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,__(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)) -> __(X1,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,__(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P))) p23: isNePal#(n____(I,__(P,I))) -> isQid#(activate(I)) p24: isNePal#(n____(I,__(P,I))) -> activate#(I) p25: isNePal#(n____(I,__(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)) -> __#(X1,X2) p30: activate#(n__isList(X)) -> isList#(X) p31: activate#(n__isNeList(X)) -> isNeList#(X) p32: activate#(n__isPal(X)) -> isPal#(X) p33: activate#(n__a()) -> a#() p34: activate#(n__e()) -> e#() p35: activate#(n__i()) -> i#() p36: activate#(n__o()) -> o#() p37: 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,__(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)) -> __(X1,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} {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,__(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: 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,__(P,I))) -> activate#(I) p22: isNePal#(n____(I,__(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,__(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)) -> __(X1,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 + 8 activate#_A(x1) = x1 + 2 n__isPal_A(x1) = x1 + 9 isNePal#_A(x1) = x1 + 3 activate_A(x1) = x1 + 4 n_____A(x1,x2) = x1 + x2 + 30 ___A(x1,x2) = x1 + x2 + 31 n__isNeList_A(x1) = x1 + 2 isNeList#_A(x1) = x1 + 3 n__isList_A(x1) = x1 + 9 isList#_A(x1) = x1 + 8 and#_A(x1,x2) = x2 + 3 isList_A(x1) = x1 + 10 tt_A() = 3 isNeList_A(x1) = x1 + 5 isQid_A(x1) = 4 isNePal_A(x1) = x1 + 5 and_A(x1,x2) = x1 + x2 + 2 nil_A() = 5 isPal_A(x1) = x1 + 10 n__nil_A() = 4 a_A() = 5 n__a_A() = 4 e_A() = 5 n__e_A() = 4 i_A() = 5 n__i_A() = 4 o_A() = 5 n__o_A() = 4 u_A() = 2 n__u_A() = 1 precedence: __ > n__isList > isList > isList# > isNePal# = activate = n____ = n__isNeList = tt = isNeList = isQid = and > nil = isPal = n__nil = a = n__a > isPal# = activate# = n__isPal = isNeList# = and# = isNePal = e = n__e = i = n__i = o = n__o = u = n__u partial status: pi(isPal#) = [1] pi(activate#) = [1] pi(n__isPal) = [] pi(isNePal#) = [1] pi(activate) = [1] pi(n____) = [2] pi(__) = [1, 2] pi(n__isNeList) = [1] pi(isNeList#) = [1] pi(n__isList) = [] pi(isList#) = [] pi(and#) = [2] pi(isList) = [] pi(tt) = [] pi(isNeList) = [1] pi(isQid) = [] pi(isNePal) = [1] pi(and) = [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: p2 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isPal#(V) -> activate#(V) p2: isPal#(V) -> isNePal#(activate(V)) p3: isNePal#(n____(I,__(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: 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)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) 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,__(P,I))) -> activate#(I) p21: isNePal#(n____(I,__(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,__(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)) -> __(X1,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, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(V) -> activate#(V) p3: activate#(n__isList(X)) -> isList#(X) p4: isList#(V) -> isNeList#(activate(V)) p5: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p6: and#(tt(),X) -> activate#(X) p7: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p8: isList#(V) -> activate#(V) p9: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2))) p10: isList#(n____(V1,V2)) -> isList#(activate(V1)) p11: isList#(n____(V1,V2)) -> activate#(V1) p12: isList#(n____(V1,V2)) -> activate#(V2) p13: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p14: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p15: isNeList#(n____(V1,V2)) -> activate#(V1) p16: