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),?z) -> +(?x,+(?y,?z)),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
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,?z_8) = +(?x,+(0,?z_8)),
     +(s(+(?x_1,?y_1)),?z_8) = +(?x_1,+(s(?y_1),?z_8)),
     +(p(+(?x_2,?y_2)),?z_8) = +(?x_2,+(p(?y_2),?z_8)),
     +(?y_3,?z_8) = +(0,+(?y_3,?z_8)),
     +(s(+(?x_4,?y_4)),?z_8) = +(s(?x_4),+(?y_4,?z_8)),
     +(p(+(?x_5,?y_5)),?z_8) = +(p(?x_5),+(?y_5,?z_8)),
     +(+(+(?x_9,?y_9),?z_9),?z_8) = +(?x_9,+(+(?y_9,?z_9),?z_8)),
     +(?x_9,?x) = +(+(?x_9,?x),0),
     +(?x_9,s(+(?x_1,?y_1))) = +(+(?x_9,?x_1),s(?y_1)),
     +(?x_9,p(+(?x_2,?y_2))) = +(+(?x_9,?x_2),p(?y_2)),
     +(?x_9,?y_3) = +(+(?x_9,0),?y_3),
     +(?x_9,s(+(?x_4,?y_4))) = +(+(?x_9,s(?x_4)),?y_4),
     +(?x_9,p(+(?x_5,?y_5))) = +(+(?x_9,p(?x_5)),?y_5),
     +(?x_9,+(?x_8,+(?y_8,?z_8))) = +(+(?x_9,+(?x_8,?y_8)),?z_8),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)),
     +(?x_1,+(+(?x,?y),?z)) = +(+(?x_1,?x),+(?y,?z)) ]
Outer CPs:
   [ 0 = 0,
     s(?x_4) = s(+(?x_4,0)),
     p(?x_5) = p(+(?x_5,0)),
     +(?x_8,?y_8) = +(?x_8,+(?y_8,0)),
     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_8,?y_8),?y_1)) = +(?x_8,+(?y_8,s(?y_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_8,?y_8),?y_2)) = +(?x_8,+(?y_8,p(?y_2))),
     +(?y_9,?z_9) = +(+(0,?y_9),?z_9),
     s(+(?x_4,+(?y_9,?z_9))) = +(+(s(?x_4),?y_9),?z_9),
     p(+(?x_5,+(?y_9,?z_9))) = +(+(p(?x_5),?y_9),?z_9),
     +(?x_8,+(?y_8,+(?y_9,?z_9))) = +(+(+(?x_8,?y_8),?y_9),?z_9) ]
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_8,+(?y_8,0)) = +(?x_8,?y_8),
     +(?x,+(0,?z_9)) = +(?x,?z_9),
     +(+(?x_10,?x),0) = +(?x_10,?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_8,+(?y_8,?x_15)) = s(+(+(?x_8,?y_8),p(?x_15))),
     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_8,+(?y_8,s(?y))) = s(+(+(?x_8,?y_8),?y)),
     +(?x,?x_7) = s(+(?x,p(?x_7))),
     +(?x,+(?x_16,?z_9)) = +(s(+(?x,p(?x_16))),?z_9),
     +(+(?x_10,?x),?x_17) = +(?x_10,s(+(?x,p(?x_17)))),
     +(?x,+(s(?y),?z_9)) = +(s(+(?x,?y)),?z_9),
     +(+(?x_10,?x),s(?y)) = +(?x_10,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,+(?y_8,?x_16)) = p(+(+(?x_8,?y_8),s(?x_16))),
     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_8,+(?y_8,p(?y))) = p(+(+(?x_8,?y_8),?y)),
     +(?x,?x_8) = p(+(?x,s(?x_8))),
     +(?x,+(?x_17,?z_9)) = +(p(+(?x,s(?x_17))),?z_9),
     +(+(?x_10,?x),?x_18) = +(?x_10,p(+(?x,s(?x_18)))),
     +(?x,+(p(?y),?z_9)) = +(p(+(?x,?y)),?z_9),
     +(+(?x_10,?x),p(?y)) = +(?x_10,p(+(?x,?y))),
     s(+(0,?y_2)) = s(?y_2),
     p(+(0,?y_3)) = p(?y_3),
     +(+(0,?y_9),?z_9) = +(?y_9,?z_9),
