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 monotone reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: a____#_A(x1,x2) = x1 + x2 + 1 ___A(x1,x2) = x1 + x2 + 1 mark_A(x1) = x1 a_____A(x1,x2) = x1 + x2 + 1 nil_A() = 1 mark#_A(x1) = x1 U81_A(x1) = x1 U72_A(x1) = x1 + 1 U71_A(x1,x2) = x1 + x2 + 1 U61_A(x1) = x1 U52_A(x1) = x1 U51_A(x1,x2) = x1 + x2 U42_A(x1) = x1 U41_A(x1,x2) = x1 + x2 U31_A(x1) = x1 + 1 U22_A(x1) = x1 U21_A(x1,x2) = x1 + x2 U11_A(x1) = x1 a__U11_A(x1) = x1 tt_A() = 2 a__U21_A(x1,x2) = x1 + x2 a__U22_A(x1) = x1 a__isList_A(x1) = x1 + 2 a__U31_A(x1) = x1 + 1 a__U41_A(x1,x2) = x1 + x2 a__U42_A(x1) = x1 a__isNeList_A(x1) = x1 + 2 a__U51_A(x1,x2) = x1 + x2 a__U52_A(x1) = x1 a__U61_A(x1) = x1 a__U71_A(x1,x2) = x1 + x2 + 1 a__U72_A(x1) = x1 + 1 a__isPal_A(x1) = x1 + 1 a__U81_A(x1) = x1 a__isQid_A(x1) = x1 + 1 a__isNePal_A(x1) = x1 + 1 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 isList_A(x1) = x1 + 2 isNeList_A(x1) = x1 + 2 isPal_A(x1) = x1 + 1 isQid_A(x1) = x1 + 1 isNePal_A(x1) = x1 + 1 The next rules are strictly ordered: p2, p4, p5, p11, p15, p16, p18, p19, p20, p21, p22 r2, r3, r7, r13, r14, r17, r18, r20, r21, r23 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#(U81(X)) -> mark#(X) 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#(U22(X)) -> mark#(X) p9: mark#(U21(X1,X2)) -> mark#(X1) p10: mark#(U11(X)) -> mark#(X) p11: mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The estimated dependency graph contains the following SCCs: {p2, p3, p4, p5, p6, p7, p8, p9, p10} {p1} -- Reduction pair. 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#(U41(X1,X2)) -> mark#(X1) p5: mark#(U42(X)) -> mark#(X) p6: mark#(U51(X1,X2)) -> mark#(X1) p7: mark#(U52(X)) -> mark#(X) p8: mark#(U61(X)) -> mark#(X) p9: mark#(U81(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U11_A(x1) = x1 U21_A(x1,x2) = x1 U22_A(x1) = x1 U41_A(x1,x2) = x1 U42_A(x1) = x1 + 1 U51_A(x1,x2) = x1 U52_A(x1) = x1 U61_A(x1) = x1 U81_A(x1) = x1 The next rules are strictly ordered: p5 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#(U41(X1,X2)) -> mark#(X1) p5: mark#(U51(X1,X2)) -> mark#(X1) p6: mark#(U52(X)) -> mark#(X) p7: mark#(U61(X)) -> mark#(X) p8: mark#(U81(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) 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#(U11(X)) -> mark#(X) p2: mark#(U81(X)) -> mark#(X) p3: mark#(U61(X)) -> mark#(X) p4: mark#(U52(X)) -> mark#(X) p5: mark#(U51(X1,X2)) -> mark#(X1) p6: mark#(U41(X1,X2)) -> mark#(X1) p7: mark#(U22(X)) -> mark#(X) p8: mark#(U21(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U11_A(x1) = x1 + 1 U81_A(x1) = x1 U61_A(x1) = x1 U52_A(x1) = x1 U51_A(x1,x2) = x1 U41_A(x1,x2) = x1 U22_A(x1) = x1 U21_A(x1,x2) = x1 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#(U81(X)) -> mark#(X) p2: mark#(U61(X)) -> mark#(X) p3: mark#(U52(X)) -> mark#(X) p4: mark#(U51(X1,X2)) -> mark#(X1) p5: mark#(U41(X1,X2)) -> mark#(X1) p6: mark#(U22(X)) -> mark#(X) p7: mark#(U21(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) 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#(U81(X)) -> mark#(X) p2: mark#(U21(X1,X2)) -> mark#(X1) p3: mark#(U22(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) p5: mark#(U51(X1,X2)) -> mark#(X1) p6: mark#(U52(X)) -> mark#(X) p7: mark#(U61(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U81_A(x1) = x1 U21_A(x1,x2) = x1 U22_A(x1) = x1 U41_A(x1,x2) = x1 U51_A(x1,x2) = x1 + 1 U52_A(x1) = x1 U61_A(x1) = x1 The next rules are strictly ordered: p5 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#(U21(X1,X2)) -> mark#(X1) p3: mark#(U22(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) p5: mark#(U52(X)) -> mark#(X) p6: mark#(U61(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) 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#(U81(X)) -> mark#(X) p2: mark#(U61(X)) -> mark#(X) p3: mark#(U52(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) p5: mark#(U22(X)) -> mark#(X) p6: mark#(U21(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U81_A(x1) = x1 + 1 U61_A(x1) = x1 U52_A(x1) = x1 U41_A(x1,x2) = x1 U22_A(x1) = x1 U21_A(x1,x2) = x1 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#(U41(X1,X2)) -> mark#(X1) p4: mark#(U22(X)) -> mark#(X) p5: mark#(U21(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) 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#(U61(X)) -> mark#(X) p2: mark#(U21(X1,X2)) -> mark#(X1) p3: mark#(U22(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) p5: mark#(U52(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U61_A(x1) = x1 U21_A(x1,x2) = x1 U22_A(x1) = x1 U41_A(x1,x2) = x1 U52_A(x1) = x1 + 1 The next rules are strictly ordered: p5 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#(U21(X1,X2)) -> mark#(X1) p3: mark#(U22(X)) -> mark#(X) p4: mark#(U41(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) 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#(U61(X)) -> mark#(X) p2: mark#(U41(X1,X2)) -> mark#(X1) p3: mark#(U22(X)) -> mark#(X) p4: mark#(U21(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U61_A(x1) = x1 + 1 U41_A(x1,x2) = x1 U22_A(x1) = x1 U21_A(x1,x2) = x1 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#(U22(X)) -> mark#(X) p3: mark#(U21(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) 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#(U41(X1,X2)) -> mark#(X1) p2: mark#(U21(X1,X2)) -> mark#(X1) p3: mark#(U22(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U41_A(x1,x2) = x1 U21_A(x1,x2) = x1 + 1 U22_A(x1) = x1 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: mark#(U41(X1,X2)) -> mark#(X1) p2: mark#(U22(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) 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#(U22(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U41_A(x1,x2) = x1 + 1 U22_A(x1) = x1 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) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U22(X)) -> mark#(X) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) The set of usable rules consists of (no rules) Take the monotone reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: mark#_A(x1) = x1 U22_A(x1) = x1 + 1 The next rules are strictly ordered: p1 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 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____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) and R consists of: r1: a__U11(tt()) -> tt() r2: a__U21(tt(),V2) -> a__U22(a__isList(V2)) r3: a__U22(tt()) -> tt() r4: a__U41(tt(),V2) -> a__U42(a__isNeList(V2)) r5: a__U42(tt()) -> tt() r6: a__U51(tt(),V2) -> a__U52(a__isList(V2)) r7: a__U52(tt()) -> tt() r8: a__U61(tt()) -> tt() r9: a__U81(tt()) -> tt() r10: a__isList(V) -> a__U11(a__isNeList(V)) r11: a__isNeList(V) -> a__U31(a__isQid(V)) r12: a__isNePal(V) -> a__U61(a__isQid(V)) r13: a__isPal(V) -> a__U81(a__isNePal(V)) r14: a__isPal(nil()) -> tt() r15: a__isQid(a()) -> tt() r16: a__isQid(e()) -> tt() r17: a__isQid(i()) -> tt() r18: a__isQid(o()) -> tt() r19: a__isQid(u()) -> tt() r20: a__U11(X) -> U11(X) r21: a__U21(X1,X2) -> U21(X1,X2) r22: a__U22(X) -> U22(X) r23: a__isList(X) -> isList(X) r24: a__U31(X) -> U31(X) r25: a__U41(X1,X2) -> U41(X1,X2) r26: a__U42(X) -> U42(X) r27: a__isNeList(X) -> isNeList(X) r28: a__U51(X1,X2) -> U51(X1,X2) r29: a__U52(X) -> U52(X) r30: a__U61(X) -> U61(X) r31: a__U71(X1,X2) -> U71(X1,X2) r32: a__U72(X) -> U72(X) r33: a__isPal(X) -> isPal(X) r34: a__U81(X) -> U81(X) r35: a__isQid(X) -> isQid(X) r36: a__isNePal(X) -> isNePal(X) r37: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) r38: mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) r39: mark(U11(X)) -> a__U11(mark(X)) r40: mark(U21(X1,X2)) -> a__U21(mark(X1),X2) r41: mark(U22(X)) -> a__U22(mark(X)) r42: mark(isList(X)) -> a__isList(X) r43: mark(U31(X)) -> a__U31(mark(X)) r44: mark(U41(X1,X2)) -> a__U41(mark(X1),X2) r45: mark(U42(X)) -> a__U42(mark(X)) r46: mark(isNeList(X)) -> a__isNeList(X) r47: mark(U51(X1,X2)) -> a__U51(mark(X1),X2) r48: mark(U52(X)) -> a__U52(mark(X)) r49: mark(U61(X)) -> a__U61(mark(X)) r50: mark(U71(X1,X2)) -> a__U71(mark(X1),X2) r51: mark(U72(X)) -> a__U72(mark(X)) r52: mark(isPal(X)) -> a__isPal(X) r53: mark(U81(X)) -> a__U81(mark(X)) r54: mark(isQid(X)) -> a__isQid(X) r55: mark(isNePal(X)) -> a__isNePal(X) r56: mark(nil()) -> nil() r57: mark(tt()) -> tt() r58: mark(a()) -> a() r59: mark(e()) -> e() r60: mark(i()) -> i() r61: mark(o()) -> o() r62: mark(u()) -> u() r63: a____(X1,X2) -> __(X1,X2) 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 Take the reduction pair: matrix interpretations: carrier: N^1 order: standard order interpretations: a____#_A(x1,x2) = x1 ___A(x1,x2) = x1 + 1 mark_A(x1) = x1 a_____A(x1,x2) = x1 + 1 a__U11_A(x1) = 0 tt_A() = 0 a__U21_A(x1,x2) = x1 + x2 a__U22_A(x1) = x1 a__isList_A(x1) = x1 a__U41_A(x1,x2) = x1 + x2 a__U42_A(x1) = x1 a__isNeList_A(x1) = 0 a__U51_A(x1,x2) = x1 + x2 a__U52_A(x1) = x1 a__U61_A(x1) = x1 a__U81_A(x1) = x1 a__U31_A(x1) = 0 a__isQid_A(x1) = x1 a__isNePal_A(x1) = x1 a__isPal_A(x1) = x1 nil_A() = 0 a_A() = 0 e_A() = 0 i_A() = 0 o_A() = 0 u_A() = 0 U11_A(x1) = 0 U21_A(x1,x2) = x1 + x2 U22_A(x1) = x1 isList_A(x1) = x1 U31_A(x1) = 0 U41_A(x1,x2) = x1 + x2 U42_A(x1) = x1 isNeList_A(x1) = 0 U51_A(x1,x2) = x1 + x2 U52_A(x1) = x1 U61_A(x1) = x1 a__U71_A(x1,x2) = x1 + x2 U71_A(x1,x2) = x1 + x2 a__U72_A(x1) = x1 U72_A(x1) = x1 isPal_A(x1) = x1 U81_A(x1) = x1 isQid_A(x1) = x1 isNePal_A(x1) = x1 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: matrix interpretations: carrier: N^1 order: standard order interpretations: a__U21#_A(x1,x2) = x2 + 1 tt_A() = 2 a__isList#_A(x1) = x1 + 1 ___A(x1,x2) = x1 + x2 a__isList_A(x1) = x1 + 1 a__isNeList#_A(x1) = x1 + 1 a__U51#_A(x1,x2) = x1 + x2 a__isNeList_A(x1) = x1 + 1 a__U41#_A(x1,x2) = x2 + 1 a__U22_A(x1) = x1 a__U42_A(x1) = 2 a__U52_A(x1) = 2 U22_A(x1) = 0 U42_A(x1) = 0 U52_A(x1) = 0 a__U11_A(x1) = x1 a__U21_A(x1,x2) = x1 + x2 a__U31_A(x1) = x1 a__U41_A(x1,x2) = x1 + x2 a__U51_A(x1,x2) = x1 + x2 a__isQid_A(x1) = x1 + 1 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 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() = 1 isList_A(x1) = 0 isNeList_A(x1) = 0 The next rules are strictly ordered: p7 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: 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__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, p2, p3, p4, p5, p7, p8, p9} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U21#(tt(),V2) -> a__isList#(V2) 