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: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: a____#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (2,27) ___A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (25,0) mark_A(x1) = ((1,0),(1,1)) x1 a_____A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (25,1) nil_A() = (2,27) mark#_A(x1) = x1 + (3,28) U81_A(x1) = x1 + (4,1) U72_A(x1) = ((1,0),(1,0)) x1 + (4,26) U71_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (15,12) U61_A(x1) = x1 + (2,1) U52_A(x1) = x1 + (4,27) U51_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (17,29) U42_A(x1) = ((1,0),(1,1)) x1 + (4,1) U41_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (12,3) U31_A(x1) = x1 + (4,1) U22_A(x1) = ((1,0),(1,1)) x1 + (4,29) U21_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (17,29) U11_A(x1) = x1 + (4,1) a__U11_A(x1) = x1 + (4,2) tt_A() = (0,28) a__U21_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (17,30) a__U22_A(x1) = ((1,0),(1,1)) x1 + (4,30) a__isList_A(x1) = x1 + (12,2) a__U31_A(x1) = x1 + (4,2) a__U41_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (12,14) a__U42_A(x1) = ((1,0),(1,1)) x1 + (4,5) a__isNeList_A(x1) = ((1,0),(0,0)) x1 + (7,29) a__U51_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (17,30) a__U52_A(x1) = x1 + (4,28) a__U61_A(x1) = x1 + (2,2) a__U71_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (15,12) a__U72_A(x1) = ((1,0),(1,0)) x1 + (4,29) a__isPal_A(x1) = ((1,0),(1,0)) x1 + (10,41) a__U81_A(x1) = x1 + (4,2) a__isQid_A(x1) = ((1,0),(0,0)) x1 + (2,30) a__isNePal_A(x1) = ((1,0),(1,0)) x1 + (5,33) a_A() = (1,29) e_A() = (1,1) i_A() = (1,1) o_A() = (1,1) u_A() = (1,27) isList_A(x1) = x1 + (12,1) isNeList_A(x1) = ((1,0),(0,0)) x1 + (7,23) isPal_A(x1) = ((1,0),(1,0)) x1 + (10,32) isQid_A(x1) = ((1,0),(0,0)) x1 + (2,29) isNePal_A(x1) = ((1,0),(1,0)) x1 + (5,32) precedence: U41 = a__U41 > U21 > mark = a____ = nil = U61 = U52 = U51 = U31 = U22 = a__U11 = tt = a__U21 = a__U22 = a__isList = a__U31 = a__U42 = a__isNeList = a__U51 = a__U52 = a__U61 = a__U71 = a__isPal = a__U81 = a__isQid = a__isNePal = o = u = isList = isNeList = isPal = isQid = isNePal > a____# > U11 > a > mark# > U42 = a__U72 > U72 > __ > U81 > U71 = e = i partial status: pi(a____#) = [1] pi(__) = [1] pi(mark) = [1] pi(a____) = [] pi(nil) = [] pi(mark#) = [1] pi(U81) = [1] pi(U72) = [] pi(U71) = [1] pi(U61) = [1] pi(U52) = [1] pi(U51) = [1] pi(U42) = [1] pi(U41) = [1, 2] pi(U31) = [1] pi(U22) = [1] pi(U21) = [1] pi(U11) = [1] pi(a__U11) = [] pi(tt) = [] pi(a__U21) = [] pi(a__U22) = [1] pi(a__isList) = [] pi(a__U31) = [1] pi(a__U41) = [1, 2] pi(a__U42) = [1] pi(a__isNeList) = [] pi(a__U51) = [1] pi(a__U52) = [1] pi(a__U61) = [1] pi(a__U71) = [] pi(a__U72) = [] pi(a__isPal) = [] pi(a__U81) = [] pi(a__isQid) = [] pi(a__isNePal) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isList) = [] pi(isNeList) = [] pi(isPal) = [] pi(isQid) = [] pi(isNePal) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: a____#_A(x1,x2) = (3,10) ___A(x1,x2) = (0,0) mark_A(x1) = (0,11) a_____A(x1,x2) = (4,12) nil_A() = (0,11) mark#_A(x1) = ((1,0),(1,1)) x1 + (2,9) U81_A(x1) = (1,1) U72_A(x1) = (0,1) U71_A(x1,x2) = x1 + (3,8) U61_A(x1) = x1 + (3,1) U52_A(x1) = x1 + (3,1) U51_A(x1,x2) = x1 + (3,1) U42_A(x1) = ((1,0),(1,1)) x1 + (3,12) U41_A(x1,x2) = x1 + x2 + (0,1) U31_A(x1) = ((1,0),(1,1)) x1 + (3,12) U22_A(x1) = x1 + (3,1) U21_A(x1,x2) = x1 + (6,2) U11_A(x1) = x1 + (3,1) a__U11_A(x1) = (0,0) tt_A() = (0,5) a__U21_A(x1,x2) = (5,3) a__U22_A(x1) = (4,2) a__isList_A(x1) = (0,4) a__U31_A(x1) = (4,13) a__U41_A(x1,x2) = (4,21) a__U42_A(x1) = ((1,0),(1,1)) x1 + (3,13) a__isNeList_A(x1) = (0,7) a__U51_A(x1,x2) = (0,5) a__U52_A(x1) = x1 + (4,2) a__U61_A(x1) = (0,6) a__U71_A(x1,x2) = (0,7) a__U72_A(x1) = (0,6) a__isPal_A(x1) = (0,6) a__U81_A(x1) = (2,12) a__isQid_A(x1) = (0,6) a__isNePal_A(x1) = (2,0) a_A() = (0,11) e_A() = (0,11) i_A() = (1,11) o_A() = (0,4) u_A() = (0,0) isList_A(x1) = (0,1) isNeList_A(x1) = (0,1) isPal_A(x1) = (0,1) isQid_A(x1) = (0,1) isNePal_A(x1) = (1,1) precedence: a > e > isQid > a__isNePal > i > mark = a__U81 > a__U51 > U41 > U31 > isPal > U81 > isList > o > U51 > a__U71 > a__U41 > a__U61 > isNePal > a__isNeList > a__U52 > a__U21 > U52 > tt > a____ = U22 > __ > a____# > mark# = U61 = a__U72 > a__U42 > U11 > U21 > a__isList > isNeList > a__isQid > a__U31 > U42 > a__U11 > u > U72 = U71 > nil > a__U22 = a__isPal partial status: pi(a____#) = [] pi(__) = [] pi(mark) = [] pi(a____) = [] pi(nil) = [] pi(mark#) = [1] pi(U81) = [] pi(U72) = [] pi(U71) = [1] pi(U61) = [1] pi(U52) = [1] pi(U51) = [1] pi(U42) = [] pi(U41) = [] pi(U31) = [] pi(U22) = [1] pi(U21) = [1] pi(U11) = [1] pi(a__U11) = [] pi(tt) = [] pi(a__U21) = [] pi(a__U22) = [] pi(a__isList) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [1] 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) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isList) = [] pi(isNeList) = [] pi(isPal) = [] pi(isQid) = [] pi(isNePal) = [] The next rules are strictly ordered: p2, p3, p4, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22 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: 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} -- 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))) 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: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: a____#_A(x1,x2) = ((0,0),(1,0)) x1 + (5,0) ___A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,4) mark_A(x1) = ((1,0),(1,0)) x1 a_____A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,5) a__U11_A(x1) = ((1,0),(1,0)) x1 