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(),V) -> a__U12(a__isNeList(V))
  a__U12(tt()) -> tt()
  a__U21(tt(),V1,V2) -> a__U22(a__isList(V1),V2)
  a__U22(tt(),V2) -> a__U23(a__isList(V2))
  a__U23(tt()) -> tt()
  a__U31(tt(),V) -> a__U32(a__isQid(V))
  a__U32(tt()) -> tt()
  a__U41(tt(),V1,V2) -> a__U42(a__isList(V1),V2)
  a__U42(tt(),V2) -> a__U43(a__isNeList(V2))
  a__U43(tt()) -> tt()
  a__U51(tt(),V1,V2) -> a__U52(a__isNeList(V1),V2)
  a__U52(tt(),V2) -> a__U53(a__isList(V2))
  a__U53(tt()) -> tt()
  a__U61(tt(),V) -> a__U62(a__isQid(V))
  a__U62(tt()) -> tt()
  a__U71(tt(),V) -> a__U72(a__isNePal(V))
  a__U72(tt()) -> tt()
  a__and(tt(),X) -> mark(X)
  a__isList(V) -> a__U11(a__isPalListKind(V),V)
  a__isList(nil()) -> tt()
  a__isList(__(V1,V2)) -> a__U21(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
  a__isNeList(V) -> a__U31(a__isPalListKind(V),V)
  a__isNeList(__(V1,V2)) -> a__U41(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
  a__isNeList(__(V1,V2)) -> a__U51(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
  a__isNePal(V) -> a__U61(a__isPalListKind(V),V)
  a__isNePal(__(I,__(P,I))) -> a__and(a__and(a__isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P)))
  a__isPal(V) -> a__U71(a__isPalListKind(V),V)
  a__isPal(nil()) -> tt()
  a__isPalListKind(a()) -> tt()
  a__isPalListKind(e()) -> tt()
  a__isPalListKind(i()) -> tt()
  a__isPalListKind(nil()) -> tt()
  a__isPalListKind(o()) -> tt()
  a__isPalListKind(u()) -> tt()
  a__isPalListKind(__(V1,V2)) -> a__and(a__isPalListKind(V1),isPalListKind(V2))
  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(X1,X2)) -> a__U11(mark(X1),X2)
  mark(U12(X)) -> a__U12(mark(X))
  mark(isNeList(X)) -> a__isNeList(X)
  mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
  mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
  mark(isList(X)) -> a__isList(X)
  mark(U23(X)) -> a__U23(mark(X))
  mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
  mark(U32(X)) -> a__U32(mark(X))
  mark(isQid(X)) -> a__isQid(X)
  mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
  mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
  mark(U43(X)) -> a__U43(mark(X))
  mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
  mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
  mark(U53(X)) -> a__U53(mark(X))
  mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
  mark(U62(X)) -> a__U62(mark(X))
  mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
  mark(U72(X)) -> a__U72(mark(X))
  mark(isNePal(X)) -> a__isNePal(X)
  mark(and(X1,X2)) -> a__and(mark(X1),X2)
  mark(isPalListKind(X)) -> a__isPalListKind(X)
  mark(isPal(X)) -> a__isPal(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(X1,X2) -> U11(X1,X2)
  a__U12(X) -> U12(X)
  a__isNeList(X) -> isNeList(X)
  a__U21(X1,X2,X3) -> U21(X1,X2,X3)
  a__U22(X1,X2) -> U22(X1,X2)
  a__isList(X) -> isList(X)
  a__U23(X) -> U23(X)
  a__U31(X1,X2) -> U31(X1,X2)
  a__U32(X) -> U32(X)
  a__isQid(X) -> isQid(X)
  a__U41(X1,X2,X3) -> U41(X1,X2,X3)
  a__U42(X1,X2) -> U42(X1,X2)
  a__U43(X) -> U43(X)
  a__U51(X1,X2,X3) -> U51(X1,X2,X3)
  a__U52(X1,X2) -> U52(X1,X2)
  a__U53(X) -> U53(X)
  a__U61(X1,X2) -> U61(X1,X2)
  a__U62(X) -> U62(X)
  a__U71(X1,X2) -> U71(X1,X2)
  a__U72(X) -> U72(X)
  a__isNePal(X) -> isNePal(X)
  a__and(X1,X2) -> and(X1,X2)
  a__isPalListKind(X) -> isPalListKind(X)
  a__isPal(X) -> isPal(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__U11#(tt(),V) -> a__U12#(a__isNeList(V))
p9: a__U11#(tt(),V) -> a__isNeList#(V)
p10: a__U21#(tt(),V1,V2) -> a__U22#(a__isList(V1),V2)
p11: a__U21#(tt(),V1,V2) -> a__isList#(V1)
p12: a__U22#(tt(),V2) -> a__U23#(a__isList(V2))
p13: a__U22#(tt(),V2) -> a__isList#(V2)
p14: a__U31#(tt(),V) -> a__U32#(a__isQid(V))
p15: a__U31#(tt(),V) -> a__isQid#(V)
p16: a__U41#(tt(),V1,V2) -> a__U42#(a__isList(V1),V2)
p17: a__U41#(tt(),V1,V2) -> a__isList#(V1)
p18: a__U42#(tt(),V2) -> a__U43#(a__isNeList(V2))
p19: a__U42#(tt(),V2) -> a__isNeList#(V2)
p20: a__U51#(tt(),V1,V2) -> a__U52#(a__isNeList(V1),V2)
p21: a__U51#(tt(),V1,V2) -> a__isNeList#(V1)
p22: a__U52#(tt(),V2) -> a__U53#(a__isList(V2))
p23: a__U52#(tt(),V2) -> a__isList#(V2)
p24: a__U61#(tt(),V) -> a__U62#(a__isQid(V))
p25: a__U61#(tt(),V) -> a__isQid#(V)
p26: a__U71#(tt(),V) -> a__U72#(a__isNePal(V))
p27: a__U71#(tt(),V) -> a__isNePal#(V)
p28: a__and#(tt(),X) -> mark#(X)
p29: a__isList#(V) -> a__U11#(a__isPalListKind(V),V)
p30: a__isList#(V) -> a__isPalListKind#(V)
p31: a__isList#(__(V1,V2)) -> a__U21#(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
p32: a__isList#(__(V1,V2)) -> a__and#(a__isPalListKind(V1),isPalListKind(V2))
p33: a__isList#(__(V1,V2)) -> a__isPalListKind#(V1)
p34: a__isNeList#(V) -> a__U31#(a__isPalListKind(V),V)
p35: a__isNeList#(V) -> a__isPalListKind#(V)
p36: a__isNeList#(__(V1,V2)) -> a__U41#(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
p37: a__isNeList#(__(V1,V2)) -> a__and#(a__isPalListKind(V1),isPalListKind(V2))
p38: a__isNeList#(__(V1,V2)) -> a__isPalListKind#(V1)
p39: a__isNeList#(__(V1,V2)) -> a__U51#(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
p40: a__isNeList#(__(V1,V2)) -> a__and#(a__isPalListKind(V1),isPalListKind(V2))
p41: a__isNeList#(__(V1,V2)) -> a__isPalListKind#(V1)
p42: a__isNePal#(V) -> a__U61#(a__isPalListKind(V),V)
p43: a__isNePal#(V) -> a__isPalListKind#(V)
p44: a__isNePal#(__(I,__(P,I))) -> a__and#(a__and(a__isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P)))
p45: a__isNePal#(__(I,__(P,I))) -> a__and#(a__isQid(I),isPalListKind(I))
p46: a__isNePal#(__(I,__(P,I))) -> a__isQid#(I)
p47: a__isPal#(V) -> a__U71#(a__isPalListKind(V),V)
p48: a__isPal#(V) -> a__isPalListKind#(V)
p49: a__isPalListKind#(__(V1,V2)) -> a__and#(a__isPalListKind(V1),isPalListKind(V2))
p50: a__isPalListKind#(__(V1,V2)) -> a__isPalListKind#(V1)
p51: mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2))
p52: mark#(__(X1,X2)) -> mark#(X1)
p53: mark#(__(X1,X2)) -> mark#(X2)
p54: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2)
p55: mark#(U11(X1,X2)) -> mark#(X1)
