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(isPalListKind(activate(V)),activate(V)) U12(tt(),V) -> U13(isNeList(activate(V))) U13(tt()) -> tt() U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) U25(tt(),V2) -> U26(isList(activate(V2))) U26(tt()) -> tt() U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) U32(tt(),V) -> U33(isQid(activate(V))) U33(tt()) -> tt() U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) U45(tt(),V2) -> U46(isNeList(activate(V2))) U46(tt()) -> tt() U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) U55(tt(),V2) -> U56(isList(activate(V2))) U56(tt()) -> tt() U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) U62(tt(),V) -> U63(isQid(activate(V))) U63(tt()) -> tt() U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) U73(tt(),P) -> U74(isPalListKind(activate(P))) U74(tt()) -> tt() U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) U82(tt(),V) -> U83(isNePal(activate(V))) U83(tt()) -> tt() U91(tt(),V2) -> U92(isPalListKind(activate(V2))) U92(tt()) -> tt() isList(V) -> U11(isPalListKind(activate(V)),activate(V)) isList(n__nil()) -> tt() isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) isPal(V) -> U81(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)) -> U91(isPalListKind(activate(V1)),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) 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__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#(isPalListKind(activate(V)),activate(V)) p4: U11#(tt(),V) -> isPalListKind#(activate(V)) p5: U11#(tt(),V) -> activate#(V) p6: U12#(tt(),V) -> U13#(isNeList(activate(V))) p7: U12#(tt(),V) -> isNeList#(activate(V)) p8: U12#(tt(),V) -> activate#(V) p9: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p10: U21#(tt(),V1,V2) -> isPalListKind#(activate(V1)) p11: U21#(tt(),V1,V2) -> activate#(V1) p12: U21#(tt(),V1,V2) -> activate#(V2) p13: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p14: U22#(tt(),V1,V2) -> isPalListKind#(activate(V2)) p15: U22#(tt(),V1,V2) -> activate#(V2) p16: U22#(tt(),V1,V2) -> activate#(V1) p17: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p18: U23#(tt(),V1,V2) -> isPalListKind#(activate(V2)) p19: U23#(tt(),V1,V2) -> activate#(V2) p20: U23#(tt(),V1,V2) -> activate#(V1) p21: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p22: U24#(tt(),V1,V2) -> isList#(activate(V1)) p23: U24#(tt(),V1,V2) -> activate#(V1) p24: U24#(tt(),V1,V2) -> activate#(V2) p25: U25#(tt(),V2) -> U26#(isList(activate(V2))) p26: U25#(tt(),V2) -> isList#(activate(V2)) p27: U25#(tt(),V2) -> activate#(V2) p28: U31#(tt(),V) -> U32#(isPalListKind(activate(V)),activate(V)) p29: U31#(tt(),V) -> isPalListKind#(activate(V)) p30: U31#(tt(),V) -> activate#(V) p31: U32#(tt(),V) -> U33#(isQid(activate(V))) p32: U32#(tt(),V) -> isQid#(activate(V)) p33: U32#(tt(),V) -> activate#(V) p34: U41#(tt(),V1,V2) -> U42#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p35: U41#(tt(),V1,V2) -> isPalListKind#(activate(V1)) p36: U41#(tt(),V1,V2) -> activate#(V1) p37: U41#(tt(),V1,V2) -> activate#(V2) p38: U42#(tt(),V1,V2) -> U43#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p39: U42#(tt(),V1,V2) -> isPalListKind#(activate(V2)) p40: U42#(tt(),V1,V2) -> activate#(V2) p41: U42#(tt(),V1,V2) -> activate#(V1) p42: U43#(tt(),V1,V2) -> U44#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p43: U43#(tt(),V1,V2) -> isPalListKind#(activate(V2)) p44: U43#(tt(),V1,V2) -> activate#(V2) p45: U43#(tt(),V1,V2) -> activate#(V1) p46: U44#(tt(),V1,V2) -> U45#(isList(activate(V1)),activate(V2)) p47: U44#(tt(),V1,V2) -> isList#(activate(V1)) p48: U44#(tt(),V1,V2) -> activate#(V1) p49: U44#(tt(),V1,V2) -> activate#(V2) p50: U45#(tt(),V2) -> U46#(isNeList(activate(V2))) p51: U45#(tt(),V2) -> isNeList#(activate(V2)) p52: U45#(tt(),V2) -> activate#(V2) p53: U51#(tt(),V1,V2) -> U52#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p54: U51#(tt(),V1,V2) -> isPalListKind#(activate(V1)) p55: U51#(tt(),V1,V2) -> activate#(V1) p56: U51#(tt(),V1,V2) -> activate#(V2) p57: U52#(tt(),V1,V2) -> U53#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p58: U52#(tt(),V1,V2) -> isPalListKind#(activate(V2)) p59: U52#(tt(),V1,V2) -> activate#(V2) p60: U52#(tt(),V1,V2) -> activate#(V1) p61: U53#(tt(),V1,V2) -> U54#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p62: U53#(tt(),V1,V2) -> isPalListKind#(activate(V2)) p63: U53#(tt(),V1,V2) -> activate#(V2) p64: U53#(tt(),V1,V2) -> activate#(V1) p65: U54#(tt(),V1,V2) -> U55#(isNeList(activate(V1)),activate(V2)) p66: U54#(tt(),V1,V2) -> isNeList#(activate(V1)) p67: U54#(tt(),V1,V2) -> activate#(V1) p68: U54#(tt(),V1,V2) -> activate#(V2) p69: U55#(tt(),V2) -> U56#(isList(activate(V2))) p70: U55#(tt(),V2) -> isList#(activate(V2)) p71: U55#(tt(),V2) -> activate#(V2) p72: U61#(tt(),V) -> U62#(isPalListKind(activate(V)),activate(V)) p73: U61#(tt(),V) -> isPalListKind#(activate(V)) p74: U61#(tt(),V) -> activate#(V) p75: U62#(tt(),V) -> U63#(isQid(activate(V))) p76: U62#(tt(),V) -> isQid#(activate(V)) p77: U62#(tt(),V) -> activate#(V) p78: U71#(tt(),I,P) -> U72#(isPalListKind(activate(I)),activate(P)) p79: U71#(tt(),I,P) -> isPalListKind#(activate(I)) p80: U71#(tt(),I,P) -> activate#(I) p81: U71#(tt(),I,P) -> activate#(P) p82: U72#(tt(),P) -> U73#(isPal(activate(P)),activate(P)) p83: U72#(tt(),P) -> isPal#(activate(P)) p84: U72#(tt(),P) -> activate#(P) p85: U73#(tt(),P) -> U74#(isPalListKind(activate(P))) p86: U73#(tt(),P) -> isPalListKind#(activate(P)) p87: U73#(tt(),P) -> activate#(P) p88: U81#(tt(),V) -> U82#(isPalListKind(activate(V)),activate(V)) p89: U81#(tt(),V) -> isPalListKind#(activate(V)) p90: U81#(tt(),V) -> activate#(V) p91: U82#(tt(),V) -> U83#(isNePal(activate(V))) p92: U82#(tt(),V) -> isNePal#(activate(V)) p93: U82#(tt(),V) -> activate#(V) p94: U91#(tt(),V2) -> U92#(isPalListKind(activate(V2))) p95: U91#(tt(),V2) -> isPalListKind#(activate(V2)) p96: U91#(tt(),V2) -> activate#(V2) p97: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V)) p98: isList#(V) -> isPalListKind#(activate(V)) p99: isList#(V) -> activate#(V) p100: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p101: isList#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p102: isList#(n____(V1,V2)) -> activate#(V1) p103: isList#(n____(V1,V2)) -> activate#(V2) p104: isNeList#(V) -> U31#(isPalListKind(activate(V)),activate(V)) p105: isNeList#(V) -> isPalListKind#(activate(V)) p106: isNeList#(V) -> activate#(V) p107: isNeList#(n____(V1,V2)) -> U41#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p108: isNeList#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p109: isNeList#(n____(V1,V2)) -> activate#(V1) p110: isNeList#(n____(V1,V2)) -> activate#(V2) p111: isNeList#(n____(V1,V2)) -> U51#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p112: isNeList#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p113: isNeList#(n____(V1,V2)) -> activate#(V1) p114: isNeList#(n____(V1,V2)) -> activate#(V2) p115: isNePal#(V) -> U61#(isPalListKind(activate(V)),activate(V)) p116: isNePal#(V) -> isPalListKind#(activate(V)) p117: isNePal#(V) -> activate#(V) p118: isNePal#(n____(I,__(P,I))) -> U71#(isQid(activate(I)),activate(I),activate(P)) p119: isNePal#(n____(I,__(P,I))) -> isQid#(activate(I)) p120: isNePal#(n____(I,__(P,I))) -> activate#(I) p121: isNePal#(n____(I,__(P,I))) -> activate#(P) p122: isPal#(V) -> U81#(isPalListKind(activate(V)),activate(V)) p123: isPal#(V) -> isPalListKind#(activate(V)) p124: isPal#(V) -> activate#(V) p125: isPalListKind#(n____(V1,V2)) -> U91#(isPalListKind(activate(V1)),activate(V2)) p126: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1)) p127: isPalListKind#(n____(V1,V2)) -> activate#(V1) p128: isPalListKind#(n____(V1,V2)) -> activate#(V2) p129: activate#(n__nil()) -> nil#() p130: activate#(n____(X1,X2)) -> __#(X1,X2) p131: activate#(n__a()) -> a#() p132: activate#(n__e()) -> e#() p133: activate#(n__i()) -> i#() p134: activate#(n__o()) -> o#() p135: 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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The estimated dependency graph contains the following SCCs: {p78, p83, p88, p92, p118, p122} {p3, p7, p9, p13, p17, p21, p22, p26, p34, p38, p42, p46, p47, p51, p53, p57, p61, p65, p66, p70, p97, p100, p107, p111} {p95, p125, p126} {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: U82#(tt(),V) -> isNePal#(activate(V)) p2: isNePal#(n____(I,__(P,I))) -> U71#(isQid(activate(I)),activate(I),activate(P)) p3: U71#(tt(),I,P) -> U72#(isPalListKind(activate(I)),activate(P)) p4: U72#(tt(),P) -> isPal#(activate(P)) p5: isPal#(V) -> U81#(isPalListKind(activate(V)),activate(V)) p6: U81#(tt(),V) -> U82#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The set of usable rules consists of r1, r2, r3, r38, r39, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: U82#_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,0)) x2 + (11,11) tt_A() = (13,12) isNePal#_A(x1) = ((1,1),(1,0)) x1 + (8,21) activate_A(x1) = x1 + (2,0) n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (0,14) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (0,14) U71#_A(x1,x2,x3) = ((0,1),(0,0)) x2 + ((1,1),(1,1)) x3 + (23,19) isQid_A(x1) = ((0,0),(1,1)) x1 + (14,27) U72#_A(x1,x2) = ((1,1),(1,1)) x2 + (20,17) isPalListKind_A(x1) = ((0,1),(0,1)) x1 + (1,0) isPal#_A(x1) = ((1,1),(1,1)) x1 + (17,15) U81#_A(x1,x2) = ((1,1),(1,1)) x2 + (14,13) U92_A(x1) = (13,12) nil_A() = (14,12) U91_A(x1,x2) = ((0,0),(0,1)) x1 + (14,1) n__nil_A() = (14,12) a_A() = (1,12) n__a_A() = (0,12) e_A() = (1,13) n__e_A() = (0,13) i_A() = (1,13) n__i_A() = (0,13) o_A() = (13,12) n__o_A() = (12,12) u_A() = (15,13) n__u_A() = (14,13) precedence: U82# = activate = i = n__i > isNePal# > __ > n____ = U71# > U72# > isPal# > U81# > isQid = isPalListKind = nil = U91 > n__nil > a > n__a = e = u > n__u > o = n__o > tt = U92 > n__e partial status: pi(U82#) = [] pi(tt) = [] pi(isNePal#) = [] pi(activate) = [1] pi(n____) = [] pi(__) = [1, 2] pi(U71#) = [] pi(isQid) = [] pi(U72#) = [] pi(isPalListKind) = [] pi(isPal#) = [] pi(U81#) = [] pi(U92) = [] pi(nil) = [] pi(U91) = [] 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: p6 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: U82#(tt(),V) -> isNePal#(activate(V)) p2: isNePal#(n____(I,__(P,I))) -> U71#(isQid(activate(I)),activate(I),activate(P)) p3: U71#(tt(),I,P) -> U72#(isPalListKind(activate(I)),activate(P)) p4: U72#(tt(),P) -> isPal#(activate(P)) p5: isPal#(V) -> U81#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: 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: U52#(tt(),V1,V2) -> U53#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p2: U53#(tt(),V1,V2) -> U54#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p3: U54#(tt(),V1,V2) -> isNeList#(activate(V1)) p4: isNeList#(n____(V1,V2)) -> U51#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p5: U51#(tt(),V1,V2) -> U52#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p6: isNeList#(n____(V1,V2)) -> U41#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p7: U41#(tt(),V1,V2) -> U42#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p8: U42#(tt(),V1,V2) -> U43#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p9: U43#(tt(),V1,V2) -> U44#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p10: U44#(tt(),V1,V2) -> isList#(activate(V1)) p11: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p12: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p13: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p14: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p15: U24#(tt(),V1,V2) -> isList#(activate(V1)) p16: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V)) p17: U11#(tt(),V) -> U12#(isPalListKind(activate(V)),activate(V)) p18: U12#(tt(),V) -> isNeList#(activate(V)) p19: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p20: U25#(tt(),V2) -> isList#(activate(V2)) p21: U44#(tt(),V1,V2) -> U45#(isList(activate(V1)),activate(V2)) p22: U45#(tt(),V2) -> isNeList#(activate(V2)) p23: U54#(tt(),V1,V2) -> U55#(isNeList(activate(V1)),activate(V2)) p24: U55#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: 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, r38, r39, r40, r41, r42, r43, r44, r45, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: U52#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (89,111) tt_A() = (108,83) U53#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (70,111) isPalListKind_A(x1) = ((1,1),(1,1)) x1 + (159,103) activate_A(x1) = x1 + (2,7) U54#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (51,111) isNeList#_A(x1) = ((1,1),(0,0)) x1 + (1,111) n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (246,110) U51#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (108,111) U41#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (167,111) U42#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (148,111) U43#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (129,111) U44#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (110,111) isList#_A(x1) = ((1,1),(0,0)) x1 + (31,111) U21#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (246,111) U22#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (227,111) U23#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (208,111) U24#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (189,111) U11#_A(x1,x2) = ((1,1),(0,0)) x2 + (21,111) U12#_A(x1,x2) = ((1,1),(0,0)) x2 + (11,111) U25#_A(x1,x2) = ((1,1),(0,0)) x2 + (100,111) isList_A(x1) = ((1,1),(1,0)) x1 + (179,84) U45#_A(x1,x2) = ((1,1),(0,0)) x2 + (100,111) U55#_A(x1,x2) = ((1,1),(0,0)) x2 + (41,111) isNeList_A(x1) = ((1,1),(1,0)) x1 + (41,6) U26_A(x1) = (109,84) U46_A(x1) = (109,84) U56_A(x1) = (109,83) U25_A(x1,x2) = (110,84) U45_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,1) U55_A(x1,x2) = ((1,0),(0,0)) x1 + (2,84) U24_A(x1,x2,x3) = ((0,1),(0,0)) x1 + (28,84) U44_A(x1,x2,x3) = ((1,1),(1,0)) x2 + ((1,0),(1,0)) x3 + (193,87) U54_A(x1,x2,x3) = ((1,1),(0,0)) x2 + (71,84) U13_A(x1) = ((1,0),(0,0)) x1 + (47,84) U23_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((1,1),(0,0)) x3 + (141,84) U33_A(x1) = ((0,1),(0,1)) x1 + (26,0) U43_A(x1,x2,x3) = ((1,1),(1,0)) x2 + ((1,1),(1,0)) x3 + (205,91) U53_A(x1,x2,x3) = ((1,1),(0,0)) x2 + (81,84) isQid_A(x1) = ((0,1),(1,0)) x1 + (108,0) n__a_A() = (83,1) n__e_A() = (83,1) n__i_A() = (109,83) n__o_A() = (107,1) n__u_A() = (109,1) U12_A(x1,x2) = ((1,1),(0,0)) x2 + (98,84) U22_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((1,1),(0,0)) x3 + (151,86) U32_A(x1,x2) = ((1,0),(1,0)) x2 + (29,2) U42_A(x1,x2,x3) = ((1,1),(1,0)) x2 + ((1,1),(1,0)) x3 + (224,95) U52_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((0,1),(0,0)) x3 + (91,84) U92_A(x1) = ((0,1),(0,0)) x1 + (56,110) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (247,111) nil_A() = (1,1) U11_A(x1,x2) = ((1,1),(0,0)) x2 + (169,84) U21_A(x1,x2,x3) = ((0,1),(1,1)) x2 + ((1,1),(0,0)) x3 + (231,102) U31_A(x1,x2) = ((1,1),(1,0)) x2 + (31,4) U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,0)) x3 + (243,99) U51_A(x1,x2,x3) = x1 + ((1,1),(0,0)) x2 + ((0,1),(0,0)) x3 + (61,1) U91_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,1),(0,0)) x2 + (60,110) n__nil_A() = (0,0) a_A() = (83,1) e_A() = (83,1) i_A() = (110,83) o_A() = (108,2) u_A() = (110,1) precedence: e > n__nil > n__i > n__u > U42 > U25 = U51 > U44 = U43 > U12# = U46 = U45 > U41# > U42# > U43# > U44# > U45# > isNeList# > isList# = U21# = U22# = U23# = U24# = U25# = U11 > isList > U11# > U12 > activate = n__a = n__e = __ = a = u > n____ > U41 > U26 > U51# > U53# = U54# = U13 > U55# > U56 > U52# > U24 = U23 = U22 = U21 > isPalListKind = U54 = U53 = U52 > tt = U33 = isQid = U32 = U31 > isNeList > nil > i > o > U55 = n__o > U92 = U91 partial status: pi(U52#) = [] pi(tt) = [] pi(U53#) = [] pi(isPalListKind) = [1] pi(activate) = [1] pi(U54#) = [] pi(isNeList#) = [] pi(n____) = [1, 2] pi(U51#) = [] pi(U41#) = [] pi(U42#) = [] pi(U43#) = [] pi(U44#) = [] pi(isList#) = [] pi(U21#) = [] pi(U22#) = [] pi(U23#) = [] pi(U24#) = [] pi(U11#) = [] pi(U12#) = [] pi(U25#) = [] pi(isList) = [] pi(U45#) = [] pi(U55#) = [] pi(isNeList) = [] pi(U26) = [] pi(U46) = [] pi(U56) = [] pi(U25) = [] pi(U45) = [] pi(U55) = [] pi(U24) = [] pi(U44) = [] pi(U54) = [] pi(U13) = [] pi(U23) = [] pi(U33) = [] pi(U43) = [] pi(U53) = [] pi(isQid) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(U12) = [] pi(U22) = [] pi(U32) = [] pi(U42) = [] pi(U52) = [] pi(U92) = [] pi(__) = [1] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(U91) = [] 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: U53#(tt(),V1,V2) -> U54#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p2: U54#(tt(),V1,V2) -> isNeList#(activate(V1)) p3: isNeList#(n____(V1,V2)) -> U51#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p4: U51#(tt(),V1,V2) -> U52#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p5: isNeList#(n____(V1,V2)) -> U41#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p6: U41#(tt(),V1,V2) -> U42#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p7: U42#(tt(),V1,V2) -> U43#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p8: U43#(tt(),V1,V2) -> U44#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p9: U44#(tt(),V1,V2) -> isList#(activate(V1)) p10: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p11: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p12: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p13: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p14: U24#(tt(),V1,V2) -> isList#(activate(V1)) p15: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V)) p16: U11#(tt(),V) -> U12#(isPalListKind(activate(V)),activate(V)) p17: U12#(tt(),V) -> isNeList#(activate(V)) p18: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p19: U25#(tt(),V2) -> isList#(activate(V2)) p20: U44#(tt(),V1,V2) -> U45#(isList(activate(V1)),activate(V2)) p21: U45#(tt(),V2) -> isNeList#(activate(V2)) p22: U54#(tt(),V1,V2) -> U55#(isNeList(activate(V1)),activate(V2)) p23: U55#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The estimated dependency graph contains the following SCCs: {p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: U45#(tt(),V2) -> isNeList#(activate(V2)) p2: isNeList#(n____(V1,V2)) -> U41#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p3: U41#(tt(),V1,V2) -> U42#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p4: U42#(tt(),V1,V2) -> U43#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p5: U43#(tt(),V1,V2) -> U44#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p6: U44#(tt(),V1,V2) -> U45#(isList(activate(V1)),activate(V2)) p7: U44#(tt(),V1,V2) -> isList#(activate(V1)) p8: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V)) p9: U11#(tt(),V) -> U12#(isPalListKind(activate(V)),activate(V)) p10: U12#(tt(),V) -> isNeList#(activate(V)) p11: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p12: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p13: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p14: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p15: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p16: U25#(tt(),V2) -> isList#(activate(V2)) p17: U24#(tt(),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: U11(tt(),V) -> U12(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: 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, r38, r39, r40, r41, r42, r43, r44, r45, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: U45#_A(x1,x2) = x2 + (30,27) tt_A() = (31,26) isNeList#_A(x1) = x1 + (0,26) activate_A(x1) = x1 + (2,1) n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (67,38) U41#_A(x1,x2,x3) = ((1,1),(0,1)) x2 + x3 + (62,36) isPalListKind_A(x1) = ((1,1),(1,1)) x1 + (26,4) U42#_A(x1,x2,x3) = ((1,1),(0,1)) x2 + x3 + (57,34) U43#_A(x1,x2,x3) = ((1,1),(0,1)) x2 + x3 + (52,32) U44#_A(x1,x2,x3) = x2 + x3 + (47,30) isList_A(x1) = ((1,1),(1,1)) x1 + (43,7) isList#_A(x1) = x1 + (7,29) U11#_A(x1,x2) = x2 + (4,28) U12#_A(x1,x2) = x2 + (2,27) U21#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + x3 + (28,39) U22#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + x3 + (23,35) U23#_A(x1,x2,x3) = x2 + x3 + (18,33) U24#_A(x1,x2,x3) = x2 + x3 + (13,31) U25#_A(x1,x2) = x2 + (10,30) U46_A(x1) = (32,26) U56_A(x1) = ((0,1),(0,0)) x1 + (6,26) U45_A(x1,x2) = (33,27) isNeList_A(x1) = ((1,1),(0,1)) x1 + (37,4) U55_A(x1,x2) = ((1,1),(0,0)) x2 + (17,27) U44_A(x1,x2,x3) = (34,27) U54_A(x1,x2,x3) = ((1,1),(0,0)) x3 + (21,27) U26_A(x1) = (32,27) U33_A(x1) = ((0,1),(0,1)) x1 + (6,0) U43_A(x1,x2,x3) = (35,27) U53_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,1),(0,0)) x3 + (25,27) isQid_A(x1) = ((1,1),(0,1)) x1 + (27,1) n__a_A() = (32,26) n__e_A() = (1,25) n__i_A() = (32,26) n__o_A() = (32,25) n__u_A() = (1,25) U25_A(x1,x2) = x1 + (2,17) U32_A(x1,x2) = ((1,1),(0,1)) x2 + (29,2) U42_A(x1,x2,x3) = (36,27) U52_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (31,27) U24_A(x1,x2,x3) = ((1,1),(1,1)) x2 + (49,27) U31_A(x1,x2) = ((1,1),(0,1)) x2 + (33,3) U41_A(x1,x2,x3) = ((0,1),(0,0)) x3 + (67,39) U51_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (62,32) U13_A(x1) = ((0,0),(0,1)) x1 + (32,0) U23_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((0,0),(1,0)) x3 + (44,30) U12_A(x1,x2) = ((0,0),(0,1)) x2 + (41,5) U22_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,0)) x3 + (55,35) U92_A(x1) = ((1,0),(0,0)) x1 + (1,27) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (68,38) nil_A() = (22,2) U11_A(x1,x2) = ((0,0),(0,1)) x2 + (42,6) U21_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,0)) x3 + (62,40) U91_A(x1,x2) = x1 + ((1,1),(0,0)) x2 + (36,32) n__nil_A() = (21,1) a_A() = (33,26) e_A() = (1,25) i_A() = (33,26) o_A() = (33,26) u_A() = (1,25) precedence: n__a > isList = u > isPalListKind > __ > U24# = U25# = n__o = o > nil > n__i > U92 > isList# = U11# = U12# = n__u > n____ = U41# = U42# = isQid > U21# > U22# > U23# > U12 = U11 > n__nil > U45# = U43# = U44# > a > i > U21 > U46 = U45 > U54 > U91 > U23 = U22 > U24 > tt = U56 = U55 > isNeList# > U52 > U51 > isNeList > U41 > U42 > U44 = U43 > activate > U13 > U31 > U32 > U33 > U53 > e > n__e > U25 > U26 partial status: pi(U45#) = [] pi(tt) = [] pi(isNeList#) = [1] pi(activate) = [] pi(n____) = [2] pi(U41#) = [] pi(isPalListKind) = [1] pi(U42#) = [] pi(U43#) = [] pi(U44#) = [] pi(isList) = [1] pi(isList#) = [1] pi(U11#) = [] pi(U12#) = [] pi(U21#) = [] pi(U22#) = [2] pi(U23#) = [] pi(U24#) = [] pi(U25#) = [] pi(U46) = [] pi(U56) = [] pi(U45) = [] pi(isNeList) = [] pi(U55) = [] pi(U44) = [] pi(U54) = [] pi(U26) = [] pi(U33) = [] pi(U43) = [] pi(U53) = [] pi(isQid) = [1] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(U25) = [] pi(U32) = [] pi(U42) = [] pi(U52) = [] pi(U24) = [] pi(U31) = [2] pi(U41) = [] pi(U51) = [] pi(U13) = [] pi(U23) = [] pi(U12) = [] pi(U22) = [] pi(U92) = [] pi(__) = [1, 2] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U91) = [1] 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)) -> U41#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p2: U41#(tt(),V1,V2) -> U42#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p3: U42#(tt(),V1,V2) -> U43#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p4: U43#(tt(),V1,V2) -> U44#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p5: U44#(tt(),V1,V2) -> U45#(isList(activate(V1)),activate(V2)) p6: U44#(tt(),V1,V2) -> isList#(activate(V1)) p7: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V)) p8: U11#(tt(),V) -> U12#(isPalListKind(activate(V)),activate(V)) p9: U12#(tt(),V) -> isNeList#(activate(V)) p10: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p11: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p12: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p13: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p14: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p15: U25#(tt(),V2) -> isList#(activate(V2)) p16: U24#(tt(),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: U11(tt(),V) -> U12(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> U41#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p2: U41#(tt(),V1,V2) -> U42#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p3: U42#(tt(),V1,V2) -> U43#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p4: U43#(tt(),V1,V2) -> U44#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p5: U44#(tt(),V1,V2) -> isList#(activate(V1)) p6: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p7: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p8: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p9: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p10: U24#(tt(),V1,V2) -> isList#(activate(V1)) p11: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V)) p12: U11#(tt(),V) -> U12#(isPalListKind(activate(V)),activate(V)) p13: U12#(tt(),V) -> isNeList#(activate(V)) p14: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p15: U25#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: 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, r38, r39, r40, r41, r42, r43, r44, r45, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isNeList#_A(x1) = ((1,0),(1,1)) x1 + (16,13) n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (16,4) U41#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,0),(1,1)) x3 + (24,21) isPalListKind_A(x1) = ((1,1),(1,1)) x1 + (2,16) activate_A(x1) = x1 + (0,1) tt_A() = (15,15) U42#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,0),(1,1)) x3 + (22,19) U43#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((0,0),(1,1)) x3 + (21,17) U44#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (20,12) isList#_A(x1) = ((1,0),(1,1)) x1 + (19,18) U21#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (23,11) U22#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (23,24) U23#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (22,22) U24#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (21,20) U11#_A(x1,x2) = ((1,0),(1,1)) x2 + (18,17) U12#_A(x1,x2) = ((1,0),(1,1)) x2 + (17,14) U25#_A(x1,x2) = ((1,0),(1,1)) x2 + (20,19) isList_A(x1) = ((1,1),(1,0)) x1 + (13,0) U46_A(x1) = (16,15) U56_A(x1) = ((0,1),(0,1)) x1 + (16,0) U45_A(x1,x2) = (17,15) isNeList_A(x1) = ((1,1),(1,0)) x1 + (6,0) U55_A(x1,x2) = ((1,0),(1,0)) x2 + (17,0) U44_A(x1,x2,x3) = (18,16) U54_A(x1,x2,x3) = ((1,0),(1,0)) x3 + (18,0) U26_A(x1) = (16,16) U33_A(x1) = ((0,1),(0,1)) x1 + (1,0) U43_A(x1,x2,x3) = (19,16) U53_A(x1,x2,x3) = ((1,0),(1,0)) x3 + (19,0) isQid_A(x1) = ((1,0),(1,0)) x1 n__a_A() = (16,15) n__e_A() = (16,15) n__i_A() = (15,0) n__o_A() = (16,0) n__u_A() = (16,15) U25_A(x1,x2) = ((0,1),(0,0)) x1 + (16,16) U32_A(x1,x2) = ((1,1),(1,0)) x2 + (2,0) U42_A(x1,x2,x3) = ((0,1),(0,0)) x2 + (20,16) U52_A(x1,x2,x3) = ((1,0),(1,0)) x3 + (20,16) U24_A(x1,x2,x3) = ((1,1),(0,0)) x2 + (17,16) U31_A(x1,x2) = ((1,1),(1,0)) x2 + (4,0) U41_A(x1,x2,x3) = ((1,1),(0,0)) x2 + (22,16) U51_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,0),(1,0)) x3 + (9,16) U13_A(x1) = x1 + (1,0) U23_A(x1,x2,x3) = ((1,1),(0,0)) x2 + (19,16) U12_A(x1,x2) = ((1,1),(1,0)) x2 + (9,0) U22_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,0),(0,0)) x3 + (21,16) U92_A(x1) = ((0,1),(0,0)) x1 + (1,16) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (16,4) nil_A() = (16,15) U11_A(x1,x2) = ((1,1),(1,0)) x2 + (11,0) U21_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,0),(0,0)) x3 + (23,16) U91_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,0)) x2 + (0,1) n__nil_A() = (16,15) a_A() = (16,16) e_A() = (16,15) i_A() = (15,0) o_A() = (16,0) u_A() = (16,15) precedence: isQid > isPalListKind > isList > U22 = U21 > U25 = U24 = U23 > n__u = u > isNeList > U53 = U52 = U51 > nil = n__nil > U13 = U12 = U11 > U54 > U31 > activate = e = i > n__o = o > U56 = U55 > U43# > U44# = isList# = U21# = U22# = U23# = U24# = U25# > isNeList# = U11# = U12# > U42 = U41 > U43 > U44 > U46 = U45 > U92 = U91 > U41# > n__i = U32 > __ > n____ > U33 > tt > U42# > a > n__a > U26 = n__e partial status: pi(isNeList#) = [] pi(n____) = [2] pi(U41#) = [] pi(isPalListKind) = [] pi(activate) = [1] pi(tt) = [] pi(U42#) = [] pi(U43#) = [] pi(U44#) = [] pi(isList#) = [] pi(U21#) = [] pi(U22#) = [] pi(U23#) = [] pi(U24#) = [] pi(U11#) = [] pi(U12#) = [] pi(U25#) = [] pi(isList) = [] pi(U46) = [] pi(U56) = [] pi(U45) = [] pi(isNeList) = [] pi(U55) = [] pi(U44) = [] pi(U54) = [] pi(U26) = [] pi(U33) = [] pi(U43) = [] pi(U53) = [] pi(isQid) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(U25) = [] pi(U32) = [] pi(U42) = [] pi(U52) = [] pi(U24) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(U13) = [] pi(U23) = [] pi(U12) = [] pi(U22) = [] pi(U92) = [] pi(__) = [1, 2] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U91) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p11 