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),?z) -> +(?x,+(?y,?z)),
     +(?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,?z_10) = +(?x,+(0,?z_10)),
     +(s(+(?x_1,?y_1)),?z_10) = +(?x_1,+(s(?y_1),?z_10)),
     +(p(+(?x_2,?y_2)),?z_10) = +(?x_2,+(p(?y_2),?z_10)),
     +(?y_3,?z_10) = +(0,+(?y_3,?z_10)),
     +(s(+(?x_4,?y_4)),?z_10) = +(s(?x_4),+(?y_4,?z_10)),
     +(p(+(?x_5,?y_5)),?z_10) = +(p(?x_5),+(?y_5,?z_10)),
     +(+(?y_11,?x_11),?z_10) = +(?x_11,+(?y_11,?z_10)),
     -(?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)),
     -(+(?x_10,+(?y_10,?z_10))) = +(-(+(?x_10,?y_10)),-(?z_10)),
     -(+(?y_11,?x_11)) = +(-(?x_11),-(?y_11)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ 0 = 0,
     s(?x_4) = s(+(?x_4,0)),
     p(?x_5) = p(+(?x_5,0)),
     +(?x_10,?y_10) = +(?x_10,+(?y_10,0)),
     ?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_10,?y_10),?y_1)) = +(?x_10,+(?y_10,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_10,?y_10),?y_2)) = +(?x_10,+(?y_10,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)),
     +(?x_10,+(?y_10,?z_10)) = +(?z_10,+(?x_10,?y_10)) ]
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),
     +(?x_10,+(?y_10,0)) = +(?x_10,?y_10),
     +(0,?x) = ?x,
     +(?x,+(0,?z_11)) = +(?x,?z_11),
     +(-(?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_10,+(?y_10,?x_17)) = s(+(+(?x_10,?y_10),p(?x_17))),
     +(?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)),
     +(?x_10,+(?y_10,s(?y))) = s(+(+(?x_10,?y_10),?y)),
     +(s(?y),?x) = s(+(?x,?y)),
     +(?x,?x_7) = s(+(?x,p(?x_7))),
     +(?x,+(?x_18,?z_11)) = +(s(+(?x,p(?x_18))),?z_11),
     +(-(?x),-(?x_7)) = -(s(+(?x,p(?x_7)))),
     +(?x,+(s(?y),?z_11)) = +(s(+(?x,?y)),?z_11),
     +(-(?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_10,+(?y_10,?x_18)) = p(+(+(?x_10,?y_10),s(?x_18))),
     +(?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)),
     +(?x_10,+(?y_10,p(?y))) = p(+(+(?x_10,?y_10),?y)),
     +(p(?y),?x) = p(+(?x,?y)),
     +(?x,?x_8) = p(+(?x,s(?x_8))),
     +(?x,+(?x_19,?z_11)) = +(p(+(?x,s(?x_19))),?z_11),
     +(-(?x),-(?x_8)) = -(p(+(?x,s(?x_8)))),
     +(?x,+(p(?y),?z_11)) = +(p(+(?x,?y)),?z_11),
     +(-(?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,?z_11)) = +(?y,?z_11),
     +(-(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_18,+(?y,?z_11)) = +(s(+(p(?x_18),?y)),?z_11),
     +(-(?x_7),-(?y)) = -(s(+(p(?x_7),?y))),
     +(s(?x),+(?y,?z_11)) = +(s(+(?x,?y)),?z_11),
     +(-(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_19,+(?y,?z_11)) = +(p(+(s(?x_19),?y)),?z_11),
     +(-(?x_8),-(?y)) = -(p(+(s(?x_8),?y))),
     +(p(?x),+(?y,?z_11)) = +(p(+(?x,?y)),?z_11),