p56: mark#(U12(X)) -> a__U12#(mark(X))
p57: mark#(U12(X)) -> mark#(X)
p58: mark#(isNeList(X)) -> a__isNeList#(X)
p59: mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3)
p60: mark#(U21(X1,X2,X3)) -> mark#(X1)
p61: mark#(U22(X1,X2)) -> a__U22#(mark(X1),X2)
p62: mark#(U22(X1,X2)) -> mark#(X1)
p63: mark#(isList(X)) -> a__isList#(X)
p64: mark#(U23(X)) -> a__U23#(mark(X))
p65: mark#(U23(X)) -> mark#(X)
p66: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p67: mark#(U31(X1,X2)) -> mark#(X1)
p68: mark#(U32(X)) -> a__U32#(mark(X))
p69: mark#(U32(X)) -> mark#(X)
p70: mark#(isQid(X)) -> a__isQid#(X)
p71: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p72: mark#(U41(X1,X2,X3)) -> mark#(X1)
p73: mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2)
p74: mark#(U42(X1,X2)) -> mark#(X1)
p75: mark#(U43(X)) -> a__U43#(mark(X))
p76: mark#(U43(X)) -> mark#(X)
p77: mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3)
p78: mark#(U51(X1,X2,X3)) -> mark#(X1)
p79: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p80: mark#(U52(X1,X2)) -> mark#(X1)
p81: mark#(U53(X)) -> a__U53#(mark(X))
p82: mark#(U53(X)) -> mark#(X)
p83: mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2)
p84: mark#(U61(X1,X2)) -> mark#(X1)
p85: mark#(U62(X)) -> a__U62#(mark(X))
p86: mark#(U62(X)) -> mark#(X)
p87: mark#(U71(X1,X2)) -> a__U71#(mark(X1),X2)
p88: mark#(U71(X1,X2)) -> mark#(X1)
p89: mark#(U72(X)) -> a__U72#(mark(X))
p90: mark#(U72(X)) -> mark#(X)
p91: mark#(isNePal(X)) -> a__isNePal#(X)
p92: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p93: mark#(and(X1,X2)) -> mark#(X1)
p94: mark#(isPalListKind(X)) -> a__isPalListKind#(X)
p95: mark#(isPal(X)) -> a__isPal#(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(),V) -> a__U12(a__isNeList(V))
r5: a__U12(tt()) -> tt()
r6: a__U21(tt(),V1,V2) -> a__U22(a__isList(V1),V2)
r7: a__U22(tt(),V2) -> a__U23(a__isList(V2))
r8: a__U23(tt()) -> tt()
r9: a__U31(tt(),V) -> a__U32(a__isQid(V))
r10: a__U32(tt()) -> tt()
r11: a__U41(tt(),V1,V2) -> a__U42(a__isList(V1),V2)
r12: a__U42(tt(),V2) -> a__U43(a__isNeList(V2))
r13: a__U43(tt()) -> tt()
r14: a__U51(tt(),V1,V2) -> a__U52(a__isNeList(V1),V2)
r15: a__U52(tt(),V2) -> a__U53(a__isList(V2))
r16: a__U53(tt()) -> tt()
r17: a__U61(tt(),V) -> a__U62(a__isQid(V))
r18: a__U62(tt()) -> tt()
r19: a__U71(tt(),V) -> a__U72(a__isNePal(V))
r20: a__U72(tt()) -> tt()
r21: a__and(tt(),X) -> mark(X)
r22: a__isList(V) -> a__U11(a__isPalListKind(V),V)
r23: a__isList(nil()) -> tt()
r24: a__isList(__(V1,V2)) -> a__U21(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r25: a__isNeList(V) -> a__U31(a__isPalListKind(V),V)
r26: a__isNeList(__(V1,V2)) -> a__U41(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r27: a__isNeList(__(V1,V2)) -> a__U51(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r28: a__isNePal(V) -> a__U61(a__isPalListKind(V),V)
r29: a__isNePal(__(I,__(P,I))) -> a__and(a__and(a__isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P)))
r30: a__isPal(V) -> a__U71(a__isPalListKind(V),V)
r31: a__isPal(nil()) -> tt()
r32: a__isPalListKind(a()) -> tt()
r33: a__isPalListKind(e()) -> tt()
r34: a__isPalListKind(i()) -> tt()
r35: a__isPalListKind(nil()) -> tt()
r36: a__isPalListKind(o()) -> tt()
r37: a__isPalListKind(u()) -> tt()
r38: a__isPalListKind(__(V1,V2)) -> a__and(a__isPalListKind(V1),isPalListKind(V2))
r39: a__isQid(a()) -> tt()
r40: a__isQid(e()) -> tt()
r41: a__isQid(i()) -> tt()
r42: a__isQid(o()) -> tt()
r43: a__isQid(u()) -> tt()
r44: mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
r45: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r46: mark(U12(X)) -> a__U12(mark(X))
r47: mark(isNeList(X)) -> a__isNeList(X)
r48: mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
r49: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r50: mark(isList(X)) -> a__isList(X)
r51: mark(U23(X)) -> a__U23(mark(X))
r52: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r53: mark(U32(X)) -> a__U32(mark(X))
r54: mark(isQid(X)) -> a__isQid(X)
r55: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r56: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r57: mark(U43(X)) -> a__U43(mark(X))
r58: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r59: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r60: mark(U53(X)) -> a__U53(mark(X))
r61: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r62: mark(U62(X)) -> a__U62(mark(X))
r63: mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
r64: mark(U72(X)) -> a__U72(mark(X))
r65: mark(isNePal(X)) -> a__isNePal(X)
r66: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r67: mark(isPalListKind(X)) -> a__isPalListKind(X)
r68: mark(isPal(X)) -> a__isPal(X)
r69: mark(nil()) -> nil()
r70: mark(tt()) -> tt()
r71: mark(a()) -> a()
r72: mark(e()) -> e()
r73: mark(i()) -> i()
r74: mark(o()) -> o()
r75: mark(u()) -> u()
r76: a____(X1,X2) -> __(X1,X2)
r77: a__U11(X1,X2) -> U11(X1,X2)
r78: a__U12(X) -> U12(X)
r79: a__isNeList(X) -> isNeList(X)
r80: a__U21(X1,X2,X3) -> U21(X1,X2,X3)
r81: a__U22(X1,X2) -> U22(X1,X2)
r82: a__isList(X) -> isList(X)
r83: a__U23(X) -> U23(X)
r84: a__U31(X1,X2) -> U31(X1,X2)
r85: a__U32(X) -> U32(X)
r86: a__isQid(X) -> isQid(X)
r87: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r88: a__U42(X1,X2) -> U42(X1,X2)
r89: a__U43(X) -> U43(X)
r90: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r91: a__U52(X1,X2) -> U52(X1,X2)
r92: a__U53(X) -> U53(X)
r93: a__U61(X1,X2) -> U61(X1,X2)
r94: a__U62(X) -> U62(X)
r95: a__U71(X1,X2) -> U71(X1,X2)
r96: a__U72(X) -> U72(X)
r97: a__isNePal(X) -> isNePal(X)
r98: a__and(X1,X2) -> and(X1,X2)
r99: a__isPalListKind(X) -> isPalListKind(X)
r100: a__isPal(X) -> isPal(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p9, p10, p11, p13, p16, p17, p19, p20, p21, p23, p27, p28, p29, p30, p31, p32, p33, p35, p36, p37, p38, p39, p40, p41, p43, p44, p45, p47, p48, p49, p50, p51, p52, p53, p54, p55, p57, p58, p59, p60, p61, p62, p63, p65, p67, p69, p71, p72, p73, p74, p76, p77, p78, p79, p80, p82, p84, p86, p87, p88, p90, p91, p92, p93, p94, p95}