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(n____(V1,V2)) -> U41#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p2: U41#(tt(),V1,V2) -> U42#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p3: U42#(tt(),V1,V2) -> U43#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p4: U43#(tt(),V1,V2) -> U44#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p5: U44#(tt(),V1,V2) -> isList#(activate(V1)) p6: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p7: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p8: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p9: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p10: U24#(tt(),V1,V2) -> isList#(activate(V1)) p11: U11#(tt(),V) -> U12#(isPalListKind(activate(V)),activate(V)) p12: U12#(tt(),V) -> isNeList#(activate(V)) p13: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p14: U25#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The estimated dependency graph contains the following SCCs: {p6, p7, p8, p9, p10, p13, p14} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p2: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p3: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p4: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p5: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p6: U25#(tt(),V2) -> isList#(activate(V2)) p7: U24#(tt(),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: U11(tt(),V) -> U12(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: 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, r38, r39, r40, r41, r42, r43, r44, r45, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isList#_A(x1) = ((1,1),(0,1)) x1 + (118,89) n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,108) U21#_A(x1,x2,x3) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(0,1)) x3 + (1,133) isPalListKind_A(x1) = ((1,0),(0,0)) x1 + (108,109) activate_A(x1) = x1 + (2,1) tt_A() = (88,89) U22#_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(0,1)) x3 + (55,20) U23#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(0,1)) x3 + (137,99) U24#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(0,1)) x3 + (130,95) U25#_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (33,1) isList_A(x1) = ((1,1),(0,1)) x1 + (176,92) U46_A(x1) = (89,90) U56_A(x1) = (89,89) U45_A(x1,x2) = ((0,1),(0,0)) x1 + (1,90) isNeList_A(x1) = ((0,1),(1,1)) x1 + (145,139) U55_A(x1,x2) = (180,89) U44_A(x1,x2,x3) = ((0,0),(1,1)) x1 + ((1,1),(0,0)) x2 + (95,0) U54_A(x1,x2,x3) = ((0,1),(0,0)) x1 + (92,89) U26_A(x1) = ((0,1),(0,0)) x1 + (1,90) U33_A(x1) = ((0,1),(0,1)) x1 + (1,0) U43_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((0,1),(1,1)) x3 + (99,219) U53_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,0),(0,1)) x2 + ((0,1),(0,0)) x3 + (113,89) isQid_A(x1) = ((0,1),(0,0)) x1 + (86,89) n__a_A() = (1,3) n__e_A() = (0,3) n__i_A() = (1,3) n__o_A() = (89,89) n__u_A() = (89,89) U25_A(x1,x2) = ((0,1),(0,0)) x2 + (95,92) U32_A(x1,x2) = ((0,1),(0,0)) x2 + (91,90) U42_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(0,0)) x2 + ((0,1),(1,1)) x3 + (104,133) U52_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(0,1)) x2 + ((0,1),(0,0)) x3 + (136,90) U24_A(x1,x2,x3) = ((1,1),(0,0)) x1 + ((0,1),(0,0)) x2 + ((0,1),(0,0)) x3 + (1,92) U31_A(x1,x2) = ((0,1),(0,1)) x2 + (111,90) U41_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((0,1),(1,1)) x3 + (109,245) U51_A(x1,x2,x3) = ((1,1),(0,1)) x2 + ((0,1),(0,0)) x3 + (248,109) U13_A(x1) = ((1,0),(0,0)) x1 + (1,90) U23_A(x1,x2,x3) = ((0,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (223,109) U12_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,1),(0,1)) x2 + (60,90) U22_A(x1,x2,x3) = ((0,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (228,109) U92_A(x1) = (88,89) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (2,109) nil_A() = (2,1) U11_A(x1,x2) = ((1,1),(0,1)) x2 + (172,91) U21_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (233,109) U91_A(x1,x2) = ((0,1),(0,1)) x1 n__nil_A() = (1,1) a_A() = (2,3) e_A() = (0,3) i_A() = (2,4) o_A() = (90,90) u_A() = (90,90) precedence: isQid > isList > e > isNeList > U42 = U41 > U44 = U43 > a > U54 = U53 = U52 = U51 > U32 = U31 > n__nil > n__a > o > i > n__u > U11 > n__i > U13 = U12 > n__o > U45 > U25 = U24 = U23 = U22 = U21 > isPalListKind = activate = tt = U46 = U56 = U55 = U26 = U33 = U92 = U91 > n__e > __ > n____ > isList# = U24# > U25# > U21# = U22# = U23# = nil = u partial status: pi(isList#) = [] pi(n____) = [1] pi(U21#) = [] pi(isPalListKind) = [] pi(activate) = [] pi(tt) = [] pi(U22#) = [] pi(U23#) = [] pi(U24#) = [2] pi(U25#) = [] pi(isList) = [] pi(U46) = [] pi(U56) = [] pi(U45) = [] pi(isNeList) = [] pi(U55) = [] pi(U44) = [] pi(U54) = [] pi(U26) = [] pi(U33) = [] pi(U43) = [] pi(U53) = [] pi(isQid) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(U25) = [] pi(U32) = [] pi(U42) = [] pi(U52) = [] pi(U24) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [2] pi(U13) = [] pi(U23) = [] pi(U12) = [] pi(U22) = [] pi(U92) = [] pi(__) = [1] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U91) = [] pi(n__nil) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p7 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p2: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p3: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p4: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p5: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p6: U25#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p2: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p3: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p4: U23#(tt(),V1,V2) -> U24#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p5: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p6: U25#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: 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, r38, r39, r40, r41, r42, r43, r44, r45, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isList#_A(x1) = ((0,1),(1,1)) x1 + (131,131) n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (711,0) U21#_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((0,1),(1,1)) x3 + (105,223) isPalListKind_A(x1) = ((0,0),(1,1)) x1 + (286,353) activate_A(x1) = x1 + (1,12) tt_A() = (141,143) U22#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,1),(1,1)) x3 + (79,196) U23#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,1),(1,1)) x3 + (53,170) U24#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,1),(1,1)) x3 + (27,144) U25#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(1,1)) x2 + (1,118) isList_A(x1) = ((1,1),(1,0)) x1 + (654,12) U46_A(x1) = ((0,0),(0,1)) x1 + (142,0) U56_A(x1) = (142,144) U45_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,7) isNeList_A(x1) = ((0,1),(1,0)) x1 + (304,6) U55_A(x1,x2) = ((0,0),(1,1)) x1 + (143,0) U44_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((0,0),(1,0)) x3 + (13,8) U54_A(x1,x2,x3) = ((0,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (144,323) U26_A(x1) = (141,143) U33_A(x1) = ((0,0),(0,1)) x1 + (142,0) U43_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((0,0),(1,0)) x3 + (26,9) U53_A(x1,x2,x3) = ((0,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (145,337) isQid_A(x1) = ((1,0),(1,0)) x1 + (139,3) n__a_A() = (140,1) n__e_A() = (140,143) n__i_A() = (140,1) n__o_A() = (140,1) n__u_A() = (142,143) U25_A(x1,x2) = ((1,0),(1,0)) x1 + (527,2) U32_A(x1,x2) = ((0,0),(1,0)) x2 + (143,4) U42_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((0,0),(1,0)) x3 + (39,10) U52_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (5,351) U24_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,1),(0,0)) x3 + (1195,669) U31_A(x1,x2) = ((0,0),(1,0)) x2 + (287,5) U41_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((0,0),(1,0)) x3 + (291,11) U51_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,0),(1,0)) x3 + (291,365) U13_A(x1) = ((0,0),(0,1)) x1 + (142,0) U23_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,1),(0,0)) x3 + (1221,682) U12_A(x1,x2) = ((0,0),(1,0)) x2 + (143,7) U22_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,0)) x3 + (1247,695) U92_A(x1) = ((0,0),(1,1)) x1 + (141,0) ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (712,1) nil_A() = (142,144) U11_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,0),(1,0)) x2 + (1,8) U21_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,0)) x3 + (1274,709) U91_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,1)) x2 + (0,652) n__nil_A() = (142,143) a_A() = (140,1) e_A() = (140,143) i_A() = (140,1) o_A() = (141,1) u_A() = (142,143) precedence: U54 > U55 > i > isNeList = U41 > activate = u > o > nil > n__o > isPalListKind = U91 > n____ = U21# = __ > U22# > U23# > U51 > U52 > U53 > isList > U21 > U22 > U24 = U23 > a > U44 = U43 = U42 > U11 > U24# > isList# = U25# > U26 = U25 > e > n__i > U92 > U45 > U56 > n__nil > n__e = U12 > isQid > tt = U46 = U13 > U31 > U32 > U33 > n__a > n__u partial status: pi(isList#) = [] pi(n____) = [2] pi(U21#) = [] pi(isPalListKind) = [] pi(activate) = [1] pi(tt) = [] pi(U22#) = [] pi(U23#) = [] pi(U24#) = [] pi(U25#) = [] pi(isList) = [] pi(U46) = [] pi(U56) = [] pi(U45) = [] pi(isNeList) = [] pi(U55) = [] pi(U44) = [] pi(U54) = [] pi(U26) = [] pi(U33) = [] pi(U43) = [] pi(U53) = [] pi(isQid) = [] pi(n__a) = [] pi(n__e) = [] pi(n__i) = [] pi(n__o) = [] pi(n__u) = [] pi(U25) = [] pi(U32) = [] pi(U42) = [] pi(U52) = [] pi(U24) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(U13) = [] pi(U23) = [] pi(U12) = [] pi(U22) = [] pi(U92) = [] pi(__) = [2] pi(nil) = [] pi(U11) = [] pi(U21) = [] pi(U91) = [] pi(n__nil) = [] 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: isList#(n____(V1,V2)) -> U21#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p2: U21#(tt(),V1,V2) -> U22#(isPalListKind(activate(V1)),activate(V1),activate(V2)) p3: U22#(tt(),V1,V2) -> U23#(isPalListKind(activate(V2)),activate(V1),activate(V2)) p4: U24#(tt(),V1,V2) -> U25#(isList(activate(V1)),activate(V2)) p5: U25#(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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: 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: isPalListKind#(n____(V1,V2)) -> U91#(isPalListKind(activate(V1)),activate(V2)) p2: U91#(tt(),V2) -> isPalListKind#(activate(V2)) p3: 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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The set of usable rules consists of r1, r2, r3, r38, r39, r50, r51, r52, r53, r54, r55, r56, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isPalListKind#_A(x1) = x1 + (5,3) n_____A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (6,4) U91#_A(x1,x2) = x2 + (8,5) isPalListKind_A(x1) = (0,1) activate_A(x1) = x1 + (2,2) tt_A() = (0,0) U92_A(x1) = (0,0) ___A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (7,5) nil_A() = (0,1) U91_A(x1,x2) = (0,0) n__nil_A() = (0,1) a_A() = (1,2) n__a_A() = (0,1) e_A() = (2,2) n__e_A() = (1,1) i_A() = (2,2) n__i_A() = (1,1) o_A() = (2,2) n__o_A() = (1,1) u_A() = (1,2) n__u_A() = (0,1) precedence: activate = nil = n__nil > U91# > __ > a = n__a = i = o = u = n__u > e = n__e > isPalListKind# = n____ = isPalListKind = tt = U92 = U91 = n__i > n__o partial status: pi(isPalListKind#) = [1] pi(n____) = [1, 2] pi(U91#) = [] pi(isPalListKind) = [] pi(activate) = [] pi(tt) = [] pi(U92) = [] pi(__) = [1, 2] pi(nil) = [] pi(U91) = [] pi(n__nil) = [] pi(a) = [] pi(n__a) = [] pi(e) = [] pi(n__e) = [] pi(i) = [] pi(n__i) = [] pi(o) = [] pi(n__o) = [] pi(u) = [] pi(n__u) = [] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: U91#(tt(),V2) -> isPalListKind#(activate(V2)) p2: 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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The estimated dependency graph contains the following SCCs: {p2} -- 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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The set of usable rules consists of r1, r2, r3, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: isPalListKind#_A(x1) = ((0,1),(1,1)) x1 + (1,0) n_____A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (1,2) activate_A(x1) = x1 + (2,1) ___A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (1,2) nil_A() = (1,1) n__nil_A() = (1,1) a_A() = (1,1) n__a_A() = (1,1) e_A() = (2,2) n__e_A() = (1,1) i_A() = (1,1) n__i_A() = (1,1) o_A() = (2,2) n__o_A() = (1,1) u_A() = (1,1) n__u_A() = (1,1) precedence: isPalListKind# = n____ = __ = nil = n__nil = a > activate = n__a = e = n__e = i = n__i = o = n__o = u = n__u partial status: pi(isPalListKind#) = [] pi(n____) = [1, 2] pi(activate) = [1] pi(__) = [1, 2] pi(nil) = [] pi(n__nil) = [] pi(a) = [] pi(n__a) = [] pi(e) = [] pi(n__e) = [] pi(i) = [] pi(n__i) = [] pi(o) = [] pi(n__o) = [] pi(u) = [] pi(n__u) = [] The next rules are strictly ordered: p1 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(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The set of usable rules consists of r1, r2, r3, r63 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: __#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (3,2) ___A(x1,x2) = x1 + x2 + (2,1) nil_A() = (1,1) n_____A(x1,x2) = (1,1) precedence: __ > __# = nil = n____ partial status: pi(__#) = [1] pi(__) = [] pi(nil) = [] pi(n____) = [] The next rules are strictly ordered: p2 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z)) and R consists of: r1: __(__(X,Y),Z) -> __(X,__(Y,Z)) r2: __(X,nil()) -> X r3: __(nil(),X) -> X r4: U11(tt(),V) -> U12(isPalListKind(activate(V)),activate(V)) r5: U12(tt(),V) -> U13(isNeList(activate(V))) r6: U13(tt()) -> tt() r7: U21(tt(),V1,V2) -> U22(isPalListKind(activate(V1)),activate(V1),activate(V2)) r8: U22(tt(),V1,V2) -> U23(isPalListKind(activate(V2)),activate(V1),activate(V2)) r9: U23(tt(),V1,V2) -> U24(isPalListKind(activate(V2)),activate(V1),activate(V2)) r10: U24(tt(),V1,V2) -> U25(isList(activate(V1)),activate(V2)) r11: U25(tt(),V2) -> U26(isList(activate(V2))) r12: U26(tt()) -> tt() r13: U31(tt(),V) -> U32(isPalListKind(activate(V)),activate(V)) r14: U32(tt(),V) -> U33(isQid(activate(V))) r15: U33(tt()) -> tt() r16: U41(tt(),V1,V2) -> U42(isPalListKind(activate(V1)),activate(V1),activate(V2)) r17: U42(tt(),V1,V2) -> U43(isPalListKind(activate(V2)),activate(V1),activate(V2)) r18: U43(tt(),V1,V2) -> U44(isPalListKind(activate(V2)),activate(V1),activate(V2)) r19: U44(tt(),V1,V2) -> U45(isList(activate(V1)),activate(V2)) r20: U45(tt(),V2) -> U46(isNeList(activate(V2))) r21: U46(tt()) -> tt() r22: U51(tt(),V1,V2) -> U52(isPalListKind(activate(V1)),activate(V1),activate(V2)) r23: U52(tt(),V1,V2) -> U53(isPalListKind(activate(V2)),activate(V1),activate(V2)) r24: U53(tt(),V1,V2) -> U54(isPalListKind(activate(V2)),activate(V1),activate(V2)) r25: U54(tt(),V1,V2) -> U55(isNeList(activate(V1)),activate(V2)) r26: U55(tt(),V2) -> U56(isList(activate(V2))) r27: U56(tt()) -> tt() r28: U61(tt(),V) -> U62(isPalListKind(activate(V)),activate(V)) r29: U62(tt(),V) -> U63(isQid(activate(V))) r30: U63(tt()) -> tt() r31: U71(tt(),I,P) -> U72(isPalListKind(activate(I)),activate(P)) r32: U72(tt(),P) -> U73(isPal(activate(P)),activate(P)) r33: U73(tt(),P) -> U74(isPalListKind(activate(P))) r34: U74(tt()) -> tt() r35: U81(tt(),V) -> U82(isPalListKind(activate(V)),activate(V)) r36: U82(tt(),V) -> U83(isNePal(activate(V))) r37: U83(tt()) -> tt() r38: U91(tt(),V2) -> U92(isPalListKind(activate(V2))) r39: U92(tt()) -> tt() r40: isList(V) -> U11(isPalListKind(activate(V)),activate(V)) r41: isList(n__nil()) -> tt() r42: isList(n____(V1,V2)) -> U21(isPalListKind(activate(V1)),activate(V1),activate(V2)) r43: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V)) r44: isNeList(n____(V1,V2)) -> U41(isPalListKind(activate(V1)),activate(V1),activate(V2)) r45: isNeList(n____(V1,V2)) -> U51(isPalListKind(activate(V1)),activate(V1),activate(V2)) r46: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V)) r47: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(I),activate(P)) r48: isPal(V) -> U81(isPalListKind(activate(V)),activate(V)) r49: isPal(n__nil()) -> tt() r50: isPalListKind(n__a()) -> tt() r51: isPalListKind(n__e()) -> tt() r52: isPalListKind(n__i()) -> tt() r53: isPalListKind(n__nil()) -> tt() r54: isPalListKind(n__o()) -> tt() r55: isPalListKind(n__u()) -> tt() r56: isPalListKind(n____(V1,V2)) -> U91(isPalListKind(activate(V1)),activate(V2)) r57: isQid(n__a()) -> tt() r58: isQid(n__e()) -> tt() r59: isQid(n__i()) -> tt() r60: isQid(n__o()) -> tt() r61: isQid(n__u()) -> tt() r62: nil() -> n__nil() r63: __(X1,X2) -> n____(X1,X2) r64: a() -> n__a() r65: e() -> n__e() r66: i() -> n__i() r67: o() -> n__o() r68: u() -> n__u() r69: activate(n__nil()) -> nil() r70: activate(n____(X1,X2)) -> __(X1,X2) r71: activate(n__a()) -> a() r72: activate(n__e()) -> e() r73: activate(n__i()) -> i() r74: activate(n__o()) -> o() r75: activate(n__u()) -> u() r76: activate(X) -> X The set of usable rules consists of r1, r2, r3, r63 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: __#_A(x1,x2) = ((1,1),(0,1)) x1 + (2,2) ___A(x1,x2) = x1 + x2 + (1,1) nil_A() = (1,1) n_____A(x1,x2) = (0,0) precedence: __ > __# = nil = n____ partial status: pi(__#) = [1] pi(__) = [] pi(nil) = [] pi(n____) = [] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains.