Section PredicativeNotationRecursion.
Variable sl : nat. (* sl = |x| *)
Hypothesis sl_not_zero : sl <> 0.
Definition comp2n h ln ls := comp 2 0 h ln ls.
Definition comp2s h ln ls := comp 2 sl h ln ls.
Definition comp2Ss h ln ls := comp 2 (S sl) h ln ls.
Definition comp2SSs h ln ls := comp 2 (S (S sl)) h ln ls.
Definition proj2n i := proj 2 0 i.
Definition proj2s i := proj 2 sl i.
Definition proj2Ss i := proj 2 (S sl) i.
Definition proj2SSs i := proj 2 (S (S sl)) i.
Definition f'_cond := comp2Ss Y_e
[comp2n (succ true) nil [proj2n 0];
proj2n 1]
[proj2Ss 3].
Opaque f'_cond.
Variables g : BC.
Hypothesis g_inf  : arities g = ok_arities 1 (sl - 1).
Definition f'_nil := BC_dummies 1 1 2 (sl - 1) g.

Lemma f'_nil_inf : arities f'_nil = ok_arities 2 (S sl).