A programming language doesn’t have to be Turing-complete though! In fact many highly formal functional and logic programming languages are deliberately *not* Turing-complete to improve their theoretical properties.
I believe Isabelle/HOL is not Turing-complete but don’t quote me on that. For a while many people believed Idris wasn’t, but a proof came out late last year
Tbf I think that is more of a bug in Idris though. It’s a pretty hard thing to get right in a complicated surface language - Agda is often found to have these kinds of bugs in its totality checker.
Complicated in terms of supporting mutual definitions/inductive recursive types, cumulative universes, etc, and trying not to get in your way as much as possible.