I'm having trouble showing that $P\neq R$. Obviously $P\subseteq R$, but is there  a decidable language which is definitely not (under all answers to open questions s.t. $P=NP$ or $NP=PSPACE$) in $P$ ?
What is an example for a decidable language not in P?
complexity theory;polynomial time
Yes, there are decidable languages that are definitely not in P. The time hierarchy theorem says that P$\,\neq\,$EXP, so P$\,\neq\,$R, independently of the P vs NP problem. Any EXP-complete problem is definitely not in P: for example determining whether white has a winning strategy from a position in generalized chess (generalized in the sense of allowing a board of any dimensions, with any arrangement of any number of pieces, but otherwise following all the rules of standard chess).