YES

We show the termination of the TRS R:

  __(__(X,Y),Z) -> __(X,__(Y,Z))
  __(X,nil()) -> X
  __(nil(),X) -> X
  U11(tt(),V) -> U12(isNeList(activate(V)))
  U12(tt()) -> tt()
  U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
  U22(tt(),V2) -> U23(isList(activate(V2)))
  U23(tt()) -> tt()
  U31(tt(),V) -> U32(isQid(activate(V)))
  U32(tt()) -> tt()
  U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
  U42(tt(),V2) -> U43(isNeList(activate(V2)))
  U43(tt()) -> tt()
  U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
  U52(tt(),V2) -> U53(isList(activate(V2)))
  U53(tt()) -> tt()
  U61(tt(),V) -> U62(isQid(activate(V)))
  U62(tt()) -> tt()
  U71(tt(),V) -> U72(isNePal(activate(V)))
  U72(tt()) -> tt()
  and(tt(),X) -> activate(X)
  isList(V) -> U11(isPalListKind(activate(V)),activate(V))
  isList(n__nil()) -> tt()
  isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
  isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
  isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
  isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
  isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
  isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P))))
  isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
  isPal(n__nil()) -> tt()
  isPalListKind(n__a()) -> tt()
  isPalListKind(n__e()) -> tt()
  isPalListKind(n__i()) -> tt()
  isPalListKind(n__nil()) -> tt()
  isPalListKind(n__o()) -> tt()
  isPalListKind(n__u()) -> tt()
  isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
  isQid(n__a()) -> tt()
  isQid(n__e()) -> tt()
  isQid(n__i()) -> tt()
  isQid(n__o()) -> tt()
  isQid(n__u()) -> tt()
  nil() -> n__nil()
  __(X1,X2) -> n____(X1,X2)
  isPalListKind(X) -> n__isPalListKind(X)
  and(X1,X2) -> n__and(X1,X2)
  a() -> n__a()
  e() -> n__e()
  i() -> n__i()
  o() -> n__o()
  u() -> n__u()
  activate(n__nil()) -> nil()
  activate(n____(X1,X2)) -> __(X1,X2)
  activate(n__isPalListKind(X)) -> isPalListKind(X)
  activate(n__and(X1,X2)) -> and(X1,X2)
  activate(n__a()) -> a()
  activate(n__e()) -> e()
  activate(n__i()) -> i()
  activate(n__o()) -> o()
  activate(n__u()) -> u()
  activate(X) -> X

-- SCC decomposition.

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

p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z))
p2: __#(__(X,Y),Z) -> __#(Y,Z)
p3: U11#(tt(),V) -> U12#(isNeList(activate(V)))
p4: U11#(tt(),V) -> isNeList#(activate(V))
p5: U11#(tt(),V) -> activate#(V)
p6: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2))
p7: U21#(tt(),V1,V2) -> isList#(activate(V1))
p8: U21#(tt(),V1,V2) -> activate#(V1)
p9: U21#(tt(),V1,V2) -> activate#(V2)
p10: U22#(tt(),V2) -> U23#(isList(activate(V2)))
p11: U22#(tt(),V2) -> isList#(activate(V2))
p12: U22#(tt(),V2) -> activate#(V2)
p13: U31#(tt(),V) -> U32#(isQid(activate(V)))
p14: U31#(tt(),V) -> isQid#(activate(V))
p15: U31#(tt(),V) -> activate#(V)
p16: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2))
p17: U41#(tt(),V1,V2) -> isList#(activate(V1))
p18: U41#(tt(),V1,V2) -> activate#(V1)
p19: U41#(tt(),V1,V2) -> activate#(V2)
p20: U42#(tt(),V2) -> U43#(isNeList(activate(V2)))
p21: U42#(tt(),V2) -> isNeList#(activate(V2))
p22: U42#(tt(),V2) -> activate#(V2)
p23: U51#(tt(),V1,V2) -> U52#(isNeList(activate(V1)),activate(V2))
p24: U51#(tt(),V1,V2) -> isNeList#(activate(V1))
p25: U51#(tt(),V1,V2) -> activate#(V1)
p26: U51#(tt(),V1,V2) -> activate#(V2)
p27: U52#(tt(),V2) -> U53#(isList(activate(V2)))
p28: U52#(tt(),V2) -> isList#(activate(V2))
p29: U52#(tt(),V2) -> activate#(V2)
p30: U61#(tt(),V) -> U62#(isQid(activate(V)))
p31: U61#(tt(),V) -> isQid#(activate(V))
p32: U61#(tt(),V) -> activate#(V)
p33: U71#(tt(),V) -> U72#(isNePal(activate(V)))
p34: U71#(tt(),V) -> isNePal#(activate(V))
p35: U71#(tt(),V) -> activate#(V)
p36: and#(tt(),X) -> activate#(X)
p37: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V))
p38: isList#(V) -> isPalListKind#(activate(V))
p39: isList#(V) -> activate#(V)
p40: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p41: isList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p42: isList#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p43: isList#(n____(V1,V2)) -> activate#(V1)
p44: isList#(n____(V1,V2)) -> activate#(V2)
p45: isNeList#(V) -> U31#(isPalListKind(activate(V)),activate(V))
p46: isNeList#(V) -> isPalListKind#(activate(V))
p47: isNeList#(V) -> activate#(V)
p48: isNeList#(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p49: isNeList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p50: isNeList#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p51: isNeList#(n____(V1,V2)) -> activate#(V1)
p52: isNeList#(n____(V1,V2)) -> activate#(V2)
p53: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p54: isNeList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p55: isNeList#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p56: isNeList#(n____(V1,V2)) -> activate#(V1)
p57: isNeList#(n____(V1,V2)) -> activate#(V2)
p58: isNePal#(V) -> U61#(isPalListKind(activate(V)),activate(V))
p59: isNePal#(V) -> isPalListKind#(activate(V))
p60: isNePal#(V) -> activate#(V)
p61: isNePal#(n____(I,__(P,I))) -> and#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P))))
p62: isNePal#(n____(I,__(P,I))) -> and#(isQid(activate(I)),n__isPalListKind(activate(I)))
p63: isNePal#(n____(I,__(P,I))) -> isQid#(activate(I))
p64: isNePal#(n____(I,__(P,I))) -> activate#(I)
p65: isNePal#(n____(I,__(P,I))) -> isPal#(activate(P))
p66: isNePal#(n____(I,__(P,I))) -> activate#(P)
p67: isPal#(V) -> U71#(isPalListKind(activate(V)),activate(V))
p68: isPal#(V) -> isPalListKind#(activate(V))
p69: isPal#(V) -> activate#(V)
p70: isPalListKind#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p71: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p72: isPalListKind#(n____(V1,V2)) -> activate#(V1)
p73: isPalListKind#(n____(V1,V2)) -> activate#(V2)
p74: activate#(n__nil()) -> nil#()
p75: activate#(n____(X1,X2)) -> __#(X1,X2)
p76: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p77: activate#(n__and(X1,X2)) -> and#(X1,X2)
p78: activate#(n__a()) -> a#()
p79: activate#(n__e()) -> e#()
p80: activate#(n__i()) -> i#()
p81: activate#(n__o()) -> o#()
p82: activate#(n__u()) -> u#()

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: a() -> n__a()
r49: e() -> n__e()
r50: i() -> n__i()
r51: o() -> n__o()
r52: u() -> n__u()
r53: activate(n__nil()) -> nil()
r54: activate(n____(X1,X2)) -> __(X1,X2)
r55: activate(n__isPalListKind(X)) -> isPalListKind(X)
r56: activate(n__and(X1,X2)) -> and(X1,X2)
r57: activate(n__a()) -> a()
r58: activate(n__e()) -> e()
r59: activate(n__i()) -> i()
r60: activate(n__o()) -> o()
r61: activate(n__u()) -> u()
r62: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p34, p65, p67}
  {p4, p6, p7, p11, p16, p17, p21, p23, p24, p28, p37, p40, p48, p53}
  {p36, p70, p71, p72, p73, p76, p77}
  {p1, p2}


-- Reduction pair.

