Impredicativity + large eliminations (with no excluded middle) in Coq It is known that impredicativity + large eliminations + excluded middle is inconsistent. Prop is impredicative and consistent with excluded middle, but does not allow large eliminations (except when constructing witnesses whose type lives in Prop). Set and Type(i) allow large eliminations and are consistent with excluded middle, but are predicative. If we use the -impredicative-set flag, then Set is impredicative, but does not allow large eliminations for large inductive types.
Is there a universe in Coq which is impredicative, allows large eliminations on any inductive type that lives in it, but is consistent in the absence of the axiom of excluded middle?
No: strong elimination of large inductive types (SELIT) is itself inconsistent because it breaks the layering of universes by trivially allowing you to smuggle a large value in a Prop box and take it back out unscathed.
In Is Impredicativity Implicitly Implicit? I proposed a restriction on SELIT which is a bit more permissive than Coq's while still enjoying consistency (I still don't have a proof for it, tho I believe I'm almost there ;-)