NO
Rewrite Rules:
   [ node(?x,?y) -> node(?y,?x),
     max(?x,0) -> ?x,
     max(0,?y) -> ?y,
     max(s(?x),s(?y)) -> s(max(?x,?y)),
     height(leaf) -> 0,
     height(node(?x,?y)) -> s(max(height(?x),height(?y))) ]
Apply Direct Methods...
Inner CPs:
   [ height(node(?y,?x)) = s(max(height(?x),height(?y))) ]
Outer CPs:
   [ 0 = 0 ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
inner CP cond (upside-parallel)
innter CP Cond (outside)
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ s(max(height(?x),height(?y))) = height(node(?y,?x)),
     0 = 0,
     height(node(?y,?x)) = s(max(height(?x),height(?y))) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <height(node(?y,?x)), s(max(height(?x),height(?y)))> by Rules <0, 5> preceded by [(height,1)]
 joinable by a reduction of rules <[([(height,1)],0),([],5)], []>
Critical Pair <0, 0> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 9 rules by 3 steps unfolding
obtain 7 candidates for checking non-joinability
check by TCAP-Approximation   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
   [ node(?x:Tree,?y:Tree) -> node(?y:Tree,?x:Tree),
     max(?x:Nat,0) -> ?x:Nat,
     max(0,?y:Nat) -> ?y:Nat,
     max(s(?x:Nat),s(?y:Nat)) -> s(max(?x:Nat,?y:Nat)),
     height(leaf) -> 0,
     height(node(?x:Tree,?y:Tree)) -> s(max(height(?x:Tree),height(?y:Tree))) ]
 (success)
Witness for Non-Confluence: <s(max(height(c_1),height(c_2))) <- height(node(c_1,c_2)) -> s(max(height(c_2),height(c_1)))>
Direct Methods: not CR

Final result: not CR
new/tree3.trs: Success(not CR)
(33 msec.)