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__isList#(__(V1,V2)) -> a__U21#(a__isList(V1),V2) p7: a__isList#(__(V1,V2)) -> a__isList#(V1) 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: matrix interpretations: carrier: N^1 order: standard order interpretations: a__U21#_A(x1,x2) = x1 + x2 + 1 tt_A() = 1 a__isList#_A(x1) = x1 + 2 a__isNeList#_A(x1) = x1 + 1 ___A(x1,x2) = x1 + x2 + 2 a__U41#_A(x1,x2) = x1 + x2 a__isList_A(x1) = 3 a__U42_A(x1) = 1 a__U52_A(x1) = 1 U42_A(x1) = 0 U52_A(x1) = 0 a__U22_A(x1) = 1 a__U31_A(x1) = 1 a__U41_A(x1,x2) = 1 a__isNeList_A(x1) = 1 a__U51_A(x1,x2) = 1 a__isQid_A(x1) = 1 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 U22_A(x1) = 0 U31_A(x1) = 0 U41_A(x1,x2) = 0 U51_A(x1,x2) = 0 isQid_A(x1) = 0 a__U11_A(x1) = 3 a__U21_A(x1,x2) = x1 U11_A(x1) = 0 U21_A(x1,x2) = 0 isNeList_A(x1) = 0 nil_A() = 0 isList_A(x1) = 0 The next rules are strictly ordered: p2, p5, p7, p8 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__U21#(tt(),V2) -> a__isList#(V2) p2: a__isNeList#(__(V1,V2)) -> a__U41#(a__isList(V1),V2) p3: a__U41#(tt(),V2) -> a__isNeList#(V2) p4: a__isList#(__(V1,V2)) -> a__U21#(a__isList(V1),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, p4} {p2, p3} -- 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__U21#(a__isList(V1),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: matrix interpretations: carrier: N^1 order: standard order interpretations: a__U21#_A(x1,x2) = x2 + 1 tt_A() = 1 a__isList#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 2 a__isList_A(x1) = 1 a__U42_A(x1) = x1 a__U52_A(x1) = x1 U42_A(x1) = 0 U52_A(x1) = 0 a__U22_A(x1) = 1 a__U31_A(x1) = x1 a__U41_A(x1,x2) = x1 + x2 a__isNeList_A(x1) = x1 + 1 a__U51_A(x1,x2) = x2 + 1 a__isQid_A(x1) = x1 + 1 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 U22_A(x1) = 0 U31_A(x1) = 0 U41_A(x1,x2) = 0 U51_A(x1,x2) = 0 isQid_A(x1) = 0 a__U11_A(x1) = 1 a__U21_A(x1,x2) = 1 U11_A(x1) = 0 U21_A(x1,x2) = 0 isNeList_A(x1) = 0 nil_A() = 1 isList_A(x1) = 0 The next rules are strictly ordered: p1, p2 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__isNeList#(__(V1,V2)) -> a__U41#(a__isList(V1),V2) p2: 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: matrix interpretations: carrier: N^1 order: standard order interpretations: a__isNeList#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 1 a__U41#_A(x1,x2) = x2 + 1 a__isList_A(x1) = 1 tt_A() = 1 a__U42_A(x1) = x1 a__U52_A(x1) = x1 U42_A(x1) = 0 U52_A(x1) = 0 a__U22_A(x1) = 1 a__U31_A(x1) = x1 a__U41_A(x1,x2) = x1 + x2 a__isNeList_A(x1) = x1 + 1 a__U51_A(x1,x2) = x2 + 1 a__isQid_A(x1) = x1 + 1 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 U22_A(x1) = 0 U31_A(x1) = 0 U41_A(x1,x2) = 0 U51_A(x1,x2) = 0 isQid_A(x1) = 0 a__U11_A(x1) = 1 a__U21_A(x1,x2) = 1 U11_A(x1) = 0 U21_A(x1,x2) = 0 isNeList_A(x1) = 0 nil_A() = 1 isList_A(x1) = 0 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__isNeList#(__(V1,V2)) -> a__U41#(a__isList(V1),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: (no SCCs) -- 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: matrix interpretations: carrier: N^1 order: standard order interpretations: a__U71#_A(x1,x2) = x1 + x2 tt_A() = 1 a__isPal#_A(x1) = x1 + 1 a__isNePal#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 1 a__isQid_A(x1) = x1 + 1 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 isQid_A(x1) = 0 The next rules are strictly ordered: p2, 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) 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)