YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ max(?x,0) -> ?x,
     max(s(?x),s(?y)) -> s(max(?x,?y)),
     max(?x,?y) -> max(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ ?x = max(0,?x),
     s(max(?x_1,?y_1)) = max(s(?y_1),s(?x_1)) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ max(0,?x) = ?x,
     max(s(?y),s(?x)) = s(max(?x,?y)),
     ?x = max(0,?x),
     s(max(?x_2,?y_2)) = max(s(?y_2),s(?x_2)) ]
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 <max(0,?x_2), ?x_2> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <max(s(?y_1),s(?x_1)), s(max(?x_1,?y_1))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],2)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 13 rules by 3 steps unfolding
strenghten max(?x_5,0) and ?x_5
strenghten max(0,?x_2) and ?x_2
strenghten max(0,0) and 0
strenghten max(?x_8,?y_8) and max(?y_8,?x_8)
strenghten max(?x_11,max(0,0)) and ?x_11
strenghten max(max(0,0),?x_11) and ?x_11
strenghten max(s(?x_1),s(?y_1)) and s(max(?x_1,?y_1))
strenghten max(s(?y_1),s(?x_1)) and s(max(?x_1,?y_1))
obtain 9 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Root-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (failure)
check by Interpretation(mod2) (failure)
check by Descendants-Approximation, check by Ordering(poly) (failure)
unknown Non-Confluence
Check relative termination:
   [ max(?x,0) -> ?x,
     max(s(?x),s(?y)) -> s(max(?x,?y)) ]
   [ max(?x,?y) -> max(?y,?x) ]
Polynomial Interpretation:
  0:= 0
  s:= (2)+(2)*x1
  max:= (8)*x1+(1)*x1*x2+(8)*x2
retract max(s(?x),s(?y)) -> s(max(?x,?y))
Polynomial Interpretation:
  0:= 0
  s:= (8)*x1
  max:= (1)+(8)*x1+(8)*x2
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ max(?x,0) -> ?x,
     max(s(?x),s(?y)) -> s(max(?x,?y)) ]
P:
   [ max(?x,?y) -> max(?y,?x) ]
S: terminating
CP(S,S):
PCP_in(symP,S):
CP(S,symP):
<?x, max(0,?x)> --> <?x, max(0,?x)> => no
<s(max(?x,?y)), max(s(?y),s(?x))> --> <s(max(?x,?y)), s(max(?y,?x))> => yes
check joinability condition:
check modulo reachablity from ?x to max(0,?x): maybe not reachable
failed
failure(Step 1)
   [ max(0,?x) -> ?x ]
Added S-Rules:
   [ max(0,?x) -> ?x ]
Added P-Rules:
   [ ]
STEP: 2 (linear)
S:
   [ max(?x,0) -> ?x,
     max(s(?x),s(?y)) -> s(max(?x,?y)) ]
P:
   [ max(?x,?y) -> max(?y,?x) ]
S: terminating
CP(S,S):
CP_in(symP,S):
CP(S,symP):
<?x, max(0,?x)> --> <?x, max(0,?x)> => no
<s(max(?x,?y)), max(s(?y),s(?x))> --> <s(max(?x,?y)), s(max(?y,?x))> => yes
check joinability condition:
check modulo reachablity from ?x to max(0,?x): maybe not reachable
failed
failure(Step 2)
   [ max(0,?x) -> ?x ]
Added S-Rules:
   [ max(0,?x) -> ?x ]
Added P-Rules:
   [ ]
STEP: 3 (relative)
S:
   [ max(?x,0) -> ?x,
     max(s(?x),s(?y)) -> s(max(?x,?y)) ]
P:
   [ max(?x,?y) -> max(?y,?x) ]
Check relative termination:
   [ max(?x,0) -> ?x,
     max(s(?x),s(?y)) -> s(max(?x,?y)) ]
   [ max(?x,?y) -> max(?y,?x) ]
Polynomial Interpretation:
  0:= 0
  s:= (2)+(2)*x1
  max:= (8)*x1+(1)*x1*x2+(8)*x2
retract max(s(?x),s(?y)) -> s(max(?x,?y))
Polynomial Interpretation:
  0:= 0
  s:= (8)*x1
  max:= (1)+(8)*x1+(8)*x2
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ max(?x,0) -> ?x,
     max(s(?x),s(?y)) -> s(max(?x,?y)),
     max(0,?x) -> ?x ]
P:
   [ max(?x,?y) -> max(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
PCP_in(symP,S):
CP(S,symP):
<?x, max(0,?x)> --> <?x, ?x> => yes
<s(max(?x,?y)), max(s(?y),s(?x))> --> <s(max(?x,?y)), s(max(?y,?x))> => yes
<?x, max(?x,0)> --> <?x, ?x> => yes
S:
   [ max(?x,0) -> ?x,
     max(s(?x),s(?y)) -> s(max(?x,?y)),
     max(0,?x) -> ?x ]
P:
   [ max(?x,?y) -> max(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
165.trs: Success(CR)
(364 msec.)