isNeList#(n____(V1,V2)) -> activate#(V2) 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,__(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)) -> __(X1,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__isNeList_A(x1) = x1 isNeList#_A(x1) = x1 n__isList_A(x1) = x1 isList#_A(x1) = x1 activate_A(x1) = x1 n_____A(x1,x2) = x1 + x2 and#_A(x1,x2) = x1 + x2 isList_A(x1) = x1 tt_A() = 1 isNeList_A(x1) = x1 isNePal_A(x1) = x1 isQid_A(x1) = x1 ___A(x1,x2) = x1 + x2 and_A(x1,x2) = x2 n__isPal_A(x1) = x1 nil_A() = 2 isPal_A(x1) = x1 n__nil_A() = 2 n__a_A() = 2 n__e_A() = 2 n__i_A() = 2 n__o_A() = 2 n__u_A() = 2 a_A() = 2 e_A() = 2 i_A() = 2 o_A() = 2 u_A() = 2 precedence: nil = n__nil = n__a = n__e = a = e > n__i = i > n____ = __ > n__isPal = isPal > tt > n__isList = isList > isList# > isNePal > and > activate = o > n__isNeList = isNeList > isNeList# = and# > activate# > n__o = u > n__u > isQid partial status: pi(activate#) = [1] pi(n__isNeList) = [1] pi(isNeList#) = [1] pi(n__isList) = [1] pi(isList#) = [1] pi(activate) = [1] pi(n____) = [1, 2] pi(and#) = [1, 2] pi(isList) = [1] pi(tt) = [] pi(isNeList) = [1] pi(isNePal) = [1] pi(isQid) = [] pi(__) = [1, 2] pi(and) = [2] pi(n__isPal) = [1] pi(nil) = [] pi(isPal) = [1] pi(n__nil) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p9 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(V) -> activate#(V) p3: activate#(n__isList(X)) -> isList#(X) p4: isList#(V) -> isNeList#(activate(V)) p5: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p6: and#(tt(),X) -> activate#(X) p7: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p8: isList#(V) -> activate#(V) p9: isList#(n____(V1,V2)) -> isList#(activate(V1)) p10: isList#(n____(V1,V2)) -> activate#(V1) p11: isList#(n____(V1,V2)) -> activate#(V2) p12: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p13: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p14: isNeList#(n____(V1,V2)) -> activate#(V1) p15: isNeList#(n____(V1,V2)) -> activate#(V2) 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,__(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)) -> __(X1,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} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(n____(V1,V2)) -> activate#(V2) p3: activate#(n__isList(X)) -> isList#(X) p4: isList#(n____(V1,V2)) -> activate#(V2) p5: isList#(n____(V1,V2)) -> activate#(V1) p6: isList#(n____(V1,V2)) -> isList#(activate(V1)) p7: isList#(V) -> activate#(V) p8: isList#(V) -> isNeList#(activate(V)) p9: isNeList#(n____(V1,V2)) -> activate#(V1) p10: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p11: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p12: and#(tt(),X) -> activate#(X) 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,__(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)) -> __(X1,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 + 14 n__isNeList_A(x1) = x1 isNeList#_A(x1) = x1 + 14 n_____A(x1,x2) = x1 + x2 + 17 n__isList_A(x1) = x1 + 6 isList#_A(x1) = x1 + 19 activate_A(x1) = x1 + 4 and#_A(x1,x2) = x2 + 15 isNeList_A(x1) = x1 + 3 tt_A() = 1 isList_A(x1) = x1 + 8 isNePal_A(x1) = x1 + 5 isQid_A(x1) = 2 ___A(x1,x2) = x1 + x2 + 20 and_A(x1,x2) = x2 + 5 n__isPal_A(x1) = x1 + 10 nil_A() = 3 isPal_A(x1) = x1 + 10 n__nil_A() = 2 n__a_A() = 2 n__e_A() = 2 n__i_A() = 2 n__o_A() = 2 n__u_A() = 2 a_A() = 3 e_A() = 3 i_A() = 3 o_A() = 3 u_A() = 3 precedence: nil > n____ = and# > tt > n__nil = e = o > isNePal = and = n__isPal = isPal > n__isList = activate = isList > activate# = isNeList# = isList# > __ = n__i = i > n__o > n__isNeList = isNeList = isQid = n__a = n__e = n__u = a = u partial status: pi(activate#) = [] pi(n__isNeList) = [] pi(isNeList#) = [] pi(n____) = [2] pi(n__isList) = [] pi(isList#) = [] pi(activate) = [1] pi(and#) = [2] pi(isNeList) = [1] pi(tt) = [] pi(isList) = [] pi(isNePal) = [] pi(isQid) = [] pi(__) = [1, 2] pi(and) = [] pi(n__isPal) = [] pi(nil) = [] pi(isPal) = [] pi(n__nil) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] 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__isNeList(X)) -> isNeList#(X) 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#(V) -> activate#(V) p7: isList#(V) -> isNeList#(activate(V)) p8: isNeList#(n____(V1,V2)) -> activate#(V1) p9: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p10: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p11: and#(tt(),X) -> activate#(X) 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,__(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)) -> __(X1,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} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(V) -> activate#(V) p3: activate#(n__isList(X)) -> isList#(X) p4: isList#(V) -> isNeList#(activate(V)) p5: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p6: and#(tt(),X) -> activate#(X) p7: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p8: isList#(V) -> activate#(V) p9: isList#(n____(V1,V2)) -> isList#(activate(V1)) p10: isList#(n____(V1,V2)) -> activate#(V1) p11: isList#(n____(V1,V2)) -> activate#(V2) p12: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p13: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p14: isNeList#(n____(V1,V2)) -> 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,__(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)) -> __(X1,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 + 2 n__isNeList_A(x1) = x1 isNeList#_A(x1) = x1 + 2 n__isList_A(x1) = x1 + 12 isList#_A(x1) = x1 + 11 activate_A(x1) = x1 + 6 n_____A(x1,x2) = x1 + x2 + 28 and#_A(x1,x2) = x1 + x2 isList_A(x1) = x1 + 12 tt_A() = 3 isNeList_A(x1) = x1 + 5 isNePal_A(x1) = x1 + 5 isQid_A(x1) = 4 ___A(x1,x2) = x1 + x2 + 29 and_A(x1,x2) = x2 + 7 n__isPal_A(x1) = x1 + 12 nil_A() = 2 isPal_A(x1) = x1 + 12 n__nil_A() = 1 n__a_A() = 4 n__e_A() = 4 n__i_A() = 4 n__o_A() = 4 n__u_A() = 4 a_A() = 5 e_A() = 5 i_A() = 5 o_A() = 5 u_A() = 5 precedence: activate# = isNeList# > n____ = nil > and# = isQid = n__nil > isList# = n__a = n__e = n__i > n__isList = isList = __ = a = e = i > tt = n__o = o > n__isPal = isPal > n__isNeList = activate = isNeList = isNePal = and = n__u = u partial status: pi(activate#) = [] pi(n__isNeList) = [] pi(isNeList#) = [] pi(n__isList) = [1] pi(isList#) = [1] pi(activate) = [1] pi(n____) = [1] pi(and#) = [] pi(isList) = [1] pi(tt) = [] pi(isNeList) = [1] pi(isNePal) = [1] pi(isQid) = [] pi(__) = [1, 2] pi(and) = [2] pi(n__isPal) = [] pi(nil) = [] pi(isPal) = [] pi(n__nil) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p3 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(V) -> activate#(V) p3: isList#(V) -> isNeList#(activate(V)) p4: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p5: and#(tt(),X) -> activate#(X) p6: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p7: isList#(V) -> activate#(V) p8: isList#(n____(V1,V2)) -> isList#(activate(V1)) p9: isList#(n____(V1,V2)) -> activate#(V1) p10: isList#(n____(V1,V2)) -> activate#(V2) p11: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p12: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p13: isNeList#(n____(V1,V2)) -> 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,__(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)) -> __(X1,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} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(n____(V1,V2)) -> activate#(V1) p3: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p4: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2))) p5: and#(tt(),X) -> activate#(X) p6: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p7: isList#(n____(V1,V2)) -> activate#(V2) p8: isList#(n____(V1,V2)) -> activate#(V1) p9: isList#(n____(V1,V2)) -> isList#(activate(V1)) p10: isList#(V) -> activate#(V) p11: isList#(V) -> isNeList#(activate(V)) p12: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p13: 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,__(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)) -> __(X1,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__isNeList_A(x1) = x1 + 2 isNeList#_A(x1) = x1 + 6 n_____A(x1,x2) = x1 + x2 activate_A(x1) = x1 and#_A(x1,x2) = x1 + x2 + 1 isNeList_A(x1) = x1 + 2 n__isList_A(x1) = x1 + 2 tt_A() = 7 isList#_A(x1) = x1 + 6 isList_A(x1) = x1 + 2 isNePal_A(x1) = x1 + 2 isQid_A(x1) = x1 + 1 ___A(x1,x2) = x1 + x2 and_A(x1,x2) = x2 n__isPal_A(x1) = x1 + 2 nil_A() = 8 isPal_A(x1) = x1 + 2 n__nil_A() = 8 n__a_A() = 8 n__e_A() = 8 n__i_A() = 8 n__o_A() = 8 n__u_A() = 8 a_A() = 8 e_A() = 8 i_A() = 8 o_A() = 8 u_A() = 8 precedence: activate# = n__isNeList = isNeList# = n____ = activate = and# = isNeList = n__isList = tt = isList# = isList = isNePal = isQid = __ = and = n__isPal = nil = isPal = n__nil = n__a = n__e = n__i = n__o = n__u = a = e = i = o = u partial status: pi(activate#) = [] pi(n__isNeList) = [] pi(isNeList#) = [] pi(n____) = [] pi(activate) = [] pi(and#) = [] pi(isNeList) = [] pi(n__isList) = [] pi(tt) = [] pi(isList#) = [] pi(isList) = [] pi(isNePal) = [] pi(isQid) = [] pi(__) = [] pi(and) = [] pi(n__isPal) = [] pi(nil) = [] pi(isPal) = [] pi(n__nil) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(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__isNeList(X)) -> isNeList#(X) p2: isNeList#(n____(V1,V2)) -> activate#(V1) p3: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p4: and#(tt(),X) -> activate#(X) p5: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p6: isList#(n____(V1,V2)) -> activate#(V2) p7: isList#(n____(V1,V2)) -> activate#(V1) p8: isList#(n____(V1,V2)) -> isList#(activate(V1)) p9: isList#(V) -> activate#(V) p10: isList#(V) -> isNeList#(activate(V)) p11: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p12: 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,__(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)) -> __(X1,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} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(V) -> activate#(V) p3: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p4: and#(tt(),X) -> activate#(X) p5: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p6: isList#(V) -> isNeList#(activate(V)) p7: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p8: isNeList#(n____(V1,V2)) -> activate#(V1) p9: isList#(V) -> activate#(V) p10: isList#(n____(V1,V2)) -> isList#(activate(V1)) p11: isList#(n____(V1,V2)) -> activate#(V1) p12: isList#(n____(V1,V2)) -> activate#(V2) 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,__(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)) -> __(X1,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 + 13 n__isNeList_A(x1) = x1 + 4 isNeList#_A(x1) = x1 + 14 n_____A(x1,x2) = x1 + x2 + 20 and#_A(x1,x2) = x1 + x2 + 4 isList_A(x1) = x1 + 8 activate_A(x1) = x1 + 2 tt_A() = 14 isList#_A(x1) = x1 + 17 isNePal_A(x1) = x1 + 5 isQid_A(x1) = x1 + 2 ___A(x1,x2) = x1 + x2 + 20 and_A(x1,x2) = x1 + x2 + 1 n__isPal_A(x1) = x1 + 32 n__a_A() = 13 n__e_A() = 15 n__i_A() = 13 n__o_A() = 15 n__u_A() = 15 nil_A() = 15 isNeList_A(x1) = x1 + 5 n__isList_A(x1) = x1 + 7 isPal_A(x1) = x1 + 33 n__nil_A() = 15 a_A() = 14 e_A() = 16 i_A() = 13 o_A() = 16 u_A() = 16 precedence: __ > isQid = n__a = n__u > e = o = u > n____ > activate# = isNeList# = and# > isList# = i > and = n__e = n__i > isList = activate = tt = n__o = isNeList > n__isNeList = n__isList > n__isPal = isPal > isNePal = nil = n__nil = a partial status: pi(activate#) = [1] pi(n__isNeList) = [1] pi(isNeList#) = [1] pi(n____) = [2] pi(and#) = [2] pi(isList) = [1] pi(activate) = [1] pi(tt) = [] pi(isList#) = [] pi(isNePal) = [1] pi(isQid) = [1] pi(__) = [] pi(and) = [] pi(n__isPal) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(nil) = [] pi(isNeList) = [1] pi(n__isList) = [1] pi(isPal) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p12 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(V) -> activate#(V) p3: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p4: and#(tt(),X) -> activate#(X) p5: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p6: isList#(V) -> isNeList#(activate(V)) p7: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p8: isNeList#(n____(V1,V2)) -> activate#(V1) p9: isList#(V) -> activate#(V) p10: isList#(n____(V1,V2)) -> isList#(activate(V1)) p11: isList#(n____(V1,V2)) -> 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,__(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)) -> __(X1,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} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__isNeList(X)) -> isNeList#(X) p2: isNeList#(n____(V1,V2)) -> activate#(V1) p3: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p4: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p5: isList#(n____(V1,V2)) -> activate#(V1) p6: isList#(n____(V1,V2)) -> isList#(activate(V1)) p7: isList#(V) -> activate#(V) p8: isList#(V) -> isNeList#(activate(V)) p9: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p10: and#(tt(),X) -> activate#(X) p11: 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,__(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)) -> __(X1,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__isNeList_A(x1) = x1 + 3 isNeList#_A(x1) = x1 + 1 n_____A(x1,x2) = x1 + x2 + 20 activate_A(x1) = x1 + 2 isList#_A(x1) = x1 + 4 and#_A(x1,x2) = x2 + 1 isList_A(x1) = x1 + 7 tt_A() = 0 isNePal_A(x1) = x1 + 4 isQid_A(x1) = x1 + 1 ___A(x1,x2) = x1 + x2 + 21 and_A(x1,x2) = x1 + x2 + 2 n__isPal_A(x1) = x1 + 6 n__a_A() = 1 n__e_A() = 1 n__i_A() = 1 n__o_A() = 1 n__u_A() = 1 nil_A() = 1 isNeList_A(x1) = x1 + 4 n__isList_A(x1) = x1 + 6 isPal_A(x1) = x1 + 7 n__nil_A() = 1 a_A() = 2 e_A() = 2 i_A() = 2 o_A() = 2 u_A() = 2 precedence: activate# = and# > isQid = a > n__e > n__u = o > isList# > isNeList# = i = u > n__isNeList = isList = isNeList = e > n__isList > activate = isNePal = __ = and = n__isPal = n__a = n__i = n__o = nil = isPal = n__nil > n____ > tt partial status: pi(activate#) = [] pi(n__isNeList) = [] pi(isNeList#) = [1] pi(n____) = [2] pi(activate) = [] pi(isList#) = [1] pi(and#) = [2] pi(isList) = [] pi(tt) = [] pi(isNePal) = [] pi(isQid) = [] pi(__) = [] pi(and) = [] pi(n__isPal) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(nil) = [] pi(isNeList) = [] pi(n__isList) = [1] pi(isPal) = [] pi(n__nil) = [] 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)) -> activate#(V1) p2: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1)) p3: isNeList#(n____(V1,V2)) -> isList#(activate(V1)) p4: isList#(n____(V1,V2)) -> activate#(V1) p5: isList#(n____(V1,V2)) -> isList#(activate(V1)) p6: isList#(V) -> activate#(V) p7: isList#(V) -> isNeList#(activate(V)) p8: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2))) p9: and#(tt(),X) -> activate#(X) p10: 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,__(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)) -> __(X1,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: {p2, p3, p5, p7} -- 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,__(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)) -> __(X1,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 + 12 activate_A(x1) = x1 + 2 