YES
(ignored inputs)COMMENT full experiments for [35] submitted by: Takahito Aoto and Yoshihito Toyama
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) -> +(?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),
+(?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)),
+(+(?y_9,?x_9),?z_8) = +(?x_9,+(?y_9,?z_8)),
+(+(?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_8,?y_8) = +(?x_8,+(?y_8,0)),
?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_8,?y_8),?y_1)) = +(?x_8,+(?y_8,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_8,?y_8),?y_2)) = +(?x_8,+(?y_8,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_8,+(?y_8,?z_8)) = +(?z_8,+(?x_8,?y_8)) ]
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),
+(0,?x) = ?x,
+(?x,+(0,?z_9)) = +(?x,?z_9),
?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))),
+(?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_8,+(?y_8,s(?y))) = s(+(+(?x_8,?y_8),?y)),
+(s(?y),?x) = s(+(?x,?y)),
+(?x,?x_7) = s(+(?x,p(?x_7))),
+(?x,+(?x_16,?z_9)) = +(s(+(?x,p(?x_16))),?z_9),
+(?x,+(s(?y),?z_9)) = +(s(+(?x,?y)),?z_9),
?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))),
+(?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_8,+(?y_8,p(?y))) = p(+(+(?x_8,?y_8),?y)),
+(p(?y),?x) = p(+(?x,?y)),
+(?x,?x_8) = p(+(?x,s(?x_8))),
+(?x,+(?x_17,?z_9)) = +(p(+(?x,s(?x_17))),?z_9),
+(?x,+(p(?y),?z_9)) = +(p(+(?x,?y)),?z_9),
s(+(0,?y_2)) = s(?y_2),
p(+(0,?y_3)) = p(?y_3),
+(?y,0) = ?y,
+(0,+(?y,?z_9)) = +(?y,?z_9),
?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_16,+(?y,?z_9)) = +(s(+(p(?x_16),?y)),?z_9),
+(s(?x),+(?y,?z_9)) = +(s(+(?x,?y)),?z_9),
?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_17,+(?y,?z_9)) = +(p(+(s(?x_17),?y)),?z_9),
+(p(?x),+(?y,?z_9)) = +(p(+(?x,?y)),?z_9),
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)),
+(?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,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
?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_9,+(?y_9,?y)) = +(?y,+(?x_9,?y_9)),
+(?x,+(?y,?z_10)) = +(+(?y,?x),?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 by Rules <7, 6> preceded by [(s,1)]
joinable by a reduction of rules <[], []>
Critical Pair
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 <[([],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 <[([],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)]>
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)]>
Critical Pair <+(+(?y_9,?x_9),?z_8), +(?x_9,+(?y_9,?z_8))> by Rules <9, 8> preceded by [(+,1)]
joinable by a reduction of rules <[([(+,1)],9),([],8)], []>
joinable by a reduction of rules <[([],8),([(+,2)],9)], [([],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 <[([],8),([(+,2)],8)], [([],8)]>
Critical Pair <0, 0> by Rules <3, 0> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <4, 0> preceded by []
joinable by a reduction of rules <[([(s,1)],0)], []>
Critical Pair
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 <+(0,?x_9), ?x_9> by Rules <9, 0> preceded by []
joinable by a reduction of rules <[([],3)], []>
Critical Pair by Rules <3, 1> preceded by []
joinable by a reduction of rules <[], [([(s,1)],3)]>
Critical Pair by Rules <4, 1> preceded by []
joinable by a reduction of rules <[([(s,1)],1)], [([(s,1)],4)]>
Critical Pair
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)]> Critical Pair <+(s(?y_1),?x_9), s(+(?x_9,?y_1))> by Rules <9, 1> preceded by [] joinable by a reduction of rules <[([],4)], [([(s,1)],9)]> Critical Pair
by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], [([(p,1)],3)]>
Critical Pair by Rules <4, 2> preceded by []
joinable by a reduction of rules <[([(s,1)],2),([],6)], [([(p,1)],4),([],7)]>
Critical Pair
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)]> Critical Pair <+(p(?y_2),?x_9), p(+(?x_9,?y_2))> by Rules <9, 2> preceded by [] joinable by a reduction of rules <[([],5)], [([(p,1)],9)]> Critical Pair <+(?y_9,0), ?y_9> by Rules <9, 3> preceded by [] joinable by a reduction of rules <[([],0)], []> Critical Pair <+(?y_9,s(?x_4)), s(+(?x_4,?y_9))> by Rules <9, 4> preceded by [] joinable by a reduction of rules <[([],1)], [([(s,1)],9)]> Critical Pair <+(?y_9,p(?x_5)), p(+(?x_5,?y_9))> by Rules <9, 5> preceded by [] joinable by a reduction of rules <[([],2)], [([(p,1)],9)]> Critical Pair <+(?y_9,+(?x_8,?y_8)), +(?x_8,+(?y_8,?y_9))> by Rules <9, 8> preceded by [] joinable by a reduction of rules <[([],9),([],8)], []> unknown Diagram Decreasing check Non-Confluence... obtain 11 rules by 3 steps unfolding obtain 100 candidates for checking non-joinability check by TCAP-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 ] [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), +(?x,?y) -> +(?y,?x) ] Polynomial Interpretation: +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2 0:= (4) p:= (6)+(4)*x1 s:= (1)*x1 retract +(?x,0) -> ?x retract +(0,?y) -> ?y retract s(p(?x)) -> ?x retract p(s(?x)) -> ?x Polynomial Interpretation: +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2 0:= (12) p:= (7)+(1)*x1 s:= (3)+(1)*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 Polynomial Interpretation: +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2 0:= (10) p:= (2)+(2)*x1 s:= (5)*x1 relatively terminating Huet (modulo AC) Direct Methods: CR Combined result: CR 176.trs: Success(CR) (2111 msec.)