YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x,p(?y)) -> p(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(p(?x),?y) -> p(+(?x,?y)),
     s(p(?x)) -> ?x,
     p(s(?x)) -> ?x,
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x_1,?x_6) = s(+(?x_1,p(?x_6))),
     +(?x_2,?x_7) = p(+(?x_2,s(?x_7))),
     +(?x_6,?y_4) = s(+(p(?x_6),?y_4)),
     +(?x_7,?y_5) = p(+(s(?x_7),?y_5)),
     s(?x_7) = s(?x_7),
     p(?x_6) = p(?x_6) ]
Outer CPs:
   [ 0 = 0,
     s(?x_4) = s(+(?x_4,0)),
     p(?x_5) = p(+(?x_5,0)),
     ?x = +(0,?x),
     s(+(0,?y_1)) = s(?y_1),
     s(+(s(?x_4),?y_1)) = s(+(?x_4,s(?y_1))),
     s(+(p(?x_5),?y_1)) = p(+(?x_5,s(?y_1))),
     s(+(?x_1,?y_1)) = +(s(?y_1),?x_1),
     p(+(0,?y_2)) = p(?y_2),
     p(+(s(?x_4),?y_2)) = s(+(?x_4,p(?y_2))),
     p(+(p(?x_5),?y_2)) = p(+(?x_5,p(?y_2))),
     p(+(?x_2,?y_2)) = +(p(?y_2),?x_2),
     ?y_3 = +(?y_3,0),
     s(+(?x_4,?y_4)) = +(?y_4,s(?x_4)),
     p(+(?x_5,?y_5)) = +(?y_5,p(?x_5)) ]
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_4,0)) = s(?x_4),
     p(+(?x_5,0)) = p(?x_5),
     +(0,?x) = ?x,
     ?x_7 = s(+(0,p(?x_7))),
     s(+(?x_4,?x_11)) = s(+(s(?x_4),p(?x_11))),
     p(+(?x_5,?x_12)) = s(+(p(?x_5),p(?x_12))),
     +(?x_7,?x) = s(+(?x,p(?x_7))),
     s(?y) = s(+(0,?y)),
     s(+(?x_4,s(?y))) = s(+(s(?x_4),?y)),
     p(+(?x_5,s(?y))) = s(+(p(?x_5),?y)),
     +(s(?y),?x) = s(+(?x,?y)),
     +(?x,?x_7) = s(+(?x,p(?x_7))),
     ?x_8 = p(+(0,s(?x_8))),
     s(+(?x_4,?x_12)) = p(+(s(?x_4),s(?x_12))),
     p(+(?x_5,?x_13)) = p(+(p(?x_5),s(?x_13))),
     +(?x_8,?x) = p(+(?x,s(?x_8))),
     p(?y) = p(+(0,?y)),
     s(+(?x_4,p(?y))) = p(+(s(?x_4),?y)),
     p(+(?x_5,p(?y))) = p(+(p(?x_5),?y)),
     +(p(?y),?x) = p(+(?x,?y)),
     +(?x,?x_8) = p(+(?x,s(?x_8))),
     s(+(0,?y_2)) = s(?y_2),
     p(+(0,?y_3)) = p(?y_3),
     +(?y,0) = ?y,
     ?x_7 = s(+(p(?x_7),0)),
     s(+(?x_9,?y_2)) = s(+(p(?x_9),s(?y_2))),
     +(?y,?x_7) = s(+(p(?x_7),?y)),
     s(?x) = s(+(?x,0)),
     s(+(s(?x),?y_2)) = s(+(?x,s(?y_2))),
     p(+(s(?x),?y_3)) = s(+(?x,p(?y_3))),
     +(?y,s(?x)) = s(+(?x,?y)),
     +(?x_7,?y) = s(+(p(?x_7),?y)),
     ?x_8 = p(+(s(?x_8),0)),
     p(+(?x_11,?y_3)) = p(+(s(?x_11),p(?y_3))),
     +(?y,?x_8) = p(+(s(?x_8),?y)),
     p(?x) = p(+(?x,0)),
     s(+(p(?x),?y_2)) = p(+(?x,s(?y_2))),
     p(+(p(?x),?y_3)) = p(+(?x,p(?y_3))),
     +(?y,p(?x)) = p(+(?x,?y)),
     +(?x_8,?y) = p(+(s(?x_8),?y)),
     s(?x_8) = s(?x_8),
     s(+(?x_3,?x_11)) = +(?x_3,s(?x_11)),
     s(+(?x_14,?y_6)) = +(s(?x_14),?y_6),
     ?x_8 = p(s(?x_8)),
     s(+(?x_3,p(?x))) = +(?x_3,?x),
     s(+(p(?x),?y_6)) = +(?x,?y_6),
     p(?x) = p(?x),
     p(+(?x_4,?x_12)) = +(?x_4,p(?x_12)),
     p(+(?x_15,?y_7)) = +(p(?x_15),?y_7),
     ?x_8 = s(p(?x_8)),
     p(+(?x_4,s(?x))) = +(?x_4,?x),
     p(+(s(?x),?y_7)) = +(?x,?y_7),
     ?x = +(0,?x),
     s(+(?x,?y_2)) = +(s(?y_2),?x),
     p(+(?x,?y_3)) = +(p(?y_3),?x),
     ?y = +(?y,0),