     +(0,+(?y,?z_9)) = +(?y,?z_9),
     +(+(?x_10,0),?y) = +(?x_10,?y),
     ?x_7 = s(+(p(?x_7),0)),
     s(+(?x_9,?y_2)) = s(+(p(?x_9),s(?y_2))),
     +(+(?x_16,?y_9),?z_9) = s(+(p(?x_16),+(?y_9,?z_9))),
     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))),
     +(+(s(?x),?y_9),?z_9) = s(+(?x,+(?y_9,?z_9))),
     +(?x_7,?y) = s(+(p(?x_7),?y)),
     +(?x_16,+(?y,?z_9)) = +(s(+(p(?x_16),?y)),?z_9),
     +(+(?x_10,?x_17),?y) = +(?x_10,s(+(p(?x_17),?y))),
     +(s(?x),+(?y,?z_9)) = +(s(+(?x,?y)),?z_9),
     +(+(?x_10,s(?x)),?y) = +(?x_10,s(+(?x,?y))),
     ?x_8 = p(+(s(?x_8),0)),
     p(+(?x_11,?y_3)) = p(+(s(?x_11),p(?y_3))),
     +(+(?x_17,?y_9),?z_9) = p(+(s(?x_17),+(?y_9,?z_9))),
     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))),
     +(+(p(?x),?y_9),?z_9) = p(+(?x,+(?y_9,?z_9))),
     +(?x_8,?y) = p(+(s(?x_8),?y)),
     +(?x_17,+(?y,?z_9)) = +(p(+(s(?x_17),?y)),?z_9),
     +(+(?x_10,?x_18),?y) = +(?x_10,p(+(s(?x_18),?y))),
     +(p(?x),+(?y,?z_9)) = +(p(+(?x,?y)),?z_9),
     +(+(?x_10,p(?x)),?y) = +(?x_10,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)),
     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_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)),
     +(+(?x,?y_10),?z_10) = +(?x,+(+(?y_10,?z_10),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(+(+(+(?x,?y_12),?z_12),?y_2)) = +(?x,+(+(?y_12,?z_12),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(+(+(+(?x,?y_13),?z_13),?y_3)) = +(?x,+(+(?y_13,?z_13),p(?y_3))),
     +(+(+(?x_10,+(?y_10,?y)),?y_9),?z_9) = +(+(?x_10,?y_10),+(?y,+(?y_9,?z_9))),
     +(+(?x,?y_9),?z_9) = +(?x,+(0,+(?y_9,?z_9))),
     +(+(s(+(?x,?y_12)),?y_9),?z_9) = +(?x,+(s(?y_12),+(?y_9,?z_9))),
     +(+(p(+(?x,?y_13)),?y_9),?z_9) = +(?x,+(p(?y_13),+(?y_9,?z_9))),
     +(+(?y,?y_9),?z_9) = +(0,+(?y,+(?y_9,?z_9))),
     +(+(s(+(?x_15,?y)),?y_9),?z_9) = +(s(?x_15),+(?y,+(?y_9,?z_9))),
     +(+(p(+(?x_16,?y)),?y_9),?z_9) = +(p(?x_16),+(?y,+(?y_9,?z_9))),
     +(+(+(+(?x,?y_19),?z_19),?y_9),?z_9) = +(?x,+(+(?y_19,?z_19),+(?y_9,?z_9))),
     +(?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))),
     +(+(+(?x,?y),?y_9),?z_9) = +(?x,+(?y,+(?y_9,?z_9))),
     +(+(?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)),
     +(+(+(?x,?y_10),?z_10),?z) = +(?x,+(+(?y_10,?z_10),?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),
     +(+(+(?x,?y_11),?z_11),+(?z,?z_1)) = +(+(?x,+(+(?y_11,?z_11),?z)),?z_1),
     +(+(?x_10,+(?x_11,+(?y_11,?y))),?z) = +(?x_10,+(+(?x_11,?y_11),+(?y,?z))),
     +(+(?x_10,?x),?z) = +(?x_10,+(?x,+(0,?z))),
     +(+(?x_10,s(+(?x,?y_13))),?z) = +(?x_10,+(?x,+(s(?y_13),?z))),
