YES We show the termination of the TRS R: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(X,nil()) -> X __(nil(),X) -> X and(tt(),X) -> activate(X) isList(V) -> isNeList(activate(V)) isList(n__nil()) -> tt() isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isNeList(V) -> isQid(activate(V)) isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) -> 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) isList(X) -> n__isList(X) isNeList(X) -> n__isNeList(X) isPal(X) -> n__isPal(X) 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__isList(X)) -> isList(X) activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) 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: and#(tt(),X) -> activate#(X) p4: isList#(V) -> isNeList#(activate(V)) p5: isList#(V) -> activate#(V) p6: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p7: isList#(n____(V1,V2)) -> isList#(activate(V1)) p8: isList#(n____(V1,V2)) -> activate#(V1) p9: isList#(n____(V1,V2)) -> activate#(V2) p10: isNeList#(V) -> isQid#(activate(V)) p11: isNeList#(V) -> activate#(V) p12: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p13: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p14: isNeList#(n____(V1,V2)) -> activate#(V1) p15: isNeList#(n____(V1,V2)) -> activate#(V2) p16: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p17: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p18: isNeList#(n____(V1,V2)) -> activate#(V1) p19: isNeList#(n____(V1,V2)) -> activate#(V2) p20: isNePal#(V) -> isQid#(activate(V)) p21: isNePal#(V) -> activate#(V) p22: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p23: isNePal#(n____(I,n____(P,I))) -> isQid#(activate(I)) p24: isNePal#(n____(I,n____(P,I))) -> activate#(I) p25: isNePal#(n____(I,n____(P,I))) -> activate#(P) p26: isPal#(V) -> isNePal#(activate(V)) p27: isPal#(V) -> activate#(V) p28: activate#(n__nil()) -> nil#() p29: activate#(n____(X1,X2)) -> __#(activate(X1),activate(X2)) p30: activate#(n____(X1,X2)) -> activate#(X1) p31: activate#(n____(X1,X2)) -> activate#(X2) p32: activate#(n__isList(X)) -> isList#(X) p33: activate#(n__isNeList(X)) -> isNeList#(X) p34: activate#(n__isPal(X)) -> isPal#(X) p35: activate#(n__a()) -> a#() p36: activate#(n__e()) -> e#() p37: activate#(n__i()) -> i#() p38: activate#(n__o()) -> o#() p39: activate#(n__u()) -> u#() and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p3, p4, p5, p6, p7, p8, p9, p11, p12, p13, p14, p15, p16, p17, p18, p19, p21, p22, p24, p25, p26, p27, p30, p31, p32, p33, p34} {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isPal#(V) -> activate#(V) p2: activate#(n__isPal(X)) -> isPal#(X) p3: isPal#(V) -> isNePal#(activate(V)) p4: isNePal#(n____(I,n____(P,I))) -> activate#(P) p5: activate#(n__isNeList(X)) -> isNeList#(X) p6: isNeList#(n____(V1,V2)) -> activate#(V2) p7: activate#(n__isList(X)) -> isList#(X) p8: isList#(n____(V1,V2)) -> activate#(V2) p9: activate#(n____(X1,X2)) -> activate#(X2) p10: activate#(n____(X1,X2)) -> activate#(X1) p11: isList#(n____(V1,V2)) -> activate#(V1) p12: isList#(n____(V1,V2)) -> isList#(activate(V1)) p13: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p14: and#(tt(),X) -> activate#(X) p15: isList#(V) -> activate#(V) p16: isList#(V) -> isNeList#(activate(V)) p17: isNeList#(n____(V1,V2)) -> activate#(V1) p18: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p19: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p20: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p21: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p22: isNeList#(V) -> activate#(V) p23: isNePal#(n____(I,n____(P,I))) -> activate#(I) p24: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p25: isNePal#(V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: isPal#_A(x1) = x1 + 2 activate#_A(x1) = x1 + 2 n__isPal_A(x1) = x1 + 1 isNePal#_A(x1) = x1 + 2 activate_A(x1) = x1 n_____A(x1,x2) = x1 + x2 