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. matrix interpretations: carrier: N^2 order: standard order interpretations: a____#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 ___A(x1,x2) = x1 + x2 + (1,5) mark_A(x1) = ((0,1),(0,1)) x1 + (2,0) a_____A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (2,5) nil_A() = (1,1) mark#_A(x1) = ((0,1),(0,0)) x1 U81_A(x1) = x1 + (1,1) U72_A(x1) = x1 + (1,1) U71_A(x1,x2) = x1 + x2 + (2,4) U61_A(x1) = x1 + (1,1) U52_A(x1) = x1 + (1,1) U51_A(x1,x2) = x1 + x2 + (1,3) U42_A(x1) = ((0,1),(0,1)) x1 + (1,1) U41_A(x1,x2) = x1 + x2 + (1,4) U31_A(x1) = x1 + (1,1) U22_A(x1) = x1 + (1,1) U21_A(x1,x2) = x1 + x2 + (1,3) U11_A(x1) = x1 + (1,1) a__U11_A(x1) = x1 + (2,1) tt_A() = (1,1) a__U21_A(x1,x2) = x1 + x2 + (3,3) a__U22_A(x1) = x1 + (2,1) a__isList_A(x1) = ((0,1),(0,1)) x1 + (3,3) a__U31_A(x1) = x1 + (2,1) a__U41_A(x1,x2) = x1 + ((0,1),(0,1)) x2 + (5,4) a__U42_A(x1) = ((0,1),(0,1)) x1 + (2,1) a__isNeList_A(x1) = ((0,1),(0,1)) x1 + (3,2) a__U51_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (2,3) a__U52_A(x1) = x1 + (2,1) a__U61_A(x1) = x1 + (2,1) a__U71_A(x1,x2) = x1 + x2 + (3,4) a__U72_A(x1) = x1 + (2,1) a__isPal_A(x1) = ((0,1),(0,1)) x1 + (3,4) a__U81_A(x1) = x1 + (2,1) a__isQid_A(x1) = ((0,1),(0,1)) x1 + (2,1) a__isNePal_A(x1) = x1 + (4,3) a_A() = (0,1) e_A() = (0,1) i_A() = (0,1) o_A() = (0,1) u_A() = (1,1) isList_A(x1) = x1 + (3,3) isNeList_A(x1) = x1 + (3,2) isPal_A(x1) = x1 + (1,4) isQid_A(x1) = x1 + (1,1) isNePal_A(x1) = x1 + (1,3) 2. matrix interpretations: carrier: N^2 order: standard order interpretations: a____#_A(x1,x2) = (1,1) ___A(x1,x2) = (5,4) mark_A(x1) = (3,2) a_____A(x1,x2) = (4,3) nil_A() = (4,3) mark#_A(x1) = (0,0) U81_A(x1) = (6,1) U72_A(x1) = (0,1) U71_A(x1,x2) = (1,1) U61_A(x1) = (6,5) U52_A(x1) = (1,1) U51_A(x1,x2) = (1,1) U42_A(x1) = (7,4) U41_A(x1,x2) = (1,1) U31_A(x1) = (6,1) U22_A(x1) = (9,1) U21_A(x1,x2) = (1,1) U11_A(x1) = (0,1) a__U11_A(x1) = (4,2) tt_A() = (5,6) a__U21_A(x1,x2) = (7,1) a__U22_A(x1) = (8,2) a__isList_A(x1) = (6,6) a__U31_A(x1) = (5,6) a__U41_A(x1,x2) = (5,1) a__U42_A(x1) = (6,3) a__isNeList_A(x1) = (5,2) a__U51_A(x1,x2) = (2,2) a__U52_A(x1) = (6,3) a__U61_A(x1) = (5,4) a__U71_A(x1,x2) = (2,4) a__U72_A(x1) = (3,5) a__isPal_A(x1) = (0,2) a__U81_A(x1) = (5,6) a__isQid_A(x1) = (4,3) a__isNePal_A(x1) = (6,3) a_A() = (4,3) e_A() = (4,3) i_A() = (4,3) o_A() = (4,0) u_A() = (4,0) isList_A(x1) = (0,1) isNeList_A(x1) = (0,0) isPal_A(x1) = (0,0) isQid_A(x1) = (0,0) isNePal_A(x1) = (7,0) The next rules are strictly ordered: p2, p3, p4, p5, 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))) 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(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. matrix interpretations: carrier: N^2 order: standard order