     +(+(?x_10,p(+(?x,?y_14))),?z) = +(?x_10,+(?x,+(p(?y_14),?z))),
     +(+(?x_10,s(+(?x_16,?y))),?z) = +(?x_10,+(s(?x_16),+(?y,?z))),
     +(+(?x_10,p(+(?x_17,?y))),?z) = +(?x_10,+(p(?x_17),+(?y,?z))),
     +(+(?x_10,+(+(?x,?y_20),?z_20)),?z) = +(?x_10,+(?x,+(+(?y_20,?z_20),?z))),
     +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(+(?x_10,+(?x,?y)),?z) = +(?x_10,+(?x,+(?y,?z))),
     +(+(?y,?y_1),?z_1) = +(+(0,?y),+(?y_1,?z_1)),
     ?y = +(+(0,?y),0),
     s(+(?y,?y_3)) = +(+(0,?y),s(?y_3)),
     p(+(?y,?y_4)) = +(+(0,?y),p(?y_4)),
     ?z = +(+(0,0),?z),
     s(+(?x_6,?z)) = +(+(0,s(?x_6)),?z),
     p(+(?x_7,?z)) = +(+(0,p(?x_7)),?z),
     s(+(?x_5,+(+(?y,?y_6),?z_6))) = +(+(s(?x_5),?y),+(?y_6,?z_6)),
     s(+(?x_5,?y)) = +(+(s(?x_5),?y),0),
     s(+(?x_5,s(+(?y,?y_8)))) = +(+(s(?x_5),?y),s(?y_8)),
     s(+(?x_5,p(+(?y,?y_9)))) = +(+(s(?x_5),?y),p(?y_9)),
     s(+(?x_5,?z)) = +(+(s(?x_5),0),?z),
     s(+(?x_5,s(+(?x_11,?z)))) = +(+(s(?x_5),s(?x_11)),?z),
     s(+(?x_5,p(+(?x_12,?z)))) = +(+(s(?x_5),p(?x_12)),?z),
     s(+(?x_5,+(?x_15,+(?y_15,?z)))) = +(+(s(?x_5),+(?x_15,?y_15)),?z),
     p(+(?x_6,+(+(?y,?y_7),?z_7))) = +(+(p(?x_6),?y),+(?y_7,?z_7)),
     p(+(?x_6,?y)) = +(+(p(?x_6),?y),0),
     p(+(?x_6,s(+(?y,?y_9)))) = +(+(p(?x_6),?y),s(?y_9)),
     p(+(?x_6,p(+(?y,?y_10)))) = +(+(p(?x_6),?y),p(?y_10)),
     p(+(?x_6,?z)) = +(+(p(?x_6),0),?z),
     p(+(?x_6,s(+(?x_12,?z)))) = +(+(p(?x_6),s(?x_12)),?z),
     p(+(?x_6,p(+(?x_13,?z)))) = +(+(p(?x_6),p(?x_13)),?z),
     p(+(?x_6,+(?x_16,+(?y_16,?z)))) = +(+(p(?x_6),+(?x_16,?y_16)),?z),
     +(?x_9,+(?y_9,+(+(?y,?y_10),?z_10))) = +(+(+(?x_9,?y_9),?y),+(?y_10,?z_10)),
     +(?x_9,+(?y_9,?y)) = +(+(+(?x_9,?y_9),?y),0),
     +(?x_9,+(?y_9,s(+(?y,?y_12)))) = +(+(+(?x_9,?y_9),?y),s(?y_12)),
     +(?x_9,+(?y_9,p(+(?y,?y_13)))) = +(+(+(?x_9,?y_9),?y),p(?y_13)),
     +(?x_9,+(?y_9,?z)) = +(+(+(?x_9,?y_9),0),?z),
     +(?x_9,+(?y_9,s(+(?x_15,?z)))) = +(+(+(?x_9,?y_9),s(?x_15)),?z),
     +(?x_9,+(?y_9,p(+(?x_16,?z)))) = +(+(+(?x_9,?y_9),p(?x_16)),?z),
     +(?x_9,+(?y_9,+(?x_19,+(?y_19,?z)))) = +(+(+(?x_9,?y_9),+(?x_19,?y_19)),?z),
     +(?y,?z) = +(+(0,?y),?z),
     s(+(?x_5,+(?y,?z))) = +(+(s(?x_5),?y),?z),
     p(+(?x_6,+(?y,?z))) = +(+(p(?x_6),?y),?z),
     +(?x_9,+(?y_9,+(?y,?z))) = +(+(+(?x_9,?y_9),?y),?z),
     +(?x,+(+(?y,?y_1),?z_1)) = +(+(?x,?y),+(?y_1,?z_1)),
     +(?x,?y) = +(+(?x,?y),0),
     +(?x,s(+(?y,?y_3))) = +(+(?x,?y),s(?y_3)),
     +(?x,p(+(?y,?y_4))) = +(+(?x,?y),p(?y_4)),
     +(?x,?z) = +(+(?x,0),?z),
     +(?x,s(+(?x_6,?z))) = +(+(?x,s(?x_6)),?z),
     +(?x,p(+(?x_7,?z))) = +(+(?x,p(?x_7)),?z),
     +(?x,+(?x_10,+(?y_10,?z))) = +(+(?x,+(?x_10,?y_10)),?z),
