MAYBE
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ sum(leaf(?x)) -> ?x,
     sum(node(?x,?yt,?zt)) -> +(?x,+(sum(?yt),sum(?zt))),
     +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     node(?x,?yt,?zt) -> node(?x,?zt,?yt) ]
constructor detection timeout; switch to an approximation.
Constructors: {0,s,leaf,node}
Defined function symbols: {+,sum}
Constructor subsystem:
   [ ]
Rule part & Conj Part:
   [ sum(leaf(?x)) -> ?x,
     sum(node(?x,?yt,?zt)) -> +(?x,+(sum(?yt),sum(?zt))),
     +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
   [ node(?x,?yt,?zt) -> node(?x,?zt,?yt) ]
Trying with:
R:
   [ sum(leaf(?x)) -> ?x,
     sum(node(?x,?yt,?zt)) -> +(?x,+(sum(?yt),sum(?zt))),
     +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
E:
   [ node(?x,?yt,?zt) = node(?x,?zt,?yt) ]
*** Ground Confluence Check by Rewriting Induction ***
Sort: {Nat,Tree}
Signature:
   [ + : Nat,Nat -> Nat,
     s : Nat -> Nat,
     0 : Nat,
     node : Nat,Tree,Tree -> Tree,
     leaf : Nat -> Tree,
     sum : Tree -> Nat ]
Rule Part:
   [ sum(leaf(?x)) -> ?x,
     sum(node(?x,?yt,?zt)) -> +(?x,+(sum(?yt),sum(?zt))),
     +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Conjecture Part:
   [ node(?x,?yt,?zt) = node(?x,?zt,?yt) ]
Precedence (by weight): {(+,5),(0,1),(s,4),(sum,6),(leaf,0),(node,2)}
Rule part is confluent.
R0 is ground confluent.
Check conj part consists of inductive theorems of R0.
Rules:
   [ sum(leaf(?x)) -> ?x,
     sum(node(?x,?yt,?zt)) -> +(?x,+(sum(?yt),sum(?zt))),
     +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Conjectures:
   [ node(?x,?yt,?zt) = node(?x,?zt,?yt) ]
STEP 0
ES:
   [ node(?x,?yt,?zt) = node(?x,?zt,?yt) ]
HS:
   [ ]
ES0:
   [ node(?x,?yt,?zt) = node(?x,?zt,?yt) ]
HS0:
   [ ]
ES1:
   [ node(?x,?yt,?zt) = node(?x,?zt,?yt) ]
HS1:
   [ ]
No equation to expand
Failed to prove conj part consists of inductive theorems of R0.
new/tree5.trs: Failure(unknown)
(1869 msec.)