     s(+(?x_5,?y)) = +(?y,s(?x_5)),
     p(+(?x_6,?y)) = +(?y,p(?x_6)) ]
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 <+(?x_1,?x_6), s(+(?x_1,p(?x_6)))> by Rules <6, 1> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([(s,1)],2),([],6)]>
Critical Pair <+(?x_2,?x_7), p(+(?x_2,s(?x_7)))> by Rules <7, 2> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([(p,1)],1),([],7)]>
Critical Pair <+(?x_6,?y_4), s(+(p(?x_6),?y_4))> by Rules <6, 4> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(s,1)],5),([],6)]>
Critical Pair <+(?x_7,?y_5), p(+(s(?x_7),?y_5))> by Rules <7, 5> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(p,1)],4),([],7)]>
Critical Pair <s(?x_7), s(?x_7)> by Rules <7, 6> preceded by [(s,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <p(?x_6), p(?x_6)> by Rules <6, 7> preceded by [(p,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <0, 0> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[], []>
Critical Pair <s(+(?x_4,0)), s(?x_4)> by Rules <4, 0> preceded by []
 joinable by a reduction of rules <[([(s,1)],0)], []>
Critical Pair <p(+(?x_5,0)), p(?x_5)> by Rules <5, 0> preceded by []
 joinable by a reduction of rules <[([(p,1)],0)], []>
Critical Pair <+(0,?x_8), ?x_8> by Rules <8, 0> preceded by []
 joinable by a reduction of rules <[([],3)], []>
Critical Pair <s(?y_1), s(+(0,?y_1))> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[], [([(s,1)],3)]>
Critical Pair <s(+(?x_4,s(?y_1))), s(+(s(?x_4),?y_1))> by Rules <4, 1> preceded by []
 joinable by a reduction of rules <[([(s,1)],1)], [([(s,1)],4)]>
Critical Pair <p(+(?x_5,s(?y_1))), s(+(p(?x_5),?y_1))> by Rules <5, 1> preceded by []
 joinable by a reduction of rules <[([(p,1)],1),([],7)], [([(s,1)],5),([],6)]>
Critical Pair <+(s(?y_1),?x_8), s(+(?x_8,?y_1))> by Rules <8, 1> preceded by []
 joinable by a reduction of rules <[([],4)], [([(s,1)],8)]>
Critical Pair <p(?y_2), p(+(0,?y_2))> by Rules <3, 2> preceded by []
 joinable by a reduction of rules <[], [([(p,1)],3)]>
Critical Pair <s(+(?x_4,p(?y_2))), p(+(s(?x_4),?y_2))> by Rules <4, 2> preceded by []
 joinable by a reduction of rules <[([(s,1)],2),([],6)], [([(p,1)],4),([],7)]>
Critical Pair <p(+(?x_5,p(?y_2))), p(+(p(?x_5),?y_2))> by Rules <5, 2> preceded by []
 joinable by a reduction of rules <[([(p,1)],2)], [([(p,1)],5)]>
Critical Pair <+(p(?y_2),?x_8), p(+(?x_8,?y_2))> by Rules <8, 2> preceded by []
 joinable by a reduction of rules <[([],5)], [([(p,1)],8)]>
Critical Pair <+(?y_8,0), ?y_8> by Rules <8, 3> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(?y_8,s(?x_4)), s(+(?x_4,?y_8))> by Rules <8, 4> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],8)]>
Critical Pair <+(?y_8,p(?x_5)), p(+(?x_5,?y_8))> by Rules <8, 5> preceded by []
 joinable by a reduction of rules <[([],2)], [([(p,1)],8)]>
Satisfiable by 3,6,7,8,9>2>4,5>1; +(1,1)p(0)s(0); 6,3>9>2>4,5>1,7,8
Diagram Decreasing
Direct Methods: CR

Final result: CR
134.trs: Success(CR)
(10 msec.)