-- 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#(isPal(X)) -> a__isPal#(X)
p4: a__isPal#(V) -> a__isPalListKind#(V)
p5: a__isPalListKind#(__(V1,V2)) -> a__isPalListKind#(V1)
p6: a__isPalListKind#(__(V1,V2)) -> a__and#(a__isPalListKind(V1),isPalListKind(V2))
p7: a__and#(tt(),X) -> mark#(X)
p8: mark#(isPalListKind(X)) -> a__isPalListKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(isNePal(X)) -> a__isNePal#(X)
p12: a__isNePal#(__(I,__(P,I))) -> a__and#(a__isQid(I),isPalListKind(I))
p13: a__isNePal#(__(I,__(P,I))) -> a__and#(a__and(a__isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P)))
p14: a__isNePal#(V) -> a__isPalListKind#(V)
p15: mark#(U72(X)) -> mark#(X)
p16: mark#(U71(X1,X2)) -> mark#(X1)
p17: mark#(U71(X1,X2)) -> a__U71#(mark(X1),X2)
p18: a__U71#(tt(),V) -> a__isNePal#(V)
p19: mark#(U62(X)) -> mark#(X)
p20: mark#(U61(X1,X2)) -> mark#(X1)
p21: mark#(U53(X)) -> mark#(X)
p22: mark#(U52(X1,X2)) -> mark#(X1)
p23: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p24: a__U52#(tt(),V2) -> a__isList#(V2)
p25: a__isList#(__(V1,V2)) -> a__isPalListKind#(V1)
p26: a__isList#(__(V1,V2)) -> a__and#(a__isPalListKind(V1),isPalListKind(V2))
p27: a__isList#(__(V1,V2)) -> a__U21#(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
p28: a__U21#(tt(),V1,V2) -> a__isList#(V1)
p29: a__isList#(V) -> a__isPalListKind#(V)
p30: a__isList#(V) -> a__U11#(a__isPalListKind(V),V)
p31: a__U11#(tt(),V) -> a__isNeList#(V)
p32: a__isNeList#(__(V1,V2)) -> a__isPalListKind#(V1)
p33: a__isNeList#(__(V1,V2)) -> a__and#(a__isPalListKind(V1),isPalListKind(V2))
p34: a__isNeList#(__(V1,V2)) -> a__U51#(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
p35: a__U51#(tt(),V1,V2) -> a__isNeList#(V1)
p36: a__isNeList#(__(V1,V2)) -> a__U41#(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
p37: a__U41#(tt(),V1,V2) -> a__isList#(V1)
p38: a__U41#(tt(),V1,V2) -> a__U42#(a__isList(V1),V2)
p39: a__U42#(tt(),V2) -> a__isNeList#(V2)
p40: a__isNeList#(V) -> a__isPalListKind#(V)
p41: a__U51#(tt(),V1,V2) -> a__U52#(a__isNeList(V1),V2)
p42: a__U21#(tt(),V1,V2) -> a__U22#(a__isList(V1),V2)
p43: a__U22#(tt(),V2) -> a__isList#(V2)
p44: mark#(U51(X1,X2,X3)) -> mark#(X1)
p45: mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3)
p46: mark#(U43(X)) -> mark#(X)
p47: mark#(U42(X1,X2)) -> mark#(X1)
p48: mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2)
p49: mark#(U41(X1,X2,X3)) -> mark#(X1)
p50: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p51: mark#(U32(X)) -> mark#(X)
p52: mark#(U31(X1,X2)) -> mark#(X1)
p53: mark#(U23(X)) -> mark#(X)
p54: mark#(isList(X)) -> a__isList#(X)
p55: mark#(U22(X1,X2)) -> mark#(X1)
p56: mark#(U22(X1,X2)) -> a__U22#(mark(X1),X2)
p57: mark#(U21(X1,X2,X3)) -> mark#(X1)
p58: mark#(U21(X1,X2,X3)) -> a__U21#(mark(X1),X2,X3)
p59: mark#(isNeList(X)) -> a__isNeList#(X)
p60: mark#(U12(X)) -> mark#(X)
p61: mark#(U11(X1,X2)) -> mark#(X1)
p62: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2)
p63: mark#(__(X1,X2)) -> mark#(X2)
p64: mark#(__(X1,X2)) -> mark#(X1)
p65: mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2))
p66: a____#(X,nil()) -> mark#(X)
p67: a____#(__(X,Y),Z) -> mark#(Z)
p68: a____#(__(X,Y),Z) -> mark#(Y)
p69: a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z))
p70: a____#(__(X,Y),Z) -> mark#(X)
p71: a__isPal#(V) -> a__U71#(a__isPalListKind(V),V)

