YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     inc(?x) -> s(?x),
     +(?x,?y) -> +(?y,?x),
     inc(+(?x,?y)) -> +(inc(?x),?y) ]
Apply Direct Methods...
Inner CPs:
   [ inc(?x) = +(inc(?x),0),
     inc(s(+(?x_1,?y_1))) = +(inc(?x_1),s(?y_1)),
     inc(?y_2) = +(inc(0),?y_2),
     inc(s(+(?x_3,?y_3))) = +(inc(s(?x_3)),?y_3),
     inc(+(?y_5,?x_5)) = +(inc(?x_5),?y_5) ]
Outer CPs:
   [ 0 = 0,
     s(?x_3) = s(+(?x_3,0)),
     ?x = +(0,?x),
     s(+(0,?y_1)) = s(?y_1),
     s(+(s(?x_3),?y_1)) = s(+(?x_3,s(?y_1))),
     s(+(?x_1,?y_1)) = +(s(?y_1),?x_1),
     ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)),
     s(+(?x_6,?y_6)) = +(inc(?x_6),?y_6) ]
not Overlay, check Termination...
unknown/not 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:
   [ 0 = 0,
     s(+(?x_3,0)) = s(?x_3),
     +(0,?x) = ?x,
     +(inc(?x),0) = inc(?x),
     s(?y) = s(+(0,?y)),
     s(+(?x_3,s(?y))) = s(+(s(?x_3),?y)),
     +(s(?y),?x) = s(+(?x,?y)),
     +(inc(?x),s(?y)) = inc(s(+(?x,?y))),
     s(+(0,?y_2)) = s(?y_2),
     +(?y,0) = ?y,
     +(inc(0),?y) = inc(?y),
     s(?x) = s(+(?x,0)),
     s(+(s(?x),?y_2)) = s(+(?x,s(?y_2))),
     +(?y,s(?x)) = s(+(?x,?y)),
     +(inc(s(?x)),?y) = inc(s(+(?x,?y))),
     +(inc(?x_6),?y_6) = s(+(?x_6,?y_6)),
     ?x = +(0,?x),
     s(+(?x,?y_2)) = +(s(?y_2),?x),
     ?y = +(?y,0),
     s(+(?x_4,?y)) = +(?y,s(?x_4)),
     +(inc(?x),?y) = inc(+(?y,?x)),
     s(?x) = +(inc(?x),0),
     s(s(+(?x,?y_3))) = +(inc(?x),s(?y_3)),
     s(?y) = +(inc(0),?y),
     s(s(+(?x_5,?y))) = +(inc(s(?x_5)),?y),
     s(+(?y,?x)) = +(inc(?x),?y),
     s(+(?x,?y)) = +(inc(?x),?y),
     inc(?x) = +(inc(?x),0),
     inc(s(+(?x,?y_3))) = +(inc(?x),s(?y_3)),
     inc(?y) = +(inc(0),?y),
     inc(s(+(?x_5,?y))) = +(inc(s(?x_5)),?y),
     inc(+(?y,?x)) = +(inc(?x),?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 <inc(?x), +(inc(?x),0)> by Rules <0, 6> preceded by [(inc,1)]
 joinable by a reduction of rules <[], [([],0)]>
Critical Pair <inc(s(+(?x_1,?y_1))), +(inc(?x_1),s(?y_1))> by Rules <1, 6> preceded by [(inc,1)]
 joinable by a reduction of rules <[([],4)], [([(+,1)],4),([],3),([(s,1)],1)]>
 joinable by a reduction of rules <[([],4)], [([(+,1)],4),([],1),([(s,1)],3)]>
 joinable by a reduction of rules <[([],4)], [([],1),([(s,1),(+,1)],4),([(s,1)],3)]>
 joinable by a reduction of rules <[([(inc,1),(s,1)],5),([],4),([(s,1),(s,1)],5)], [([(+,1)],4),([],3),([(s,1)],1)]>
 joinable by a reduction of rules <[([(inc,1),(s,1)],5),([],4),([(s,1),(s,1)],5)], [([(+,1)],4),([],1),([(s,1)],3)]>
 joinable by a reduction of rules <[([(inc,1),(s,1)],5),([],4),([(s,1),(s,1)],5)], [([],1),([(s,1),(+,1)],4),([(s,1)],3)]>
Critical Pair <inc(?y_2), +(inc(0),?y_2)> by Rules <2, 6> preceded by [(inc,1)]
 joinable by a reduction of rules <[([],4)], [([(+,1)],4),([],3),([(s,1)],2)]>
Critical Pair <inc(s(+(?x_3,?y_3))), +(inc(s(?x_3)),?y_3)> by Rules <3, 6> preceded by [(inc,1)]
 joinable by a reduction of rules <[([],4)], [([(+,1)],4),([],3),([(s,1)],3)]>
 joinable by a reduction of rules <[([(inc,1),(s,1)],5),([],4),([(s,1),(s,1)],5)], [([(+,1)],4),([],3),([(s,1)],3)]>
Critical Pair <inc(+(?y_5,?x_5)), +(inc(?x_5),?y_5)> by Rules <5, 6> preceded by [(inc,1)]
 joinable by a reduction of rules <[([(inc,1)],5),([],6)], []>
 joinable by a reduction of rules <[([(inc,1)],5),([],4)], [([(+,1)],4),([],3)]>
 joinable by a reduction of rules <[([],4),([(s,1)],5)], [([(+,1)],4),([],3)]>
Critical Pair <0, 0> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[], []>
Critical Pair <s(+(?x_3,0)), s(?x_3)> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([(s,1)],0)], []>
Critical Pair <+(0,?x_5), ?x_5> by Rules <5, 0> preceded by []
 joinable by a reduction of rules <[([],2)], []>
Critical Pair <s(?y_1), s(+(0,?y_1))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[], [([(s,1)],2)]>
Critical Pair <s(+(?x_3,s(?y_1))), s(+(s(?x_3),?y_1))> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[([(s,1)],1)], [([(s,1)],3)]>
Critical Pair <+(s(?y_1),?x_5), s(+(?x_5,?y_1))> by Rules <5, 1> preceded by []
 joinable by a reduction of rules <[([],3)], [([(s,1)],5)]>
Critical Pair <+(?y_5,0), ?y_5> by Rules <5, 2> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(?y_5,s(?x_3)), s(+(?x_3,?y_5))> by Rules <5, 3> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],5)]>
Critical Pair <+(inc(?x_6),?y_6), s(+(?x_6,?y_6))> by Rules <6, 4> preceded by []
 joinable by a reduction of rules <[([(+,1)],4),([],3)], []>
Satisfiable by 6>2,1>3>4,7>5; +(1,1)s(0)inc(2); 2>6>7>1>3>5>4
Diagram Decreasing
Direct Methods: CR

Final result: CR
159.trs: Success(CR)
(8 msec.)