n__isNeList_A(x1) = x1 isNeList#_A(x1) = x1 + 2 n__isList_A(x1) = x1 isList#_A(x1) = x1 + 2 and#_A(x1,x2) = x1 + x2 isList_A(x1) = x1 tt_A() = 3 isNeList_A(x1) = x1 isQid_A(x1) = x1 isNePal_A(x1) = x1 + 1 and_A(x1,x2) = x1 + x2 ___A(x1,x2) = x1 + x2 nil_A() = 4 isPal_A(x1) = x1 + 1 n__nil_A() = 4 a_A() = 4 n__a_A() = 4 e_A() = 4 n__e_A() = 4 i_A() = 4 n__i_A() = 4 o_A() = 4 n__o_A() = 4 u_A() = 4 n__u_A() = 4 precedence: a = n__a > activate = isList > nil > n__isPal = isNeList = isNePal = isPal = i > and > isPal# = activate# = isNePal# = n____ = isNeList# = isList# = and# = __ = e = n__e > n__isList > tt = isQid > n__nil = n__i = u = n__u > o = n__o > n__isNeList partial status: pi(isPal#) = [1] pi(activate#) = [] pi(n__isPal) = [] pi(isNePal#) = [] pi(activate) = [1] pi(n____) = [1, 2] pi(n__isNeList) = [] pi(isNeList#) = [] pi(n__isList) = [1] pi(isList#) = [] pi(and#) = [] pi(isList) = [1] pi(tt) = [] pi(isNeList) = [] pi(isQid) = [] pi(isNePal) = [] pi(and) = [] pi(__) = [1, 2] pi(nil) = [] pi(isPal) = [] 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 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(n____(I,n____(P,I))) -> activate#(P) p4: activate#(n__isNeList(X)) -> isNeList#(X) p5: isNeList#(n____(V1,V2)) -> activate#(V2) p6: activate#(n__isList(X)) -> isList#(X) p7: isList#(n____(V1,V2)) -> activate#(V2) p8: activate#(n____(X1,X2)) -> activate#(X2) p9: activate#(n____(X1,X2)) -> activate#(X1) p10: isList#(n____(V1,V2)) -> activate#(V1) p11: isList#(n____(V1,V2)) -> isList#(activate(V1)) p12: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p13: and#(tt(),X) -> activate#(X) p14: isList#(V) -> activate#(V) p15: isList#(V) -> isNeList#(activate(V)) p16: isNeList#(n____(V1,V2)) -> activate#(V1) p17: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p18: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p19: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p20: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p21: isNeList#(V) -> activate#(V) p22: isNePal#(n____(I,n____(P,I))) -> activate#(I) p23: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p24: isNePal#(V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(V) -> activate#(V) p4: activate#(n____(X1,X2)) -> activate#(X1) p5: activate#(n____(X1,X2)) -> activate#(X2) p6: activate#(n__isList(X)) -> isList#(X) p7: isList#(V) -> isNeList#(activate(V)) p8: isNeList#(V) -> activate#(V) p9: activate#(n__isNeList(X)) -> isNeList#(X) p10: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p11: and#(tt(),X) -> activate#(X) p12: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p13: isList#(V) -> activate#(V) p14: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p15: isList#(n____(V1,V2)) -> isList#(activate(V1)) p16: isList#(n____(V1,V2)) -> activate#(V1) p17: isList#(n____(V1,V2)) -> activate#(V2) p18: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p19: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p20: isNeList#(n____(V1,V2)) -> activate#(V1) p21: isNeList#(n____(V1,V2)) -> activate#(V2) p22: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p23: isNePal#(n____(I,n____(P,I))) -> activate#(I) p24: isNePal#(n____(I,n____(P,I))) -> activate#(P) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: activate#_A(x1) = x1 n__isPal_A(x1) = x1 + 4 isPal#_A(x1) = x1 + 2 isNePal#_A(x1) = x1 + 1 activate_A(x1) = x1 n_____A(x1,x2) = x1 + x2 + 10 n__isList_A(x1) = x1 + 4 isList#_A(x1) = x1 + 3 isNeList#_A(x1) = x1 + 1 n__isNeList_A(x1) = x1 + 3 and#_A(x1,x2) = x1 + x2 + 1 isList_A(x1) = x1 + 4 tt_A() = 1 isNeList_A(x1) = x1 + 3 isQid_A(x1) = 2 isNePal_A(x1) = x1 + 3 and_A(x1,x2) = x2 + 1 ___A(x1,x2) = x1 + x2 + 10 nil_A() = 2 isPal_A(x1) = x1 + 4 n__nil_A() = 2 a_A() = 2 n__a_A() = 2 e_A() = 2 n__e_A() = 2 i_A() = 2 n__i_A() = 2 o_A() = 2 n__o_A() = 2 u_A() = 2 n__u_A() = 2 precedence: activate = n____ = isList = __ = e = u > and# > and > tt > n__isList = isQid = nil = n__nil = o > n__o > n__isPal = isPal = n__u > i > isList# = isNePal > activate# = isPal# = isNePal# = isNeList# = n__isNeList = isNeList = a = n__a = n__e = n__i partial status: pi(activate#) = [] pi(n__isPal) = [] pi(isPal#) = [] pi(isNePal#) = [] pi(activate) = [] pi(n____) = [] pi(n__isList) = [] pi(isList#) = [1] pi(isNeList#) = [] pi(n__isNeList) = [1] pi(and#) = [] pi(isList) = [] pi(tt) = [] pi(isNeList) = [1] pi(isQid) = [] pi(isNePal) = [1] pi(and) = [2] pi(__) = [] pi(nil) = [] pi(isPal) = [1] 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: p4 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(V) -> activate#(V) p4: activate#(n____(X1,X2)) -> activate#(X2) p5: activate#(n__isList(X)) -> isList#(X) p6: isList#(V) -> isNeList#(activate(V)) p7: isNeList#(V) -> activate#(V) p8: activate#(n__isNeList(X)) -> isNeList#(X) p9: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p10: and#(tt(),X) -> activate#(X) p11: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p12: isList#(V) -> activate#(V) p13: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p14: isList#(n____(V1,V2)) -> isList#(activate(V1)) p15: isList#(n____(V1,V2)) -> activate#(V1) p16: isList#(n____(V1,V2)) -> activate#(V2) p17: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p18: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p19: isNeList#(n____(V1,V2)) -> activate#(V1) p20: isNeList#(n____(V1,V2)) -> activate#(V2) p21: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p22: isNePal#(n____(I,n____(P,I))) -> activate#(I) p23: isNePal#(n____(I,n____(P,I))) -> activate#(P) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(n____(I,n____(P,I))) -> activate#(P) p4: activate#(n__isNeList(X)) -> isNeList#(X) p5: isNeList#(n____(V1,V2)) -> activate#(V2) p6: activate#(n__isList(X)) -> isList#(X) p7: isList#(n____(V1,V2)) -> activate#(V2) p8: activate#(n____(X1,X2)) -> activate#(X2) p9: isList#(n____(V1,V2)) -> activate#(V1) p10: isList#(n____(V1,V2)) -> isList#(activate(V1)) p11: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p12: and#(tt(),X) -> activate#(X) p13: isList#(V) -> activate#(V) p14: isList#(V) -> isNeList#(activate(V)) p15: isNeList#(n____(V1,V2)) -> activate#(V1) p16: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p17: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p18: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p19: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p20: isNeList#(V) -> activate#(V) p21: isNePal#(n____(I,n____(P,I))) -> activate#(I) p22: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p23: isNePal#(V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: activate#_A(x1) = x1 + 4 n__isPal_A(x1) = x1 isPal#_A(x1) = x1 + 4 isNePal#_A(x1) = x1 + 4 activate_A(x1) = x1 n_____A(x1,x2) = x1 + x2 n__isNeList_A(x1) = x1 + 2 isNeList#_A(x1) = x1 + 5 n__isList_A(x1) = x1 + 2 isList#_A(x1) = x1 + 5 and#_A(x1,x2) = x1 + x2 isList_A(x1) = x1 + 2 tt_A() = 4 isNeList_A(x1) = x1 + 2 isQid_A(x1) = x1 isNePal_A(x1) = x1 and_A(x1,x2) = x2 ___A(x1,x2) = x1 + x2 nil_A() = 5 isPal_A(x1) = x1 n__nil_A() = 5 a_A() = 5 n__a_A() = 5 e_A() = 5 n__e_A() = 5 i_A() = 5 n__i_A() = 5 o_A() = 5 n__o_A() = 5 u_A() = 5 n__u_A() = 5 precedence: activate# = n__isPal = isPal# = isNePal# = activate = n____ = n__isNeList = isNeList# = n__isList = isList# = and# = isList = tt = isNeList = isQid = isNePal = and = __ = nil = isPal = n__nil = a = n__a = e = n__e = i = n__i = o = n__o = u = n__u partial status: pi(activate#) = [] pi(n__isPal) = [] pi(isPal#) = [] pi(isNePal#) = [] pi(activate) = [] pi(n____) = [] pi(n__isNeList) = [] pi(isNeList#) = [] pi(n__isList) = [] pi(isList#) = [] pi(and#) = [] pi(isList) = [] pi(tt) = [] pi(isNeList) = [] pi(isQid) = [] pi(isNePal) = [] pi(and) = [] pi(__) = [] pi(nil) = [] pi(isPal) = [] 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: p17 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(n____(I,n____(P,I))) -> activate#(P) p4: activate#(n__isNeList(X)) -> isNeList#(X) p5: isNeList#(n____(V1,V2)) -> activate#(V2) p6: activate#(n__isList(X)) -> isList#(X) p7: isList#(n____(V1,V2)) -> activate#(V2) p8: activate#(n____(X1,X2)) -> activate#(X2) p9: isList#(n____(V1,V2)) -> activate#(V1) p10: isList#(n____(V1,V2)) -> isList#(activate(V1)) p11: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p12: and#(tt(),X) -> activate#(X) p13: isList#(V) -> activate#(V) p14: isList#(V) -> isNeList#(activate(V)) p15: isNeList#(n____(V1,V2)) -> activate#(V1) p16: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p17: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p18: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p19: isNeList#(V) -> activate#(V) p20: isNePal#(n____(I,n____(P,I))) -> activate#(I) p21: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p22: isNePal#(V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(V) -> activate#(V) p4: activate#(n____(X1,X2)) -> activate#(X2) p5: activate#(n__isList(X)) -> isList#(X) p6: isList#(V) -> isNeList#(activate(V)) p7: isNeList#(V) -> activate#(V) p8: activate#(n__isNeList(X)) -> isNeList#(X) p9: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p10: and#(tt(),X) -> activate#(X) p11: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p12: isList#(V) -> activate#(V) p13: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p14: isList#(n____(V1,V2)) -> isList#(activate(V1)) p15: isList#(n____(V1,V2)) -> activate#(V1) p16: isList#(n____(V1,V2)) -> activate#(V2) p17: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p18: isNeList#(n____(V1,V2)) -> activate#(V1) p19: isNeList#(n____(V1,V2)) -> activate#(V2) p20: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p21: isNePal#(n____(I,n____(P,I))) -> activate#(I) p22: isNePal#(n____(I,n____(P,I))) -> activate#(P) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: activate#_A(x1) = x1 n__isPal_A(x1) = x1 + 1 isPal#_A(x1) = x1 + 1 isNePal#_A(x1) = x1 + 1 activate_A(x1) = x1 n_____A(x1,x2) = x1 + x2 n__isList_A(x1) = x1 isList#_A(x1) = x1 isNeList#_A(x1) = x1 n__isNeList_A(x1) = x1 and#_A(x1,x2) = x2 isList_A(x1) = x1 tt_A() = 1 isQid_A(x1) = x1 isNePal_A(x1) = x1 + 1 and_A(x1,x2) = x1 + x2 ___A(x1,x2) = x1 + x2 nil_A() = 2 isNeList_A(x1) = x1 isPal_A(x1) = x1 + 1 n__nil_A() = 2 a_A() = 2 n__a_A() = 2 e_A() = 2 n__e_A() = 2 i_A() = 2 n__i_A() = 2 o_A() = 2 n__o_A() = 2 u_A() = 2 n__u_A() = 2 precedence: a = n__a > activate = n__isList = n__isNeList = isList = __ = nil = isNeList = isPal = n__nil = i = n__i = o = n__o = u = n__u > n__isPal > isNePal = and > n____ > e = n__e > activate# = isPal# = isNePal# = isList# = isNeList# = and# = tt = isQid partial status: pi(activate#) = [] pi(n__isPal) = [] pi(isPal#) = [] pi(isNePal#) = [] pi(activate) = [1] pi(n____) = [] pi(n__isList) = [] pi(isList#) = [] pi(isNeList#) = [] pi(n__isNeList) = [] pi(and#) = [] pi(isList) = [] pi(tt) = [] pi(isQid) = [] pi(isNePal) = [] pi(and) = [] pi(__) = [] pi(nil) = [] pi(isNeList) = [] pi(isPal) = [] 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: p22 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(V) -> activate#(V) p4: activate#(n____(X1,X2)) -> activate#(X2) p5: activate#(n__isList(X)) -> isList#(X) p6: isList#(V) -> isNeList#(activate(V)) p7: isNeList#(V) -> activate#(V) p8: activate#(n__isNeList(X)) -> isNeList#(X) p9: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p10: and#(tt(),X) -> activate#(X) p11: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p12: isList#(V) -> activate#(V) p13: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p14: isList#(n____(V1,V2)) -> isList#(activate(V1)) p15: isList#(n____(V1,V2)) -> activate#(V1) p16: isList#(n____(V1,V2)) -> activate#(V2) p17: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p18: isNeList#(n____(V1,V2)) -> activate#(V1) p19: isNeList#(n____(V1,V2)) -> activate#(V2) p20: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p21: isNePal#(n____(I,n____(P,I))) -> activate#(I) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(n____(I,n____(P,I))) -> activate#(I) p4: activate#(n__isNeList(X)) -> isNeList#(X) p5: isNeList#(n____(V1,V2)) -> activate#(V2) p6: activate#(n__isList(X)) -> isList#(X) p7: isList#(n____(V1,V2)) -> activate#(V2) p8: activate#(n____(X1,X2)) -> activate#(X2) p9: isList#(n____(V1,V2)) -> activate#(V1) p10: isList#(n____(V1,V2)) -> isList#(activate(V1)) p11: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p12: and#(tt(),X) -> activate#(X) p13: isList#(V) -> activate#(V) p14: isList#(V) -> isNeList#(activate(V)) p15: isNeList#(n____(V1,V2)) -> activate#(V1) p16: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p17: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p18: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p19: isNeList#(V) -> activate#(V) p20: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p21: isNePal#(V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: activate#_A(x1) = x1 + 6 n__isPal_A(x1) = x1 + 3 isPal#_A(x1) = x1 + 8 isNePal#_A(x1) = x1 + 7 activate_A(x1) = x1 n_____A(x1,x2) = x1 + x2 n__isNeList_A(x1) = x1 + 3 isNeList#_A(x1) = x1 + 8 n__isList_A(x1) = x1 + 3 isList#_A(x1) = x1 + 8 and#_A(x1,x2) = x1 + x2 + 1 isList_A(x1) = x1 + 3 tt_A() = 7 isQid_A(x1) = x1 + 2 isNePal_A(x1) = x1 + 3 and_A(x1,x2) = x2 ___A(x1,x2) = x1 + x2 nil_A() = 5 isNeList_A(x1) = x1 + 3 isPal_A(x1) = x1 + 3 n__nil_A() = 5 a_A() = 8 n__a_A() = 8 e_A() = 8 n__e_A() = 8 i_A() = 8 n__i_A() = 8 o_A() = 8 n__o_A() = 8 u_A() = 6 n__u_A() = 6 precedence: activate# = n__isPal = isPal# = isNePal# = activate = n____ = n__isNeList = isNeList# = n__isList = isList# = and# = isList = tt = isQid = isNePal = and = __ = nil = isNeList = isPal = n__nil = a = n__a = e = n__e = i = n__i = o = n__o = u = n__u partial status: pi(activate#) = [] pi(n__isPal) = [] pi(isPal#) = [] pi(isNePal#) = [] pi(activate) = [] pi(n____) = [] pi(n__isNeList) = [] pi(isNeList#) = [] pi(n__isList) = [] pi(isList#) = [] pi(and#) = [] pi(isList) = [] pi(tt) = [] pi(isQid) = [] pi(isNePal) = [] pi(and) = [] pi(__) = [] pi(nil) = [] pi(isNeList) = [] pi(isPal) = [] 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: p4 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(n____(I,n____(P,I))) -> activate#(I) p4: isNeList#(n____(V1,V2)) -> activate#(V2) p5: activate#(n__isList(X)) -> isList#(X) p6: isList#(n____(V1,V2)) -> activate#(V2) p7: activate#(n____(X1,X2)) -> activate#(X2) p8: isList#(n____(V1,V2)) -> activate#(V1) p9: isList#(n____(V1,V2)) -> isList#(activate(V1)) p10: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p11: and#(tt(),X) -> activate#(X) p12: isList#(V) -> activate#(V) p13: isList#(V) -> isNeList#(activate(V)) p14: isNeList#(n____(V1,V2)) -> activate#(V1) p15: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p16: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p17: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p18: isNeList#(V) -> activate#(V) p19: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p20: isNePal#(V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isPal(X)) -> isPal#(X) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(V) -> activate#(V) p4: activate#(n____(X1,X2)) -> activate#(X2) p5: activate#(n__isList(X)) -> isList#(X) p6: isList#(V) -> isNeList#(activate(V)) p7: isNeList#(V) -> activate#(V) p8: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p9: and#(tt(),X) -> activate#(X) p10: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p11: isList#(V) -> activate#(V) p12: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p13: isList#(n____(V1,V2)) -> isList#(activate(V1)) p14: isList#(n____(V1,V2)) -> activate#(V1) p15: isList#(n____(V1,V2)) -> activate#(V2) p16: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p17: isNeList#(n____(V1,V2)) -> activate#(V1) p18: isNeList#(n____(V1,V2)) -> activate#(V2) p19: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p20: isNePal#(n____(I,n____(P,I))) -> activate#(I) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: activate#_A(x1) = x1 n__isPal_A(x1) = x1 + 3 isPal#_A(x1) = x1 + 2 isNePal#_A(x1) = x1 + 1 activate_A(x1) = x1 n_____A(x1,x2) = x1 + x2 + 8 n__isList_A(x1) = x1 + 3 isList#_A(x1) = x1 + 2 isNeList#_A(x1) = x1 + 1 and#_A(x1,x2) = x1 + x2 isList_A(x1) = x1 + 3 n__isNeList_A(x1) = x1 tt_A() = 1 isQid_A(x1) = x1 isNePal_A(x1) = x1 + 1 and_A(x1,x2) = x1 + x2 + 1 ___A(x1,x2) = x1 + x2 + 8 nil_A() = 2 isNeList_A(x1) = x1 isPal_A(x1) = x1 + 3 n__nil_A() = 2 a_A() = 2 n__a_A() = 2 e_A() = 2 n__e_A() = 2 i_A() = 2 n__i_A() = 2 o_A() = 2 n__o_A() = 2 u_A() = 2 n__u_A() = 2 precedence: activate# = isNeList# > isNePal# = n__isList = isList > n__isPal = isPal > activate = n____ = and# = n__isNeList = isQid = isNePal = __ = isNeList = a = n__a > i > isList# = tt = and = nil = n__nil = n__i = o = n__o > e > n__e > u = n__u > isPal# partial status: pi(activate#) = [] pi(n__isPal) = [1] pi(isPal#) = [] pi(isNePal#) = [] pi(activate) = [1] pi(n____) = [] pi(n__isList) = [] pi(isList#) = [] pi(isNeList#) = [] pi(and#) = [2] pi(isList) = [] pi(n__isNeList) = [] pi(tt) = [] pi(isQid) = [] pi(isNePal) = [1] pi(and) = [2] pi(__) = [1, 2] pi(nil) = [] pi(isNeList) = [] pi(isPal) = [1] 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 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isPal#(V) -> isNePal#(activate(V)) p2: isNePal#(V) -> activate#(V) p3: activate#(n____(X1,X2)) -> activate#(X2) p4: activate#(n__isList(X)) -> isList#(X) p5: isList#(V) -> isNeList#(activate(V)) p6: isNeList#(V) -> activate#(V) p7: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p8: and#(tt(),X) -> activate#(X) p9: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p10: isList#(V) -> activate#(V) p11: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p12: isList#(n____(V1,V2)) -> isList#(activate(V1)) p13: isList#(n____(V1,V2)) -> activate#(V1) p14: isList#(n____(V1,V2)) -> activate#(V2) p15: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p16: isNeList#(n____(V1,V2)) -> activate#(V1) p17: isNeList#(n____(V1,V2)) -> activate#(V2) p18: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p19: isNePal#(n____(I,n____(P,I))) -> activate#(I) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n____(X1,X2)) -> activate#(X2) p2: activate#(n__isList(X)) -> isList#(X) p3: isList#(n____(V1,V2)) -> activate#(V2) p4: isList#(n____(V1,V2)) -> activate#(V1) p5: isList#(n____(V1,V2)) -> isList#(activate(V1)) p6: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p7: and#(tt(),X) -> activate#(X) p8: isList#(V) -> activate#(V) p9: isList#(V) -> isNeList#(activate(V)) p10: isNeList#(n____(V1,V2)) -> activate#(V2) p11: isNeList#(n____(V1,V2)) -> activate#(V1) p12: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p13: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p14: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p15: isNeList#(V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: activate#_A(x1) = x1 + 1 n_____A(x1,x2) = x1 + x2 + 8 n__isList_A(x1) = x1 + 5 isList#_A(x1) = x1 + 3 activate_A(x1) = x1 and#_A(x1,x2) = x2 + 2 isList_A(x1) = x1 + 5 tt_A() = 2 isNeList#_A(x1) = x1 + 2 n__isNeList_A(x1) = x1 + 4 isNePal_A(x1) = x1 + 4 isQid_A(x1) = 3 and_A(x1,x2) = x2 + 2 n__isPal_A(x1) = x1 + 5 n__a_A() = 3 n__e_A() = 3 n__i_A() = 3 n__o_A() = 3 n__u_A() = 3 ___A(x1,x2) = x1 + x2 + 8 nil_A() = 1 isNeList_A(x1) = x1 + 4 isPal_A(x1) = x1 + 5 n__nil_A() = 1 a_A() = 3 e_A() = 3 i_A() = 3 o_A() = 3 u_A() = 3 precedence: and > n__a = n__e = nil = n__nil = a = e > isNeList# > n__isNeList = isQid = isNeList > isList# > activate# = activate = and# = isList = n__isPal = __ = isPal > n____ > n__isList = tt > isNePal = n__i = n__o = n__u = i = o = u partial status: pi(activate#) = [] pi(n____) = [1, 2] pi(n__isList) = [] pi(isList#) = [] pi(activate) = [1] pi(and#) = [] pi(isList) = [] pi(tt) = [] pi(isNeList#) = [1] pi(n__isNeList) = [1] pi(isNePal) = [1] pi(isQid) = [] pi(and) = [] pi(n__isPal) = [1] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(__) = [1, 2] pi(nil) = [] pi(isNeList) = [1] pi(isPal) = [1] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] 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: activate#(n____(X1,X2)) -> activate#(X2) p2: isList#(n____(V1,V2)) -> activate#(V2) p3: isList#(n____(V1,V2)) -> activate#(V1) p4: isList#(n____(V1,V2)) -> isList#(activate(V1)) p5: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p6: and#(tt(),X) -> activate#(X) p7: isList#(V) -> activate#(V) p8: isList#(V) -> isNeList#(activate(V)) p9: isNeList#(n____(V1,V2)) -> activate#(V2) p10: isNeList#(n____(V1,V2)) -> activate#(V1) p11: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p12: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p13: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p14: isNeList#(V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p4, p8, p11, p12} {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p2: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p3: isList#(V) -> isNeList#(activate(V)) p4: isList#(n____(V1,V2)) -> isList#(activate(V1)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: isNeList#_A(x1) = x1 + 1 n_____A(x1,x2) = x1 + x2 + 5 activate_A(x1) = x1 isList#_A(x1) = x1 + 2 and_A(x1,x2) = x2 tt_A() = 1 isNePal_A(x1) = 3 isQid_A(x1) = 2 n__isPal_A(x1) = 3 n__a_A() = 2 n__e_A() = 2 n__i_A() = 2 n__o_A() = 2 n__u_A() = 2 ___A(x1,x2) = x1 + x2 + 5 nil_A() = 2 isList_A(x1) = x1 + 4 isNeList_A(x1) = x1 + 3 n__nil_A() = 2 n__isList_A(x1) = x1 + 4 n__isNeList_A(x1) = x1 + 3 isPal_A(x1) = 3 a_A() = 2 e_A() = 2 i_A() = 2 o_A() = 2 u_A() = 2 precedence: isNeList# = activate = isList# = and = tt = isNePal = n__isPal = n__a = n__e = n__o = n__u = __ = nil = isList = isNeList = n__nil = n__isList = isPal = a = e = i = o = u > n__i > n____ > isQid > n__isNeList partial status: pi(isNeList#) = [] pi(n____) = [2] pi(activate) = [] pi(isList#) = [] pi(and) = [] pi(tt) = [] pi(isNePal) = [] pi(isQid) = [] pi(n__isPal) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(__) = [] pi(nil) = [] pi(isList) = [] pi(isNeList) = [] pi(n__nil) = [] pi(n__isList) = [] pi(n__isNeList) = [1] pi(isPal) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p2: isList#(V) -> isNeList#(activate(V)) p3: isList#(n____(V1,V2)) -> isList#(activate(V1)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2, p3} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p2: isList#(n____(V1,V2)) -> isList#(activate(V1)) p3: isList#(V) -> isNeList#(activate(V)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: isNeList#_A(x1) = x1 + 1 n_____A(x1,x2) = x1 + x2 + 7 isList#_A(x1) = x1 + 2 activate_A(x1) = x1 and_A(x1,x2) = x2 + 1 tt_A() = 2 isNePal_A(x1) = x1 + 4 isQid_A(x1) = 3 n__isPal_A(x1) = x1 + 5 n__a_A() = 1 n__e_A() = 1 n__i_A() = 1 n__o_A() = 1 n__u_A() = 1 ___A(x1,x2) = x1 + x2 + 7 nil_A() = 3 isList_A(x1) = x1 + 5 isNeList_A(x1) = x1 + 4 n__nil_A() = 3 n__isList_A(x1) = x1 + 5 n__isNeList_A(x1) = x1 + 4 isPal_A(x1) = x1 + 5 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 precedence: isNeList# = isList# = tt = n__isPal = nil = isList = isNeList = n__nil = n__isList = n__isNeList = isPal > activate = and = isNePal = __ > n____ = a > n__a > n__e = e > i > n__i > isQid = n__o = n__u = o = u partial status: pi(isNeList#) = [] pi(n____) = [2] pi(isList#) = [1] pi(activate) = [1] pi(and) = [2] pi(tt) = [] pi(isNePal) = [1] pi(isQid) = [] pi(n__isPal) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(__) = [1, 