     +(-(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_1,+(?y_1,?y)) = +(+(?x_1,?y_1),+(?y,0)),
     ?x = +(?x,+(0,0)),
     s(+(?x,?y_3)) = +(?x,+(s(?y_3),0)),
     p(+(?x,?y_4)) = +(?x,+(p(?y_4),0)),
     ?y = +(0,+(?y,0)),
     s(+(?x_6,?y)) = +(s(?x_6),+(?y,0)),
     p(+(?x_7,?y)) = +(p(?x_7),+(?y,0)),
     +(?y,?x) = +(?x,+(?y,0)),
     s(+(+(?x_3,+(?y_3,?y)),?y_2)) = +(+(?x_3,?y_3),+(?y,s(?y_2))),
     s(+(?x,?y_2)) = +(?x,+(0,s(?y_2))),
     s(+(s(+(?x,?y_5)),?y_2)) = +(?x,+(s(?y_5),s(?y_2))),
     s(+(p(+(?x,?y_6)),?y_2)) = +(?x,+(p(?y_6),s(?y_2))),
     s(+(?y,?y_2)) = +(0,+(?y,s(?y_2))),
     s(+(s(+(?x_8,?y)),?y_2)) = +(s(?x_8),+(?y,s(?y_2))),
     s(+(p(+(?x_9,?y)),?y_2)) = +(p(?x_9),+(?y,s(?y_2))),
     s(+(+(?y,?x),?y_2)) = +(?x,+(?y,s(?y_2))),
     p(+(+(?x_4,+(?y_4,?y)),?y_3)) = +(+(?x_4,?y_4),+(?y,p(?y_3))),
     p(+(?x,?y_3)) = +(?x,+(0,p(?y_3))),
     p(+(s(+(?x,?y_6)),?y_3)) = +(?x,+(s(?y_6),p(?y_3))),
     p(+(p(+(?x,?y_7)),?y_3)) = +(?x,+(p(?y_7),p(?y_3))),
     p(+(?y,?y_3)) = +(0,+(?y,p(?y_3))),
     p(+(s(+(?x_9,?y)),?y_3)) = +(s(?x_9),+(?y,p(?y_3))),
     p(+(p(+(?x_10,?y)),?y_3)) = +(p(?x_10),+(?y,p(?y_3))),
     p(+(+(?y,?x),?y_3)) = +(?x,+(?y,p(?y_3))),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,?x) = +(?x,+(0,?z)),
     +(?z,s(+(?x,?y_3))) = +(?x,+(s(?y_3),?z)),
     +(?z,p(+(?x,?y_4))) = +(?x,+(p(?y_4),?z)),
     +(?z,?y) = +(0,+(?y,?z)),
     +(?z,s(+(?x_6,?y))) = +(s(?x_6),+(?y,?z)),
     +(?z,p(+(?x_7,?y))) = +(p(?x_7),+(?y,?z)),
     +(?z,+(?y,?x)) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?x,+(?y,0)),
     s(+(+(?x,?y),?y_2)) = +(?x,+(?y,s(?y_2))),
     p(+(+(?x,?y),?y_3)) = +(?x,+(?y,p(?y_3))),
     +(?z,+(?x,?y)) = +(?x,+(?y,?z)),
     +(+(?x_1,+(?y_1,?y)),?z) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?x,?z) = +(?x,+(0,?z)),
     +(s(+(?x,?y_3)),?z) = +(?x,+(s(?y_3),?z)),
     +(p(+(?x,?y_4)),?z) = +(?x,+(p(?y_4),?z)),
     +(?y,?z) = +(0,+(?y,?z)),
     +(s(+(?x_6,?y)),?z) = +(s(?x_6),+(?y,?z)),
     +(p(+(?x_7,?y)),?z) = +(p(?x_7),+(?y,?z)),
     +(+(?y,?x),?z) = +(?x,+(?y,?z)),
     +(+(?x_2,+(?y_2,?y)),+(?z,?z_1)) = +(+(+(?x_2,?y_2),+(?y,?z)),?z_1),
     +(?x,+(?z,?z_1)) = +(+(?x,+(0,?z)),?z_1),
     +(s(+(?x,?y_4)),+(?z,?z_1)) = +(+(?x,+(s(?y_4),?z)),?z_1),
     +(p(+(?x,?y_5)),+(?z,?z_1)) = +(+(?x,+(p(?y_5),?z)),?z_1),
     +(?y,+(?z,?z_1)) = +(+(0,+(?y,?z)),?z_1),
     +(s(+(?x_7,?y)),+(?z,?z_1)) = +(+(s(?x_7),+(?y,?z)),?z_1),
     +(p(+(?x_8,?y)),+(?z,?z_1)) = +(+(p(?x_8),+(?y,?z)),?z_1),
     +(+(?y,?x),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(-(+(?x_1,+(?y_1,?y))),-(?z)) = -(+(+(?x_1,?y_1),+(?y,?z))),
     +(-(?x),-(?z)) = -(+(?x,+(0,?z))),
     +(-(s(+(?x,?y_3))),-(?z)) = -(+(?x,+(s(?y_3),?z))),