     +(+(?x_1,?x),+(+(?y,?y_2),?z_2)) = +(?x_1,+(+(?x,?y),+(?y_2,?z_2))),
     +(+(?x_1,?x),s(+(?y,?y_4))) = +(?x_1,+(+(?x,?y),s(?y_4))),
     +(+(?x_1,?x),p(+(?y,?y_5))) = +(?x_1,+(+(?x,?y),p(?y_5))),
     +(+(?x_1,?x),?z) = +(?x_1,+(+(?x,0),?z)),
     +(+(?x_1,?x),s(+(?x_7,?z))) = +(?x_1,+(+(?x,s(?x_7)),?z)),
     +(+(?x_1,?x),p(+(?x_8,?z))) = +(?x_1,+(+(?x,p(?x_8)),?z)),
     +(+(?x_1,?x),+(?x_11,+(?y_11,?z))) = +(?x_1,+(+(?x,+(?x_11,?y_11)),?z)),
     +(?x,+(+(+(?y,?y_11),?z_11),?z_10)) = +(+(+(?x,?y),+(?y_11,?z_11)),?z_10),
     +(?x,+(s(+(?y,?y_13)),?z_10)) = +(+(+(?x,?y),s(?y_13)),?z_10),
     +(?x,+(p(+(?y,?y_14)),?z_10)) = +(+(+(?x,?y),p(?y_14)),?z_10),
     +(?x,+(?z,?z_10)) = +(+(+(?x,0),?z),?z_10),
     +(?x,+(s(+(?x_16,?z)),?z_10)) = +(+(+(?x,s(?x_16)),?z),?z_10),
     +(?x,+(p(+(?x_17,?z)),?z_10)) = +(+(+(?x,p(?x_17)),?z),?z_10),
     +(?x,+(+(?x_20,+(?y_20,?z)),?z_10)) = +(+(+(?x,+(?x_20,?y_20)),?z),?z_10),
     +(+(?x_1,?x),+(?y,?z)) = +(?x_1,+(+(?x,?y),?z)),
     +(?x,+(+(?y,?z),?z_10)) = +(+(+(?x,?y),?z),?z_10) ]
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,?z_8), +(?x,+(0,?z_8))> by Rules <0, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(+,2)],3)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_8), +(?x_1,+(s(?y_1),?z_8))> by Rules <1, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],9),([(+,1)],1)]>
 joinable by a reduction of rules <[([],4),([(s,1)],8)], [([(+,2)],4),([],1)]>
Critical Pair <+(p(+(?x_2,?y_2)),?z_8), +(?x_2,+(p(?y_2),?z_8))> by Rules <2, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],9),([(+,1)],2)]>
 joinable by a reduction of rules <[([],5),([(p,1)],8)], [([(+,2)],5),([],2)]>
Critical Pair <+(?y_3,?z_8), +(0,+(?y_3,?z_8))> by Rules <3, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],3)]>
Critical Pair <+(s(+(?x_4,?y_4)),?z_8), +(s(?x_4),+(?y_4,?z_8))> by Rules <4, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[([],4),([(s,1)],8)], [([],4)]>
 joinable by a reduction of rules <[], [([],9),([(+,1)],4)]>
 joinable by a reduction of rules <[([],4)], [([],4),([(s,1)],9)]>
Critical Pair <+(p(+(?x_5,?y_5)),?z_8), +(p(?x_5),+(?y_5,?z_8))> by Rules <5, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[([],5),([(p,1)],8)], [([],5)]>
 joinable by a reduction of rules <[], [([],9),([(+,1)],5)]>
 joinable by a reduction of rules <[([],5)], [([],5),([(p,1)],9)]>
Critical Pair <+(+(+(?x_9,?y_9),?z_9),?z_8), +(?x_9,+(+(?y_9,?z_9),?z_8))> by Rules <9, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],8)], [([],9)]>
Critical Pair <+(?x_9,?x), +(+(?x_9,?x),0)> by Rules <0, 9> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([],0)]>
Critical Pair <+(?x_9,s(+(?x_1,?y_1))), +(+(?x_9,?x_1),s(?y_1))> by Rules <1, 9> preceded by [(+,2)]
 joinable by a reduction of rules <[([],1),([(s,1)],9)], [([],1)]>
 joinable by a reduction of rules <[], [([],8),([(+,2)],1)]>
 joinable by a reduction of rules <[([],1)], [([],1),([(s,1)],8)]>
Critical Pair <+(?x_9,p(+(?x_2,?y_2))), +(+(?x_9,?x_2),p(?y_2))> by Rules <2, 9> preceded by [(+,2)]
 joinable by a reduction of rules <[([],2),([(p,1)],9)], [([],2)]>
 joinable by a reduction of rules <[], [([],8),([(+,2)],2)]>
 joinable by a reduction of rules <[([],2)], [([],2),([(p,1)],8)]>
Critical Pair <+(?x_9,?y_3), +(+(?x_9,0),?y_3)> by Rules <3, 9> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([(+,1)],0)]>
Critical Pair <+(?x_9,s(+(?x_4,?y_4))), +(+(?x_9,s(?x_4)),?y_4)> by Rules <4, 9> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([],8),([(+,2)],4)]>
 joinable by a reduction of rules <[([],1),([(s,1)],9)], [([(+,1)],1),([],4)]>
Critical Pair <+(?x_9,p(+(?x_5,?y_5))), +(+(?x_9,p(?x_5)),?y_5)> by Rules <5, 9> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([],8),([(+,2)],5)]>
 joinable by a reduction of rules <[([],2),([(p,1)],9)], [([(+,1)],2),([],5)]>
Critical Pair <+(?x_9,+(?x_8,+(?y_8,?z_8))), +(+(?x_9,+(?x_8,?y_8)),?z_8)> by Rules <8, 9> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],9)], [([],8)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <8, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],9)], [([],9)]>
Critical Pair <+(?x_1,+(+(?x,?y),?z)), +(+(?x_1,?x),+(?y,?z))> by Rules <9, 9> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],8)], [([],8)]>
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_8,+(?y_8,0)), +(?x_8,?y_8)> by Rules <8, 0> preceded by []
 joinable by a reduction of rules <[([(+,2)],0)], []>
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_8,+(?y_8,s(?y_1))), s(+(+(?x_8,?y_8),?y_1))> by Rules <8, 1> preceded by []
 joinable by a reduction of rules <[([(+,2)],1),([],1)], [([(s,1)],8)]>
 joinable by a reduction of rules <[([],9),([],1)], []>
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_8,+(?y_8,p(?y_2))), p(+(+(?x_8,?y_8),?y_2))> by Rules <8, 2> preceded by []
 joinable by a reduction of rules <[([(+,2)],2),([],2)], [([(p,1)],8)]>
 joinable by a reduction of rules <[([],9),([],2)], []>
Critical Pair <+(+(0,?y_9),?z_9), +(?y_9,?z_9)> by Rules <9, 3> preceded by []
 joinable by a reduction of rules <[([(+,1)],3)], []>
Critical Pair <+(+(s(?x_4),?y_9),?z_9), s(+(?x_4,+(?y_9,?z_9)))> by Rules <9, 4> preceded by []
 joinable by a reduction of rules <[([(+,1)],4),([],4)], [([(s,1)],9)]>
 joinable by a reduction of rules <[([],8),([],4)], []>
Critical Pair <+(+(p(?x_5),?y_9),?z_9), p(+(?x_5,+(?y_9,?z_9)))> by Rules <9, 5> preceded by []
 joinable by a reduction of rules <[([(+,1)],5),([],5)], [([(p,1)],9)]>
 joinable by a reduction of rules <[([],8),([],5)], []>
Critical Pair <+(+(+(?x_8,?y_8),?y_9),?z_9), +(?x_8,+(?y_8,+(?y_9,?z_9)))> by Rules <9, 8> preceded by []
 joinable by a reduction of rules <[([],8)], [([],9)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 20 rules by 3 steps unfolding
strenghten p(s(?x_7)) and +(?x_7,0)
strenghten p(s(?x_7)) and +(0,?x_7)
strenghten s(p(?x_6)) and +(0,?x_6)
strenghten s(p(0)) and p(s(0))
strenghten p(+(?x_5,0)) and p(?x_5)
strenghten p(+(0,?y_2)) and p(?y_2)
strenghten s(+(?x_4,0)) and s(?x_4)
strenghten s(+(0,?y_1)) and s(?y_1)
strenghten +(?x,+(0,?z_8)) and +(?x,?z_8)
strenghten +(?x_8,+(?y_8,0)) and +(?x_8,?y_8)
strenghten +(+(?x_9,?x),0) and +(?x_9,?x)
strenghten +(+(?x_9,0),?y_3) and +(?x_9,?y_3)
strenghten +(+(0,?y_9),?z_9) and +(?y_9,?z_9)
strenghten +(0,+(?y_3,?z_8)) and +(?y_3,?z_8)
strenghten p(+(?x_2,s(?x_7))) and +(?x_2,?x_7)
strenghten p(+(?x_5,p(0))) and p(p(?x_5))
strenghten p(+(?x_5,s(0))) and s(p(?x_5))
strenghten p(+(p(0),?y_2)) and p(p(?y_2))
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(+(?x_4,p(0))) and p(s(?x_4))
strenghten s(+(?x_4,s(0))) and s(s(?x_4))
strenghten s(+(p(?x_6),?y_4)) and +(?x_6,?y_4)
strenghten s(+(p(0),?y_1)) and p(s(?y_1))
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_8,+(?y_8,p(0))) and p(+(?x_8,?y_8))
strenghten +(?x_8,+(?y_8,s(0))) and s(+(?x_8,?y_8))
strenghten +(?x_14,+(p(0),?z_8)) and +(p(?x_14),?z_8)
strenghten +(?x_15,+(s(0),?z_8)) and +(s(?x_15),?z_8)
strenghten +(+(?x_9,?x_14),p(0)) and +(?x_9,p(?x_14))
strenghten +(+(?x_9,?x_15),s(0)) and +(?x_9,s(?x_15))
strenghten +(+(?x_9,0),p(?y_16)) and +(?x_9,p(?y_16))
strenghten +(+(?x_9,0),s(?y_17)) and +(?x_9,s(?y_17))
strenghten +(+(?x_9,p(?x_18)),0) and +(?x_9,p(?x_18))
strenghten +(+(?x_9,p(0)),?y_19) and +(?x_9,p(?y_19))
strenghten +(+(p(0),?y_9),?z_9) and p(+(?y_9,?z_9))
strenghten +(0,+(p(?y_16),?z_8)) and +(p(?y_16),?z_8)
strenghten +(0,+(s(?y_17),?z_8)) and +(s(?y_17),?z_8)
strenghten +(p(?x_18),+(0,?z_8)) and +(p(?x_18),?z_8)
strenghten +(p(0),+(?y_19,?z_8)) and +(p(?y_19),?z_8)
strenghten +(?x_1,+(s(?y_1),?z_8)) and +(s(+(?x_1,?y_1)),?z_8)
strenghten +(?x_2,+(p(?y_2),?z_8)) and +(p(+(?x_2,?y_2)),?z_8)
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
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(?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 ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(?x_4,0)), s(?x_4)> --> <s(?x_4), s(?x_4)> => yes