interpretations: a____#_A(x1,x2) = ((0,1),(0,0)) x1 ___A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (1,3) mark_A(x1) = ((0,1),(0,1)) x1 a_____A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (2,3) a__U11_A(x1) = x1 + (2,2) tt_A() = (1,2) a__U21_A(x1,x2) = x2 + (3,4) a__U22_A(x1) = (2,3) a__isList_A(x1) = ((0,0),(1,1)) x1 + (4,7) a__U31_A(x1) = x1 + (1,1) a__U41_A(x1,x2) = ((0,1),(0,1)) x2 + (3,4) a__U42_A(x1) = (2,2) a__isNeList_A(x1) = ((0,1),(1,1)) x1 + (4,5) a__U51_A(x1,x2) = (3,4) a__U52_A(x1) = (2,3) a__U61_A(x1) = (2,3) a__U71_A(x1,x2) = ((0,1),(0,1)) x2 + (7,8) a__U72_A(x1) = ((0,1),(0,1)) x1 + (2,3) a__isPal_A(x1) = x1 + (3,4) a__U81_A(x1) = (2,3) nil_A() = (1,2) a__isQid_A(x1) = ((0,0),(1,1)) x1 + (2,3) a__isNePal_A(x1) = ((0,1),(0,1)) x1 + (3,4) a_A() = (1,2) e_A() = (1,2) i_A() = (1,2) o_A() = (1,2) u_A() = (1,2) U11_A(x1) = x1 + (2,2) U21_A(x1,x2) = x2 + (1,4) U22_A(x1) = (1,3) isList_A(x1) = ((0,0),(1,1)) x1 + (1,7) U31_A(x1) = x1 + (1,1) U41_A(x1,x2) = x2 + (1,4) U42_A(x1) = (2,2) isNeList_A(x1) = ((0,0),(1,1)) x1 + (1,5) U51_A(x1,x2) = (1,4) U52_A(x1) = (1,3) U61_A(x1) = (1,3) U71_A(x1,x2) = x2 + (1,8) U72_A(x1) = x1 + (1,3) isPal_A(x1) = x1 + (1,4) U81_A(x1) = (1,3) isQid_A(x1) = ((0,0),(1,1)) x1 + (1,3) isNePal_A(x1) = x1 + (1,4) 2. matrix interpretations: carrier: N^2 order: standard order interpretations: a____#_A(x1,x2) = (0,0) ___A(x1,x2) = (1,2) mark_A(x1) = (5,2) a_____A(x1,x2) = (6,1) a__U11_A(x1) = (1,2) tt_A() = (9,6) a__U21_A(x1,x2) = (6,3) a__U22_A(x1) = (7,4) a__isList_A(x1) = (0,1) a__U31_A(x1) = ((0,1),(0,0)) x1 + (2,2) a__U41_A(x1,x2) = (4,4) a__U42_A(x1) = (2,2) a__isNeList_A(x1) = (3,3) a__U51_A(x1,x2) = (2,4) a__U52_A(x1) = (1,5) a__U61_A(x1) = (7,4) a__U71_A(x1,x2) = (11,4) a__U72_A(x1) = (10,5) a__isPal_A(x1) = (7,3) a__U81_A(x1) = (6,4) nil_A() = (6,3) a__isQid_A(x1) = (1,5) a__isNePal_A(x1) = (6,3) a_A() = (6,3) e_A() = (0,3) i_A() = (6,3) o_A() = (1,3) u_A() = (6,3) U11_A(x1) = (1,1) U21_A(x1,x2) = (7,4) U22_A(x1) = (8,5) isList_A(x1) = (1,2) U31_A(x1) = (1,1) U41_A(x1,x2) = (5,5) U42_A(x1) = (1,1) isNeList_A(x1) = (4,4) U51_A(x1,x2) = (0,5) U52_A(x1) = (0,6) U61_A(x1) = (8,5) U71_A(x1,x2) = (0,5) U72_A(x1) = (0,6) isPal_A(x1) = (0,4) U81_A(x1) = (0,5) isQid_A(x1) = (0,6) isNePal_A(x1) = (0,4) 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. matrix interpretations: carrier: N^2 order: standard order interpretations: a__U21#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,1),(1,1)) x2 + (3,0) tt_A() = (1,1) a__isList#_A(x1) = ((1,1),(1,1)) x1 + (1,0) ___A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (6,2) a__isList_A(x1) = ((1,1),(0,0)) x1 + (5,1) a__isNeList#_A(x1) = ((1,0),(1,0)) x1 a__U51#_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,1),(1,1)) x2 + (1,0) a__isNeList_A(x1) = ((1,1),(1,1)) x1 + (3,1) a__U41#_A(x1,x2) = x1 + ((1,1),(1,0)) x2 + (0,6) a__U22_A(x1) = ((1,1),(0,0)) x1 + (1,1) a__U42_A(x1) = ((0,1),(1,0)) x1 + (1,0) a__U52_A(x1) = ((1,0),(1,0)) x1 + (1,0) U22_A(x1) = (0,0) U42_A(x1) = (0,0) U52_A(x1) = (0,0) a__U11_A(x1) = x1 + (1,1) a__U21_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,0)) x2 + (6,0) a__U31_A(x1) = ((0,1),(0,1)) x1 + (1,0) a__U41_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (2,1) a__U51_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (5,3) a__isQid_A(x1) = ((0,0),(1,1)) x1 + (2,1) a_A() = (1,1) e_A() = (1,1) i_A() = (1,1) o_A() = (1,1) u_A() = (1,1) U11_A(x1) = x1 U21_A(x1,x2) = x1 + x2 U31_A(x1) = (0,0) U41_A(x1,x2) = x1 U51_A(x1,x2) = (0,0) isQid_A(x1) = (0,0) nil_A() = (1,1) isList_A(x1) = (5,0) isNeList_A(x1) = x1 2. matrix interpretations: carrier: N^2 order: standard order interpretations: a__U21#_A(x1,x2) = ((0,1),(1,1)) x1 + (0,3) tt_A() = (7,5) a__isList#_A(x1) = ((1,1),(1,1)) x1 + (11,16) ___A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(0,1)) x2 + (1,1) a__isList_A(x1) = ((1,1),(1,1)) x1 + (4,12) a__isNeList#_A(x1) = ((0,1),(1,0)) x1 + (12,13) a__U51#_A(x1,x2) = (10,15) a__isNeList_A(x1) = ((0,1),(1,0)) x1 + (1,1) a__U41#_A(x1,x2) = ((0,1),(1,1)) x1 + (6,0) a__U22_A(x1) = ((0,0),(1,1)) x1 + (1,0) a__U42_A(x1) = (6,4) a__U52_A(x1) = ((1,0),(1,1)) x1 + (1,0) U22_A(x1) = (2,1) U42_A(x1) = (7,5) U52_A(x1) = (0,1) a__U11_A(x1) = (5,13) a__U21_A(x1,x2) = (0,15) a__U31_A(x1) = (8,2) a__U41_A(x1,x2) = ((0,1),(0,0)) x1 + (0,3) a__U51_A(x1,x2) = ((0,0),(1,0)) x2 + (6,3) a__isQid_A(x1) = (8,4) a_A() = (1,1) e_A() = (1,1) i_A() = (1,1) o_A() = (1,1) u_A() = (1,1) U11_A(x1) = ((0,1),(0,1)) x1 + (6,14) U21_A(x1,x2) = ((1,1),(1,0)) x1 + ((1,1),(0,1)) x2 + (1,16) U31_A(x1) = (7,3) U41_A(x1,x2) = ((1,0),(1,0)) x1 + (1,4) U51_A(x1,x2) = (7,4) isQid_A(x1) = (9,5) nil_A() = (1,1) isList_A(x1) = (0,0) isNeList_A(x1) = (0,2) 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. matrix interpretations: carrier: N^2 order: standard order interpretations: a__U71#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 tt_A() = (1,4) a__isPal#_A(x1) = ((0,1),(0,0)) x1 + (3,0) a__isNePal#_A(x1) = ((0,1),(0,0)) x1 + (2,0) ___A(x1,x2) = x1 + ((1,1),(1,1)) x2 a__isQid_A(x1) = ((0,0),(1,0)) x1 + (2,1) a_A() = (3,1) e_A() = (3,1) i_A() = (3,1) o_A() = (3,1) u_A() = (3,1) isQid_A(x1) = (0,0) 2. matrix interpretations: carrier: N^2 order: standard order interpretations: a__U71#_A(x1,x2) = (1,0) tt_A() = (2,2) a__isPal#_A(x1) = (2,1) a__isNePal#_A(x1) = (0,2) ___A(x1,x2) = ((1,1),(1,0)) x2 + (1,1) a__isQid_A(x1) = (1,1) a_A() = (1,1) e_A() = (1,1) i_A() = (1,1) o_A() = (1,1) u_A() = (1,1) isQid_A(x1) = (0,2) The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. Then no dependency pair remains.