MAYBE
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ size(leaf) -> s(0),
     size(node(?x,?y)) -> +(size(?x),size(?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     node(?x,?y) -> node(?y,?x) ]
Constructors: {0,s,leaf,node}
Defined function symbols: {+,size}
Constructor subsystem:
   [ ]
Rule part & Conj Part:
   [ size(leaf) -> s(0),
     size(node(?x,?y)) -> +(size(?x),size(?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ node(?x,?y) -> node(?y,?x) ]
Trying with:
R:
   [ size(leaf) -> s(0),
     size(node(?x,?y)) -> +(size(?x),size(?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
E:
   [ node(?x,?y) = node(?y,?x) ]
*** Ground Confluence Check by Rewriting Induction ***
Sort: {Nat,Tree}
Signature:
   [ + : Nat,Nat -> Nat,
     s : Nat -> Nat,
     0 : Nat,
     node : Tree,Tree -> Tree,
     leaf : Tree,
     size : Tree -> Nat ]
Rule Part:
   [ size(leaf) -> s(0),
     size(node(?x,?y)) -> +(size(?x),size(?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
Conjecture Part:
   [ node(?x,?y) = node(?y,?x) ]
Precedence (by weight): {(+,6),(0,1),(s,0),(leaf,3),(node,7),(size,4)}
Rule part is confluent.
R0 is ground confluent.
Check conj part consists of inductive theorems of R0.
Rules:
   [ size(leaf) -> s(0),
     size(node(?x,?y)) -> +(size(?x),size(?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
Conjectures:
   [ node(?x,?y) = node(?y,?x) ]
STEP 0
ES:
   [ node(?x,?y) = node(?y,?x) ]
HS:
   [ ]
ES0:
   [ node(?x,?y) = node(?y,?x) ]
HS0:
   [ ]
ES1:
   [ node(?x,?y) = node(?y,?x) ]
HS1:
   [ ]
No equation to expand
Failed to prove conj part consists of inductive theorems of R0.
new/tree1.trs: Failure(unknown)
(11 msec.)