2] pi(nil) = [] pi(isList) = [1] pi(isNeList) = [] pi(n__nil) = [] pi(n__isList) = [1] pi(n__isNeList) = [] pi(isPal) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isList#(n____(V1,V2)) -> isList#(activate(V1)) p2: isList#(V) -> isNeList#(activate(V)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: 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: isList#(n____(V1,V2)) -> isList#(activate(V1)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: isList#_A(x1) = x1 + 1 n_____A(x1,x2) = x1 + x2 + 6 activate_A(x1) = x1 and_A(x1,x2) = x2 tt_A() = 2 isNePal_A(x1) = x1 + 4 isQid_A(x1) = 3 n__isPal_A(x1) = x1 + 5 n__a_A() = 0 n__e_A() = 0 n__i_A() = 1 n__o_A() = 0 n__u_A() = 3 ___A(x1,x2) = x1 + x2 + 6 nil_A() = 1 isList_A(x1) = 4 isNeList_A(x1) = 4 n__nil_A() = 1 n__isList_A(x1) = 4 n__isNeList_A(x1) = 4 isPal_A(x1) = x1 + 5 a_A() = 0 e_A() = 0 i_A() = 1 o_A() = 0 u_A() = 3 precedence: activate = and = isNePal = isQid = n__isPal = n__a = n__u = nil = isList = isNeList = n__isNeList = isPal = a = i = u > isList# > tt > n__i > n____ = __ = n__nil = n__isList > o > n__o > n__e = e partial status: pi(isList#) = [1] pi(n____) = [] pi(activate) = [] pi(and) = [] pi(tt) = [] pi(isNePal) = [] pi(isQid) = [] pi(n__isPal) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(__) = [1, 2] pi(nil) = [] pi(isList) = [] pi(isNeList) = [] pi(n__nil) = [] pi(n__isList) = [] pi(n__isNeList) = [] pi(isPal) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p1 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) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: activate#_A(x1) = x1 n_____A(x1,x2) = x1 + x2 + 1 precedence: activate# = n____ partial status: pi(activate#) = [] pi(n____) = [2] The next rules are strictly ordered: p1 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: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r21 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: __#_A(x1,x2) = x1 + 1 ___A(x1,x2) = x1 + x2 + 2 nil_A() = 1 n_____A(x1,x2) = x2 precedence: __# = __ = nil = n____ partial status: pi(__#) = [1] pi(__) = [1, 2] 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: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: 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: and(tt(),X) -> activate(X) r5: isList(V) -> isNeList(activate(V)) r6: isList(n__nil()) -> tt() r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) r8: isNeList(V) -> isQid(activate(V)) r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) r11: isNePal(V) -> isQid(activate(V)) r12: isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) r13: isPal(V) -> isNePal(activate(V)) r14: isPal(n__nil()) -> tt() r15: isQid(n__a()) -> tt() r16: isQid(n__e()) -> tt() r17: isQid(n__i()) -> tt() r18: isQid(n__o()) -> tt() r19: isQid(n__u()) -> tt() r20: nil() -> n__nil() r21: __(X1,X2) -> n____(X1,X2) r22: isList(X) -> n__isList(X) r23: isNeList(X) -> n__isNeList(X) r24: isPal(X) -> n__isPal(X) r25: a() -> n__a() r26: e() -> n__e() r27: i() -> n__i() r28: o() -> n__o() r29: u() -> n__u() r30: activate(n__nil()) -> nil() r31: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r32: activate(n__isList(X)) -> isList(X) r33: activate(n__isNeList(X)) -> isNeList(X) r34: activate(n__isPal(X)) -> isPal(X) r35: activate(n__a()) -> a() r36: activate(n__e()) -> e() r37: activate(n__i()) -> i() r38: activate(n__o()) -> o() r39: activate(n__u()) -> u() r40: activate(X) -> X The set of usable rules consists of r1, r2, r3, r21 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: __#_A(x1,x2) = x1 ___A(x1,x2) = x1 + x2 + 1 nil_A() = 1 n_____A(x1,x2) = 0 precedence: __# = __ = nil = n____ partial status: pi(__#) = [1] pi(__) = [1, 2] pi(nil) = [] pi(n____) = [] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains.