YES We show the termination of the TRS R: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> tt() a__U21(tt(),V2) -> a__U22(a__isList(V2)) a__U22(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isList(V2)) a__U52(tt()) -> tt() a__U61(tt()) -> tt() a__U71(tt(),P) -> a__U72(a__isPal(P)) a__U72(tt()) -> tt() a__U81(tt()) -> tt() a__isList(V) -> a__U11(a__isNeList(V)) a__isList(nil()) -> tt() a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) a__isNeList(V) -> a__U31(a__isQid(V)) a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) a__isNePal(V) -> a__U61(a__isQid(V)) a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) a__isPal(V) -> a__U81(a__isNePal(V)) a__isPal(nil()) -> tt() a__isQid(a()) -> tt() a__isQid(e()) -> tt() a__isQid(i()) -> tt() a__isQid(o()) -> tt() a__isQid(u()) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isList(X)) -> a__isList(X) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(U72(X)) -> a__U72(mark(X)) mark(isPal(X)) -> a__isPal(X) mark(U81(X)) -> a__U81(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(nil()) -> nil() mark(tt()) -> tt() mark(a()) -> a() mark(e()) -> e() mark(i()) -> i() mark(o()) -> o() mark(u()) -> u() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isList(X) -> isList(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNeList(X) -> isNeList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X1,X2) -> U71(X1,X2) a__U72(X) -> U72(X) a__isPal(X) -> isPal(X) a__U81(X) -> U81(X) a__isQid(X) -> isQid(X) a__isNePal(X) -> isNePal(X) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) p2: a____#(__(X,Y),Z) -> mark#(X) p3: a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) p4: a____#(__(X,Y),Z) -> mark#(Y) p5: a____#(__(X,Y),Z) -> mark#(Z) p6: a____#(X,nil()) -> mark#(X) p7: a____#(nil(),X) -> mark#(X) p8: a__U21#(tt(),V2) -> a__U22#(a__isList(V2)) p9: a__U21#(tt(),V2) -> a__isList#(V2) p10: a__U41#(tt(),V2) -> a__U42#(a__isNeList(V2)) p11: a__U41#(tt(),V2) -> a__isNeList#(V2) p12: a__U51#(tt(),V2) -> a__U52#(a__isList(V2)) p13: a__U51#(tt(),V2) -> a__isList#(V2) p14: a__U71#(tt(),P) -> a__U72#(a__isPal(P)) p15: a__U71#(tt(),P) -> a__isPal#(P) p16: a__isList#(V) -> a__U11#(a__isNeList(V)) p17: a__isList#(V) -> a__isNeList#(V) p18: a__isList#(__(V1,V2)) -> a__U21#(a__isList(V1),V2) p19: a__isList#(__(V1,V2)) -> a__isList#(V1) p20: a__isNeList#(V) -> a__U31#(a__isQid(V)) p21: a__isNeList#(V) -> a__isQid#(V) p22: a__isNeList#(__(V1,V2)) -> a__U41#(a__isList(V1),V2) p23: a__isNeList#(__(V1,V2)) -> a__isList#(V1) p24: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p25: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) p26: a__isNePal#(V) -> a__U61#(a__isQid(V)) p27: a__isNePal#(V) -> a__isQid#(V) p28: a__isNePal#(__(I,__(P,I))) -> a__U71#(a__isQid(I),P) p29: a__isNePal#(__(I,__(P,I))) -> a__isQid#(I) p30: a__isPal#(V) -> a__U81#(a__isNePal(V)) p31: a__isPal#(V) -> a__isNePal#(V) p32: mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) p33: mark#(__(X1,X2)) -> mark#(X1) p34: mark#(__(X1,X2)) -> mark#(X2) p35: mark#(U11(X)) -> a__U11#(mark(X)) p36: mark#(U11(X)) -> mark#(X) p37: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) p38: mark#(U21(X1,X2)) -> mark#(X1) p39: mark#(U22(X)) -> a__U22#(mark(X)) p40: mark#(U22(X)) -> mark#(X) p41: mark#(isList(X)) -> a__isList#(X) p42: mark#(U31(X)) -> a__U31#(mark(X)) p43: mark#(U31(X)) -> mark#(X) p44: mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) p45: mark#(U41(X1,X2)) -> mark#(X1) p46: mark#(U42(X)) -> a__U42#(mark(X)) p47: mark#(U42(X)) -> mark#(X) p48: mark#(isNeList(X)) -> a__isNeList#(X) p49: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) p50: mark#(U51(X1,X2)) -> mark#(X1) p51: mark#(U52(X)) -> a__U52#(mark(X)) p52: mark#(U52(X)) -> mark#(X) p53: mark#(U61(X)) -> a__U61#(mark(X)) p54: mark#(U61(X)) -> mark#(X) p55: mark#(U71(X1,X2)) -> a__U71#(mark(X1),X2) p56: mark#(U71(X1,X2)) -> mark#(X1) p57: mark#(U72(X)) -> a__U72#(mark(X)) p58: mark#(U72(X)) -> mark#(X) p59: mark#(isPal(X)) -> a__isPal#(X) p60: mark#(U81(X)) -> a__U81#(mark(X)) p61: mark#(U81(X)) -> mark#(X) p62: mark#(isQid(X)) -> a__isQid#(X) p63: mark#(isNePal(X)) -> a__isNePal#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p32, p33, p34, p36, p38, p40, p43, p45, p47, p50, p52, p54, p56, p58, p61} {p9, p11, p13, p17, p18, p19, p22, p23, p24, p25} {p15, p28, p31} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) p2: a____#(nil(),X) -> mark#(X) p3: mark#(U81(X)) -> mark#(X) p4: mark#(U72(X)) -> mark#(X) p5: mark#(U71(X1,X2)) -> mark#(X1) p6: mark#(U61(X)) -> mark#(X) p7: mark#(U52(X)) -> mark#(X) p8: mark#(U51(X1,X2)) -> mark#(X1) p9: mark#(U42(X)) -> mark#(X) p10: mark#(U41(X1,X2)) -> mark#(X1) p11: mark#(U31(X)) -> mark#(X) p12: mark#(U22(X)) -> mark#(X) p13: mark#(U21(X1,X2)) -> mark#(X1) p14: mark#(U11(X)) -> mark#(X) p15: mark#(__(X1,X2)) -> mark#(X2) p16: mark#(__(X1,X2)) -> mark#(X1) p17: mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) p18: a____#(X,nil()) -> mark#(X) p19: a____#(__(X,Y),Z) -> mark#(Z) p20: a____#(__(X,Y),Z) -> mark#(Y) p21: a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) p22: a____#(__(X,Y),Z) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a____#_A(x1,x2) = x1 + x2 + 1 ___A(x1,x2) = x1 + x2 + 9 mark_A(x1) = x1 a_____A(x1,x2) = x1 + x2 + 9 nil_A() = 1 mark#_A(x1) = x1 U81_A(x1) = x1 + 1 U72_A(x1) = x1 + 1 U71_A(x1,x2) = x1 + x2 + 7 U61_A(x1) = x1 + 1 U52_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 5 U42_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 4 U31_A(x1) = x1 + 1 U22_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 5 U11_A(x1) = x1 + 1 a__U11_A(x1) = x1 + 1 tt_A() = 0 a__U21_A(x1,x2) = x1 + x2 + 5 a__U22_A(x1) = x1 + 1 a__isList_A(x1) = x1 + 3 a__U31_A(x1) = x1 + 1 a__U41_A(x1,x2) = x1 + x2 + 4 a__U42_A(x1) = x1 + 1 a__isNeList_A(x1) = x1 + 2 a__U51_A(x1,x2) = x1 + x2 + 5 a__U52_A(x1) = x1 + 1 a__U61_A(x1) = x1 + 1 a__U71_A(x1,x2) = x1 + x2 + 7 a__U72_A(x1) = x1 + 1 a__isPal_A(x1) = x1 + 5 a__U81_A(x1) = x1 + 1 a__isQid_A(x1) = 1 a__isNePal_A(x1) = x1 + 3 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 isList_A(x1) = x1 + 3 isNeList_A(x1) = x1 + 2 isPal_A(x1) = x1 + 5 isQid_A(x1) = 1 isNePal_A(x1) = x1 + 3 precedence: mark = a____ > __ > a__isPal > nil = U81 = U52 = U51 = U42 = U41 = U31 = a__U21 = a__isList = a__U31 = a__U41 = a__U42 = a__isNeList = a__U51 = a__U52 = a__U81 = a = isPal > U21 = U11 = a__U11 > a____# = e > U72 = a__U61 = a__U71 = a__U72 = a__isQid = a__isNePal = o = isNePal > u > U71 = U61 = U22 = a__U22 = isList = isQid > tt > isNeList > mark# = i partial status: pi(a____#) = [] pi(__) = [] pi(mark) = [] pi(a____) = [] pi(nil) = [] pi(mark#) = [1] pi(U81) = [] pi(U72) = [] pi(U71) = [] pi(U61) = [] pi(U52) = [] pi(U51) = [] pi(U42) = [] pi(U41) = [] pi(U31) = [] pi(U22) = [] pi(U21) = [] pi(U11) = [] pi(a__U11) = [1] pi(tt) = [] pi(a__U21) = [] pi(a__U22) = [] pi(a__isList) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [] pi(a__isNeList) = [] pi(a__U51) = [] pi(a__U52) = [] pi(a__U61) = [] pi(a__U71) = [] pi(a__U72) = [] pi(a__isPal) = [] pi(a__U81) = [] pi(a__isQid) = [] pi(a__isNePal) = [1] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isList) = [] pi(isNeList) = [] pi(isPal) = [] pi(isQid) = [] pi(isNePal) = [1] The next rules are strictly ordered: p17 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) p2: a____#(nil(),X) -> mark#(X) p3: mark#(U81(X)) -> mark#(X) p4: mark#(U72(X)) -> mark#(X) p5: mark#(U71(X1,X2)) -> mark#(X1) p6: mark#(U61(X)) -> mark#(X) p7: mark#(U52(X)) -> mark#(X) p8: mark#(U51(X1,X2)) -> mark#(X1) p9: mark#(U42(X)) -> mark#(X) p10: mark#(U41(X1,X2)) -> mark#(X1) p11: mark#(U31(X)) -> mark#(X) p12: mark#(U22(X)) -> mark#(X) p13: mark#(U21(X1,X2)) -> mark#(X1) p14: mark#(U11(X)) -> mark#(X) p15: mark#(__(X1,X2)) -> mark#(X2) p16: mark#(__(X1,X2)) -> mark#(X1) p17: a____#(X,nil()) -> mark#(X) p18: a____#(__(X,Y),Z) -> mark#(Z) p19: a____#(__(X,Y),Z) -> mark#(Y) p20: a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) p21: a____#(__(X,Y),Z) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p20} {p3, p4, p5, 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: a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) p2: a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a____#_A(x1,x2) = x1 + 1 ___A(x1,x2) = x1 + x2 + 4 mark_A(x1) = x1 a_____A(x1,x2) = x1 + x2 + 4 a__U11_A(x1) = 0 tt_A() = 0 a__U21_A(x1,x2) = x1 + x2 + 1 a__U22_A(x1) = 1 a__isList_A(x1) = x1 + 1 a__U31_A(x1) = 1 a__U41_A(x1,x2) = 1 a__U42_A(x1) = 0 a__isNeList_A(x1) = 5 a__U51_A(x1,x2) = 0 a__U52_A(x1) = 0 a__U61_A(x1) = 1 a__U71_A(x1,x2) = 1 a__U72_A(x1) = 0 a__isPal_A(x1) = 3 a__U81_A(x1) = 1 nil_A() = 1 a__isQid_A(x1) = 3 a__isNePal_A(x1) = 2 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 U11_A(x1) = 0 U21_A(x1,x2) = x1 + x2 + 1 U22_A(x1) = 1 isList_A(x1) = x1 + 1 U31_A(x1) = 1 U41_A(x1,x2) = 1 U42_A(x1) = 0 isNeList_A(x1) = 5 U51_A(x1,x2) = 0 U52_A(x1) = 0 U61_A(x1) = 1 U71_A(x1,x2) = 1 U72_A(x1) = 0 isPal_A(x1) = 3 U81_A(x1) = 1 isQid_A(x1) = 3 isNePal_A(x1) = 2 precedence: a__isNeList = i = isNeList > __ = mark = a____ > a____# > a__U42 = U42 > a__U11 = a__isList > a__U41 > a__U31 = a__U61 = U31 > a > a__U51 = a__U52 = U41 = U52 > a__U21 = a__U22 = a__U71 = a__isPal = a__U81 = nil = a__isNePal > tt = a__U72 = a__isQid > e = o = u = U11 = U21 = U22 = U61 = U71 > isList = U51 = U72 = isPal = U81 = isQid = isNePal partial status: pi(a____#) = [1] pi(__) = [1, 2] pi(mark) = [1] pi(a____) = [1, 2] pi(a__U11) = [] pi(tt) = [] pi(a__U21) = [] pi(a__U22) = [] pi(a__isList) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [] pi(a__isNeList) = [] pi(a__U51) = [] pi(a__U52) = [] pi(a__U61) = [] pi(a__U71) = [] pi(a__U72) = [] pi(a__isPal) = [] pi(a__U81) = [] pi(nil) = [] pi(a__isQid) = [] pi(a__isNePal) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(U11) = [] pi(U21) = [] pi(U22) = [] pi(isList) = [] pi(U31) = [] pi(U41) = [] pi(U42) = [] pi(isNeList) = [] pi(U51) = [] pi(U52) = [] pi(U61) = [] pi(U71) = [] pi(U72) = [] pi(isPal) = [] pi(U81) = [] pi(isQid) = [] pi(isNePal) = [] 