Fixpoint case_L_R' {n m} : forall (P : t (n + m) -> Type) (p : t (n + m)),
  (forall q, P (L m q)) -> (forall q, P (R n q)) -> P p :=
match n with
| 0 => fun P p HL HR => HR p
| S n' => fun P p HL HR => Fin.caseS' p P
    (HL Fin.F1)
    (fun p' => case_L_R' (fun q => P (Fin.FS q)) p' (fun q => HL (Fin.FS q)) HR)
end.Definition case_L_R (P : forall n m, t (n + m) -> Type) {n m} (p : t (n + m)) :
  (forall n m (q : t n), P n m (L m q)) -> (forall n m (q : t m), P n m (R n q)) -> P n m p :=
fun HL HR => case_L_R' _ p (HL _ _) (HR _ _).

Lemma case_L_R_L (P : forall n m, t (n + m) -> Type) {n m} (p : Fin.t n) HL HR : case_L_R P (Fin.L m p) HL HR = HL _ _ p.