tt_A() = (1,0) a__U21_A(x1,x2) = ((1,0),(0,0)) x2 + (7,2) a__U22_A(x1) = ((1,0),(0,0)) x1 a__isList_A(x1) = ((1,0),(1,0)) x1 + (3,3) a__U31_A(x1) = ((1,0),(0,0)) x1 a__U41_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 a__U42_A(x1) = ((1,0),(0,0)) x1 a__isNeList_A(x1) = ((1,0),(1,0)) x1 a__U51_A(x1,x2) = ((1,0),(1,0)) x1 a__U52_A(x1) = (1,0) a__U61_A(x1) = ((1,0),(0,0)) x1 a__U71_A(x1,x2) = ((1,0),(0,0)) x2 + (12,12) a__U72_A(x1) = ((1,0),(0,0)) x1 a__isPal_A(x1) = ((1,0),(0,0)) x1 + (11,10) a__U81_A(x1) = ((1,0),(0,0)) x1 nil_A() = (2,1) a__isQid_A(x1) = ((1,0),(0,0)) x1 a__isNePal_A(x1) = ((1,0),(1,0)) x1 a_A() = (2,2) e_A() = (2,2) i_A() = (2,2) o_A() = (2,2) u_A() = (2,2) U11_A(x1) = ((1,0),(0,0)) x1 U21_A(x1,x2) = ((1,0),(0,0)) x2 + (7,1) U22_A(x1) = ((1,0),(0,0)) x1 isList_A(x1) = ((1,0),(0,0)) x1 + (3,3) U31_A(x1) = ((1,0),(0,0)) x1 U41_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 U42_A(x1) = ((1,0),(0,0)) x1 isNeList_A(x1) = ((1,0),(1,0)) x1 U51_A(x1,x2) = ((1,0),(0,0)) x1 U52_A(x1) = (1,0) U61_A(x1) = ((1,0),(0,0)) x1 U71_A(x1,x2) = ((1,0),(0,0)) x2 + (12,12) U72_A(x1) = ((1,0),(0,0)) x1 isPal_A(x1) = ((1,0),(0,0)) x1 + (11,1) U81_A(x1) = ((1,0),(0,0)) x1 isQid_A(x1) = ((1,0),(0,0)) x1 isNePal_A(x1) = ((1,0),(0,0)) x1 precedence: mark = a__U21 = a__isList = a__U41 = a__U42 = a__U81 = nil = a = e = i = o = U21 = isList > a__U22 > a__U51 > a__U52 = a__U71 = a__isNePal = U41 = U71 > a____ = a__U11 = tt = a__U31 = a__isNeList = a__U61 = a__U72 = a__isPal = a__isQid = u = U22 = U31 = U42 = isPal > U11 = U52 = U61 = U72 = isQid > a____# = __ = isNeList > isNePal > U51 = U81 partial status: pi(a____#) = [] pi(__) = [] pi(mark) = [] pi(a____) = [] 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) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: a____#_A(x1,x2) = (1,1) ___A(x1,x2) = (0,0) mark_A(x1) = (0,9) a_____A(x1,x2) = (0,2) a__U11_A(x1) = (0,3) tt_A() = (0,2) a__U21_A(x1,x2) = (0,4) a__U22_A(x1) = (0,3) a__isList_A(x1) = (0,5) a__U31_A(x1) = (0,3) a__U41_A(x1,x2) = (0,8) a__U42_A(x1) = (0,1) a__isNeList_A(x1) = (0,7) a__U51_A(x1,x2) = (0,6) a__U52_A(x1) = (0,3) a__U61_A(x1) = (0,5) a__U71_A(x1,x2) = (0,5) a__U72_A(x1) = (0,3) a__isPal_A(x1) = (0,4) a__U81_A(x1) = (0,3) nil_A() = (0,0) a__isQid_A(x1) = (0,3) a__isNePal_A(x1) = (0,6) a_A() = (0,8) e_A() = (0,8) i_A() = (0,3) o_A() = (0,9) u_A() = (0,3) U11_A(x1) = (0,2) U21_A(x1,x2) = (0,3) U22_A(x1) = (0,2) isList_A(x1) = (0,5) U31_A(x1) = (0,2) U41_A(x1,x2) = (0,7) U42_A(x1) = (0,0) isNeList_A(x1) = (0,0) U51_A(x1,x2) = (0,1) U52_A(x1) = (0,2) U61_A(x1) = (0,4) U71_A(x1,x2) = (0,4) U72_A(x1) = (1,4) isPal_A(x1) = (0,0) U81_A(x1) = (0,1) isQid_A(x1) = (0,1) isNePal_A(x1) = (0,1) precedence: U22 > mark = a__U21 = a__isList > a__U72 > a__isPal > U41 > U81 > a____ > i = u > U31 > U21 > a__U52 > U52 > isPal > a__U41 > a__U42 > a__isQid > U42 > a__U61 > U61 > a__U31 > a__isNeList > isList > a____# > a__U11 > a__U51 > U51 > isNeList > isNePal > U71 > isQid > U72 > tt = a__U81 > a__U22 > a__isNePal > a__U71 > U11 > __ = nil = a = e = o partial status: pi(a____#) = [] pi(__) = [] pi(mark) = [] pi(a____) = [] 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. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: 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 set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: mark#_A(x1) = x1 + (1,2) U71_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,1) precedence: U71 > mark# partial status: pi(mark#) = [1] pi(U71) = [2] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: mark#_A(x1) = (0,0) U71_A(x1,x2) = (1,1) precedence: mark# > U71 partial status: pi(mark#) = [] pi(U71) = [] 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: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: a__U21#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,1) tt_A() = (14,3) a__isList#_A(x1) = x1 + (0,13) ___A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (19,12) a__isList_A(x1) = ((1,0),(1,1)) x1 + (15,11) a__isNeList#_A(x1) = x1 a__U51#_A(x1,x2) = ((0,0),(1,0)) x1 + x2 a__isNeList_A(x1) = ((1,0),(1,1)) x1 + (5,4) a__U41#_A(x1,x2) = x1 + x2 + (1,2) a__U22_A(x1) = ((1,0),(0,0)) x1 + (1,1) a__U42_A(x1) = (15,0) a__U52_A(x1) = (15,0) U22_A(x1) = (0,0) U42_A(x1) = (0,0) U52_A(x1) = (0,0) a__U11_A(x1) = ((1,0),(1,1)) x1 + (1,1) a__U21_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,1) a__U31_A(x1) = ((1,0),(0,0)) x1 + (2,2) a__U41_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (2,1) a__U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,4) a__isQid_A(x1) = ((1,0),(0,0)) x1 + (2,2) a_A() = (15,4) e_A() = (15,4) i_A() = (15,1) o_A() = (13,1) u_A() = (15,1) U11_A(x1) = x1 U21_A(x1,x2) = x2 U31_A(x1) = (1,1) U41_A(x1,x2) = (0,0) U51_A(x1,x2) = (0,0) isQid_A(x1) = (1,1) nil_A() = (15,4) isList_A(x1) = (1,1) isNeList_A(x1) = (0,0) precedence: a__U21# = tt = a__isList# = __ = a__isList = a__isNeList# = a__U51# = a__isNeList = a__U41# = a__U22 = a__U42 = a__U52 = U22 = U42 = U52 = a__U11 = a__U21 = a__U31 = a__U41 = a__U51 = a__isQid = a = e = i = o = u = U11 = U21 = U31 = U41 = U51 = isQid = nil = isList = isNeList partial status: pi(a__U21#) = [] pi(tt) = [] pi(a__isList#) = [] pi(__) = [] pi(a__isList) = [] pi(a__isNeList#) = [] pi(a__U51#) = [] pi(a__isNeList) = [] pi(a__U41#) = [] 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) = [] 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) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: a__U21#_A(x1,x2) = ((1,0),(1,1)) x1 + (3,1) tt_A() = (2,2) a__isList#_A(x1) = x1 + (6,6) ___A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (10,8) a__isList_A(x1) = ((0,0),(1,0)) x1 + (8,6) a__isNeList#_A(x1) = ((1,0),(1,1)) x1 + (4,7) a__U51#_A(x1,x2) = (0,0) a__isNeList_A(x1) = ((1,0),(1,1)) x1 + (9,7) a__U41#_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (1,1) a__U22_A(x1) = (2,1) a__U42_A(x1) = (21,27) a__U52_A(x1) = (3,3) U22_A(x1) = (0,0) U42_A(x1) = (0,0) U52_A(x1) = (0,0) a__U11_A(x1) = ((1,0),(1,1)) x1 + (1,1) a__U21_A(x1,x2) = ((1,0),(1,1)) x1 + (0,1) a__U31_A(x1) = (3,8) a__U41_A(x1,x2) = (20,26) a__U51_A(x1,x2) = ((1,0),(1,1)) x1 + (2,1) a__isQid_A(x1) = (3,3) a_A() = (1,1) e_A() = (3,3) i_A() = (3,3) o_A() = (3,3) u_A() = (3,3) U11_A(x1) = (0,0) U21_A(x1,x2) = (0,0) U31_A(x1) = (0,0) U41_A(x1,x2) = (0,0) U51_A(x1,x2) = (0,0) isQid_A(x1) = (0,0) nil_A() = (1,1) isList_A(x1) = (9,7) isNeList_A(x1) = (0,0) precedence: a__U51 > a__U41 > i > e > U42 > a > u > a__isQid > a__U51# > a__isList > __ > a__U21 > U52 > a__isNeList# > a__U42 > a__U31 > o > a__U22 = a__U11 > a__U52 > a__U21# = U22 > U21 = U41 = U51 = isQid = nil > tt > a__isList# = a__U41# > U11 = isList > U31 = isNeList > a__isNeList partial status: pi(a__U21#) = [] pi(tt) = [] pi(a__isList#) = [1] pi(__) = [1] pi(a__isList) = [] pi(a__isNeList#) = [1] pi(a__U51#) = [] pi(a__isNeList) = [1] pi(a__U41#) = [] pi(a__U22) = [] pi(a__U42) = [] pi(a__U52) = [] pi(U22) = [] pi(U42) = [] pi(U52) = [] pi(a__U11) = [1] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U51) = [1] 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: p1, p2, p3, p4, p5, p6, p7, p8, p9, p10 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: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: a__U71#_A(x1,x2) = x1 + x2 + (2,2) tt_A() = (2,2) a__isPal#_A(x1) = ((1,0),(0,0)) x1 + (3,3) a__isNePal#_A(x1) = x1 + (2,4) ___A(x1,x2) = x1 + x2 + (2,2) a__isQid_A(x1) = ((1,0),(1,1)) x1 + (3,1) a_A() = (1,1) e_A() = (3,3) i_A() = (3,3) o_A() = (3,3) u_A() = (3,3) isQid_A(x1) = (0,0) precedence: a__U71# = tt = a__isPal# = a__isNePal# = __ = a__isQid = a = e = i = o = u = isQid partial status: pi(a__U71#) = [] pi(tt) = [] pi(a__isPal#) = [] pi(a__isNePal#) = [] pi(__) = [] pi(a__isQid) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isQid) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: a__U71#_A(x1,x2) = x2 + (3,3) tt_A() = (2,2) a__isPal#_A(x1) = (0,0) a__isNePal#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + (2,2) a__isQid_A(x1) = (1,1) a_A() = (3,3) e_A() = (3,3) i_A() = (3,3) o_A() = (3,3) u_A() = (3,3) isQid_A(x1) = (0,0) precedence: o > e > __ > u > tt = a__isPal# > a__U71# > a__isQid > a > i > isQid > a__isNePal# partial status: pi(a__U71#) = [2] pi(tt) = [] pi(a__isPal#) = [] pi(a__isNePal#) = [] pi(__) = [] pi(a__isQid) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] pi(isQid) = [] The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. Then no dependency pair remains.