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: a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a____#_A(x1,x2) = x1 ___A(x1,x2) = x1 + x2 + 6 mark_A(x1) = x1 a_____A(x1,x2) = x1 + x2 + 6 nil_A() = 1 a__U11_A(x1) = 1 tt_A() = 0 a__U21_A(x1,x2) = x1 + 2 a__U22_A(x1) = 1 a__isList_A(x1) = x1 + 3 a__U31_A(x1) = 8 a__U41_A(x1,x2) = x2 + 2 a__U42_A(x1) = 1 a__isNeList_A(x1) = x1 + 8 a__U51_A(x1,x2) = x2 + 4 a__U52_A(x1) = 1 a__U61_A(x1) = 1 a__U71_A(x1,x2) = x2 + 5 a__U72_A(x1) = x1 + 1 a__isPal_A(x1) = x1 + 3 a__U81_A(x1) = x1 a__isQid_A(x1) = 7 a__isNePal_A(x1) = x1 + 2 a_A() = 1 e_A() = 0 i_A() = 1 o_A() = 1 u_A() = 1 U11_A(x1) = 1 U21_A(x1,x2) = x1 + 2 U22_A(x1) = 1 isList_A(x1) = x1 + 3 U31_A(x1) = 8 U41_A(x1,x2) = x2 + 2 U42_A(x1) = 1 isNeList_A(x1) = x1 + 8 U51_A(x1,x2) = x2 + 4 U52_A(x1) = 1 U61_A(x1) = 1 U71_A(x1,x2) = x2 + 5 U72_A(x1) = x1 + 1 isPal_A(x1) = x1 + 3 U81_A(x1) = x1 isQid_A(x1) = 7 isNePal_A(x1) = x1 + 2 precedence: mark = a____ = a__U11 = a__isList = a__U31 = a__U41 = a__U42 = a__isNeList = isList = U41 > U31 > nil > __ = a__U51 = a__U52 = a__U71 > U11 > a__isPal > a__U21 = a__U61 = a__U81 = a__isNePal = U21 > a__U22 = U22 > e > a__isQid > tt = a__U72 = a = o = u > i > U52 = U61 = U71 = U72 = isPal = isNePal > a____# = U42 = isNeList = U51 = U81 = isQid partial status: pi(a____#) = [] pi(__) = [2] pi(mark) = [] pi(a____) = [] pi(nil) = [] pi(a__U11) = [] pi(tt) = [] pi(a__U21) = [] pi(a__U22) = [] pi(a__isList) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [] pi(a__isNeList) = [] pi(a__U51) = [] pi(a__U52) = [] pi(a__U61) = [] pi(a__U71) = [] pi(a__U72) = [1] pi(a__isPal) = [1] pi(a__U81) = [] pi(a__isQid) = [] pi(a__isNePal) = [1] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(U11) = [] pi(U21) = [] pi(U22) = [] pi(isList) = [] pi(U31) = [] pi(U41) = [] pi(U42) = [] pi(isNeList) = [1] pi(U51) = [] pi(U52) = [] pi(U61) = [] pi(U71) = [2] pi(U72) = [] pi(isPal) = [1] pi(U81) = [] pi(isQid) = [] pi(isNePal) = [1] 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: mark#(U81(X)) -> mark#(X) p2: mark#(__(X1,X2)) -> mark#(X1) p3: mark#(__(X1,X2)) -> mark#(X2) p4: mark#(U11(X)) -> mark#(X) p5: mark#(U21(X1,X2)) -> mark#(X1) p6: mark#(U22(X)) -> mark#(X) p7: mark#(U31(X)) -> mark#(X) p8: mark#(U41(X1,X2)) -> mark#(X1) p9: mark#(U42(X)) -> mark#(X) p10: mark#(U51(X1,X2)) -> mark#(X1) p11: mark#(U52(X)) -> mark#(X) p12: mark#(U61(X)) -> mark#(X) p13: mark#(U71(X1,X2)) -> mark#(X1) p14: mark#(U72(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U81_A(x1) = x1 + 1 ___A(x1,x2) = x1 + x2 + 1 U11_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 1 U22_A(x1) = x1 + 1 U31_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 1 U52_A(x1) = x1 + 1 U61_A(x1) = x1 + 1 U71_A(x1,x2) = x1 + x2 + 1 U72_A(x1) = x1 + 1 precedence: mark# = U81 = __ = U11 = U21 = U22 = U31 = U41 = U42 = U51 = U52 = U61 > U71 = U72 partial status: pi(mark#) = [1] pi(U81) = [1] pi(__) = [1, 2] pi(U11) = [1] pi(U21) = [1, 2] pi(U22) = [1] pi(U31) = [1] pi(U41) = [1, 2] pi(U42) = [1] pi(U51) = [1, 2] pi(U52) = [1] pi(U61) = [1] pi(U71) = [] pi(U72) = [1] The next rules are strictly ordered: p14 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U81(X)) -> mark#(X) p2: mark#(__(X1,X2)) -> mark#(X1) p3: mark#(__(X1,X2)) -> mark#(X2) p4: mark#(U11(X)) -> mark#(X) p5: mark#(U21(X1,X2)) -> mark#(X1) p6: mark#(U22(X)) -> mark#(X) p7: mark#(U31(X)) -> mark#(X) p8: mark#(U41(X1,X2)) -> mark#(X1) p9: mark#(U42(X)) -> mark#(X) p10: mark#(U51(X1,X2)) -> mark#(X1) p11: mark#(U52(X)) -> mark#(X) p12: mark#(U61(X)) -> mark#(X) p13: mark#(U71(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U81(X)) -> mark#(X) p2: mark#(U71(X1,X2)) -> mark#(X1) p3: mark#(U61(X)) -> mark#(X) p4: mark#(U52(X)) -> mark#(X) p5: mark#(U51(X1,X2)) -> mark#(X1) p6: mark#(U42(X)) -> mark#(X) p7: mark#(U41(X1,X2)) -> mark#(X1) p8: mark#(U31(X)) -> mark#(X) p9: mark#(U22(X)) -> mark#(X) p10: mark#(U21(X1,X2)) -> mark#(X1) p11: mark#(U11(X)) -> mark#(X) p12: mark#(__(X1,X2)) -> mark#(X2) p13: mark#(__(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U81_A(x1) = x1 + 1 U71_A(x1,x2) = x1 + x2 + 1 U61_A(x1) = x1 + 1 U52_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U31_A(x1) = x1 + 1 U22_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 1 U11_A(x1) = x1 + 1 ___A(x1,x2) = x1 + x2 + 1 precedence: mark# = U81 = U71 = U61 = U52 = U51 = U42 = U41 = U31 = U22 = U21 = U11 = __ partial status: pi(mark#) = [] pi(U81) = [1] pi(U71) = [2] pi(U61) = [1] pi(U52) = [1] pi(U51) = [2] pi(U42) = [1] pi(U41) = [2] pi(U31) = [1] pi(U22) = [1] pi(U21) = [2] pi(U11) = [1] pi(__) = [2] 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: mark#(U71(X1,X2)) -> mark#(X1) p2: mark#(U61(X)) -> mark#(X) p3: mark#(U52(X)) -> mark#(X) p4: mark#(U51(X1,X2)) -> mark#(X1) p5: mark#(U42(X)) -> mark#(X) p6: mark#(U41(X1,X2)) -> mark#(X1) p7: mark#(U31(X)) -> mark#(X) p8: mark#(U22(X)) -> mark#(X) p9: mark#(U21(X1,X2)) -> mark#(X1) p10: mark#(U11(X)) -> mark#(X) p11: mark#(__(X1,X2)) -> mark#(X2) p12: mark#(__(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U71(X1,X2)) -> mark#(X1) p2: mark#(__(X1,X2)) -> mark#(X1) p3: mark#(__(X1,X2)) -> mark#(X2) p4: mark#(U11(X)) -> mark#(X) p5: mark#(U21(X1,X2)) -> mark#(X1) p6: mark#(U22(X)) -> mark#(X) p7: mark#(U31(X)) -> mark#(X) p8: mark#(U41(X1,X2)) -> mark#(X1) p9: mark#(U42(X)) -> mark#(X) p10: mark#(U51(X1,X2)) -> mark#(X1) p11: mark#(U52(X)) -> mark#(X) p12: mark#(U61(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U71_A(x1,x2) = x1 + x2 + 1 ___A(x1,x2) = x1 + x2 + 1 U11_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 1 U22_A(x1) = x1 + 1 U31_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 1 U52_A(x1) = x1 + 1 U61_A(x1) = x1 + 1 precedence: mark# = U71 = __ > U21 = U22 > U11 = U31 = U41 = U42 > U51 = U52 = U61 partial status: pi(mark#) = [] pi(U71) = [2] pi(__) = [2] pi(U11) = [1] pi(U21) = [2] pi(U22) = [] pi(U31) = [1] pi(U41) = [2] pi(U42) = [] pi(U51) = [2] pi(U52) = [1] pi(U61) = [1] 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: mark#(__(X1,X2)) -> mark#(X1) p2: mark#(__(X1,X2)) -> mark#(X2) p3: mark#(U11(X)) -> mark#(X) p4: mark#(U21(X1,X2)) -> mark#(X1) p5: mark#(U22(X)) -> mark#(X) p6: mark#(U31(X)) -> mark#(X) p7: mark#(U41(X1,X2)) -> mark#(X1) p8: mark#(U42(X)) -> mark#(X) p9: mark#(U51(X1,X2)) -> mark#(X1) p10: mark#(U52(X)) -> mark#(X) p11: mark#(U61(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(__(X1,X2)) -> mark#(X1) p2: mark#(U61(X)) -> mark#(X) p3: mark#(U52(X)) -> mark#(X) p4: mark#(U51(X1,X2)) -> mark#(X1) p5: mark#(U42(X)) -> mark#(X) p6: mark#(U41(X1,X2)) -> mark#(X1) p7: mark#(U31(X)) -> mark#(X) p8: mark#(U22(X)) -> mark#(X) p9: mark#(U21(X1,X2)) -> mark#(X1) p10: mark#(U11(X)) -> mark#(X) p11: mark#(__(X1,X2)) -> mark#(X2) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 1 U61_A(x1) = x1 + 1 U52_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U31_A(x1) = x1 + 1 U22_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 1 U11_A(x1) = x1 + 1 precedence: mark# = __ = U61 = U52 = U51 = U42 = U41 = U31 = U22 = U21 = U11 partial status: pi(mark#) = [] pi(__) = [2] pi(U61) = [1] pi(U52) = [1] pi(U51) = [2] pi(U42) = [1] pi(U41) = [2] pi(U31) = [1] pi(U22) = [1] pi(U21) = [2] pi(U11) = [1] 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: mark#(U61(X)) -> mark#(X) p2: mark#(U52(X)) -> mark#(X) p3: mark#(U51(X1,X2)) -> mark#(X1) p4: mark#(U42(X)) -> mark#(X) p5: mark#(U41(X1,X2)) -> mark#(X1) p6: mark#(U31(X)) -> mark#(X) p7: mark#(U22(X)) -> mark#(X) p8: mark#(U21(X1,X2)) -> mark#(X1) p9: mark#(U11(X)) -> mark#(X) p10: mark#(__(X1,X2)) -> mark#(X2) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U61(X)) -> mark#(X) p2: mark#(__(X1,X2)) -> mark#(X2) p3: mark#(U11(X)) -> mark#(X) p4: mark#(U21(X1,X2)) -> mark#(X1) p5: mark#(U22(X)) -> mark#(X) p6: mark#(U31(X)) -> mark#(X) p7: mark#(U41(X1,X2)) -> mark#(X1) p8: mark#(U42(X)) -> mark#(X) p9: mark#(U51(X1,X2)) -> mark#(X1) p10: mark#(U52(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U61_A(x1) = x1 + 1 ___A(x1,x2) = x1 + x2 + 1 U11_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 1 U22_A(x1) = x1 + 1 U31_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 1 U52_A(x1) = x1 + 1 precedence: mark# = U61 = __ = U11 = U21 = U22 = U31 = U41 = U42 = U51 = U52 partial status: pi(mark#) = [1] pi(U61) = [1] pi(__) = [2] pi(U11) = [1] pi(U21) = [1, 2] pi(U22) = [1] pi(U31) = [1] pi(U41) = [1, 2] pi(U42) = [1] pi(U51) = [1, 2] pi(U52) = [1] 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: mark#(__(X1,X2)) -> mark#(X2) p2: mark#(U11(X)) -> mark#(X) p3: mark#(U21(X1,X2)) -> mark#(X1) p4: mark#(U22(X)) -> mark#(X) p5: mark#(U31(X)) -> mark#(X) p6: mark#(U41(X1,X2)) -> mark#(X1) p7: mark#(U42(X)) -> mark#(X) p8: mark#(U51(X1,X2)) -> mark#(X1) p9: mark#(U52(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(__(X1,X2)) -> mark#(X2) p2: mark#(U52(X)) -> mark#(X) p3: mark#(U51(X1,X2)) -> mark#(X1) p4: mark#(U42(X)) -> mark#(X) p5: mark#(U41(X1,X2)) -> mark#(X1) p6: mark#(U31(X)) -> mark#(X) p7: mark#(U22(X)) -> mark#(X) p8: mark#(U21(X1,X2)) -> mark#(X1) p9: mark#(U11(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 1 U52_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U31_A(x1) = x1 + 1 U22_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 1 U11_A(x1) = x1 + 1 precedence: mark# = __ = U52 = U51 = U42 = U41 = U31 = U22 = U21 = U11 partial status: pi(mark#) = [1] pi(__) = [2] pi(U52) = [1] pi(U51) = [1, 2] pi(U42) = [1] pi(U41) = [1, 2] pi(U31) = [1] pi(U22) = [1] pi(U21) = [1, 2] pi(U11) = [1] 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: mark#(U52(X)) -> mark#(X) p2: mark#(U51(X1,X2)) -> mark#(X1) p3: mark#(U42(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) p5: mark#(U31(X)) -> mark#(X) p6: mark#(U22(X)) -> mark#(X) p7: mark#(U21(X1,X2)) -> mark#(X1) p8: mark#(U11(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U52(X)) -> mark#(X) p2: mark#(U11(X)) -> mark#(X) p3: mark#(U21(X1,X2)) -> mark#(X1) p4: mark#(U22(X)) -> mark#(X) p5: mark#(U31(X)) -> mark#(X) p6: mark#(U41(X1,X2)) -> mark#(X1) p7: mark#(U42(X)) -> mark#(X) p8: mark#(U51(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U52_A(x1) = x1 + 1 U11_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 1 U22_A(x1) = x1 + 1 U31_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 1 precedence: mark# = U52 = U11 = U21 = U22 = U31 = U41 = U42 = U51 partial status: pi(mark#) = [] pi(U52) = [1] pi(U11) = [1] pi(U21) = [2] pi(U22) = [1] pi(U31) = [1] pi(U41) = [2] pi(U42) = [1] pi(U51) = [2] 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: mark#(U11(X)) -> mark#(X) p2: mark#(U21(X1,X2)) -> mark#(X1) p3: mark#(U22(X)) -> mark#(X) p4: mark#(U31(X)) -> mark#(X) p5: mark#(U41(X1,X2)) -> mark#(X1) p6: mark#(U42(X)) -> mark#(X) p7: mark#(U51(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U11(X)) -> mark#(X) p2: mark#(U51(X1,X2)) -> mark#(X1) p3: mark#(U42(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) p5: mark#(U31(X)) -> mark#(X) p6: mark#(U22(X)) -> mark#(X) p7: mark#(U21(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U11_A(x1) = x1 + 1 U51_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U31_A(x1) = x1 + 1 U22_A(x1) = x1 + 1 U21_A(x1,x2) = x1 + x2 + 1 precedence: mark# = U11 = U51 = U42 = U41 = U31 = U22 = U21 partial status: pi(mark#) = [] pi(U11) = [1] pi(U51) = [2] pi(U42) = [1] pi(U41) = [2] pi(U31) = [1] pi(U22) = [1] pi(U21) = [2] 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: mark#(U51(X1,X2)) -> mark#(X1) p2: mark#(U42(X)) -> mark#(X) p3: mark#(U41(X1,X2)) -> mark#(X1) p4: mark#(U31(X)) -> mark#(X) p5: mark#(U22(X)) -> mark#(X) p6: mark#(U21(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(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: mark#(U51(X1,X2)) -> mark#(X1) p2: mark#(U21(X1,X2)) -> mark#(X1) p3: mark#(U22(X)) -> mark#(X) p4: mark#(U31(X)) -> mark#(X) p5: mark#(U41(X1,X2)) -> mark#(X1) p6: mark#(U42(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U51_A(x1,x2) = x1 + x2 + 1 U21_A(x1,x2) = x1 + x2 + 1 U22_A(x1) = x1 + 1 U31_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 precedence: mark# = U51 = U21 = U22 = U31 = U41 = U42 partial status: pi(mark#) = [1] pi(U51) = [1, 2] pi(U21) = [1, 2] pi(U22) = [1] pi(U31) = [1] pi(U41) = [1, 2] pi(U42) = [1] 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: mark#(U21(X1,X2)) -> mark#(X1) p2: mark#(U22(X)) -> mark#(X) p3: mark#(U31(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) p5: mark#(U42(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U21(X1,X2)) -> mark#(X1) p2: mark#(U42(X)) -> mark#(X) p3: mark#(U41(X1,X2)) -> mark#(X1) p4: mark#(U31(X)) -> mark#(X) p5: mark#(U22(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U21_A(x1,x2) = x1 + x2 + 1 U42_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U31_A(x1) = x1 + 1 U22_A(x1) = x1 + 1 precedence: mark# = U21 = U42 = U41 = U31 = U22 partial status: pi(mark#) = [] pi(U21) = [2] pi(U42) = [1] pi(U41) = [2] pi(U31) = [1] pi(U22) = [1] 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: mark#(U42(X)) -> mark#(X) p2: mark#(U41(X1,X2)) -> mark#(X1) p3: mark#(U31(X)) -> mark#(X) p4: mark#(U22(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U42(X)) -> mark#(X) p2: mark#(U22(X)) -> mark#(X) p3: mark#(U31(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U42_A(x1) = x1 + 1 U22_A(x1) = x1 + 1 U31_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 precedence: mark# = U42 = U22 = U31 = U41 partial status: pi(mark#) = [] pi(U42) = [1] pi(U22) = [1] pi(U31) = [1] pi(U41) = [2] 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: mark#(U22(X)) -> mark#(X) p2: mark#(U31(X)) -> mark#(X) p3: mark#(U41(X1,X2)) -> mark#(X1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U22(X)) -> mark#(X) p2: mark#(U41(X1,X2)) -> mark#(X1) p3: mark#(U31(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U22_A(x1) = x1 + 1 U41_A(x1,x2) = x1 + x2 + 1 U31_A(x1) = x1 + 1 precedence: mark# = U22 = U41 = U31 partial status: pi(mark#) = [1] pi(U22) = [1] pi(U41) = [1, 2] pi(U31) = [1] 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: mark#(U41(X1,X2)) -> mark#(X1) p2: mark#(U31(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U41(X1,X2)) -> mark#(X1) p2: mark#(U31(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U41_A(x1,x2) = x1 + x2 + 1 U31_A(x1) = x1 + 1 precedence: U41 > U31 > mark# partial status: pi(mark#) = [1] pi(U41) = [1, 2] pi(U31) = [] 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: mark#(U31(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U31(X)) -> mark#(X) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U31_A(x1) = x1 + 1 precedence: mark# = U31 partial status: pi(mark#) = [1] pi(U31) = [1] 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: a__U21#(tt(),V2) -> a__isList#(V2) p2: a__isList#(__(V1,V2)) -> a__isList#(V1) p3: a__isList#(__(V1,V2)) -> a__U21#(a__isList(V1),V2) p4: a__isList#(V) -> a__isNeList#(V) p5: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) p6: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p7: a__U51#(tt(),V2) -> a__isList#(V2) p8: a__isNeList#(__(V1,V2)) -> a__isList#(V1) p9: a__isNeList#(__(V1,V2)) -> a__U41#(a__isList(V1),V2) p10: a__U41#(tt(),V2) -> a__isNeList#(V2) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r4, r5, r6, r7, r8, r9, r10, r11, r16, r17, r18, r19, r20, r21, r26, r27, r28, r29, r30, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r72 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U21#_A(x1,x2) = x1 + x2 + 3 tt_A() = 8 a__isList#_A(x1) = x1 + 10 ___A(x1,x2) = x1 + x2 + 12 a__isList_A(x1) = x1 + 6 a__isNeList#_A(x1) = x1 + 9 a__U51#_A(x1,x2) = x2 + 11 a__isNeList_A(x1) = x1 + 4 a__U41#_A(x1,x2) = x2 + 13 a__U22_A(x1) = x1 + 1 a__U42_A(x1) = 9 a__U52_A(x1) = x1 + 3 U22_A(x1) = 0 U42_A(x1) = 0 U52_A(x1) = 0 a__U11_A(x1) = x1 + 1 a__U21_A(x1,x2) = x1 + x2 + 1 a__U31_A(x1) = x1 + 1 a__U41_A(x1,x2) = x1 + 7 a__U51_A(x1,x2) = x1 + x2 + 2 a__isQid_A(x1) = x1 + 2 a_A() = 7 e_A() = 7 i_A() = 9 o_A() = 9 u_A() = 7 U11_A(x1) = 0 U21_A(x1,x2) = 0 U31_A(x1) = 0 U41_A(x1,x2) = 0 U51_A(x1,x2) = 0 isQid_A(x1) = 0 nil_A() = 3 isList_A(x1) = 0 isNeList_A(x1) = 1 precedence: a__U51# = a__U22 = U42 = U52 = a__U11 = a__U21 = a__U41 > U22 = a__U51 > __ > a__U21# = a__U42 = a__isQid = a = e = i = o = u = U11 = U21 > a__U52 = U31 > U41 = U51 = isQid > a__isList# > nil = isList = isNeList > tt = a__isList = a__isNeList# = a__isNeList = a__U41# = a__U31 partial status: pi(a__U21#) = [] pi(tt) = [] pi(a__isList#) = [] pi(__) = [] pi(a__isList) = [1] pi(a__isNeList#) = [1] pi(a__U51#) = [2] pi(a__isNeList) = [] pi(a__U41#) = [2] pi(a__U22) = [] pi(a__U42) = [] pi(a__U52) = [] pi(U22) = [] pi(U42) = [] pi(U52) = [] pi(a__U11) = [1] pi(a__U21) = [2] pi(a__U31) = [] pi(a__U41) = [1] pi(a__U51) = [2] pi(a__isQid) = [1] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(U11) = [] pi(U21) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isQid) = [] pi(nil) = [] pi(isList) = [] pi(isNeList) = [] 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: a__isList#(__(V1,V2)) -> a__isList#(V1) p2: a__isList#(__(V1,V2)) -> a__U21#(a__isList(V1),V2) p3: a__isList#(V) -> a__isNeList#(V) p4: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) p5: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p6: a__U51#(tt(),V2) -> a__isList#(V2) p7: a__isNeList#(__(V1,V2)) -> a__isList#(V1) p8: a__isNeList#(__(V1,V2)) -> a__U41#(a__isList(V1),V2) p9: a__U41#(tt(),V2) -> a__isNeList#(V2) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p3, p4, p5, p6, p7, p8, p9} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__isList#(__(V1,V2)) -> a__isList#(V1) p2: a__isList#(V) -> a__isNeList#(V) p3: a__isNeList#(__(V1,V2)) -> a__U41#(a__isList(V1),V2) p4: a__U41#(tt(),V2) -> a__isNeList#(V2) p5: a__isNeList#(__(V1,V2)) -> a__isList#(V1) p6: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p7: a__U51#(tt(),V2) -> a__isList#(V2) p8: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r4, r5, r6, r7, r8, r9, r10, r11, r16, r17, r18, r19, r20, r21, r26, r27, r28, r29, r30, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r72 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__isList#_A(x1) = x1 ___A(x1,x2) = x1 + x2 a__isNeList#_A(x1) = x1 a__U41#_A(x1,x2) = x2 a__isList_A(x1) = x1 + 4 tt_A() = 2 a__U51#_A(x1,x2) = x2 a__isNeList_A(x1) = x1 + 6 a__U22_A(x1) = x1 a__U42_A(x1) = 3 a__U52_A(x1) = 5 U22_A(x1) = 0 U42_A(x1) = 0 U52_A(x1) = 0 a__U11_A(x1) = 3 a__U21_A(x1,x2) = x2 + 4 a__U31_A(x1) = x1 + 1 a__U41_A(x1,x2) = 5 a__U51_A(x1,x2) = x2 + 5 a__isQid_A(x1) = 3 a_A() = 0 e_A() = 0 i_A() = 0 o_A() = 0 u_A() = 0 U11_A(x1) = 1 U21_A(x1,x2) = 4 U31_A(x1) = 0 U41_A(x1,x2) = 1 U51_A(x1,x2) = 1 isQid_A(x1) = 1 nil_A() = 1 isList_A(x1) = 1 isNeList_A(x1) = 0 precedence: a__isList = a__U11 > __ = a__isNeList = a__U52 = U52 = a__U51 > a__U41# = U42 > a__isList# = a__isNeList# = a__U51# = a__U41 > a__U42 > a__U22 = U22 = a__U21 > a__U31 > tt = a__isQid = a = e = i = u = U11 > o = U21 = U31 = U41 = U51 = isQid = nil = isList = isNeList partial status: pi(a__isList#) = [1] pi(__) = [1, 2] pi(a__isNeList#) = [1] pi(a__U41#) = [2] pi(a__isList) = [1] pi(tt) = [] pi(a__U51#) = [2] pi(a__isNeList) = [1] pi(a__U22) = [] pi(a__U42) = [] pi(a__U52) = [] pi(U22) = [] pi(U42) = [] pi(U52) = [] pi(a__U11) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U51) = [2] pi(a__isQid) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(U11) = [] pi(U21) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isQid) = [] pi(nil) = [] pi(isList) = [] pi(isNeList) = [] The next rules are strictly ordered: p3 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__isList#(__(V1,V2)) -> a__isList#(V1) p2: a__isList#(V) -> a__isNeList#(V) p3: a__U41#(tt(),V2) -> a__isNeList#(V2) p4: a__isNeList#(__(V1,V2)) -> a__isList#(V1) p5: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p6: a__U51#(tt(),V2) -> a__isList#(V2) p7: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p4, p5, p6, p7} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__isList#(__(V1,V2)) -> a__isList#(V1) p2: a__isList#(V) -> a__isNeList#(V) p3: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) p4: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p5: a__U51#(tt(),V2) -> a__isList#(V2) p6: a__isNeList#(__(V1,V2)) -> a__isList#(V1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r4, r5, r6, r7, r8, r9, r10, r11, r16, r17, r18, r19, r20, r21, r26, r27, r28, r29, r30, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r72 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__isList#_A(x1) = x1 + 3 ___A(x1,x2) = x1 + x2 + 12 a__isNeList#_A(x1) = x1 a__U51#_A(x1,x2) = x2 + 4 a__isNeList_A(x1) = x1 + 7 tt_A() = 2 a__U22_A(x1) = 3 U22_A(x1) = 0 a__U11_A(x1) = x1 + 1 a__U21_A(x1,x2) = x1 + 2 a__isList_A(x1) = x1 + 9 a__U42_A(x1) = 3 a__U52_A(x1) = 3 U11_A(x1) = 0 U21_A(x1,x2) = 0 U42_A(x1) = 0 U52_A(x1) = 0 a__U31_A(x1) = x1 + 1 a__U41_A(x1,x2) = x1 + 2 a__U51_A(x1,x2) = x1 + 6 nil_A() = 1 a__isQid_A(x1) = x1 + 2 a_A() = 3 e_A() = 1 i_A() = 1 o_A() = 3 u_A() = 1 isList_A(x1) = 0 U31_A(x1) = 0 U41_A(x1,x2) = 0 U51_A(x1,x2) = 0 isQid_A(x1) = 1 isNeList_A(x1) = 0 precedence: __ = a__U22 = a__U21 = a__isList = U42 = U52 > a__U31 > a__U11 > a__U42 = a__U52 = a__U41 = a__U51 > a__U51# > nil > a__isQid = a = e > a__isList# = a__isNeList# = tt > a__isNeList > U21 > U11 = i = o = isList > u = U31 = U41 = U51 = isQid = isNeList > U22 partial status: pi(a__isList#) = [] pi(__) = [2] pi(a__isNeList#) = [] pi(a__U51#) = [] pi(a__isNeList) = [1] pi(tt) = [] pi(a__U22) = [] pi(U22) = [] pi(a__U11) = [] pi(a__U21) = [] pi(a__isList) = [] pi(a__U42) = [] pi(a__U52) = [] pi(U11) = [] pi(U21) = [] pi(U42) = [] pi(U52) = [] pi(a__U31) = [] pi(a__U41) = [1] pi(a__U51) = [1] pi(nil) = [] pi(a__isQid) = [1] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isList) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isQid) = [] pi(isNeList) = [] 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: a__isList#(V) -> a__isNeList#(V) p2: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) p3: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p4: a__U51#(tt(),V2) -> a__isList#(V2) p5: a__isNeList#(__(V1,V2)) -> a__isList#(V1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__isList#(V) -> a__isNeList#(V) p2: a__isNeList#(__(V1,V2)) -> a__isList#(V1) p3: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p4: a__U51#(tt(),V2) -> a__isList#(V2) p5: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r4, r5, r6, r7, r8, r9, r10, r11, r16, r17, r18, r19, r20, r21, r26, r27, r28, r29, r30, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r72 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__isList#_A(x1) = x1 + 3 a__isNeList#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 15 a__U51#_A(x1,x2) = x2 + 4 a__isNeList_A(x1) = x1 + 5 tt_A() = 2 a__U22_A(x1) = x1 + 1 U22_A(x1) = 0 a__U11_A(x1) = x1 + 1 a__U21_A(x1,x2) = x1 + x2 + 7 a__isList_A(x1) = x1 + 7 a__U42_A(x1) = 3 a__U52_A(x1) = 3 U11_A(x1) = 0 U21_A(x1,x2) = 0 U42_A(x1) = 0 U52_A(x1) = 0 a__U31_A(x1) = x1 + 1 a__U41_A(x1,x2) = x1 + 2 a__U51_A(x1,x2) = x1 + 2 nil_A() = 1 a__isQid_A(x1) = x1 + 2 a_A() = 3 e_A() = 3 i_A() = 3 o_A() = 1 u_A() = 1 isList_A(x1) = 0 U31_A(x1) = 0 U41_A(x1,x2) = 0 U51_A(x1,x2) = 0 isQid_A(x1) = 1 isNeList_A(x1) = 0 precedence: U22 = U52 > a__U22 = a__U21 = a__isList = a__U52 = U11 = a__U31 = a__U51 > __ = a__U42 = a__U41 > a__isNeList > a__U51# = U21 > U42 > a__isList# = a__isNeList# = tt = nil = a__isQid > a__U11 = a = e = i = o = u = isList = U31 = U41 = U51 = isQid = isNeList partial status: pi(a__isList#) = [] pi(a__isNeList#) = [] pi(__) = [] pi(a__U51#) = [] pi(a__isNeList) = [1] pi(tt) = [] pi(a__U22) = [] pi(U22) = [] pi(a__U11) = [1] pi(a__U21) = [] pi(a__isList) = [] pi(a__U42) = [] pi(a__U52) = [] pi(U11) = [] pi(U21) = [] pi(U42) = [] pi(U52) = [] pi(a__U31) = [] pi(a__U41) = [1] pi(a__U51) = [] pi(nil) = [] pi(a__isQid) = [1] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isList) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isQid) = [] pi(isNeList) = [] 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: a__isList#(V) -> a__isNeList#(V) p2: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p3: a__U51#(tt(),V2) -> a__isList#(V2) p4: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__isList#(V) -> a__isNeList#(V) p2: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) p3: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p4: a__U51#(tt(),V2) -> a__isList#(V2) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r4, r5, r6, r7, r8, r9, r10, r11, r16, r17, r18, r19, r20, r21, r26, r27, r28, r29, r30, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r72 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__isList#_A(x1) = x1 a__isNeList#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 13 a__U51#_A(x1,x2) = x2 + 1 a__isNeList_A(x1) = x1 + 5 tt_A() = 4 a__U22_A(x1) = x1 + 1 U22_A(x1) = 0 a__U11_A(x1) = x1 + 1 a__U21_A(x1,x2) = x1 + x2 + 5 a__isList_A(x1) = x1 + 7 a__U42_A(x1) = x1 + 1 a__U52_A(x1) = 4 U11_A(x1) = 0 U21_A(x1,x2) = 0 U42_A(x1) = 0 U52_A(x1) = 0 a__U31_A(x1) = x1 + 2 a__U41_A(x1,x2) = x1 + x2 + 3 a__U51_A(x1,x2) = x1 + 1 nil_A() = 1 a__isQid_A(x1) = x1 + 2 a_A() = 3 e_A() = 3 i_A() = 3 o_A() = 3 u_A() = 3 isList_A(x1) = 0 U31_A(x1) = 1 U41_A(x1,x2) = 0 U51_A(x1,x2) = 0 isQid_A(x1) = 1 isNeList_A(x1) = 0 precedence: __ = a__U51# > a__isList# = a__U11 = a__U21 = a__isList = U11 = U42 = U52 > a__isNeList# = a__isNeList = tt = a__U52 = U21 > a__U42 = a__U31 = a__U41 = a__U51 = nil = a__isQid = a = e = i = o = u = isList = U31 = U41 = U51 = isQid = isNeList > a__U22 = U22 partial status: pi(a__isList#) = [1] pi(a__isNeList#) = [1] pi(__) = [2] pi(a__U51#) = [] pi(a__isNeList) = [] pi(tt) = [] pi(a__U22) = [] pi(U22) = [] pi(a__U11) = [] pi(a__U21) = [] pi(a__isList) = [1] pi(a__U42) = [] pi(a__U52) = [] pi(U11) = [] pi(U21) = [] pi(U42) = [] pi(U52) = [] pi(a__U31) = [1] pi(a__U41) = [] pi(a__U51) = [1] pi(nil) = [] pi(a__isQid) = [1] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isList) = [] pi(U31) = [] pi(U41) = [] pi(U51) = [] pi(isQid) = [] pi(isNeList) = [] 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: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) p2: a__isNeList#(__(V1,V2)) -> a__U51#(a__isNeList(V1),V2) p3: a__U51#(tt(),V2) -> a__isList#(V2) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__isNeList#(__(V1,V2)) -> a__isNeList#(V1) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__isNeList#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 1 precedence: a__isNeList# = __ partial status: pi(a__isNeList#) = [] pi(__) = [2] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U71#(tt(),P) -> a__isPal#(P) p2: a__isPal#(V) -> a__isNePal#(V) p3: a__isNePal#(__(I,__(P,I))) -> a__U71#(a__isQid(I),P) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The set of usable rules consists of r26, r27, r28, r29, r30, r72 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U71#_A(x1,x2) = x1 + x2 + 1 tt_A() = 6 a__isPal#_A(x1) = x1 + 5 a__isNePal#_A(x1) = x1 + 4 ___A(x1,x2) = x1 + x2 a__isQid_A(x1) = x1 + 2 a_A() = 7 e_A() = 7 i_A() = 7 o_A() = 5 u_A() = 7 isQid_A(x1) = 0 precedence: a = isQid > a__U71# = a__isPal# = a__isNePal# > e = i > __ = a__isQid = o = u > tt partial status: pi(a__U71#) = [2] pi(tt) = [] pi(a__isPal#) = [1] pi(a__isNePal#) = [1] pi(__) = [1, 2] pi(a__isQid) = [1] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isQid) = [] The next rules are strictly ordered: p3 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__U71#(tt(),P) -> a__isPal#(P) p2: a__isPal#(V) -> a__isNePal#(V) and R consists of: r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r2: a____(X,nil()) -> mark(X) r3: a____(nil(),X) -> mark(X) r4: a__U11(tt()) -> tt() r5: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r6: a__U22(tt()) -> tt() r7: a__U31(tt()) -> tt() r8: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r9: a__U42(tt()) -> tt() r10: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r11: a__U52(tt()) -> tt() r12: a__U61(tt()) -> tt() r13: a__U71(tt(),P) -> a__U72(a__isPal(P)) r14: a__U72(tt()) -> tt() r15: a__U81(tt()) -> tt() r16: a__isList(V) -> a__U11(a__isNeList(V)) r17: a__isList(nil()) -> tt() r18: a__isList(__(V1,V2)) -> a__U21(a__isList(V1),V2) r19: a__isNeList(V) -> a__U31(a__isQid(V)) r20: a__isNeList(__(V1,V2)) -> a__U41(a__isList(V1),V2) r21: a__isNeList(__(V1,V2)) -> a__U51(a__isNeList(V1),V2) r22: a__isNePal(V) -> a__U61(a__isQid(V)) r23: a__isNePal(__(I,__(P,I))) -> a__U71(a__isQid(I),P) r24: a__isPal(V) -> a__U81(a__isNePal(V)) r25: a__isPal(nil()) -> tt() r26: a__isQid(a()) -> tt() r27: a__isQid(e()) -> tt() r28: a__isQid(i()) -> tt() r29: a__isQid(o()) -> tt() r30: a__isQid(u()) -> tt() r31: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r32: mark(U11(X)) -> a__U11(mark(X)) r33: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r34: mark(U22(X)) -> a__U22(mark(X)) r35: mark(isList(X)) -> a__isList(X) r36: mark(U31(X)) -> a__U31(mark(X)) r37: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r38: mark(U42(X)) -> a__U42(mark(X)) r39: mark(isNeList(X)) -> a__isNeList(X) r40: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r41: mark(U52(X)) -> a__U52(mark(X)) r42: mark(U61(X)) -> a__U61(mark(X)) r43: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r44: mark(U72(X)) -> a__U72(mark(X)) r45: mark(isPal(X)) -> a__isPal(X) r46: mark(U81(X)) -> a__U81(mark(X)) r47: mark(isQid(X)) -> a__isQid(X) r48: mark(isNePal(X)) -> a__isNePal(X) r49: mark(nil()) -> nil() r50: mark(tt()) -> tt() r51: mark(a()) -> a() r52: mark(e()) -> e() r53: mark(i()) -> i() r54: mark(o()) -> o() r55: mark(u()) -> u() r56: a____(X1,X2) -> __(X1,X2) r57: a__U11(X) -> U11(X) r58: a__U21(X1,X2) -> U21(X1,X2) r59: a__U22(X) -> U22(X) r60: a__isList(X) -> isList(X) r61: a__U31(X) -> U31(X) r62: a__U41(X1,X2) -> U41(X1,X2) r63: a__U42(X) -> U42(X) r64: a__isNeList(X) -> isNeList(X) r65: a__U51(X1,X2) -> U51(X1,X2) r66: a__U52(X) -> U52(X) r67: a__U61(X) -> U61(X) r68: a__U71(X1,X2) -> U71(X1,X2) r69: a__U72(X) -> U72(X) r70: a__isPal(X) -> isPal(X) r71: a__U81(X) -> U81(X) r72: a__isQid(X) -> isQid(X) r73: a__isNePal(X) -> isNePal(X) The estimated dependency graph contains the following SCCs: (no SCCs)