and R consists of:

r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
r2: a____(X,nil()) -> mark(X)
r3: a____(nil(),X) -> mark(X)
r4: a__U11(tt(),V) -> a__U12(a__isNeList(V))
r5: a__U12(tt()) -> tt()
r6: a__U21(tt(),V1,V2) -> a__U22(a__isList(V1),V2)
r7: a__U22(tt(),V2) -> a__U23(a__isList(V2))
r8: a__U23(tt()) -> tt()
r9: a__U31(tt(),V) -> a__U32(a__isQid(V))
r10: a__U32(tt()) -> tt()
r11: a__U41(tt(),V1,V2) -> a__U42(a__isList(V1),V2)
r12: a__U42(tt(),V2) -> a__U43(a__isNeList(V2))
r13: a__U43(tt()) -> tt()
r14: a__U51(tt(),V1,V2) -> a__U52(a__isNeList(V1),V2)
r15: a__U52(tt(),V2) -> a__U53(a__isList(V2))
r16: a__U53(tt()) -> tt()
r17: a__U61(tt(),V) -> a__U62(a__isQid(V))
r18: a__U62(tt()) -> tt()
r19: a__U71(tt(),V) -> a__U72(a__isNePal(V))
r20: a__U72(tt()) -> tt()
r21: a__and(tt(),X) -> mark(X)
r22: a__isList(V) -> a__U11(a__isPalListKind(V),V)
r23: a__isList(nil()) -> tt()
r24: a__isList(__(V1,V2)) -> a__U21(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r25: a__isNeList(V) -> a__U31(a__isPalListKind(V),V)
r26: a__isNeList(__(V1,V2)) -> a__U41(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r27: a__isNeList(__(V1,V2)) -> a__U51(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r28: a__isNePal(V) -> a__U61(a__isPalListKind(V),V)
r29: a__isNePal(__(I,__(P,I))) -> a__and(a__and(a__isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P)))
r30: a__isPal(V) -> a__U71(a__isPalListKind(V),V)
r31: a__isPal(nil()) -> tt()
r32: a__isPalListKind(a()) -> tt()
r33: a__isPalListKind(e()) -> tt()
r34: a__isPalListKind(i()) -> tt()
r35: a__isPalListKind(nil()) -> tt()
r36: a__isPalListKind(o()) -> tt()
r37: a__isPalListKind(u()) -> tt()
r38: a__isPalListKind(__(V1,V2)) -> a__and(a__isPalListKind(V1),isPalListKind(V2))
r39: a__isQid(a()) -> tt()
r40: a__isQid(e()) -> tt()
r41: a__isQid(i()) -> tt()
r42: a__isQid(o()) -> tt()
r43: a__isQid(u()) -> tt()
r44: mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
r45: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r46: mark(U12(X)) -> a__U12(mark(X))
r47: mark(isNeList(X)) -> a__isNeList(X)
r48: mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
r49: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r50: mark(isList(X)) -> a__isList(X)
r51: mark(U23(X)) -> a__U23(mark(X))
r52: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r53: mark(U32(X)) -> a__U32(mark(X))
r54: mark(isQid(X)) -> a__isQid(X)
r55: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r56: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r57: mark(U43(X)) -> a__U43(mark(X))
r58: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r59: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r60: mark(U53(X)) -> a__U53(mark(X))
r61: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r62: mark(U62(X)) -> a__U62(mark(X))
r63: mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
r64: mark(U72(X)) -> a__U72(mark(X))
r65: mark(isNePal(X)) -> a__isNePal(X)
r66: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r67: mark(isPalListKind(X)) -> a__isPalListKind(X)
r68: mark(isPal(X)) -> a__isPal(X)
r69: mark(nil()) -> nil()
r70: mark(tt()) -> tt()
r71: mark(a()) -> a()
r72: mark(e()) -> e()
r73: mark(i()) -> i()
r74: mark(o()) -> o()
r75: mark(u()) -> u()
r76: a____(X1,X2) -> __(X1,X2)
r77: a__U11(X1,X2) -> U11(X1,X2)
r78: a__U12(X) -> U12(X)
r79: a__isNeList(X) -> isNeList(X)
r80: a__U21(X1,X2,X3) -> U21(X1,X2,X3)
r81: a__U22(X1,X2) -> U22(X1,X2)
r82: a__isList(X) -> isList(X)
r83: a__U23(X) -> U23(X)
r84: a__U31(X1,X2) -> U31(X1,X2)
r85: a__U32(X) -> U32(X)
r86: a__isQid(X) -> isQid(X)
r87: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r88: a__U42(X1,X2) -> U42(X1,X2)
r89: a__U43(X) -> U43(X)
r90: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r91: a__U52(X1,X2) -> U52(X1,X2)
r92: a__U53(X) -> U53(X)
r93: a__U61(X1,X2) -> U61(X1,X2)
r94: a__U62(X) -> U62(X)
r95: a__U71(X1,X2) -> U71(X1,X2)
r96: a__U72(X) -> U72(X)
r97: a__isNePal(X) -> isNePal(X)
r98: a__and(X1,X2) -> and(X1,X2)
r99: a__isPalListKind(X) -> isPalListKind(X)
r100: a__isPal(X) -> isPal(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, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: lexicographic order
      interpretations:
        a____#_A(x1,x2) = x1 + x2 + (1,116)
        ___A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (51,66)
        mark_A(x1) = x1 + (0,14)
        a_____A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (51,66)
        nil_A() = (0,1)
        mark#_A(x1) = ((1,0),(1,1)) x1
        isPal_A(x1) = ((1,0),(1,0)) x1 + (6,30)
        a__isPal#_A(x1) = ((1,0),(1,1)) x1 + (3,76)
        a__isPalListKind#_A(x1) = ((0,0),(1,0)) x1 + (0,37)
        a__and#_A(x1,x2) = ((1,0),(1,1)) x2 + (0,1)
        a__isPalListKind_A(x1) = ((0,0),(1,0)) x1 + (0,39)
        isPalListKind_A(x1) = ((0,0),(1,0)) x1 + (0,38)
        tt_A() = (0,3)
        and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,12)
        isNePal_A(x1) = x1 + (3,36)
        a__isNePal#_A(x1) = x1 + (1,38)
        a__isQid_A(x1) = x1 + (0,1)
        a__and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,12)
        U72_A(x1) = ((1,0),(1,0)) x1 + (1,1)
        U71_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (5,6)
        a__U71#_A(x1,x2) = x1 + x2 + (2,36)
        U62_A(x1) = ((1,0),(1,0)) x1 + (1,1)
        U61_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,2)
        U53_A(x1) = x1
        U52_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (7,43)
        a__U52#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (4,51)
        a__isList#_A(x1) = ((1,0),(1,0)) x1 + (3,36)
        a__U21#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (3,88)
        a__U11#_A(x1,x2) = ((1,0),(1,0)) x2 + (2,35)
        a__isNeList#_A(x1) = ((1,0),(1,0)) x1 + (1,34)
        a__U51#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (8,84)
        a__U41#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (3,44)
        a__U42#_A(x1,x2) = ((1,0),(1,0)) x2 + (1,35)
        a__isList_A(x1) = ((1,0),(1,1)) x1 + (6,42)
        a__isNeList_A(x1) = x1 + (3,5)
        a__U22#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (3,43)
        U51_A(x1,x2,x3) = x1 + ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (11,72)
        U43_A(x1) = x1
        U42_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (35,1)
        U41_A(x1,x2,x3) = x1 + x2 + ((1,0),(1,0)) x3 + (42,1)
        U32_A(x1) = ((1,0),(1,0)) x1 + (1,0)
        U31_A(x1,x2) = x1 + x2 + (2,1)
        U23_A(x1) = x1 + (1,1)
        isList_A(x1) = ((1,0),(1,1)) x1 + (6,29)
        U22_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (8,34)
        U21_A(x1,x2,x3) = ((1,0),(1,0)) x1 + x2 + x3 + (15,74)
        isNeList_A(x1) = x1 + (3,1)
        U12_A(x1) = x1 + (1,3)
        U11_A(x1,x2) = x1 + x2 + (5,2)
        a__U11_A(x1,x2) = x1 + x2 + (5,2)
        a__U12_A(x1) = x1 + (1,4)
        a__U21_A(x1,x2,x3) = ((1,0),(1,0)) x1 + x2 + x3 + (15,75)
        a__U22_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (8,34)
        a__U23_A(x1) = x1 + (1,4)
        a__U31_A(x1,x2) = x1 + x2 + (2,2)
        a__U32_A(x1) = ((1,0),(1,0)) x1 + (1,4)
        a__U41_A(x1,x2,x3) = x1 + x2 + ((1,0),(1,0)) x3 + (42,2)
        a__U42_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (35,1)
        a__U43_A(x1) = x1
        a__U51_A(x1,x2,x3) = x1 + ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (11,72)
        a__U52_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (7,43)
        a__U53_A(x1) = x1
        a__U61_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,2)
        a__U62_A(x1) = ((1,0),(1,0)) x1 + (1,4)
        a__U71_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (5,19)
        a__U72_A(x1) = ((1,0),(1,0)) x1 + (1,15)
        a__isNePal_A(x1) = x1 + (3,37)
        a__isPal_A(x1) = ((1,0),(1,0)) x1 + (6,31)
        a_A() = (0,3)
        e_A() = (1,1)
        i_A() = (1,1)
        o_A() = (1,1)
        u_A() = (1,1)
        isQid_A(x1) = x1
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        isQid > a__U31 > a__U51 > a > i > and > a__isPal > a__isPalListKind > isPalListKind > isNeList > U31 > tt > a__U41# > a__U21 > U21 > mark > a__U32 > U32 > a__and > a__U12 > isPal > a__U22 > a__U23 > U23 > a__U11 > a__U71 > a__U72 > a__isPal# > U11 > a__U43 > U43 > u > nil > a__U41 > U41 > a__U71# > a__isNePal# > mark# > a__U62 > e > U62 > a__isNePal > a__U61 > isNePal > a__isNeList > a__U42# > isList > a__isPalListKind# > a__and# > U52 > a__isList > a__U51# > a__U21# > a__U52# > a__U22# > a__isList# > a__U11# > a__isNeList# > a____ > __ > a____# > a__U42 > U72 > o > a__U52 > U61 > U12 > U22 > a__isQid > a__U53 > U53 > U42 > U51 > U71
      
      argument filter:
    
        pi(a____#) = []
        pi(__) = [1]
        pi(mark) = [1]
        pi(a____) = [1]
        pi(nil) = []
        pi(mark#) = 1
        pi(isPal) = []
        pi(a__isPal#) = [1]
        pi(a__isPalListKind#) = []
        pi(a__and#) = []
        pi(a__isPalListKind) = []
        pi(isPalListKind) = []
        pi(tt) = []
        pi(and) = 2
        pi(isNePal) = 1
        pi(a__isNePal#) = []
        pi(a__isQid) = 1
        pi(a__and) = [2]
        pi(U72) = []
        pi(U71) = []
        pi(a__U71#) = [1, 2]
        pi(U62) = []
        pi(U61) = 1
        pi(U53) = 1
        pi(U52) = 2
        pi(a__U52#) = []
        pi(a__isList#) = []
        pi(a__U21#) = []
        pi(a__U11#) = []
        pi(a__isNeList#) = []
        pi(a__U51#) = []
        pi(a__U41#) = []
        pi(a__U42#) = []
        pi(a__isList) = 1
        pi(a__isNeList) = 1
        pi(a__U22#) = []
        pi(U51) = 3
        pi(U43) = 1
        pi(U42) = []
        pi(U41) = 2
        pi(U32) = []
        pi(U31) = 2
        pi(U23) = []
        pi(isList) = 1
        pi(U22) = 1
        pi(U21) = 2
        pi(isNeList) = [1]
        pi(U12) = []
        pi(U11) = 2
        pi(a__U11) = 2
        pi(a__U12) = []
        pi(a__U21) = 2
        pi(a__U22) = 1
        pi(a__U23) = []
        pi(a__U31) = 2
        pi(a__U32) = []
        pi(a__U41) = 2
        pi(a__U42) = []
        pi(a__U43) = 1
        pi(a__U51) = 3
        pi(a__U52) = 2
        pi(a__U53) = 1
        pi(a__U61) = 1
        pi(a__U62) = []
        pi(a__U71) = []
        pi(a__U72) = []
        pi(a__isNePal) = 1
        pi(a__isPal) = []
        pi(a) = []
        pi(e) = []
        pi(i) = []
        pi(o) = []
        pi(u) = []
        pi(isQid) = 1
    

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, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71

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#(U53(X)) -> mark#(X)
p3: mark#(U43(X)) -> mark#(X)

and R consists of:

