Earlier quoted context omitted.
In a language like Haskell you wouldn't want to prove the absence of recursion, but that all recursions in the program fit into a handful of patterns. (Eg 'structural-recursion' or 'tail-recursion'.) Some type systems are strong enough to put that kind of analysis / constraints directly into the language. (Haskell might already be strong enough with GADTs and other language extensions enabled.) In any case, the Adden…
Tee hee, Haskell doesn't have tail recursion (e.g. foldl takes linear space), and structural recursion in Haskell isn't guaranteed to terminate (e.g. if you're given an infinite list). If I were in charge of developing a safety critical system, and someone came to me with a proposal to write it in Haskell, I'd be very skeptical.
Structural recursion can't be guaranteed to terminate in any language that supports codata unless you have some sort of totality checker (e.g. via a monotonically structurally decreasing requirement imposed at the type or value level). I don't think any mainstream language supports this out of the box. Liquid Haskell does offer this, though.
I agree that standard Haskell is inappropriate for safety critical software, but only because it allows dynamic allocation. Any program using dynamic allocation is probably unsuitable for safety critical software. Now, a terminating and fixed-memory subset of Haskell a la Clash would be interesting for safety critical software...