  We investigate the proof complexity of systems based on positive branching
programs, i.e. non-deterministic branching programs (NBPs) where, for any
0-transition between two nodes, there is also a 1-transition. Positive NBPs
compute monotone Boolean functions, just like negation-free circuits or
formulas, but constitute a positive version of (non-uniform) NL, rather than P
or NC1, respectively.
  The proof complexity of NBPs was investigated in previous work by Buss, Das
and Knop, using extension variables to represent the dag-structure, over a
language of (non-deterministic) decision trees, yielding the system eLNDT. Our
system eLNDT+ is obtained by restricting their systems to a positive syntax,
similarly to how the 'monotone sequent calculus' MLK is obtained from the usual
sequent calculus LK by restricting to negation-free formulas.
  Our main result is that eLNDT+ polynomially simulates eLNDT over positive
sequents. Our proof method is inspired by a similar result for MLK by Atserias,
Galesi and Pudl\'ak, that was recently improved to a bona fide polynomial
simulation via works of Je\v{r}\'abek and Buss, Kabanets, Kolokolova and
Kouck\'y. Along the way we formalise several properties of counting functions
within eLNDT+ by polynomial-size proofs and, as a case study, give explicit
polynomial-size poofs of the propositional pigeonhole principle.