     +(-(p(+(?x,?y_4))),-(?z)) = -(+(?x,+(p(?y_4),?z))),
     +(-(?y),-(?z)) = -(+(0,+(?y,?z))),
     +(-(s(+(?x_6,?y))),-(?z)) = -(+(s(?x_6),+(?y,?z))),
     +(-(p(+(?x_7,?y))),-(?z)) = -(+(p(?x_7),+(?y,?z))),
     +(-(+(?y,?x)),-(?z)) = -(+(?x,+(?y,?z))),
     +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(-(+(?x,?y)),-(?z)) = -(+(?x,+(?y,?z))),
     ?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_11,+(?y_11,?y)) = +(?y,+(?x_11,?y_11)),
     +(?x,+(?y,?z_12)) = +(+(?y,?x),?z_12),
     +(-(?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)),
     -(+(?x_12,+(?y_12,?y))) = +(-(+(?x_12,?y_12)),-(?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,?z_10), +(?x,+(0,?z_10))> by Rules <0, 11> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(+,2)],3)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_10), +(?x_1,+(s(?y_1),?z_10))> by Rules <1, 11> preceded by [(+,1)]
 joinable by a reduction of rules <[([],4),([(s,1)],11)], [([(+,2)],4),([],1)]>
Critical Pair <+(p(+(?x_2,?y_2)),?z_10), +(?x_2,+(p(?y_2),?z_10))> by Rules <2, 11> preceded by [(+,1)]
 joinable by a reduction of rules <[([],5),([(p,1)],11)], [([(+,2)],5),([],2)]>
Critical Pair <+(?y_3,?z_10), +(0,+(?y_3,?z_10))> by Rules <3, 11> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],3)]>
Critical Pair <+(s(+(?x_4,?y_4)),?z_10), +(s(?x_4),+(?y_4,?z_10))> by Rules <4, 11> preceded by [(+,1)]
 joinable by a reduction of rules <[([],4),([(s,1)],11)], [([],4)]>
Critical Pair <+(p(+(?x_5,?y_5)),?z_10), +(p(?x_5),+(?y_5,?z_10))> by Rules <5, 11> preceded by [(+,1)]
 joinable by a reduction of rules <[([],5),([(p,1)],11)], [([],5)]>
Critical Pair <+(+(?y_11,?x_11),?z_10), +(?x_11,+(?y_11,?z_10))> by Rules <12, 11> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],12),([],11)], []>
 joinable by a reduction of rules <[([],11),([(+,2)],12)], [([],12),([],11)]>
Critical Pair <-(?x), +(-(?x),-(0))> by Rules <0, 13> 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, 13> preceded by [(-,1)]
 joinable by a reduction of rules <[([],9),([(p,1)],13)], [([(+,2)],9),([],2)]>
Critical Pair <-(p(+(?x_2,?y_2))), +(-(?x_2),-(p(?y_2)))> by Rules <2, 13> preceded by [(-,1)]
 joinable by a reduction of rules <[([],10),([(s,1)],13)], [([(+,2)],10),([],1)]>
Critical Pair <-(?y_3), +(-(0),-(?y_3))> by Rules <3, 13> 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, 13> preceded by [(-,1)]
 joinable by a reduction of rules <[([],9),([(p,1)],13)], [([(+,1)],9),([],5)]>
Critical Pair <-(p(+(?x_5,?y_5))), +(-(p(?x_5)),-(?y_5))> by Rules <5, 13> preceded by [(-,1)]
 joinable by a reduction of rules <[([],10),([(s,1)],13)], [([(+,1)],10),([],4)]>
Critical Pair <-(+(?x_10,+(?y_10,?z_10))), +(-(+(?x_10,?y_10)),-(?z_10))> by Rules <11, 13> preceded by [(-,1)]
