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,
     -(0) -> 0,
     -(s(?x)) -> p(-(?x)),
     -(p(?x)) -> s(-(?x)),
     +(?x,?y) -> +(?y,?x),
     -(+(?x,?y)) -> +(-(?x),-(?y)) ]
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),
     -(?x_6) = p(-(p(?x_6))),
     -(?x_7) = s(-(s(?x_7))),
     -(?x) = +(-(?x),-(0)),
     -(s(+(?x_1,?y_1))) = +(-(?x_1),-(s(?y_1))),
     -(p(+(?x_2,?y_2))) = +(-(?x_2),-(p(?y_2))),
     -(?y_3) = +(-(0),-(?y_3)),
     -(s(+(?x_4,?y_4))) = +(-(s(?x_4)),-(?y_4)),
     -(p(+(?x_5,?y_5))) = +(-(p(?x_5)),-(?y_5)),
     -(+(?y_10,?x_10)) = +(-(?x_10),-(?y_10)) ]
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),-(0)) = -(?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),-(?x_7)) = -(s(+(?x,p(?x_7)))),
     +(-(?x),-(s(?y))) = -(s(+(?x,?y))),
     ?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))),
     +(-(?x),-(?x_8)) = -(p(+(?x,s(?x_8)))),
     +(-(?x),-(p(?y))) = -(p(+(?x,?y))),
     s(+(0,?y_2)) = s(?y_2),
     p(+(0,?y_3)) = p(?y_3),
     +(?y,0) = ?y,
     +(-(0),-(?y)) = -(?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_7),-(?y)) = -(s(+(p(?x_7),?y))),
     +(-(s(?x)),-(?y)) = -(s(+(?x,?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)),
     +(-(?x_8),-(?y)) = -(p(+(s(?x_8),?y))),
     +(-(p(?x)),-(?y)) = -(p(+(?x,?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)),
     p(-(?x_8)) = -(s(?x_8)),
     s(+(?x_3,p(?x))) = +(?x_3,?x),
     s(+(p(?x),?y_6)) = +(?x,?y_6),
     p(?x) = p(?x),
     p(-(p(?x))) = -(?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)),
     s(-(?x_8)) = -(p(?x_8)),
     p(+(?x_4,s(?x))) = +(?x_4,?x),
     p(+(s(?x),?y_7)) = +(?x,?y_7),
     s(-(s(?x))) = -(?x),
     -(?x_8) = p(-(p(?x_8))),
     -(?x_9) = s(-(s(?x_9))),
     ?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)),
     +(-(?x),-(?y)) = -(+(?y,?x)),
     -(?x) = +(-(?x),-(0)),
     -(s(+(?x,?y_3))) = +(-(?x),-(s(?y_3))),
     -(p(+(?x,?y_4))) = +(-(?x),-(p(?y_4))),
     -(?y) = +(-(0),-(?y)),
     -(s(+(?x_6,?y))) = +(-(s(?x_6)),-(?y)),
     -(p(+(?x_7,?y))) = +(-(p(?x_7)),-(?y)),
     -(+(?y,?x)) = +(-(?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 <+(?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 <-(?x_6), p(-(p(?x_6)))> by Rules <6, 9> preceded by [(-,1)]
 joinable by a reduction of rules <[], [([(p,1)],10),([],7)]>
Critical Pair <-(?x_7), s(-(s(?x_7)))> by Rules <7, 10> preceded by [(-,1)]
 joinable by a reduction of rules <[], [([(s,1)],9),([],6)]>
Critical Pair <-(?x), +(-(?x),-(0))> by Rules <0, 12> preceded by [(-,1)]
 joinable by a reduction of rules <[], [([(+,2)],8),([],0)]>
Critical Pair <-(s(+(?x_1,?y_1))), +(-(?x_1),-(s(?y_1)))> by Rules <1, 12> preceded by [(-,1)]
 joinable by a reduction of rules <[([],9),([(p,1)],12)], [([(+,2)],9),([],2)]>
Critical Pair <-(p(+(?x_2,?y_2))), +(-(?x_2),-(p(?y_2)))> by Rules <2, 12> preceded by [(-,1)]
 joinable by a reduction of rules <[([],10),([(s,1)],12)], [([(+,2)],10),([],1)]>
Critical Pair <-(?y_3), +(-(0),-(?y_3))> by Rules <3, 12> preceded by [(-,1)]
 joinable by a reduction of rules <[], [([(+,1)],8),([],3)]>
Critical Pair <-(s(+(?x_4,?y_4))), +(-(s(?x_4)),-(?y_4))> by Rules <4, 12> preceded by [(-,1)]
 joinable by a reduction of rules <[([],9),([(p,1)],12)], [([(+,1)],9),([],5)]>
Critical Pair <-(p(+(?x_5,?y_5))), +(-(p(?x_5)),-(?y_5))> by Rules <5, 12> preceded by [(-,1)]
 joinable by a reduction of rules <[([],10),([(s,1)],12)], [([(+,1)],10),([],4)]>
Critical Pair <-(+(?y_10,?x_10)), +(-(?x_10),-(?y_10))> by Rules <11, 12> preceded by [(-,1)]
 joinable by a reduction of rules <[([],12)], [([],11)]>
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_10), ?x_10> by Rules <11, 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_10), s(+(?x_10,?y_1))> by Rules <11, 1> preceded by []
 joinable by a reduction of rules <[([],4)], [([(s,1)],11)]>
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_10), p(+(?x_10,?y_2))> by Rules <11, 2> preceded by []
 joinable by a reduction of rules <[([],5)], [([(p,1)],11)]>
Critical Pair <+(?y_10,0), ?y_10> by Rules <11, 3> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(?y_10,s(?x_4)), s(+(?x_4,?y_10))> by Rules <11, 4> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],11)]>
Critical Pair <+(?y_10,p(?x_5)), p(+(?x_5,?y_10))> by Rules <11, 5> preceded by []
 joinable by a reduction of rules <[([],2)], [([(p,1)],11)]>
Satisfiable by 13>5,8,10,2>3,6,12>4,7>9,11>1; +(1,1)-(2)p(0)s(0); 10>11>13>5,2>6,3>8,12>7>4>9>1
Diagram Decreasing
Direct Methods: CR

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