YES We show the termination of the TRS R: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(X,nil()) -> X __(nil(),X) -> X U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() isList(V) -> U11(isNeList(activate(V))) isList(n__nil()) -> tt() isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(isNePal(activate(V))) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() __(X1,X2) -> n____(X1,X2) a() -> n__a() e() -> n__e() i() -> n__i() o() -> n__o() u() -> n__u() activate(n__nil()) -> nil() activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__o()) -> o() activate(n__u()) -> u() activate(X) -> X -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) p2: __#(__(X,Y),Z) -> __#(Y,Z) p3: U21#(tt(),V2) -> U22#(isList(activate(V2))) p4: U21#(tt(),V2) -> isList#(activate(V2)) p5: U21#(tt(),V2) -> activate#(V2) p6: U41#(tt(),V2) -> U42#(isNeList(activate(V2))) p7: U41#(tt(),V2) -> isNeList#(activate(V2)) p8: U41#(tt(),V2) -> activate#(V2) p9: U51#(tt(),V2) -> U52#(isList(activate(V2))) p10: U51#(tt(),V2) -> isList#(activate(V2)) p11: U51#(tt(),V2) -> activate#(V2) p12: U71#(tt(),P) -> U72#(isPal(activate(P))) p13: U71#(tt(),P) -> isPal#(activate(P)) p14: U71#(tt(),P) -> activate#(P) p15: isList#(V) -> U11#(isNeList(activate(V))) p16: isList#(V) -> isNeList#(activate(V)) p17: isList#(V) -> activate#(V) p18: isList#(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2)) p19: isList#(n____(V1,V2)) -> isList#(activate(V1)) p20: isList#(n____(V1,V2)) -> activate#(V1) p21: isList#(n____(V1,V2)) -> activate#(V2) p22: isNeList#(V) -> U31#(isQid(activate(V))) p23: isNeList#(V) -> isQid#(activate(V)) p24: isNeList#(V) -> activate#(V) p25: isNeList#(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) p26: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p27: isNeList#(n____(V1,V2)) -> activate#(V1) p28: isNeList#(n____(V1,V2)) -> activate#(V2) p29: isNeList#(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) p30: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p31: isNeList#(n____(V1,V2)) -> activate#(V1) p32: isNeList#(n____(V1,V2)) -> activate#(V2) p33: isNePal#(V) -> U61#(isQid(activate(V))) p34: isNePal#(V) -> isQid#(activate(V)) p35: isNePal#(V) -> activate#(V) p36: isNePal#(n____(I,n____(P,I))) -> U71#(isQid(activate(I)),activate(P)) p37: isNePal#(n____(I,n____(P,I))) -> isQid#(activate(I)) p38: isNePal#(n____(I,n____(P,I))) -> activate#(I) p39: isNePal#(n____(I,n____(P,I))) -> activate#(P) p40: isPal#(V) -> U81#(isNePal(activate(V))) p41: isPal#(V) -> isNePal#(activate(V)) p42: isPal#(V) -> activate#(V) p43: activate#(n__nil()) -> nil#() p44: activate#(n____(X1,X2)) -> __#(activate(X1),activate(X2)) p45: activate#(n____(X1,X2)) -> activate#(X1) p46: activate#(n____(X1,X2)) -> activate#(X2) p47: activate#(n__a()) -> a#() p48: activate#(n__e()) -> e#() p49: activate#(n__i()) -> i#() p50: activate#(n__o()) -> o#() p51: activate#(n__u()) -> u#() and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The estimated dependency graph contains the following SCCs: {p13, p36, p41} {p4, p7, p10, p16, p18, p19, p25, p26, p29, p30} {p45, p46} {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isPal#(V) -> isNePal#(activate(V)) p2: isNePal#(n____(I,n____(P,I))) -> U71#(isQid(activate(I)),activate(P)) p3: U71#(tt(),P) -> isPal#(activate(P)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The set of usable rules consists of r1, r2, r3, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: isPal#_A(x1) = ((1,0),(1,0)) x1 + (3,3) isNePal#_A(x1) = ((1,0),(0,0)) x1 activate_A(x1) = ((1,0),(1,1)) x1 + (0,4) n_____A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (5,5) U71#_A(x1,x2) = ((1,0),(1,1)) x2 + (4,2) isQid_A(x1) = ((1,0),(0,0)) x1 + (9,6) tt_A() = (2,1) ___A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (5,6) nil_A() = (1,0) n__nil_A() = (1,0) a_A() = (3,3) n__a_A() = (3,2) e_A() = (3,3) n__e_A() = (3,2) i_A() = (3,3) n__i_A() = (3,2) o_A() = (3,3) n__o_A() = (3,2) u_A() = (1,3) n__u_A() = (1,2) precedence: isPal# = isNePal# = activate = n____ = U71# = isQid = tt = __ = nil = n__nil = a = n__a = e = n__e = i = n__i = o = n__o = u = n__u partial status: pi(isPal#) = [] pi(isNePal#) = [] pi(activate) = [] pi(n____) = [] pi(U71#) = [] pi(isQid) = [] pi(tt) = [] pi(__) = [] pi(nil) = [] pi(n__nil) = [] pi(a) = [] pi(n__a) = [] pi(e) = [] pi(n__e) = [] pi(i) = [] pi(n__i) = [] pi(o) = [] pi(n__o) = [] pi(u) = [] pi(n__u) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: isPal#_A(x1) = (4,2) isNePal#_A(x1) = (3,1) activate_A(x1) = ((1,0),(1,0)) x1 + (2,4) n_____A(x1,x2) = (1,1) U71#_A(x1,x2) = (0,0) isQid_A(x1) = (0,0) tt_A() = (5,3) ___A(x1,x2) = (2,2) nil_A() = (1,1) n__nil_A() = (1,1) a_A() = (7,5) n__a_A() = (6,4) e_A() = (7,5) n__e_A() = (6,4) i_A() = (7,5) n__i_A() = (6,4) o_A() = (7,5) n__o_A() = (6,4) u_A() = (1,2) n__u_A() = (1,1) precedence: n__o > n__u > o > u > nil > i > n__e > n__a > n__i > e > a > n__nil > isPal# = isNePal# = activate = n____ = isQid > tt > __ > U71# partial status: pi(isPal#) = [] pi(isNePal#) = [] pi(activate) = [] pi(n____) = [] pi(U71#) = [] pi(isQid) = [] pi(tt) = [] pi(__) = [] pi(nil) = [] pi(n__nil) = [] pi(a) = [] pi(n__a) = [] pi(e) = [] pi(n__e) = [] pi(i) = [] pi(n__i) = [] pi(o) = [] pi(n__o) = [] pi(u) = [] pi(n__u) = [] The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2)) p2: U51#(tt(),V2) -> isList#(activate(V2)) p3: isList#(n____(V1,V2)) -> isList#(activate(V1)) p4: isList#(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2)) p5: U21#(tt(),V2) -> isList#(activate(V2)) p6: isList#(V) -> isNeList#(activate(V)) p7: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p8: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p9: isNeList#(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) p10: U41#(tt(),V2) -> isNeList#(activate(V2)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r16, r17, r18, r19, r20, r21, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: isNeList#_A(x1) = x1 + (20,1) n_____A(x1,x2) = x1 + x2 + (19,15) U51#_A(x1,x2) = ((1,0),(1,1)) x2 + (22,1) isNeList_A(x1) = ((1,0),(0,0)) x1 + (3,23) activate_A(x1) = ((1,0),(1,1)) x1 + (0,17) tt_A() = (4,6) isList#_A(x1) = ((1,0),(1,1)) x1 + (21,19) U21#_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (18,42) isList_A(x1) = (9,12) U41#_A(x1,x2) = x1 + x2 + (17,10) U22_A(x1) = ((0,0),(1,0)) x1 + (5,1) U42_A(x1) = (5,7) U52_A(x1) = x1 + (1,1) ___A(x1,x2) = x1 + x2 + (19,16) nil_A() = (1,1) U11_A(x1) = (5,1) U21_A(x1,x2) = ((0,0),(1,0)) x1 + (6,7) U31_A(x1) = ((1,0),(0,0)) x1 + (1,1) U41_A(x1,x2) = ((1,0),(0,0)) x2 + (20,8) U51_A(x1,x2) = ((1,0),(0,0)) x2 + (11,14) isQid_A(x1) = x1 + (1,5) n__a_A() = (5,2) n__e_A() = (5,0) n__i_A() = (5,2) n__o_A() = (5,7) n__u_A() = (5,0) n__nil_A() = (1,0) a_A() = (5,3) e_A() = (5,1) i_A() = (5,3) o_A() = (5,8) u_A() = (5,1) precedence: isNeList# = n____ = U51# = isNeList = activate = tt = isList# = U21# = isList = U41# = U22 = U42 = U52 = __ = nil = U11 = U21 = U31 = U41 = U51 = isQid = n__a = n__e = n__i = n__o = n__u = n__nil = a = e = i = o = u partial status: pi(isNeList#) = [] pi(n____) = [] pi(U51#) = [] pi(isNeList) = [] pi(activate) = [] pi(tt) = [] pi(isList#) = [] pi(U21#) = [] pi(isList) = [] pi(U41#) = [] pi(U22) = [] pi(U42) = [] pi(U52) = [] pi(__) = [] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isQid) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: isNeList#_A(x1) = ((0,0),(1,0)) x1 + (5,1) n_____A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (14,36) U51#_A(x1,x2) = (4,1) isNeList_A(x1) = (13,16) activate_A(x1) = (3,28) tt_A() = (10,2) isList#_A(x1) = ((1,0),(1,0)) x1 + (6,13) U21#_A(x1,x2) = (0,0) isList_A(x1) = (1,3) U41#_A(x1,x2) = ((1,0),(0,0)) x1 + (1,5) U22_A(x1) = (0,0) U42_A(x1) = (11,0) U52_A(x1) = (11,1) ___A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (15,1) nil_A() = (1,1) U11_A(x1) = (11,4) U21_A(x1,x2) = (0,0) U31_A(x1) = (11,3) U41_A(x1,x2) = (12,15) U51_A(x1,x2) = (12,4) isQid_A(x1) = (11,3) n__a_A() = (0,0) n__e_A() = (1,0) n__i_A() = (0,0) n__o_A() = (1,0) n__u_A() = (1,0) n__nil_A() = (1,0) a_A() = (1,1) e_A() = (2,1) i_A() = (1,1) o_A() = (2,1) u_A() = (2,1) precedence: U42 = __ > U41 > n____ = U51# = activate = tt = isList# = isList = U52 = U11 = U31 = n__e = n__o = n__u = a = e > u > n__i > nil > isQid > U21 > isNeList# = U21# = U41# = n__a = n__nil > o > i > U51 > isNeList > U22 partial status: pi(isNeList#) = [] pi(n____) = [] pi(U51#) = [] pi(isNeList) = [] pi(activate) = [] pi(tt) = [] pi(isList#) = [] pi(U21#) = [] pi(isList) = [] pi(U41#) = [] pi(U22) = [] pi(U42) = [] pi(U52) = [] pi(__) = [] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isQid) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p1, p2, p3, p4, p5, p6, p7, p8 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) p2: U41#(tt(),V2) -> isNeList#(activate(V2)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2)) p2: U41#(tt(),V2) -> isNeList#(activate(V2)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r16, r17, r18, r19, r20, r21, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: isNeList#_A(x1) = ((1,0),(1,0)) x1 + (9,17) n_____A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (11,7) U41#_A(x1,x2) = ((1,0),(1,1)) x2 + (10,16) isList_A(x1) = x1 + (19,14) activate_A(x1) = ((1,0),(1,1)) x1 + (0,13) tt_A() = (2,4) U42_A(x1) = (3,3) U52_A(x1) = (3,5) U22_A(x1) = (3,5) U31_A(x1) = ((0,0),(1,0)) x1 + (3,1) U41_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (4,10) isNeList_A(x1) = ((0,0),(1,0)) x1 + (5,17) U51_A(x1,x2) = (4,8) isQid_A(x1) = x1 + (2,2) n__a_A() = (3,5) n__e_A() = (1,1) n__i_A() = (3,1) n__o_A() = (3,1) n__u_A() = (3,5) ___A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (11,12) nil_A() = (3,6) U11_A(x1) = (3,1) U21_A(x1,x2) = ((1,0),(0,0)) x1 + (1,6) n__nil_A() = (3,5) a_A() = (3,6) e_A() = (1,2) i_A() = (3,2) o_A() = (3,2) u_A() = (3,6) precedence: activate = a = e = u > isList = i > __ > n____ = U41 = isNeList = n__e = n__i = o > n__o > U41# = U52 = U51 = n__a = n__u > isNeList# = U11 > U42 = nil > U31 = U21 = n__nil > tt = isQid > U22 partial status: pi(isNeList#) = [] pi(n____) = [1] pi(U41#) = [2] pi(isList) = [1] pi(activate) = [1] pi(tt) = [] pi(U42) = [] pi(U52) = [] pi(U22) = [] pi(U31) = [] pi(U41) = [] pi(isNeList) = [] pi(U51) = [] pi(isQid) = [1] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(__) = [1] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: isNeList#_A(x1) = (0,0) n_____A(x1,x2) = ((1,0),(0,0)) x1 + (6,2) U41#_A(x1,x2) = x2 + (4,1) isList_A(x1) = ((1,0),(0,0)) x1 + (2,0) activate_A(x1) = ((1,0),(0,0)) x1 + (3,4) tt_A() = (0,2) U42_A(x1) = (1,4) U52_A(x1) = (1,1) U22_A(x1) = (1,3) U31_A(x1) = (1,1) U41_A(x1,x2) = (8,5) isNeList_A(x1) = (8,5) U51_A(x1,x2) = (7,6) isQid_A(x1) = (1,3) n__a_A() = (1,0) n__e_A() = (1,1) n__i_A() = (1,0) n__o_A() = (1,0) n__u_A() = (1,0) ___A(x1,x2) = ((1,0),(0,0)) x1 + (6,3) nil_A() = (2,1) U11_A(x1) = (1,3) U21_A(x1,x2) = (2,1) n__nil_A() = (1,0) a_A() = (2,1) e_A() = (2,2) i_A() = (2,1) o_A() = (2,1) u_A() = (1,0) precedence: u > o > i > e > a > U21 > U11 > n__nil > n__e > n__u > U51 = n__o > isQid > tt = U52 = U22 > nil > activate > __ > n__i > isNeList > U31 > n__a > isNeList# = n____ = U41# > isList = U42 = U41 partial status: pi(isNeList#) = [] pi(n____) = [] pi(U41#) = [] pi(isList) = [] pi(activate) = [] pi(tt) = [] pi(U42) = [] pi(U52) = [] pi(U22) = [] pi(U31) = [] pi(U41) = [] pi(isNeList) = [] pi(U51) = [] pi(isQid) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(__) = [] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n____(X1,X2)) -> activate#(X2) p2: activate#(n____(X1,X2)) -> activate#(X1) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: activate#_A(x1) = ((1,0),(1,1)) x1 n_____A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,1) precedence: activate# > n____ partial status: pi(activate#) = [1] pi(n____) = [1, 2] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: activate#_A(x1) = x1 n_____A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1) precedence: n____ > activate# partial status: pi(activate#) = [1] pi(n____) = [] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) p2: __#(__(X,Y),Z) -> __#(Y,Z) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The set of usable rules consists of r1, r2, r3, r32 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: __#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,1) ___A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,2) nil_A() = (1,1) n_____A(x1,x2) = (0,0) precedence: __# = __ = nil = n____ partial status: pi(__#) = [] pi(__) = [] pi(nil) = [] pi(n____) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: __#_A(x1,x2) = (0,0) ___A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,1) nil_A() = (1,1) n_____A(x1,x2) = (0,0) precedence: __# = __ = nil = n____ partial status: pi(__#) = [] pi(__) = [] pi(nil) = [] pi(n____) = [] The next rules are strictly ordered: p2 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt()) -> tt() r5: U21(tt(),V2) -> U22(isList(activate(V2))) r6: U22(tt()) -> tt() r7: U31(tt()) -> tt() r8: U41(tt(),V2) -> U42(isNeList(activate(V2))) r9: U42(tt()) -> tt() r10: U51(tt(),V2) -> U52(isList(activate(V2))) r11: U52(tt()) -> tt() r12: U61(tt()) -> tt() r13: U71(tt(),P) -> U72(isPal(activate(P))) r14: U72(tt()) -> tt() r15: U81(tt()) -> tt() r16: isList(V) -> U11(isNeList(activate(V))) r17: isList(n__nil()) -> tt() r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) r19: isNeList(V) -> U31(isQid(activate(V))) r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) r22: isNePal(V) -> U61(isQid(activate(V))) r23: isNePal(n____(I,n____(P,I))) -> U71(isQid(activate(I)),activate(P)) r24: isPal(V) -> U81(isNePal(activate(V))) r25: isPal(n__nil()) -> tt() r26: isQid(n__a()) -> tt() r27: isQid(n__e()) -> tt() r28: isQid(n__i()) -> tt() r29: isQid(n__o()) -> tt() r30: isQid(n__u()) -> tt() r31: nil() -> n__nil() r32: __(X1,X2) -> n____(X1,X2) r33: a() -> n__a() r34: e() -> n__e() r35: i() -> n__i() r36: o() -> n__o() r37: u() -> n__u() r38: activate(n__nil()) -> nil() r39: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r40: activate(n__a()) -> a() r41: activate(n__e()) -> e() r42: activate(n__i()) -> i() r43: activate(n__o()) -> o() r44: activate(n__u()) -> u() r45: activate(X) -> X The set of usable rules consists of r1, r2, r3, r32 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: __#_A(x1,x2) = ((1,0),(1,0)) x1 ___A(x1,x2) = x1 + x2 + (2,3) nil_A() = (1,1) n_____A(x1,x2) = (1,1) precedence: __ > __# = nil = n____ partial status: pi(__#) = [] pi(__) = [1, 2] pi(nil) = [] pi(n____) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: __#_A(x1,x2) = (0,0) ___A(x1,x2) = x1 + x2 + (1,1) nil_A() = (1,1) n_____A(x1,x2) = (0,0) precedence: __# = __ = nil = n____ partial status: pi(__#) = [] pi(__) = [] pi(nil) = [] pi(n____) = [] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains.