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: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^2 order: standard order interpretations: isPal#_A(x1) = ((1,1),(1,1)) x1 + (5,3) activate#_A(x1) = ((0,1),(0,1)) x1 n__isPal_A(x1) = ((1,1),(1,1)) x1 + (7,6) isNePal#_A(x1) = ((1,1),(1,1)) x1 + (1,0) activate_A(x1) = x1 + (2,1) n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (5,9) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (6,9) n__isNeList_A(x1) = ((0,1),(1,1)) x1 + (4,3) isNeList#_A(x1) = ((1,1),(1,1)) x1 + (1,0) n__isList_A(x1) = ((0,1),(1,1)) x1 + (6,6) isList#_A(x1) = ((1,1),(1,1)) x1 + (5,3) and#_A(x1,x2) = ((0,1),(0,1)) x2 + (5,0) isList_A(x1) = ((0,1),(1,1)) x1 + (7,7) tt_A() = (3,1) isNeList_A(x1) = ((0,1),(1,1)) x1 + (5,4) isQid_A(x1) = (4,1) isNePal_A(x1) = ((1,0),(1,0)) x1 + (5,5) and_A(x1,x2) = x1 + x2 + (0,1) nil_A() = (2,1) isPal_A(x1) = ((1,1),(1,1)) x1 + (8,7) n__nil_A() = (1,1) a_A() = (2,1) n__a_A() = (1,1) e_A() = (2,1) n__e_A() = (1,1) i_A() = (2,1) n__i_A() = (1,1) o_A() = (2,1) n__o_A() = (1,1) u_A() = (2,1) n__u_A() = (1,1) 2. matrix interpretations: carrier: N^2 order: standard order interpretations: isPal#_A(x1) = (0,0) activate#_A(x1) = (1,0) n__isPal_A(x1) = ((0,0),(1,1)) x1 + (1,4) isNePal#_A(x1) = ((0,1),(0,0)) x1 + (2,0) activate_A(x1) = x1 + (2,2) n_____A(x1,x2) = ((1,0),(1,0)) x1 + ((1,1),(0,1)) x2 + (2,4) ___A(x1,x2) = ((1,1),(0,0)) x1 + (5,3) n__isNeList_A(x1) = (1,1) isNeList#_A(x1) = ((0,1),(1,0)) x1 + (2,1) n__isList_A(x1) = (2,1) isList#_A(x1) = ((1,1),(0,1)) x1 + (0,1) and#_A(x1,x2) = (5,0) isList_A(x1) = (1,0) tt_A() = (11,6) isNeList_A(x1) = (2,0) isQid_A(x1) = (4,5) isNePal_A(x1) = x1 + (3,4) and_A(x1,x2) = x1 + (2,1) nil_A() = (1,3) isPal_A(x1) = ((1,1),(1,0)) x1 + (4,3) n__nil_A() = (2,4) a_A() = (4,3) n__a_A() = (1,4) e_A() = (4,3) n__e_A() = (1,4) i_A() = (4,3) n__i_A() = (1,4) o_A() = (4,3) n__o_A() = (1,4) u_A() = (4,3) n__u_A() = (1,4) The next rules are strictly ordered: p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23 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: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^2 order: standard order interpretations: __#_A(x1,x2) = ((1,1),(1,1)) x1 + x2 ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,1) nil_A() = (1,1) n_____A(x1,x2) = (0,0) 2. matrix interpretations: carrier: N^2 order: standard order interpretations: __#_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,0),(1,1)) x2 ___A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(0,0)) x2 + (1,2) nil_A() = (1,1) n_____A(x1,x2) = (2,3) The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains.