YES We show the termination of the TRS R: __(__(X,Y),Z) -> __(X,__(Y,Z)) __(X,nil()) -> X __(nil(),X) -> X U11(tt(),V) -> U12(isNeList(activate(V))) U12(tt()) -> tt() U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) U22(tt(),V2) -> U23(isList(activate(V2))) U23(tt()) -> tt() U31(tt(),V) -> U32(isQid(activate(V))) U32(tt()) -> tt() U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) U42(tt(),V2) -> U43(isNeList(activate(V2))) U43(tt()) -> tt() U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) U52(tt(),V2) -> U53(isList(activate(V2))) U53(tt()) -> tt() U61(tt(),V) -> U62(isQid(activate(V))) U62(tt()) -> tt() U71(tt(),V) -> U72(isNePal(activate(V))) U72(tt()) -> tt() and(tt(),X) -> activate(X) isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isList(n__nil()) -> tt() isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) isPal(n__nil()) -> tt() isPalListKind(n__a()) -> tt() isPalListKind(n__e()) -> tt() isPalListKind(n__i()) -> tt() isPalListKind(n__nil()) -> tt() isPalListKind(n__o()) -> tt() isPalListKind(n__u()) -> tt() isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) 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) isPalListKind(X) -> n__isPalListKind(X) and(X1,X2) -> n__and(X1,X2) 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__isPalListKind(X)) -> isPalListKind(X) activate(n__and(X1,X2)) -> and(activate(X1),X2) 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: U11#(tt(),V) -> U12#(isNeList(activate(V))) p4: U11#(tt(),V) -> isNeList#(activate(V)) p5: U11#(tt(),V) -> activate#(V) p6: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2)) p7: U21#(tt(),V1,V2) -> isList#(activate(V1)) p8: U21#(tt(),V1,V2) -> activate#(V1) p9: U21#(tt(),V1,V2) -> activate#(V2) p10: U22#(tt(),V2) -> U23#(isList(activate(V2))) p11: U22#(tt(),V2) -> isList#(activate(V2)) p12: U22#(tt(),V2) -> activate#(V2) p13: U31#(tt(),V) -> U32#(isQid(activate(V))) p14: U31#(tt(),V) -> isQid#(activate(V)) p15: U31#(tt(),V) -> activate#(V) p16: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2)) p17: U41#(tt(),V1,V2) -> isList#(activate(V1)) p18: U41#(tt(),V1,V2) -> activate#(V1) p19: U41#(tt(),V1,V2) -> activate#(V2) p20: U42#(tt(),V2) -> U43#(isNeList(activate(V2))) p21: U42#(tt(),V2) -> isNeList#(activate(V2)) p22: U42#(tt(),V2) -> activate#(V2) p23: U51#(tt(),V1,V2) -> U52#(isNeList(activate(V1)),activate(V2)) p24: U51#(tt(),V1,V2) -> isNeList#(activate(V1)) p25: U51#(tt(),V1,V2) -> activate#(V1) p26: U51#(tt(),V1,V2) -> activate#(V2) p27: U52#(tt(),V2) -> U53#(isList(activate(V2))) p28: U52#(tt(),V2) -> isList#(activate(V2)) p29: U52#(tt(),V2) -> activate#(V2) p30: U61#(tt(),V) -> U62#(isQid(activate(V))) p31: U61#(tt(),V) -> isQid#(activate(V)) p32: U61#(tt(),V) -> activate#(V) p33: U71#(tt(),V) -> U72#(isNePal(activate(V))) p34: U71#(tt(),V) -> isNePal#(activate(V)) p35: U71#(tt(),V) -> activate#(V) p36: and#(tt(),X) -> activate#(X) p37: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V)) p38: isList#(V) -> isPalListKind#(activate(V)) p39: isList#(V) -> activate#(V) p40: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) p41: isList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) p42: isList#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p43: isList#(n____(V1,V2)) -> activate#(V1) p44: isList#(n____(V1,V2)) -> activate#(V2) p45: isNeList#(V) -> U31#(isPalListKind(activate(V)),activate(V)) p46: isNeList#(V) -> isPalListKind#(activate(V)) p47: isNeList#(V) -> activate#(V) p48: isNeList#(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) p49: isNeList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) p50: isNeList#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p51: isNeList#(n____(V1,V2)) -> activate#(V1) p52: isNeList#(n____(V1,V2)) -> activate#(V2) p53: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) p54: isNeList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) p55: isNeList#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p56: isNeList#(n____(V1,V2)) -> activate#(V1) p57: isNeList#(n____(V1,V2)) -> activate#(V2) p58: isNePal#(V) -> U61#(isPalListKind(activate(V)),activate(V)) p59: isNePal#(V) -> isPalListKind#(activate(V)) p60: isNePal#(V) -> activate#(V) p61: isNePal#(n____(I,n____(P,I))) -> and#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) p62: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPalListKind(activate(I))) p63: isNePal#(n____(I,n____(P,I))) -> isQid#(activate(I)) p64: isNePal#(n____(I,n____(P,I))) -> activate#(I) p65: isNePal#(n____(I,n____(P,I))) -> activate#(P) p66: isPal#(V) -> U71#(isPalListKind(activate(V)),activate(V)) p67: isPal#(V) -> isPalListKind#(activate(V)) p68: isPal#(V) -> activate#(V) p69: isPalListKind#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) p70: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p71: isPalListKind#(n____(V1,V2)) -> activate#(V1) p72: isPalListKind#(n____(V1,V2)) -> activate#(V2) p73: activate#(n__nil()) -> nil#() p74: activate#(n____(X1,X2)) -> __#(activate(X1),activate(X2)) p75: activate#(n____(X1,X2)) -> activate#(X1) p76: activate#(n____(X1,X2)) -> activate#(X2) p77: activate#(n__isPalListKind(X)) -> isPalListKind#(X) p78: activate#(n__and(X1,X2)) -> and#(activate(X1),X2) p79: activate#(n__and(X1,X2)) -> activate#(X1) p80: activate#(n__isPal(X)) -> isPal#(X) p81: activate#(n__a()) -> a#() p82: activate#(n__e()) -> e#() p83: activate#(n__i()) -> i#() p84: activate#(n__o()) -> o#() p85: activate#(n__u()) -> u#() and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isNeList(activate(V))) r5: U12(tt()) -> tt() r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) r7: U22(tt(),V2) -> U23(isList(activate(V2))) r8: U23(tt()) -> tt() r9: U31(tt(),V) -> U32(isQid(activate(V))) r10: U32(tt()) -> tt() r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) r12: U42(tt(),V2) -> U43(isNeList(activate(V2))) r13: U43(tt()) -> tt() r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) r15: U52(tt(),V2) -> U53(isList(activate(V2))) r16: U53(tt()) -> tt() r17: U61(tt(),V) -> U62(isQid(activate(V))) r18: U62(tt()) -> tt() r19: U71(tt(),V) -> U72(isNePal(activate(V))) r20: U72(tt()) -> tt() r21: and(tt(),X) -> activate(X) r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r23: isList(n__nil()) -> tt() r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) r31: isPal(n__nil()) -> tt() r32: isPalListKind(n__a()) -> tt() r33: isPalListKind(n__e()) -> tt() r34: isPalListKind(n__i()) -> tt() r35: isPalListKind(n__nil()) -> tt() r36: isPalListKind(n__o()) -> tt() r37: isPalListKind(n__u()) -> tt() r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) r39: isQid(n__a()) -> tt() r40: isQid(n__e()) -> tt() r41: isQid(n__i()) -> tt() r42: isQid(n__o()) -> tt() r43: isQid(n__u()) -> tt() r44: nil() -> n__nil() r45: __(X1,X2) -> n____(X1,X2) r46: isPalListKind(X) -> n__isPalListKind(X) r47: and(X1,X2) -> n__and(X1,X2) r48: isPal(X) -> n__isPal(X) r49: a() -> n__a() r50: e() -> n__e() r51: i() -> n__i() r52: o() -> n__o() r53: u() -> n__u() r54: activate(n__nil()) -> nil() r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r56: activate(n__isPalListKind(X)) -> isPalListKind(X) r57: activate(n__and(X1,X2)) -> and(activate(X1),X2) r58: activate(n__isPal(X)) -> isPal(X) r59: activate(n__a()) -> a() r60: activate(n__e()) -> e() r61: activate(n__i()) -> i() r62: activate(n__o()) -> o() r63: activate(n__u()) -> u() r64: activate(X) -> X The estimated dependency graph contains the following SCCs: {p4, p6, p7, p11, p16, p17, p21, p23, p24, p28, p37, p40, p48, p53} {p32, p34, p35, p36, p58, p59, p60, p61, p62, p64, p65, p66, p67, p68, p69, p70, p71, p72, p75, p76, p77, p78, p79, p80} {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: U52#(tt(),V2) -> isList#(activate(V2)) p2: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) p3: U21#(tt(),V1,V2) -> isList#(activate(V1)) p4: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V)) p5: U11#(tt(),V) -> isNeList#(activate(V)) p6: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) p7: U51#(tt(),V1,V2) -> isNeList#(activate(V1)) p8: isNeList#(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) p9: U41#(tt(),V1,V2) -> isList#(activate(V1)) p10: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2)) p11: U42#(tt(),V2) -> isNeList#(activate(V2)) p12: U51#(tt(),V1,V2) -> U52#(isNeList(activate(V1)),activate(V2)) p13: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2)) p14: U22#(tt(),V2) -> isList#(activate(V2)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isNeList(activate(V))) r5: U12(tt()) -> tt() r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) r7: U22(tt(),V2) -> U23(isList(activate(V2))) r8: U23(tt()) -> tt() r9: U31(tt(),V) -> U32(isQid(activate(V))) r10: U32(tt()) -> tt() r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) r12: U42(tt(),V2) -> U43(isNeList(activate(V2))) r13: U43(tt()) -> tt() r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) r15: U52(tt(),V2) -> U53(isList(activate(V2))) r16: U53(tt()) -> tt() r17: U61(tt(),V) -> U62(isQid(activate(V))) r18: U62(tt()) -> tt() r19: U71(tt(),V) -> U72(isNePal(activate(V))) r20: U72(tt()) -> tt() r21: and(tt(),X) -> activate(X) r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r23: isList(n__nil()) -> tt() r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) r31: isPal(n__nil()) -> tt() r32: isPalListKind(n__a()) -> tt() r33: isPalListKind(n__e()) -> tt() r34: isPalListKind(n__i()) -> tt() r35: isPalListKind(n__nil()) -> tt() r36: isPalListKind(n__o()) -> tt() r37: isPalListKind(n__u()) -> tt() r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) r39: isQid(n__a()) -> tt() r40: isQid(n__e()) -> tt() r41: isQid(n__i()) -> tt() r42: isQid(n__o()) -> tt() r43: isQid(n__u()) -> tt() r44: nil() -> n__nil() r45: __(X1,X2) -> n____(X1,X2) r46: isPalListKind(X) -> n__isPalListKind(X) r47: and(X1,X2) -> n__and(X1,X2) r48: isPal(X) -> n__isPal(X) r49: a() -> n__a() r50: e() -> n__e() r51: i() -> n__i() r52: o() -> n__o() r53: u() -> n__u() r54: activate(n__nil()) -> nil() r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r56: activate(n__isPalListKind(X)) -> isPalListKind(X) r57: activate(n__and(X1,X2)) -> and(activate(X1),X2) r58: activate(n__isPal(X)) -> isPal(X) r59: activate(n__a()) -> a() r60: activate(n__e()) -> e() r61: activate(n__i()) -> i() r62: activate(n__o()) -> o() r63: activate(n__u()) -> u() r64: 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, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: U52#_A(x1,x2) = ((1,1),(0,0)) x2 + (4,3) tt_A() = (2,1) isList#_A(x1) = ((1,1),(0,0)) x1 + (3,3) activate_A(x1) = x1 n_____A(x1,x2) = x1 + x2 + (21,0) U21#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (5,3) and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1) isPalListKind_A(x1) = ((1,1),(1,1)) x1 + (12,3) n__isPalListKind_A(x1) = ((1,1),(1,1)) x1 + (12,3) U11#_A(x1,x2) = ((1,1),(0,0)) x2 + (0,3) isNeList#_A(x1) = ((1,1),(0,0)) x1 + (0,3) U51#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (14,3) U41#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (20,3) U42#_A(x1,x2) = ((1,1),(0,0)) x2 + (0,3) isList_A(x1) = ((1,1),(1,1)) x1 + (23,4) isNeList_A(x1) = ((1,1),(1,1)) x1 + (13,4) U22#_A(x1,x2) = ((1,1),(0,0)) x2 + (4,3) U62_A(x1) = (3,2) U61_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(0,0)) x2 + (4,1) isQid_A(x1) = ((0,1),(0,1)) x1 + (9,1) U23_A(x1) = (3,1) U43_A(x1) = (3,2) U53_A(x1) = (3,2) U72_A(x1) = (3,2) isNePal_A(x1) = ((1,1),(1,1)) x1 + (12,4) n__and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1) n__isPal_A(x1) = ((0,0),(1,1)) x1 + (5,2) U12_A(x1) = (2,1) U22_A(x1,x2) = (4,2) U32_A(x1) = (3,1) U42_A(x1,x2) = (4,2) U52_A(x1,x2) = (4,2) U71_A(x1,x2) = ((0,0),(1,0)) x2 + (4,2) n__a_A() = (0,0) n__e_A() = (1,0) n__i_A() = (0,0) n__o_A() = (1,0) n__u_A() = (0,0) ___A(x1,x2) = x1 + x2 + (21,0) nil_A() = (1,0) U11_A(x1,x2) = x1 + (1,0) U21_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x3 + (5,0) U31_A(x1,x2) = ((1,0),(0,0)) x2 + (10,1) U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(0,0)) x3 + (22,1) U51_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,0),(0,0)) x2 + (22,1) isPal_A(x1) = ((0,0),(1,1)) x1 + (5,2) n__nil_A() = (1,0) a_A() = (0,0) e_A() = (1,0) i_A() = (0,0) o_A() = (1,0) u_A() = (0,0) precedence: nil = n__nil > U31 > U22 > isList = U21 > isPalListKind = n__isPalListKind > U42# > isQid > U11 > U12 > tt = U43 = U32 = U42 > U41 > U53 > and = U62 = U61 = isNePal = n__and > activate = u > __ > U11# > U23 = isPal = o > U22# > n__o > isNeList# > U41# > U51# > U52# > n__u > e > n__isPal > i > n__i > isList# > n__e > n____ = a > isNeList = U52 = U51 > n__a > U72 > U71 > U21# partial status: pi(U52#) = [] pi(tt) = [] pi(isList#) = [] pi(activate) = [1] pi(n____) = [1] pi(U21#) = [] pi(and) = [] pi(isPalListKind) = [] pi(n__isPalListKind) = [] pi(U11#) = [] pi(isNeList#) = [] pi(U51#) = [] pi(U41#) = [] pi(U42#) = [] pi(isList) = [] pi(isNeList) = [1] pi(U22#) = [] pi(U62) = [] pi(U61) = [] pi(isQid) = [] pi(U23) = [] pi(U43) = [] pi(U53) = [] pi(U72) = [] pi(isNePal) = [1] pi(n__and) = [] pi(n__isPal) = [] pi(U12) = [] pi(U22) = [] pi(U32) = [] pi(U42) = [] pi(U52) = [] pi(U71) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(__) = [1] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isPal) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: U52#_A(x1,x2) = (9,7) tt_A() = (0,2) isList#_A(x1) = (8,7) activate_A(x1) = x1 + (4,5) n_____A(x1,x2) = ((0,0),(1,1)) x1 + (0,2) U21#_A(x1,x2,x3) = (7,7) and_A(x1,x2) = (5,6) isPalListKind_A(x1) = (4,5) n__isPalListKind_A(x1) = (0,0) U11#_A(x1,x2) = (3,4) isNeList#_A(x1) = (10,6) U51#_A(x1,x2,x3) = (11,7) U41#_A(x1,x2,x3) = (9,6) U42#_A(x1,x2) = (3,6) isList_A(x1) = (6,3) isNeList_A(x1) = (3,4) U22#_A(x1,x2) = (0,0) U62_A(x1) = (1,3) U61_A(x1,x2) = (1,3) isQid_A(x1) = (0,0) U23_A(x1) = (1,3) U43_A(x1) = (0,2) U53_A(x1) = (0,2) U72_A(x1) = (1,2) isNePal_A(x1) = (6,6) n__and_A(x1,x2) = (0,1) n__isPal_A(x1) = (1,5) U12_A(x1) = (1,2) U22_A(x1,x2) = (0,0) U32_A(x1) = (1,2) U42_A(x1,x2) = (0,2) U52_A(x1,x2) = (1,2) U71_A(x1,x2) = (1,2) n__a_A() = (1,2) n__e_A() = (1,0) n__i_A() = (1,0) n__o_A() = (1,2) n__u_A() = (1,1) ___A(x1,x2) = ((0,0),(1,1)) x1 + (4,2) nil_A() = (2,0) U11_A(x1,x2) = (2,3) U21_A(x1,x2,x3) = (0,0) U31_A(x1,x2) = (2,2) U41_A(x1,x2,x3) = (0,2) U51_A(x1,x2,x3) = (1,3) isPal_A(x1) = (2,5) n__nil_A() = (1,0) a_A() = (2,3) e_A() = (2,1) i_A() = (2,1) o_A() = (2,3) u_A() = (2,2) precedence: u > isNePal > U61 > a > U42# > nil > n__nil > U51 > U11# > U51# > isNeList# > n__i > n__u > n__isPal > isPal > U41# > o > isList = U21 > isNeList > e > U12 = U11 > n__e > U31 > U53 > U52 > activate = and = isPalListKind = U72 = U71 = __ = U41 > n__and > n____ > U21# > U42 > U43 > tt > isList# = U23 = i > n__o > n__a > isQid > U52# = n__isPalListKind = U22# = U62 = U22 = U32 partial status: pi(U52#) = [] pi(tt) = [] pi(isList#) = [] pi(activate) = [] pi(n____) = [] pi(U21#) = [] pi(and) = [] pi(isPalListKind) = [] pi(n__isPalListKind) = [] pi(U11#) = [] pi(isNeList#) = [] pi(U51#) = [] pi(U41#) = [] pi(U42#) = [] pi(isList) = [] pi(isNeList) = [] pi(U22#) = [] pi(U62) = [] pi(U61) = [] pi(isQid) = [] pi(U23) = [] pi(U43) = [] pi(U53) = [] pi(U72) = [] pi(isNePal) = [] pi(n__and) = [] pi(n__isPal) = [] pi(U12) = [] pi(U22) = [] pi(U32) = [] pi(U42) = [] pi(U52) = [] pi(U71) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(__) = [] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isPal) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p2, p3, p4, p5, p6, p8, p9, p10, p11, p12, p14 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: U52#(tt(),V2) -> isList#(activate(V2)) p2: U51#(tt(),V1,V2) -> isNeList#(activate(V1)) p3: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isNeList(activate(V))) r5: U12(tt()) -> tt() r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) r7: U22(tt(),V2) -> U23(isList(activate(V2))) r8: U23(tt()) -> tt() r9: U31(tt(),V) -> U32(isQid(activate(V))) r10: U32(tt()) -> tt() r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) r12: U42(tt(),V2) -> U43(isNeList(activate(V2))) r13: U43(tt()) -> tt() r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) r15: U52(tt(),V2) -> U53(isList(activate(V2))) r16: U53(tt()) -> tt() r17: U61(tt(),V) -> U62(isQid(activate(V))) r18: U62(tt()) -> tt() r19: U71(tt(),V) -> U72(isNePal(activate(V))) r20: U72(tt()) -> tt() r21: and(tt(),X) -> activate(X) r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r23: isList(n__nil()) -> tt() r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) r31: isPal(n__nil()) -> tt() r32: isPalListKind(n__a()) -> tt() r33: isPalListKind(n__e()) -> tt() r34: isPalListKind(n__i()) -> tt() r35: isPalListKind(n__nil()) -> tt() r36: isPalListKind(n__o()) -> tt() r37: isPalListKind(n__u()) -> tt() r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) r39: isQid(n__a()) -> tt() r40: isQid(n__e()) -> tt() r41: isQid(n__i()) -> tt() r42: isQid(n__o()) -> tt() r43: isQid(n__u()) -> tt() r44: nil() -> n__nil() r45: __(X1,X2) -> n____(X1,X2) r46: isPalListKind(X) -> n__isPalListKind(X) r47: and(X1,X2) -> n__and(X1,X2) r48: isPal(X) -> n__isPal(X) r49: a() -> n__a() r50: e() -> n__e() r51: i() -> n__i() r52: o() -> n__o() r53: u() -> n__u() r54: activate(n__nil()) -> nil() r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r56: activate(n__isPalListKind(X)) -> isPalListKind(X) r57: activate(n__and(X1,X2)) -> and(activate(X1),X2) r58: activate(n__isPal(X)) -> isPal(X) r59: activate(n__a()) -> a() r60: activate(n__e()) -> e() r61: activate(n__i()) -> i() r62: activate(n__o()) -> o() r63: activate(n__u()) -> u() r64: activate(X) -> X The estimated dependency graph contains the following SCCs: (no SCCs) -- 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) -> isPalListKind#(activate(V)) p4: isPalListKind#(n____(V1,V2)) -> activate#(V2) p5: activate#(n__and(X1,X2)) -> activate#(X1) p6: activate#(n__and(X1,X2)) -> and#(activate(X1),X2) p7: and#(tt(),X) -> activate#(X) p8: activate#(n__isPalListKind(X)) -> isPalListKind#(X) p9: isPalListKind#(n____(V1,V2)) -> activate#(V1) p10: activate#(n____(X1,X2)) -> activate#(X2) p11: activate#(n____(X1,X2)) -> activate#(X1) p12: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p13: isPalListKind#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) p14: isPal#(V) -> U71#(isPalListKind(activate(V)),activate(V)) p15: U71#(tt(),V) -> activate#(V) p16: U71#(tt(),V) -> isNePal#(activate(V)) p17: isNePal#(n____(I,n____(P,I))) -> activate#(P) p18: isNePal#(n____(I,n____(P,I))) -> activate#(I) p19: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPalListKind(activate(I))) p20: isNePal#(n____(I,n____(P,I))) -> and#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) p21: isNePal#(V) -> activate#(V) p22: isNePal#(V) -> isPalListKind#(activate(V)) p23: isNePal#(V) -> U61#(isPalListKind(activate(V)),activate(V)) p24: U61#(tt(),V) -> activate#(V) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isNeList(activate(V))) r5: U12(tt()) -> tt() r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) r7: U22(tt(),V2) -> U23(isList(activate(V2))) r8: U23(tt()) -> tt() r9: U31(tt(),V) -> U32(isQid(activate(V))) r10: U32(tt()) -> tt() r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) r12: U42(tt(),V2) -> U43(isNeList(activate(V2))) r13: U43(tt()) -> tt() r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) r15: U52(tt(),V2) -> U53(isList(activate(V2))) r16: U53(tt()) -> tt() r17: U61(tt(),V) -> U62(isQid(activate(V))) r18: U62(tt()) -> tt() r19: U71(tt(),V) -> U72(isNePal(activate(V))) r20: U72(tt()) -> tt() r21: and(tt(),X) -> activate(X) r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r23: isList(n__nil()) -> tt() r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) r31: isPal(n__nil()) -> tt() r32: isPalListKind(n__a()) -> tt() r33: isPalListKind(n__e()) -> tt() r34: isPalListKind(n__i()) -> tt() r35: isPalListKind(n__nil()) -> tt() r36: isPalListKind(n__o()) -> tt() r37: isPalListKind(n__u()) -> tt() r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) r39: isQid(n__a()) -> tt() r40: isQid(n__e()) -> tt() r41: isQid(n__i()) -> tt() r42: isQid(n__o()) -> tt() r43: isQid(n__u()) -> tt() r44: nil() -> n__nil() r45: __(X1,X2) -> n____(X1,X2) r46: isPalListKind(X) -> n__isPalListKind(X) r47: and(X1,X2) -> n__and(X1,X2) r48: isPal(X) -> n__isPal(X) r49: a() -> n__a() r50: e() -> n__e() r51: i() -> n__i() r52: o() -> n__o() r53: u() -> n__u() r54: activate(n__nil()) -> nil() r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r56: activate(n__isPalListKind(X)) -> isPalListKind(X) r57: activate(n__and(X1,X2)) -> and(activate(X1),X2) r58: activate(n__isPal(X)) -> isPal(X) r59: activate(n__a()) -> a() r60: activate(n__e()) -> e() r61: activate(n__i()) -> i() r62: activate(n__o()) -> o() r63: activate(n__u()) -> u() r64: activate(X) -> X The set of usable rules consists of r1, r2, r3, r17, r18, r19, r20, r21, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isPal#_A(x1) = ((1,1),(0,0)) x1 + (5,41) activate#_A(x1) = ((0,1),(0,0)) x1 + (1,41) n__isPal_A(x1) = ((0,1),(1,1)) x1 + (29,5) isPalListKind#_A(x1) = ((0,1),(0,0)) x1 + (2,41) activate_A(x1) = x1 n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (14,20) n__and_A(x1,x2) = x1 + x2 + (2,8) and#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (1,41) tt_A() = (0,3) n__isPalListKind_A(x1) = ((1,1),(0,1)) x1 + (0,6) isPalListKind_A(x1) = ((1,1),(0,1)) x1 + (0,6) U71#_A(x1,x2) = ((1,1),(0,0)) x2 + (4,41) isNePal#_A(x1) = ((1,1),(0,0)) x1 + (3,41) isQid_A(x1) = ((0,0),(1,0)) x1 + (25,0) and_A(x1,x2) = x1 + x2 + (2,8) U61#_A(x1,x2) = ((0,1),(0,0)) x2 + (2,41) U62_A(x1) = (1,5) U61_A(x1,x2) = ((0,0),(1,0)) x2 + (2,5) U72_A(x1) = (1,4) isNePal_A(x1) = ((1,1),(1,1)) x1 + (3,5) U71_A(x1,x2) = (2,4) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (14,20) nil_A() = (0,0) isPal_A(x1) = ((0,1),(1,1)) x1 + (29,5) n__nil_A() = (0,0) a_A() = (3,0) n__a_A() = (3,0) e_A() = (3,0) n__e_A() = (3,0) i_A() = (3,3) n__i_A() = (3,3) o_A() = (3,1) n__o_A() = (3,1) u_A() = (3,1) n__u_A() = (3,1) precedence: U71# = e = n__e > isNePal# = U61# > n__isPal = isQid = U72 = U71 = isPal = u = n__u > activate = __ = a = n__a = i = o = n__o > isPalListKind# = n__and = tt = n__isPalListKind = isPalListKind = and = U62 = U61 = isNePal = n__i > and# > nil > n__nil > n____ > activate# > isPal# partial status: pi(isPal#) = [] pi(activate#) = [] pi(n__isPal) = [] pi(isPalListKind#) = [] pi(activate) = [1] pi(n____) = [1, 2] pi(n__and) = [] pi(and#) = [] pi(tt) = [] pi(n__isPalListKind) = [1] pi(isPalListKind) = [1] pi(U71#) = [] pi(isNePal#) = [] pi(isQid) = [] pi(and) = [1] pi(U61#) = [] pi(U62) = [] pi(U61) = [] pi(U72) = [] pi(isNePal) = [1] pi(U71) = [] 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) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isPal#_A(x1) = (8,0) activate#_A(x1) = (2,0) n__isPal_A(x1) = (7,0) isPalListKind#_A(x1) = (3,0) activate_A(x1) = ((1,0),(0,0)) x1 + (12,3) n_____A(x1,x2) = ((0,1),(1,1)) x2 + (0,11) n__and_A(x1,x2) = (0,0) and#_A(x1,x2) = (1,0) tt_A() = (3,1) n__isPalListKind_A(x1) = (4,2) isPalListKind_A(x1) = (9,3) U71#_A(x1,x2) = (9,1) isNePal#_A(x1) = (5,1) isQid_A(x1) = (6,3) and_A(x1,x2) = (10,2) U61#_A(x1,x2) = (3,1) U62_A(x1) = (0,0) U61_A(x1,x2) = (13,3) U72_A(x1) = (3,1) isNePal_A(x1) = ((0,1),(0,0)) x1 + (14,3) U71_A(x1,x2) = (18,3) ___A(x1,x2) = ((0,1),(0,0)) x2 + (9,0) nil_A() = (1,1) isPal_A(x1) = (19,3) n__nil_A() = (0,0) a_A() = (2,1) n__a_A() = (1,0) e_A() = (12,2) n__e_A() = (0,1) i_A() = (2,1) n__i_A() = (1,0) o_A() = (13,1) n__o_A() = (1,0) u_A() = (1,1) n__u_A() = (0,0) precedence: isNePal > U61 > isQid > n____ > activate = isPalListKind = and > isPal > U71 > n__and > o > tt = U72 = a = n__a = e = n__e = n__o = u > nil = n__nil = n__u > __ > isPal# > U71# > isPalListKind# > isNePal# > U61# > activate# > n__isPalListKind > and# > n__isPal = U62 = i = n__i partial status: pi(isPal#) = [] pi(activate#) = [] pi(n__isPal) = [] pi(isPalListKind#) = [] pi(activate) = [] pi(n____) = [] pi(n__and) = [] pi(and#) = [] pi(tt) = [] pi(n__isPalListKind) = [] pi(isPalListKind) = [] pi(U71#) = [] pi(isNePal#) = [] pi(isQid) = [] pi(and) = [] pi(U61#) = [] pi(U62) = [] pi(U61) = [] pi(U72) = [] pi(isNePal) = [] pi(U71) = [] 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: p1, p2, p3, p6, p7, p8, p9, p13, p14, p15, p16, p17, p18, p19, p20, p21, p24 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isPalListKind#(n____(V1,V2)) -> activate#(V2) p2: activate#(n__and(X1,X2)) -> activate#(X1) p3: activate#(n____(X1,X2)) -> activate#(X2) p4: activate#(n____(X1,X2)) -> activate#(X1) p5: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p6: isNePal#(V) -> isPalListKind#(activate(V)) p7: isNePal#(V) -> U61#(isPalListKind(activate(V)),activate(V)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isNeList(activate(V))) r5: U12(tt()) -> tt() r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) r7: U22(tt(),V2) -> U23(isList(activate(V2))) r8: U23(tt()) -> tt() r9: U31(tt(),V) -> U32(isQid(activate(V))) r10: U32(tt()) -> tt() r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) r12: U42(tt(),V2) -> U43(isNeList(activate(V2))) r13: U43(tt()) -> tt() r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) r15: U52(tt(),V2) -> U53(isList(activate(V2))) r16: U53(tt()) -> tt() r17: U61(tt(),V) -> U62(isQid(activate(V))) r18: U62(tt()) -> tt() r19: U71(tt(),V) -> U72(isNePal(activate(V))) r20: U72(tt()) -> tt() r21: and(tt(),X) -> activate(X) r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r23: isList(n__nil()) -> tt() r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) r31: isPal(n__nil()) -> tt() r32: isPalListKind(n__a()) -> tt() r33: isPalListKind(n__e()) -> tt() r34: isPalListKind(n__i()) -> tt() r35: isPalListKind(n__nil()) -> tt() r36: isPalListKind(n__o()) -> tt() r37: isPalListKind(n__u()) -> tt() r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) r39: isQid(n__a()) -> tt() r40: isQid(n__e()) -> tt() r41: isQid(n__i()) -> tt() r42: isQid(n__o()) -> tt() r43: isQid(n__u()) -> tt() r44: nil() -> n__nil() r45: __(X1,X2) -> n____(X1,X2) r46: isPalListKind(X) -> n__isPalListKind(X) r47: and(X1,X2) -> n__and(X1,X2) r48: isPal(X) -> n__isPal(X) r49: a() -> n__a() r50: e() -> n__e() r51: i() -> n__i() r52: o() -> n__o() r53: u() -> n__u() r54: activate(n__nil()) -> nil() r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r56: activate(n__isPalListKind(X)) -> isPalListKind(X) r57: activate(n__and(X1,X2)) -> and(activate(X1),X2) r58: activate(n__isPal(X)) -> isPal(X) r59: activate(n__a()) -> a() r60: activate(n__e()) -> e() r61: activate(n__i()) -> i() r62: activate(n__o()) -> o() r63: activate(n__u()) -> u() r64: activate(X) -> X The estimated dependency graph contains the following SCCs: {p5} {p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isNeList(activate(V))) r5: U12(tt()) -> tt() r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) r7: U22(tt(),V2) -> U23(isList(activate(V2))) r8: U23(tt()) -> tt() r9: U31(tt(),V) -> U32(isQid(activate(V))) r10: U32(tt()) -> tt() r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) r12: U42(tt(),V2) -> U43(isNeList(activate(V2))) r13: U43(tt()) -> tt() r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) r15: U52(tt(),V2) -> U53(isList(activate(V2))) r16: U53(tt()) -> tt() r17: U61(tt(),V) -> U62(isQid(activate(V))) r18: U62(tt()) -> tt() r19: U71(tt(),V) -> U72(isNePal(activate(V))) r20: U72(tt()) -> tt() r21: and(tt(),X) -> activate(X) r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r23: isList(n__nil()) -> tt() r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) r31: isPal(n__nil()) -> tt() r32: isPalListKind(n__a()) -> tt() r33: isPalListKind(n__e()) -> tt() r34: isPalListKind(n__i()) -> tt() r35: isPalListKind(n__nil()) -> tt() r36: isPalListKind(n__o()) -> tt() r37: isPalListKind(n__u()) -> tt() r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) r39: isQid(n__a()) -> tt() r40: isQid(n__e()) -> tt() r41: isQid(n__i()) -> tt() r42: isQid(n__o()) -> tt() r43: isQid(n__u()) -> tt() r44: nil() -> n__nil() r45: __(X1,X2) -> n____(X1,X2) r46: isPalListKind(X) -> n__isPalListKind(X) r47: and(X1,X2) -> n__and(X1,X2) r48: isPal(X) -> n__isPal(X) r49: a() -> n__a() r50: e() -> n__e() r51: i() -> n__i() r52: o() -> n__o() r53: u() -> n__u() r54: activate(n__nil()) -> nil() r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r56: activate(n__isPalListKind(X)) -> isPalListKind(X) r57: activate(n__and(X1,X2)) -> and(activate(X1),X2) r58: activate(n__isPal(X)) -> isPal(X) r59: activate(n__a()) -> a() r60: activate(n__e()) -> e() r61: activate(n__i()) -> i() r62: activate(n__o()) -> o() r63: activate(n__u()) -> u() r64: activate(X) -> X The set of usable rules consists of r1, r2, r3, r17, r18, r19, r20, r21, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isPalListKind#_A(x1) = ((1,1),(0,1)) x1 + (1,0) n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (12,0) activate_A(x1) = x1 U62_A(x1) = ((0,1),(0,0)) x1 + (1,4) tt_A() = (2,4) U61_A(x1,x2) = ((1,1),(0,0)) x2 + (8,4) isQid_A(x1) = ((1,0),(1,1)) x1 + (13,6) n__a_A() = (1,3) n__e_A() = (1,3) n__i_A() = (1,3) n__o_A() = (1,3) n__u_A() = (1,3) U72_A(x1) = ((0,1),(0,0)) x1 + (0,5) isNePal_A(x1) = ((1,1),(0,1)) x1 + (9,5) isPalListKind_A(x1) = ((0,1),(0,0)) x1 + (0,4) and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 n__isPalListKind_A(x1) = ((0,1),(0,0)) x1 + (0,4) n__and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 n__isPal_A(x1) = ((1,1),(1,1)) x1 + (10,3) U71_A(x1,x2) = ((0,1),(1,0)) x1 + ((0,1),(0,0)) x2 + (6,3) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (12,0) nil_A() = (3,3) isPal_A(x1) = ((1,1),(1,1)) x1 + (10,3) n__nil_A() = (3,3) a_A() = (1,3) e_A() = (1,3) i_A() = (1,3) o_A() = (1,3) u_A() = (1,3) precedence: isPalListKind# = U62 = tt = U61 = isQid = n__e = n__i = isNePal = isPalListKind = n__isPalListKind = nil = n__nil = e = i > and = n__and > n____ = activate = n__a = n__o = n__u = U72 = n__isPal = U71 = __ = isPal = a = o = u partial status: pi(isPalListKind#) = [] pi(n____) = [] pi(activate) = [1] pi(U62) = [] pi(tt) = [] pi(U61) = [] pi(isQid) = [1] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(U72) = [] pi(isNePal) = [1] pi(isPalListKind) = [] pi(and) = [] pi(n__isPalListKind) = [] pi(n__and) = [] pi(n__isPal) = [] pi(U71) = [] pi(__) = [] pi(nil) = [] pi(isPal) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isPalListKind#_A(x1) = (6,4) n_____A(x1,x2) = (5,2) activate_A(x1) = ((1,1),(0,0)) x1 + (3,4) U62_A(x1) = (0,0) tt_A() = (0,0) U61_A(x1,x2) = (1,1) isQid_A(x1) = x1 + (3,0) n__a_A() = (1,1) n__e_A() = (0,1) n__i_A() = (0,1) n__o_A() = (0,1) n__u_A() = (0,0) U72_A(x1) = (1,1) isNePal_A(x1) = ((1,1),(1,0)) x1 + (5,4) isPalListKind_A(x1) = (4,2) and_A(x1,x2) = (2,2) n__isPalListKind_A(x1) = (4,2) n__and_A(x1,x2) = (1,1) n__isPal_A(x1) = (2,3) U71_A(x1,x2) = (2,1) ___A(x1,x2) = (6,3) nil_A() = (0,0) isPal_A(x1) = (2,3) n__nil_A() = (0,0) a_A() = (1,1) e_A() = (1,2) i_A() = (1,2) o_A() = (1,2) u_A() = (1,1) precedence: __ > i > e > n__e > U62 > tt = U61 = n__u = isNePal = isPalListKind = n__isPalListKind > isQid > isPalListKind# = and > u > n__i > o > n__o > isPal > n____ = activate = n__a = U72 = n__and = n__isPal = U71 = nil = n__nil = a partial status: pi(isPalListKind#) = [] pi(n____) = [] pi(activate) = [] pi(U62) = [] pi(tt) = [] pi(U61) = [] pi(isQid) = [1] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(U72) = [] pi(isNePal) = [] pi(isPalListKind) = [] pi(and) = [] pi(n__isPalListKind) = [] pi(n__and) = [] pi(n__isPal) = [] pi(U71) = [] pi(__) = [] pi(nil) = [] 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. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: activate#(n__and(X1,X2)) -> activate#(X1) p2: activate#(n____(X1,X2)) -> activate#(X1) p3: 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: U11(tt(),V) -> U12(isNeList(activate(V))) r5: U12(tt()) -> tt() r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) r7: U22(tt(),V2) -> U23(isList(activate(V2))) r8: U23(tt()) -> tt() r9: U31(tt(),V) -> U32(isQid(activate(V))) r10: U32(tt()) -> tt() r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) r12: U42(tt(),V2) -> U43(isNeList(activate(V2))) r13: U43(tt()) -> tt() r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) r15: U52(tt(),V2) -> U53(isList(activate(V2))) r16: U53(tt()) -> tt() r17: U61(tt(),V) -> U62(isQid(activate(V))) r18: U62(tt()) -> tt() r19: U71(tt(),V) -> U72(isNePal(activate(V))) r20: U72(tt()) -> tt() r21: and(tt(),X) -> activate(X) r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r23: isList(n__nil()) -> tt() r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) r31: isPal(n__nil()) -> tt() r32: isPalListKind(n__a()) -> tt() r33: isPalListKind(n__e()) -> tt() r34: isPalListKind(n__i()) -> tt() r35: isPalListKind(n__nil()) -> tt() r36: isPalListKind(n__o()) -> tt() r37: isPalListKind(n__u()) -> tt() r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) r39: isQid(n__a()) -> tt() r40: isQid(n__e()) -> tt() r41: isQid(n__i()) -> tt() r42: isQid(n__o()) -> tt() r43: isQid(n__u()) -> tt() r44: nil() -> n__nil() r45: __(X1,X2) -> n____(X1,X2) r46: isPalListKind(X) -> n__isPalListKind(X) r47: and(X1,X2) -> n__and(X1,X2) r48: isPal(X) -> n__isPal(X) r49: a() -> n__a() r50: e() -> n__e() r51: i() -> n__i() r52: o() -> n__o() r53: u() -> n__u() r54: activate(n__nil()) -> nil() r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r56: activate(n__isPalListKind(X)) -> isPalListKind(X) r57: activate(n__and(X1,X2)) -> and(activate(X1),X2) r58: activate(n__isPal(X)) -> isPal(X) r59: activate(n__a()) -> a() r60: activate(n__e()) -> e() r61: activate(n__i()) -> i() r62: activate(n__o()) -> o() r63: activate(n__u()) -> u() r64: activate(X) -> X The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: activate#_A(x1) = max{2, x1 + 1} n__and_A(x1,x2) = max{x1, x2} n_____A(x1,x2) = max{1, x1, x2} precedence: activate# = n__and = n____ partial status: pi(activate#) = [1] pi(n__and) = [1, 2] pi(n____) = [1, 2] 2. weighted path order base order: max/plus interpretations on natural numbers: activate#_A(x1) = x1 + 3 n__and_A(x1,x2) = max{x1 + 1, x2} n_____A(x1,x2) = max{x1 + 1, x2 + 1} precedence: activate# = n__and = n____ partial status: pi(activate#) = [] pi(n__and) = [2] pi(n____) = [2] The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) p2: __#(__(X,Y),Z) -> __#(Y,Z) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isNeList(activate(V))) r5: U12(tt()) -> tt() r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2)) r7: U22(tt(),V2) -> U23(isList(activate(V2))) r8: U23(tt()) -> tt() r9: U31(tt(),V) -> U32(isQid(activate(V))) r10: U32(tt()) -> tt() r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2)) r12: U42(tt(),V2) -> U43(isNeList(activate(V2))) r13: U43(tt()) -> tt() r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2)) r15: U52(tt(),V2) -> U53(isList(activate(V2))) r16: U53(tt()) -> tt() r17: U61(tt(),V) -> U62(isQid(activate(V))) r18: U62(tt()) -> tt() r19: U71(tt(),V) -> U72(isNePal(activate(V))) r20: U72(tt()) -> tt() r21: and(tt(),X) -> activate(X) r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r23: isList(n__nil()) -> tt() r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2)) r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P)))) r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V)) r31: isPal(n__nil()) -> tt() r32: isPalListKind(n__a()) -> tt() r33: isPalListKind(n__e()) -> tt() r34: isPalListKind(n__i()) -> tt() r35: isPalListKind(n__nil()) -> tt() r36: isPalListKind(n__o()) -> tt() r37: isPalListKind(n__u()) -> tt() r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))) r39: isQid(n__a()) -> tt() r40: isQid(n__e()) -> tt() r41: isQid(n__i()) -> tt() r42: isQid(n__o()) -> tt() r43: isQid(n__u()) -> tt() r44: nil() -> n__nil() r45: __(X1,X2) -> n____(X1,X2) r46: isPalListKind(X) -> n__isPalListKind(X) r47: and(X1,X2) -> n__and(X1,X2) r48: isPal(X) -> n__isPal(X) r49: a() -> n__a() r50: e() -> n__e() r51: i() -> n__i() r52: o() -> n__o() r53: u() -> n__u() r54: activate(n__nil()) -> nil() r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) r56: activate(n__isPalListKind(X)) -> isPalListKind(X) r57: activate(n__and(X1,X2)) -> and(activate(X1),X2) r58: activate(n__isPal(X)) -> isPal(X) r59: activate(n__a()) -> a() r60: activate(n__e()) -> e() r61: activate(n__i()) -> i() r62: activate(n__o()) -> o() r63: activate(n__u()) -> u() r64: activate(X) -> X The set of usable rules consists of r1, r2, r3, r45 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: __#_A(x1,x2) = max{x1 + 4, x2 + 2} ___A(x1,x2) = max{x1 + 2, x2} nil_A = 0 n_____A(x1,x2) = max{x1 + 2, x2} precedence: __# = __ > nil = n____ partial status: pi(__#) = [1] pi(__) = [1, 2] pi(nil) = [] pi(n____) = [1, 2] 2. weighted path order base order: max/plus interpretations on natural numbers: __#_A(x1,x2) = max{0, x1 - 3} ___A(x1,x2) = max{x1 + 5, x2 + 7} nil_A = 0 n_____A(x1,x2) = max{x1 + 6, x2 + 8} precedence: __# = __ = nil = n____ partial status: pi(__#) = [] pi(__) = [2] pi(nil) = [] pi(n____) = [] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains.