YES
LCTRS
  Theories
    Core, Ints
  Signature
    cmod: (Int, Int) -> Int
    if_mod: (Int, Int) -> Int
    minus: (Int, Int) -> Int
  Rules
    cmod(?48, ?49) -> if_mod(?48, ?49) [and(>(?48, 0), >(?49, 0))]
    cmod(?50, ?51) -> 0 [or(?50 = 0, and(?51 = 0, >(?50, 0)))]
    if_mod(?52, ?53) -> ?52 [and(not(<=(-(?53, 1), -(?52, 1))), and(>(?52, 1), >(?53, 1)))]
    if_mod(?54, ?55) -> cmod(minus(-(?54, 1), -(?55, 1)), ?55) [and(<=(-(?55, 1), -(?54, 1)), and(>(?54, 1), >(?55, 1)))]
    minus(?56, ?57) -> minus(0, -(?57, ?56)) [and(>=(?56, 0), >(?57, ?56))]
    minus(?58, ?59) -> -(?58, ?59) [and(>=(?59, 0), >=(?58, ?59))]
Confluent by Orthogonality with proof:
  no critical pairs

Elapsed Time:  48.33 ms