YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ ?x = +(0,?x),
     s(+(?x_1,?y_1)) = +(s(?y_1),?x_1) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Left-Linear, not Right-Linear
unknown Development 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:
   [ +(0,?x) = ?x,
     +(s(?y),?x) = s(+(?x,?y)),
     ?x = +(0,?x),
     s(+(?x,?y_2)) = +(s(?y_2),?x) ]
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 <+(0,?x_4), ?x_4> by Rules <4, 0> preceded by []
 joinable by a reduction of rules <[([],4),([],0)], []>
Critical Pair <+(s(?y_1),?x_4), s(+(?x_4,?y_1))> by Rules <4, 1> preceded by []
 joinable by a reduction of rules <[([],4),([],1)], []>
unknown Diagram Decreasing
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     *(?x_2,0) -> 0,
     *(?x_3,s(?y_3)) -> +(*(?x_3,?y_3),?x_3),
     +(?x_4,?y_4) -> +(?y_4,?x_4) ]
Sort Assignment:
 * : 18*18=>18
 + : 18*18=>18
 0 : =>18
 s : 18=>18
non-linear variables: {?x_3}
non-linear types: {18}
types leq non-linear types: {18}
rules applicable to terms of non-linear types:
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     *(?x_2,0) -> 0,
     *(?x_3,s(?y_3)) -> +(*(?x_3,?y_3),?x_3),
     +(?x_4,?y_4) -> +(?y_4,?x_4) ]
NLR:
0: {}
1: {}
2: {}
3: {0,1,2,3,4}
4: {}
unknown innermost-termination for terms of non-linear types
unknown Quasi-Linear & Linearized-Decreasing
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     *(?x_2,0) -> 0,
     *(?x_3,s(?y_3)) -> +(*(?x_3,?y_3),?x_3),
     +(?x_4,?y_4) -> +(?y_4,?x_4) ]
Sort Assignment:
 * : 18*18=>18
 + : 18*18=>18
 0 : =>18
 s : 18=>18
non-linear variables: {?x_3}
non-linear types: {18}
types leq non-linear types: {18}
rules applicable to terms of non-linear types:
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     *(?x_2,0) -> 0,
     *(?x_3,s(?y_3)) -> +(*(?x_3,?y_3),?x_3),
     +(?x_4,?y_4) -> +(?y_4,?x_4) ]
unknown innermost-termination for terms of non-linear types
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 15 rules by 3 steps unfolding
strenghten +(?x_8,0) and ?x_8
strenghten +(0,?x_4) and ?x_4
strenghten +(?x_13,?y_13) and +(?y_13,?x_13)
strenghten s(+(0,?y_1)) and s(?y_1)
strenghten +(*(?x_10,0),?x_10) and ?x_10
strenghten +(*(0,0),0) and 0
strenghten s(+(?x_4,?y_1)) and +(s(?y_1),?x_4)
strenghten s(+(?x_12,?y_1)) and +(?x_12,s(?y_1))
obtain 16 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:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  *:= (3)*x1+(2)*x1*x1+(1)*x1*x2+(1)*x2*x2
  +:= (2)+(1)*x1+(1)*x2
  0:= (11)
  s:= (8)+(1)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  *:= (3)*x1+(1)*x1*x2+(1)*x2+(1)*x2*x2
  +:= (1)*x1+(1)*x2
  0:= (13)
  s:= (8)+(1)*x1
retract +(?x,0) -> ?x
retract *(?x,s(?y)) -> +(*(?x,?y),?x)
Polynomial Interpretation:
  *:= (1)+(1)*x1+(2)*x2
  +:= (1)*x1+(1)*x1*x2+(1)*x2
  0:= 0
  s:= (8)+(5)*x1
retract +(?x,0) -> ?x
retract *(?x,0) -> 0
retract *(?x,s(?y)) -> +(*(?x,?y),?x)
Polynomial Interpretation:
  *:= (1)*x1+(2)*x2
  +:= (1)+(2)*x1+(2)*x2
  0:= 0
  s:= (1)+(1)*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
PCP_in(symP,S):
CP(S,symP):
<?x, +(0,?x)> --> <?x, +(0,?x)> => no
<s(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), +(s(?y),?x)> => no
check joinability condition:
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(s(?y),?x): maybe not reachable
failed
failure(Step 1)
   [ +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?x,?y)) ]
Added P-Rules:
   [ ]
replace: +(?x,s(?y)) -> s(+(?x,?y)) => +(?x,s(?y)) -> s(+(?y,?x))
STEP: 2 (linear)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: not linear
failure(Step 2)
STEP: 3 (relative)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  *:= (3)*x1+(2)*x1*x1+(1)*x1*x2+(1)*x2*x2
  +:= (2)+(1)*x1+(1)*x2
  0:= (11)
  s:= (8)+(1)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  *:= (3)*x1+(1)*x1*x2+(1)*x2+(1)*x2*x2
  +:= (1)*x1+(1)*x2
  0:= (13)
  s:= (8)+(1)*x1
retract +(?x,0) -> ?x
retract *(?x,s(?y)) -> +(*(?x,?y),?x)
Polynomial Interpretation:
  *:= (1)+(1)*x1+(2)*x2
  +:= (1)*x1+(1)*x1*x2+(1)*x2
  0:= 0
  s:= (8)+(5)*x1
retract +(?x,0) -> ?x
retract *(?x,0) -> 0
retract *(?x,s(?y)) -> +(*(?x,?y),?x)
Polynomial Interpretation:
  *:= (1)*x1+(2)*x2
  +:= (1)+(2)*x1+(2)*x2
  0:= 0
  s:= (1)+(1)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x),
     +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(0,?y_5)), s(?y_5)> --> <s(?y_5), s(?y_5)> => yes
<s(?y_1), s(+(0,?y_1))> --> <s(?y_1), s(?y_1)> => yes
<s(+(s(?y_1),?y_5)), s(+(s(?y_5),?y_1))> --> <s(s(+(?y_5,?y_1))), s(s(+(?y_1,?y_5)))> => yes
PCP_in(symP,S):
CP(S,symP):
<?x, +(0,?x)> --> <?x, ?x> => yes
<s(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<?x, +(?x,0)> --> <?x, ?x> => yes
<s(+(?x,?y)), +(?x,s(?y))> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x),
     +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
194.trs: Success(CR)
(613 msec.)