES there are two abstracts that show how this can be used to give a universal reduction from Inductive-Inductive types to indexed inductive types and hence W-types.
I have discussed this with Jesper when he was in Nottingham and I think our tentative conclusion was that this could be combined with his approach to provide a reduction for IITs. However, this needs to be checked.
Thorsten
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