r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
r2: a____(X,nil()) -> mark(X)
r3: a____(nil(),X) -> mark(X)
r4: a__U11(tt(),V) -> a__U12(a__isNeList(V))
r5: a__U12(tt()) -> tt()
r6: a__U21(tt(),V1,V2) -> a__U22(a__isList(V1),V2)
r7: a__U22(tt(),V2) -> a__U23(a__isList(V2))
r8: a__U23(tt()) -> tt()
r9: a__U31(tt(),V) -> a__U32(a__isQid(V))
r10: a__U32(tt()) -> tt()
r11: a__U41(tt(),V1,V2) -> a__U42(a__isList(V1),V2)
r12: a__U42(tt(),V2) -> a__U43(a__isNeList(V2))
r13: a__U43(tt()) -> tt()
r14: a__U51(tt(),V1,V2) -> a__U52(a__isNeList(V1),V2)
r15: a__U52(tt(),V2) -> a__U53(a__isList(V2))
r16: a__U53(tt()) -> tt()
r17: a__U61(tt(),V) -> a__U62(a__isQid(V))
r18: a__U62(tt()) -> tt()
r19: a__U71(tt(),V) -> a__U72(a__isNePal(V))
r20: a__U72(tt()) -> tt()
r21: a__and(tt(),X) -> mark(X)
r22: a__isList(V) -> a__U11(a__isPalListKind(V),V)
r23: a__isList(nil()) -> tt()
r24: a__isList(__(V1,V2)) -> a__U21(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r25: a__isNeList(V) -> a__U31(a__isPalListKind(V),V)
r26: a__isNeList(__(V1,V2)) -> a__U41(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r27: a__isNeList(__(V1,V2)) -> a__U51(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r28: a__isNePal(V) -> a__U61(a__isPalListKind(V),V)
r29: a__isNePal(__(I,__(P,I))) -> a__and(a__and(a__isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P)))
r30: a__isPal(V) -> a__U71(a__isPalListKind(V),V)
r31: a__isPal(nil()) -> tt()
r32: a__isPalListKind(a()) -> tt()
r33: a__isPalListKind(e()) -> tt()
r34: a__isPalListKind(i()) -> tt()
r35: a__isPalListKind(nil()) -> tt()
r36: a__isPalListKind(o()) -> tt()
r37: a__isPalListKind(u()) -> tt()
r38: a__isPalListKind(__(V1,V2)) -> a__and(a__isPalListKind(V1),isPalListKind(V2))
r39: a__isQid(a()) -> tt()
r40: a__isQid(e()) -> tt()
r41: a__isQid(i()) -> tt()
r42: a__isQid(o()) -> tt()
r43: a__isQid(u()) -> tt()
r44: mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
r45: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r46: mark(U12(X)) -> a__U12(mark(X))
r47: mark(isNeList(X)) -> a__isNeList(X)
r48: mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
r49: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r50: mark(isList(X)) -> a__isList(X)
r51: mark(U23(X)) -> a__U23(mark(X))
r52: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r53: mark(U32(X)) -> a__U32(mark(X))
r54: mark(isQid(X)) -> a__isQid(X)
r55: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r56: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r57: mark(U43(X)) -> a__U43(mark(X))
r58: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r59: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r60: mark(U53(X)) -> a__U53(mark(X))
r61: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r62: mark(U62(X)) -> a__U62(mark(X))
r63: mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
r64: mark(U72(X)) -> a__U72(mark(X))
r65: mark(isNePal(X)) -> a__isNePal(X)
r66: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r67: mark(isPalListKind(X)) -> a__isPalListKind(X)
r68: mark(isPal(X)) -> a__isPal(X)
r69: mark(nil()) -> nil()
r70: mark(tt()) -> tt()
r71: mark(a()) -> a()
r72: mark(e()) -> e()
r73: mark(i()) -> i()
r74: mark(o()) -> o()
r75: mark(u()) -> u()
r76: a____(X1,X2) -> __(X1,X2)
r77: a__U11(X1,X2) -> U11(X1,X2)
r78: a__U12(X) -> U12(X)
r79: a__isNeList(X) -> isNeList(X)
r80: a__U21(X1,X2,X3) -> U21(X1,X2,X3)
r81: a__U22(X1,X2) -> U22(X1,X2)
r82: a__isList(X) -> isList(X)
r83: a__U23(X) -> U23(X)
r84: a__U31(X1,X2) -> U31(X1,X2)
r85: a__U32(X) -> U32(X)
r86: a__isQid(X) -> isQid(X)
r87: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r88: a__U42(X1,X2) -> U42(X1,X2)
r89: a__U43(X) -> U43(X)
r90: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r91: a__U52(X1,X2) -> U52(X1,X2)
r92: a__U53(X) -> U53(X)
r93: a__U61(X1,X2) -> U61(X1,X2)
r94: a__U62(X) -> U62(X)
r95: a__U71(X1,X2) -> U71(X1,X2)
r96: a__U72(X) -> U72(X)
r97: a__isNePal(X) -> isNePal(X)
r98: a__and(X1,X2) -> and(X1,X2)
r99: a__isPalListKind(X) -> isPalListKind(X)
r100: a__isPal(X) -> isPal(X)

The estimated dependency graph contains the following SCCs:

  {p1}
  {p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: 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(),V) -> a__U12(a__isNeList(V))
r5: a__U12(tt()) -> tt()
r6: a__U21(tt(),V1,V2) -> a__U22(a__isList(V1),V2)
r7: a__U22(tt(),V2) -> a__U23(a__isList(V2))
r8: a__U23(tt()) -> tt()
r9: a__U31(tt(),V) -> a__U32(a__isQid(V))
r10: a__U32(tt()) -> tt()
r11: a__U41(tt(),V1,V2) -> a__U42(a__isList(V1),V2)
r12: a__U42(tt(),V2) -> a__U43(a__isNeList(V2))
r13: a__U43(tt()) -> tt()
r14: a__U51(tt(),V1,V2) -> a__U52(a__isNeList(V1),V2)
r15: a__U52(tt(),V2) -> a__U53(a__isList(V2))
r16: a__U53(tt()) -> tt()
r17: a__U61(tt(),V) -> a__U62(a__isQid(V))
r18: a__U62(tt()) -> tt()
r19: a__U71(tt(),V) -> a__U72(a__isNePal(V))
r20: a__U72(tt()) -> tt()
r21: a__and(tt(),X) -> mark(X)
r22: a__isList(V) -> a__U11(a__isPalListKind(V),V)
r23: a__isList(nil()) -> tt()
r24: a__isList(__(V1,V2)) -> a__U21(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r25: a__isNeList(V) -> a__U31(a__isPalListKind(V),V)
r26: a__isNeList(__(V1,V2)) -> a__U41(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r27: a__isNeList(__(V1,V2)) -> a__U51(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r28: a__isNePal(V) -> a__U61(a__isPalListKind(V),V)
r29: a__isNePal(__(I,__(P,I))) -> a__and(a__and(a__isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P)))
r30: a__isPal(V) -> a__U71(a__isPalListKind(V),V)
r31: a__isPal(nil()) -> tt()
r32: a__isPalListKind(a()) -> tt()
r33: a__isPalListKind(e()) -> tt()
r34: a__isPalListKind(i()) -> tt()
r35: a__isPalListKind(nil()) -> tt()
r36: a__isPalListKind(o()) -> tt()
r37: a__isPalListKind(u()) -> tt()
r38: a__isPalListKind(__(V1,V2)) -> a__and(a__isPalListKind(V1),isPalListKind(V2))
r39: a__isQid(a()) -> tt()
r40: a__isQid(e()) -> tt()
r41: a__isQid(i()) -> tt()
r42: a__isQid(o()) -> tt()
r43: a__isQid(u()) -> tt()
r44: mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
r45: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r46: mark(U12(X)) -> a__U12(mark(X))
r47: mark(isNeList(X)) -> a__isNeList(X)
r48: mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
r49: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r50: mark(isList(X)) -> a__isList(X)
r51: mark(U23(X)) -> a__U23(mark(X))
r52: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r53: mark(U32(X)) -> a__U32(mark(X))
r54: mark(isQid(X)) -> a__isQid(X)
r55: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r56: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r57: mark(U43(X)) -> a__U43(mark(X))
r58: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r59: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r60: mark(U53(X)) -> a__U53(mark(X))
r61: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r62: mark(U62(X)) -> a__U62(mark(X))
r63: mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
r64: mark(U72(X)) -> a__U72(mark(X))
r65: mark(isNePal(X)) -> a__isNePal(X)
r66: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r67: mark(isPalListKind(X)) -> a__isPalListKind(X)
r68: mark(isPal(X)) -> a__isPal(X)
r69: mark(nil()) -> nil()
r70: mark(tt()) -> tt()
r71: mark(a()) -> a()
r72: mark(e()) -> e()
r73: mark(i()) -> i()
r74: mark(o()) -> o()
r75: mark(u()) -> u()
r76: a____(X1,X2) -> __(X1,X2)
r77: a__U11(X1,X2) -> U11(X1,X2)
r78: a__U12(X) -> U12(X)
r79: a__isNeList(X) -> isNeList(X)
r80: a__U21(X1,X2,X3) -> U21(X1,X2,X3)
r81: a__U22(X1,X2) -> U22(X1,X2)
r82: a__isList(X) -> isList(X)
r83: a__U23(X) -> U23(X)
r84: a__U31(X1,X2) -> U31(X1,X2)
r85: a__U32(X) -> U32(X)
r86: a__isQid(X) -> isQid(X)
r87: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r88: a__U42(X1,X2) -> U42(X1,X2)
r89: a__U43(X) -> U43(X)
r90: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r91: a__U52(X1,X2) -> U52(X1,X2)
r92: a__U53(X) -> U53(X)
r93: a__U61(X1,X2) -> U61(X1,X2)
r94: a__U62(X) -> U62(X)
r95: a__U71(X1,X2) -> U71(X1,X2)
r96: a__U72(X) -> U72(X)
r97: a__isNePal(X) -> isNePal(X)
r98: a__and(X1,X2) -> and(X1,X2)
r99: a__isPalListKind(X) -> isPalListKind(X)
r100: a__isPal(X) -> isPal(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, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: lexicographic order
      interpretations:
        a____#_A(x1,x2) = x1 + ((0,0),(1,0)) x2
        ___A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (6,1)
        mark_A(x1) = ((1,0),(1,1)) x1 + (0,2)
        a_____A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (6,2)
        a__U11_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (5,2)
        tt_A() = (5,8)
        a__U12_A(x1) = ((1,0),(1,0)) x1 + (1,5)
        a__isNeList_A(x1) = x1 + (3,2)
        a__U21_A(x1,x2,x3) = ((1,0),(1,0)) x2 + x3 + (9,4)
        a__U22_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (2,1)
        a__isList_A(x1) = x1 + (6,2)
        a__U23_A(x1) = ((0,0),(1,0)) x1 + (6,4)
        a__U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (2,2)
        a__U32_A(x1) = x1 + (0,6)
        a__isQid_A(x1) = ((1,0),(1,1)) x1 + (1,2)
        a__U41_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (8,2)
        a__U42_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (7,5)
        a__U43_A(x1) = (6,9)
        a__U51_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (6,3)
        a__U52_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,2)
        a__U53_A(x1) = ((1,0),(1,0)) x1 + (0,4)
        a__U61_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (0,10)
        a__U62_A(x1) = x1 + (1,9)
        a__U71_A(x1,x2) = ((0,0),(1,0)) x1 + (6,4)
        a__U72_A(x1) = (5,10)
        a__isNePal_A(x1) = ((1,0),(1,0)) x1 + (7,2)
        a__and_A(x1,x2) = ((1,0),(1,1)) x2 + (0,2)
        a__isPalListKind_A(x1) = ((0,0),(1,0)) x1 + (6,9)
        nil_A() = (0,7)
        isPalListKind_A(x1) = ((0,0),(1,0)) x1 + (6,6)
        and_A(x1,x2) = x2 + (0,1)
        isPal_A(x1) = x1 + (8,0)
        a__isPal_A(x1) = x1 + (8,9)
        a_A() = (5,2)
        e_A() = (7,0)
        i_A() = (5,2)
        o_A() = (5,0)
        u_A() = (7,0)
        U11_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (5,1)
        U12_A(x1) = ((1,0),(1,0)) x1 + (1,3)
        isNeList_A(x1) = x1 + (3,1)
        U21_A(x1,x2,x3) = ((1,0),(1,0)) x2 + x3 + (9,1)
        U22_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (2,1)
        isList_A(x1) = x1 + (6,1)
        U23_A(x1) = ((0,0),(1,0)) x1 + (6,1)
        U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (2,1)
        U32_A(x1) = x1 + (0,6)
        isQid_A(x1) = ((1,0),(1,1)) x1 + (1,1)
        U41_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (8,1)
        U42_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (7,4)
        U43_A(x1) = (6,2)
        U51_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (6,1)
        U52_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
        U53_A(x1) = ((1,0),(1,0)) x1 + (0,3)
        U61_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (0,9)
        U62_A(x1) = x1 + (1,8)
        U71_A(x1,x2) = ((0,0),(1,0)) x1 + (6,3)
        U72_A(x1) = (5,9)
        isNePal_A(x1) = ((1,0),(1,0)) x1 + (7,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        isQid > u > e > a > and > a__and > a__U71 > a__U72 > a__U52 > a____# > isNePal > U41 > a__U31 > nil > a__U61 > a____ > __ > U62 > a__U62 > mark > U71 > a__U53 > a__U42 > a__U43 > a__U51 > U51 > U12 > a__isPal > isPal > a__U11 > U61 > U11 > i > a__U21 > a__isPalListKind > a__U41 > a__U22 > a__U23 > a__U12 > a__isNePal > isPalListKind > U52 > U53 > U42 > U21 > isNeList > U22 > a__isList > U23 > o > U31 > a__isNeList > U72 > a__U32 > U43 > tt > a__isQid > U32 > isList
      
      argument filter:
    