 joinable by a reduction of rules <[([],13),([(+,2)],13)], [([(+,1)],13),([],11)]>
Critical Pair <-(+(?y_11,?x_11)), +(-(?x_11),-(?y_11))> by Rules <12, 13> preceded by [(-,1)]
 joinable by a reduction of rules <[([],13)], [([],12)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <11, 11> preceded by [(+,1)]
 joinable by a reduction of rules <[([],11),([(+,2)],11)], [([],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 <+(?x_10,+(?y_10,0)), +(?x_10,?y_10)> by Rules <11, 0> preceded by []
 joinable by a reduction of rules <[([(+,2)],0)], []>
Critical Pair <+(0,?x_11), ?x_11> by Rules <12, 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 <+(?x_10,+(?y_10,s(?y_1))), s(+(+(?x_10,?y_10),?y_1))> by Rules <11, 1> preceded by []
 joinable by a reduction of rules <[([(+,2)],1),([],1)], [([(s,1)],11)]>
Critical Pair <+(s(?y_1),?x_11), s(+(?x_11,?y_1))> by Rules <12, 1> preceded by []
 joinable by a reduction of rules <[([],4)], [([(s,1)],12)]>
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 <+(?x_10,+(?y_10,p(?y_2))), p(+(+(?x_10,?y_10),?y_2))> by Rules <11, 2> preceded by []
 joinable by a reduction of rules <[([(+,2)],2),([],2)], [([(p,1)],11)]>
Critical Pair <+(p(?y_2),?x_11), p(+(?x_11,?y_2))> by Rules <12, 2> preceded by []
 joinable by a reduction of rules <[([],5)], [([(p,1)],12)]>
Critical Pair <+(?y_11,0), ?y_11> by Rules <12, 3> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(?y_11,s(?x_4)), s(+(?x_4,?y_11))> by Rules <12, 4> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],12)]>
Critical Pair <+(?y_11,p(?x_5)), p(+(?x_5,?y_11))> by Rules <12, 5> preceded by []
 joinable by a reduction of rules <[([],2)], [([(p,1)],12)]>
Critical Pair <+(?y_11,+(?x_10,?y_10)), +(?x_10,+(?y_10,?y_11))> by Rules <12, 11> preceded by []
 joinable by a reduction of rules <[([],12),([],11)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 24 rules by 3 steps unfolding
strenghten -(0) and 0
strenghten +(?x_15,0) and ?x_15
strenghten +(0,?x_11) and ?x_11
strenghten p(-(0)) and p(0)
strenghten s(-(0)) and s(0)
strenghten p(+(?x_5,0)) and p(?x_5)
strenghten p(+(0,?y_2)) and p(?y_2)
strenghten p(-(p(?x_6))) and -(?x_6)
strenghten s(+(?x_4,0)) and s(?x_4)
strenghten s(+(0,?y_1)) and s(?y_1)
strenghten s(-(s(?x_7))) and -(?x_7)
strenghten +(-(?x),-(0)) and -(?x)
strenghten +(-(0),-(?x_15)) and -(?x_15)
strenghten p(+(?x_5,?y_11)) and +(?y_11,p(?x_5))
strenghten p(+(?x_11,?y_2)) and +(p(?y_2),?x_11)
strenghten s(+(?x_4,?y_11)) and +(?y_11,s(?x_4))
strenghten s(+(?x_11,?y_1)) and +(s(?y_1),?x_11)
strenghten +(?x,+(0,?z_10)) and +(?x,?z_10)
strenghten +(?x_10,+(?y_10,0)) and +(?x_10,?y_10)
strenghten +(0,+(?x_15,?z_10)) and +(?x_15,?z_10)