isList#_A(x1) = x1 + 4 and_A(x1,x2) = x2 + 2 tt_A() = 3 isNePal_A(x1) = 5 isQid_A(x1) = 4 ___A(x1,x2) = x1 + x2 + 13 n__isPal_A(x1) = 3 n__a_A() = 4 n__e_A() = 4 n__i_A() = 4 n__o_A() = 4 n__u_A() = 4 nil_A() = 5 isList_A(x1) = x1 + 8 isNeList_A(x1) = x1 + 5 n__nil_A() = 4 n__isList_A(x1) = x1 + 7 n__isNeList_A(x1) = x1 + 4 isPal_A(x1) = 5 a_A() = 5 e_A() = 5 i_A() = 5 o_A() = 5 u_A() = 5 precedence: isList# > isNeList# = n____ = activate = and = isNePal = isQid = __ = n__isPal = n__o = isList = isNeList = n__nil = n__isList = n__isNeList = isPal > tt = n__a = n__e = n__i = nil > n__u = a = i = o = u > e partial status: pi(isNeList#) = [1] pi(n____) = [] pi(activate) = [] pi(isList#) = [] pi(and) = [] pi(tt) = [] pi(isNePal) = [] pi(isQid) = [] pi(__) = [1, 2] pi(n__isPal) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(nil) = [] pi(isList) = [1] pi(isNeList) = [1] 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: p2 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> isNeList#(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,__(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)) -> __(X1,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} {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,__(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)) -> __(X1,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 + 14 activate_A(x1) = x1 + 3 and_A(x1,x2) = x2 + 4 tt_A() = 2 isNePal_A(x1) = x1 + 4 isQid_A(x1) = 3 ___A(x1,x2) = x1 + x2 + 15 n__isPal_A(x1) = x1 + 6 n__a_A() = 3 n__e_A() = 1 n__i_A() = 3 n__o_A() = 3 n__u_A() = 3 nil_A() = 4 isList_A(x1) = x1 + 8 isNeList_A(x1) = x1 + 4 n__nil_A() = 3 n__isList_A(x1) = x1 + 6 n__isNeList_A(x1) = x1 + 2 isPal_A(x1) = x1 + 8 a_A() = 4 e_A() = 2 i_A() = 4 o_A() = 4 u_A() = 4 precedence: activate = and = tt = isNePal = isQid = __ = n__a = n__i = n__o = n__u = nil = isList = isNeList = n__nil = n__isList = n__isNeList = isPal = i > isList# > n__isPal > n__e = e > u > n____ > a = o partial status: pi(isList#) = [1] pi(n____) = [] pi(activate) = [] pi(and) = [] pi(tt) = [] pi(isNePal) = [] pi(isQid) = [] pi(__) = [] pi(n__isPal) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] 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: isNeList#(n____(V1,V2)) -> isNeList#(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,__(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)) -> __(X1,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 + 16 activate_A(x1) = x1 + 3 and_A(x1,x2) = x2 + 4 tt_A() = 2 isNePal_A(x1) = x1 + 6 isQid_A(x1) = x1 + 2 ___A(x1,x2) = x1 + x2 + 17 n__isPal_A(x1) = x1 + 8 n__a_A() = 3 n__e_A() = 1 n__i_A() = 3 n__o_A() = 3 n__u_A() = 3 nil_A() = 2 isList_A(x1) = x1 + 10 isNeList_A(x1) = x1 + 6 n__nil_A() = 1 n__isList_A(x1) = x1 + 8 n__isNeList_A(x1) = x1 + 4 isPal_A(x1) = x1 + 10 a_A() = 4 e_A() = 2 i_A() = 4 o_A() = 4 u_A() = 4 precedence: isNeList# = n____ = activate = and = tt = isNePal = isQid = __ = n__isPal = n__a = n__e = n__i = n__o = n__u = nil = isList = isNeList = n__nil = n__isList = n__isNeList = isPal = a = e = i = o = u partial status: pi(isNeList#) = [] pi(n____) = [] pi(activate) = [] pi(and) = [] pi(tt) = [] pi(isNePal) = [] pi(isQid) = [] pi(__) = [] pi(n__isPal) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] 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: __#(__(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,__(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)) -> __(X1,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,__(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)) -> __(X1,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,__(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)) -> __(X1,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.