        pi(a____#) = 1
        pi(__) = 1
        pi(mark) = [1]
        pi(a____) = [1]
        pi(a__U11) = 2
        pi(tt) = []
        pi(a__U12) = []
        pi(a__isNeList) = 1
        pi(a__U21) = [3]
        pi(a__U22) = [1]
        pi(a__isList) = 1
        pi(a__U23) = []
        pi(a__U31) = 2
        pi(a__U32) = 1
        pi(a__isQid) = 1
        pi(a__U41) = 3
        pi(a__U42) = []
        pi(a__U43) = []
        pi(a__U51) = 2
        pi(a__U52) = 1
        pi(a__U53) = []
        pi(a__U61) = [2]
        pi(a__U62) = []
        pi(a__U71) = []
        pi(a__U72) = []
        pi(a__isNePal) = []
        pi(a__and) = [2]
        pi(a__isPalListKind) = []
        pi(nil) = []
        pi(isPalListKind) = []
        pi(and) = [2]
        pi(isPal) = []
        pi(a__isPal) = []
        pi(a) = []
        pi(e) = []
        pi(i) = []
        pi(o) = []
        pi(u) = []
        pi(U11) = 2
        pi(U12) = []
        pi(isNeList) = 1
        pi(U21) = [3]
        pi(U22) = 1
        pi(isList) = 1
        pi(U23) = []
        pi(U31) = 2
        pi(U32) = 1
        pi(isQid) = 1
        pi(U41) = 3
        pi(U42) = []
        pi(U43) = []
        pi(U51) = 2
        pi(U52) = 1
        pi(U53) = []
        pi(U61) = [2]
        pi(U62) = []
        pi(U71) = []
        pi(U72) = []
        pi(isNePal) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U53(X)) -> mark#(X)
p2: mark#(U43(X)) -> mark#(X)

and R consists of:

r1: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
r2: a____(X,nil()) -> mark(X)
r3: a____(nil(),X) -> mark(X)
r4: a__U11(tt(),V) -> a__U12(a__isNeList(V))
r5: a__U12(tt()) -> tt()
r6: a__U21(tt(),V1,V2) -> a__U22(a__isList(V1),V2)
r7: a__U22(tt(),V2) -> a__U23(a__isList(V2))
r8: a__U23(tt()) -> tt()
r9: a__U31(tt(),V) -> a__U32(a__isQid(V))
r10: a__U32(tt()) -> tt()
r11: a__U41(tt(),V1,V2) -> a__U42(a__isList(V1),V2)
r12: a__U42(tt(),V2) -> a__U43(a__isNeList(V2))
r13: a__U43(tt()) -> tt()
r14: a__U51(tt(),V1,V2) -> a__U52(a__isNeList(V1),V2)
r15: a__U52(tt(),V2) -> a__U53(a__isList(V2))
r16: a__U53(tt()) -> tt()
r17: a__U61(tt(),V) -> a__U62(a__isQid(V))
r18: a__U62(tt()) -> tt()
r19: a__U71(tt(),V) -> a__U72(a__isNePal(V))
r20: a__U72(tt()) -> tt()
r21: a__and(tt(),X) -> mark(X)
r22: a__isList(V) -> a__U11(a__isPalListKind(V),V)
r23: a__isList(nil()) -> tt()
r24: a__isList(__(V1,V2)) -> a__U21(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r25: a__isNeList(V) -> a__U31(a__isPalListKind(V),V)
r26: a__isNeList(__(V1,V2)) -> a__U41(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r27: a__isNeList(__(V1,V2)) -> a__U51(a__and(a__isPalListKind(V1),isPalListKind(V2)),V1,V2)
r28: a__isNePal(V) -> a__U61(a__isPalListKind(V),V)
r29: a__isNePal(__(I,__(P,I))) -> a__and(a__and(a__isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P)))
r30: a__isPal(V) -> a__U71(a__isPalListKind(V),V)
r31: a__isPal(nil()) -> tt()
r32: a__isPalListKind(a()) -> tt()
r33: a__isPalListKind(e()) -> tt()
r34: a__isPalListKind(i()) -> tt()
r35: a__isPalListKind(nil()) -> tt()
r36: a__isPalListKind(o()) -> tt()
r37: a__isPalListKind(u()) -> tt()
r38: a__isPalListKind(__(V1,V2)) -> a__and(a__isPalListKind(V1),isPalListKind(V2))
r39: a__isQid(a()) -> tt()
r40: a__isQid(e()) -> tt()
r41: a__isQid(i()) -> tt()
r42: a__isQid(o()) -> tt()
r43: a__isQid(u()) -> tt()
r44: mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
r45: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r46: mark(U12(X)) -> a__U12(mark(X))
r47: mark(isNeList(X)) -> a__isNeList(X)
r48: mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
r49: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r50: mark(isList(X)) -> a__isList(X)
r51: mark(U23(X)) -> a__U23(mark(X))
r52: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r53: mark(U32(X)) -> a__U32(mark(X))
r54: mark(isQid(X)) -> a__isQid(X)
r55: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r56: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r57: mark(U43(X)) -> a__U43(mark(X))
r58: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r59: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r60: mark(U53(X)) -> a__U53(mark(X))
r61: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r62: mark(U62(X)) -> a__U62(mark(X))
r63: mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
r64: mark(U72(X)) -> a__U72(mark(X))
r65: mark(isNePal(X)) -> a__isNePal(X)
r66: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r67: mark(isPalListKind(X)) -> a__isPalListKind(X)
r68: mark(isPal(X)) -> a__isPal(X)
r69: mark(nil()) -> nil()
r70: mark(tt()) -> tt()
r71: mark(a()) -> a()
r72: mark(e()) -> e()
r73: mark(i()) -> i()
r74: mark(o()) -> o()
r75: mark(u()) -> u()
r76: a____(X1,X2) -> __(X1,X2)
r77: a__U11(X1,X2) -> U11(X1,X2)
r78: a__U12(X) -> U12(X)
r79: a__isNeList(X) -> isNeList(X)
r80: a__U21(X1,X2,X3) -> U21(X1,X2,X3)
r81: a__U22(X1,X2) -> U22(X1,X2)
r82: a__isList(X) -> isList(X)
r83: a__U23(X) -> U23(X)
r84: a__U31(X1,X2) -> U31(X1,X2)
r85: a__U32(X) -> U32(X)
r86: a__isQid(X) -> isQid(X)
r87: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r88: a__U42(X1,X2) -> U42(X1,X2)
r89: a__U43(X) -> U43(X)
r90: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r91: a__U52(X1,X2) -> U52(X1,X2)
r92: a__U53(X) -> U53(X)
r93: a__U61(X1,X2) -> U61(X1,X2)
r94: a__U62(X) -> U62(X)
r95: a__U71(X1,X2) -> U71(X1,X2)
r96: a__U72(X) -> U72(X)
r97: a__isNePal(X) -> isNePal(X)
r98: a__and(X1,X2) -> and(X1,X2)
r99: a__isPalListKind(X) -> isPalListKind(X)
r100: a__isPal(X) -> isPal(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: lexicographic order
      interpretations:
        mark#_A(x1) = ((1,0),(1,1)) x1
        U53_A(x1) = ((1,0),(1,1)) x1 + (1,1)
        U43_A(x1) = ((1,0),(1,1)) x1 + (1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        mark# > U43 > U53
      
      argument filter:
    
        pi(mark#) = 1
        pi(U53) = [1]
        pi(U43) = [1]
    

The next rules are strictly ordered:

  p1, p2
  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, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100

We remove them from the problem.  Then no dependency pair remains.