strenghten p(+(?x_2,s(?x_7))) and +(?x_2,?x_7)
strenghten p(+(s(?x_7),?y_5)) and +(?x_7,?y_5)
strenghten s(+(?x_1,p(?x_6))) and +(?x_1,?x_6)
strenghten s(+(p(?x_6),?y_4)) and +(?x_6,?y_4)
strenghten +(-(?x_11),-(?y_11)) and -(+(?y_11,?x_11))
strenghten +(?x_10,+(?y_10,?y_11)) and +(?y_11,+(?x_10,?y_10))
strenghten +(?x_11,+(?y_11,?z_10)) and +(+(?y_11,?x_11),?z_10)
strenghten p(+(p(?x_5),?y_2)) and p(+(?x_5,p(?y_2)))
strenghten p(+(s(?x_4),?y_2)) and s(+(?x_4,p(?y_2)))
strenghten s(+(p(?x_5),?y_1)) and p(+(?x_5,s(?y_1)))
strenghten s(+(s(?x_4),?y_1)) and s(+(?x_4,s(?y_1)))
strenghten +(-(?x_1),-(s(?y_1))) and -(s(+(?x_1,?y_1)))
strenghten +(-(?x_2),-(p(?y_2))) and -(p(+(?x_2,?y_2)))
strenghten +(-(p(?x_5)),-(?y_5)) and -(p(+(?x_5,?y_5)))
strenghten +(-(s(?x_4)),-(?y_4)) and -(s(+(?x_4,?y_4)))
strenghten +(?x_1,+(s(?y_1),?z_10)) and +(s(+(?x_1,?y_1)),?z_10)
obtain 100 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,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)) -> +(-(?x),-(?y)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (1)*x1
  0:= (2)
  p:= (1)*x1
  s:= (1)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (1)+(1)*x1
  0:= (1)
  p:= (1)*x1
  s:= (1)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract -(0) -> 0
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (1)*x1
  0:= (2)
  p:= (1)*x1
  s:= (4)+(1)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract s(p(?x)) -> ?x
retract p(s(?x)) -> ?x
retract -(0) -> 0
retract -(s(?x)) -> p(-(?x))
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (3)*x1
  0:= 0
  p:= (8)+(1)*x1
  s:= (2)+(1)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract s(p(?x)) -> ?x
retract p(s(?x)) -> ?x
retract -(0) -> 0
retract -(s(?x)) -> p(-(?x))
retract -(p(?x)) -> s(-(?x))
Polynomial Interpretation:
  +:= (1)+(1)*x1+(1)*x2
  -:= (2)*x1*x1
  0:= 0
  p:= (1)*x1
  s:= (3)+(1)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract s(p(?x)) -> ?x
retract p(s(?x)) -> ?x
retract -(0) -> 0
retract -(s(?x)) -> p(-(?x))
retract -(p(?x)) -> s(-(?x))
retract -(+(?x,?y)) -> +(-(?x),-(?y))
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  -:= (1)*x1*x1
  0:= 0
  p:= (4)+(5)*x1
  s:= (4)+(4)*x1
retract +(?x,0) -> ?x
retract +(?x,s(?y)) -> s(+(?x,?y))
retract +(0,?y) -> ?y
retract +(s(?x),?y) -> s(+(?x,?y))
retract s(p(?x)) -> ?x
retract p(s(?x)) -> ?x
retract -(0) -> 0
retract -(s(?x)) -> p(-(?x))
retract -(p(?x)) -> s(-(?x))
retract -(+(?x,?y)) -> +(-(?x),-(?y))
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  -:= (2)*x1
  0:= (8)
  p:= (4)+(1)*x1
  s:= (6)+(4)*x1+(1)*x1*x1
relatively terminating
Huet (modulo AC)
Direct Methods: CR

Final result: CR
139.trs: Success(CR)
(6496 msec.)