<p(+(?x_5,0)), p(?x_5)> --> <p(?x_5), p(?x_5)> => yes
<s(?y_1), s(+(0,?y_1))> --> <s(?y_1), s(?y_1)> => yes
<s(+(?x_4,s(?y_1))), s(+(s(?x_4),?y_1))> --> <s(s(+(?x_4,?y_1))), s(s(+(?x_4,?y_1)))> => yes
<p(+(?x_5,s(?y_1))), s(+(p(?x_5),?y_1))> --> <+(?x_5,?y_1), +(?x_5,?y_1)> => yes
<+(?x_1,?x_6), s(+(?x_1,p(?x_6)))> --> <+(?x_1,?x_6), +(?x_1,?x_6)> => yes
<p(?y_2), p(+(0,?y_2))> --> <p(?y_2), p(?y_2)> => yes
<s(+(?x_4,p(?y_2))), p(+(s(?x_4),?y_2))> --> <+(?x_4,?y_2), +(?x_4,?y_2)> => yes
<p(+(?x_5,p(?y_2))), p(+(p(?x_5),?y_2))> --> <p(p(+(?x_5,?y_2))), p(p(+(?x_5,?y_2)))> => yes
<+(?x_2,?x_7), p(+(?x_2,s(?x_7)))> --> <+(?x_2,?x_7), +(?x_2,?x_7)> => yes
<+(?x_6,?y_4), s(+(p(?x_6),?y_4))> --> <+(?x_6,?y_4), +(?x_6,?y_4)> => yes
<+(?x_7,?y_5), p(+(s(?x_7),?y_5))> --> <+(?x_7,?y_5), +(?x_7,?y_5)> => yes
<s(?x_7), s(?x_7)> --> <s(?x_7), s(?x_7)> => yes
<p(?x_6), p(?x_6)> --> <p(?x_6), p(?x_6)> => yes
PCP_in(symP,S):
CP(S,symP):
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<+(?x_1,?y_1), +(?x_1,+(?y_1,0))> --> <+(?x_1,?y_1), +(?x_1,?y_1)> => yes
<+(?x,?z_1), +(?x,+(0,?z_1))> --> <+(?x,?z_1), +(?x,?z_1)> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,?x),s(?y))> --> <s(+(?x_1,+(?x,?y))), s(+(+(?x_1,?x),?y))> => yes
<s(+(+(?x_1,?y_1),?y)), +(?x_1,+(?y_1,s(?y)))> --> <s(+(+(?x_1,?y_1),?y)), s(+(?x_1,+(?y_1,?y)))> => yes
<+(s(+(?x,?y)),?z_1), +(?x,+(s(?y),?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(?x,+(?y,?z_1)))> => yes
<+(?x_1,p(+(?x,?y))), +(+(?x_1,?x),p(?y))> --> <p(+(?x_1,+(?x,?y))), p(+(+(?x_1,?x),?y))> => yes
<p(+(+(?x_1,?y_1),?y)), +(?x_1,+(?y_1,p(?y)))> --> <p(+(+(?x_1,?y_1),?y)), p(+(?x_1,+(?y_1,?y)))> => yes
<+(p(+(?x,?y)),?z_1), +(?x,+(p(?y),?z_1))> --> <p(+(+(?x,?y),?z_1)), p(+(?x,+(?y,?z_1)))> => yes
<+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes
<+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(?x_1,?y)> => yes
<+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes
<s(+(?x,+(?y_1,?z_1))), +(+(s(?x),?y_1),?z_1)> --> <s(+(?x,+(?y_1,?z_1))), s(+(+(?x,?y_1),?z_1))> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> <s(+(?x_1,+(?x,?y))), s(+(+(?x_1,?x),?y))> => yes
<+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(?x,+(?y,?z_1)))> => yes
<p(+(?x,+(?y_1,?z_1))), +(+(p(?x),?y_1),?z_1)> --> <p(+(?x,+(?y_1,?z_1))), p(+(+(?x,?y_1),?z_1))> => yes
<+(?x_1,p(+(?x,?y))), +(+(?x_1,p(?x)),?y)> --> <p(+(?x_1,+(?x,?y))), p(+(+(?x_1,?x),?y))> => yes
<+(p(+(?x,?y)),?z_1), +(p(?x),+(?y,?z_1))> --> <p(+(+(?x,?y),?z_1)), p(+(?x,+(?y,?z_1)))> => yes
S:
   [ +(?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 ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
168.trs: Success(CR)
(3610 msec.)