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

p1: isNePal#(n____(I,__(P,I))) -> isPal#(activate(P))
p2: isPal#(V) -> U71#(isPalListKind(activate(V)),activate(V))
p3: U71#(tt(),V) -> isNePal#(activate(V))

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: a() -> n__a()
r49: e() -> n__e()
r50: i() -> n__i()
r51: o() -> n__o()
r52: u() -> n__u()
r53: activate(n__nil()) -> nil()
r54: activate(n____(X1,X2)) -> __(X1,X2)
r55: activate(n__isPalListKind(X)) -> isPalListKind(X)
r56: activate(n__and(X1,X2)) -> and(X1,X2)
r57: activate(n__a()) -> a()
r58: activate(n__e()) -> e()
r59: activate(n__i()) -> i()
r60: activate(n__o()) -> o()
r61: activate(n__u()) -> u()
r62: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r21, r32, r33, r34, r35, r36, r37, r38, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^4
    order: lexicographic order
    interpretations:
      isNePal#_A(x1) = ((1,0,0,0),(0,1,0,0),(0,1,1,0),(1,1,1,0)) x1 + (0,0,0,5)
      n_____A(x1,x2) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(0,1,0,0)) x1 + ((1,0,0,0),(1,0,0,0),(1,1,0,0),(1,0,1,0)) x2 + (10,4,1,5)
      ___A(x1,x2) = ((1,0,0,0),(1,1,0,0),(0,1,1,0),(0,1,0,0)) x1 + ((1,0,0,0),(0,1,0,0),(1,1,0,0),(0,0,0,0)) x2 + (11,2,4,4)
      isPal#_A(x1) = ((1,0,0,0),(0,1,0,0),(0,1,1,0),(0,0,0,0)) x1 + (6,8,21,0)
      activate_A(x1) = x1 + (2,7,2,3)
      U71#_A(x1,x2) = ((0,0,0,0),(1,0,0,0),(0,0,0,0),(1,1,0,0)) x1 + ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,0,0)) x2 + (3,4,0,0)
      isPalListKind_A(x1) = ((1,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + (2,15,15,4)
      tt_A() = (1,14,15,6)
      nil_A() = (4,9,0,1)
      and_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(0,0,0,0),(0,0,0,0)) x1 + ((1,0,0,0),(0,1,0,0),(0,1,0,0),(0,0,0,0)) x2 + (3,0,4,4)
      n__nil_A() = (3,1,1,2)
      n__and_A(x1,x2) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(0,0,0,0)) x1 + ((1,0,0,0),(1,1,0,0),(1,1,0,0),(0,0,0,0)) x2 + (2,1,1,5)
      a_A() = (4,9,4,4)
      n__a_A() = (3,1,1,1)
      e_A() = (4,0,0,0)
      n__e_A() = (3,1,1,1)
      i_A() = (4,0,4,4)
      n__i_A() = (3,1,1,5)
      o_A() = (4,9,0,4)
      n__o_A() = (3,1,1,5)
      u_A() = (2,2,2,2)
      n__u_A() = (1,1,1,1)
      n__isPalListKind_A(x1) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(0,0,0,0)) x1 + (1,1,1,1)

The next rules are strictly ordered:

  p1, p2, p3

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: U52#(tt(),V2) -> isList#(activate(V2))
p2: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p3: U21#(tt(),V1,V2) -> isList#(activate(V1))
p4: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V))
p5: U11#(tt(),V) -> isNeList#(activate(V))
p6: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p7: U51#(tt(),V1,V2) -> isNeList#(activate(V1))
p8: isNeList#(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p9: U41#(tt(),V1,V2) -> isList#(activate(V1))
p10: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2))
p11: U42#(tt(),V2) -> isNeList#(activate(V2))
p12: U51#(tt(),V1,V2) -> U52#(isNeList(activate(V1)),activate(V2))
p13: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2))
p14: U22#(tt(),V2) -> isList#(activate(V2))

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: a() -> n__a()
r49: e() -> n__e()
r50: i() -> n__i()
r51: o() -> n__o()
r52: u() -> n__u()
r53: activate(n__nil()) -> nil()
r54: activate(n____(X1,X2)) -> __(X1,X2)
r55: activate(n__isPalListKind(X)) -> isPalListKind(X)
r56: activate(n__and(X1,X2)) -> and(X1,X2)
r57: activate(n__a()) -> a()
r58: activate(n__e()) -> e()
r59: activate(n__i()) -> i()
r60: activate(n__o()) -> o()
r61: activate(n__u()) -> u()
r62: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r21, r22, r23, r24, r25, r26, r27, 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

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^4
    order: lexicographic order
    interpretations:
      U52#_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + ((1,0,0,0),(0,1,0,0),(0,1,0,0),(1,1,1,0)) x2 + (38,0,0,5)
      tt_A() = (15,3,55,16)
      isList#_A(x1) = ((1,0,0,0),(1,1,0,0),(0,1,0,0),(1,1,0,0)) x1 + (36,38,41,39)
      activate_A(x1) = ((1,0,0,0),(1,1,0,0),(0,0,1,0),(0,0,1,0)) x1 + (16,16,4,57)
      n_____A(x1,x2) = ((1,0,0,0),(1,1,0,0),(1,0,0,0),(0,1,1,0)) x1 + ((1,0,0,0),(0,1,0,0),(1,0,0,0),(0,0,0,0)) x2 + (122,3,0,88)
      U21#_A(x1,x2,x3) = ((1,0,0,0),(0,1,0,0),(0,0,0,0),(1,0,1,0)) x1 + x2 + ((1,0,0,0),(0,1,0,0),(0,1,1,0),(1,0,1,1)) x3 + (55,66,25,0)
      and_A(x1,x2) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0)) x1 + ((1,0,0,0),(1,1,0,0),(0,0,0,0),(0,0,1,0)) x2 + (16,60,0,0)
      isPalListKind_A(x1) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0)) x1 + (17,1,15,77)
      n__isPalListKind_A(x1) = ((0,0,0,0),(1,0,0,0),(1,1,0,0),(1,1,1,0)) x1 + (1,2,16,78)
      U11#_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,0,0,0),(1,0,0,0)) x1 + x2 + (2,0,0,23)
      isNeList#_A(x1) = ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,1,0,1)) x1 + (0,0,53,0)
      U51#_A(x1,x2,x3) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(0,0,0,0)) x2 + ((1,0,0,0),(1,0,0,0),(1,0,0,0),(1,0,0,0)) x3 + (89,0,0,74)
      U41#_A(x1,x2,x3) = x1 + ((1,0,0,0),(1,1,0,0),(0,0,1,0),(1,0,0,1)) x2 + ((1,0,0,0),(0,1,0,0),(1,1,0,0),(1,1,1,0)) x3 + (38,66,1,56)
      U42#_A(x1,x2) = ((0,0,0,0),(1,0,0,0),(0,0,0,0),(0,0,0,0)) x1 + ((1,0,0,0),(1,0,0,0),(0,1,0,0),(1,0,1,0)) x2 + (17,0,0,0)
      isList_A(x1) = ((1,0,0,0),(0,0,0,0),(0,1,0,0),(1,0,0,0)) x1 + (36,4,49,0)
      isNeList_A(x1) = ((1,0,0,0),(1,0,0,0),(0,1,0,0),(1,0,0,0)) x1 + (18,0,77,34)
      U22#_A(x1,x2) = ((1,0,0,0),(1,1,0,0),(0,1,1,0),(0,1,1,0)) x2 + (53,38,6,0)
      U23_A(x1) = (16,0,0,1)
      U43_A(x1) = ((0,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + (16,1,53,0)
      U53_A(x1) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,1,0,0)) x1 + (16,0,0,14)
      U12_A(x1) = (16,4,56,1)
      U22_A(x1,x2) = x1 + ((0,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x2 + (2,1,1,0)
      U32_A(x1) = ((0,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + (16,2,56,0)
      U42_A(x1,x2) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,1,0,0)) x2 + (17,0,51,33)
      U52_A(x1,x2) = x1 + ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,0,0)) x2 + (2,76,0,1)
      isQid_A(x1) = ((0,0,0,0),(1,0,0,0),(0,0,0,0),(0,1,0,0)) x1 + (16,0,56,0)
      n__a_A() = (1,40,1,60)
      n__e_A() = (1,40,1,1)
      n__i_A() = (4,40,1,1)
      n__o_A() = (2,40,1,1)
      n__u_A() = (4,40,1,1)
      ___A(x1,x2) = ((1,0,0,0),(1,0,0,0),(0,0,0,0),(0,1,0,0)) x1 + x2 + (123,1,0,58)
      nil_A() = (2,2,1,60)
      U11_A(x1,x2) = x1 + ((1,0,0,0),(0,1,0,0),(0,1,0,0),(0,1,0,0)) x2 + (2,0,0,0)
      U21_A(x1,x2,x3) = ((1,0,0,0),(0,1,0,0),(0,1,0,0),(0,0,1,0)) x2 + ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,0,0)) x3 + (55,2,0,119)
      U31_A(x1,x2) = ((0,0,0,0),(0,0,0,0),(1,0,0,0),(0,0,0,0)) x1 + ((0,0,0,0),(0,0,0,0),(1,0,0,0),(0,1,0,0)) x2 + (17,1,42,17)
      U41_A(x1,x2,x3) = ((0,0,0,0),(1,0,0,0),(0,0,0,0),(0,1,0,0)) x1 + ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,0,0)) x2 + ((0,0,0,0),(1,0,0,0),(0,0,0,0),(0,1,0,0)) x3 + (18,2,65,45)
      U51_A(x1,x2,x3) = x2 + ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,1,0,0)) x3 + (123,107,65,0)
      n__nil_A() = (1,4,2,61)
      a_A() = (2,39,0,59)
      e_A() = (2,0,6,59)
      i_A() = (5,39,0,59)
      o_A() = (3,0,1,0)
      u_A() = (5,61,0,59)
      n__and_A(x1,x2) = ((0,0,0,0),(1,0,0,0),(1,1,0,0),(0,1,1,0)) x1 + ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,0,0)) x2 + (1,61,1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14

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: isPalListKind#(n____(V1,V2)) -> activate#(V2)
p2: activate#(n__and(X1,X2)) -> and#(X1,X2)
p3: and#(tt(),X) -> activate#(X)
p4: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p5: isPalListKind#(n____(V1,V2)) -> activate#(V1)
p6: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p7: isPalListKind#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: a() -> n__a()
r49: e() -> n__e()
r50: i() -> n__i()
r51: o() -> n__o()
r52: u() -> n__u()
r53: activate(n__nil()) -> nil()
r54: activate(n____(X1,X2)) -> __(X1,X2)
r55: activate(n__isPalListKind(X)) -> isPalListKind(X)
r56: activate(n__and(X1,X2)) -> and(X1,X2)
r57: activate(n__a()) -> a()
r58: activate(n__e()) -> e()
r59: activate(n__i()) -> i()
r60: activate(n__o()) -> o()
r61: activate(n__u()) -> u()
r62: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r21, r32, r33, r34, r35, r36, r37, r38, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^4
    order: lexicographic order
    interpretations:
      isPalListKind#_A(x1) = ((1,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0)) x1 + (0,0,17,3)
      n_____A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,0,0,0),(1,0,0,0)) x1 + ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,1,0,0)) x2 + (8,1,5,1)
      activate#_A(x1) = ((1,0,0,0),(0,1,0,0),(1,1,1,0),(1,0,1,0)) x1 + (0,9,19,0)
      n__and_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,1,0,0),(1,0,0,0)) x1 + ((1,0,0,0),(1,1,0,0),(0,0,1,0),(1,0,1,1)) x2 + (1,1,3,1)
      and#_A(x1,x2) = ((1,0,0,0),(0,0,0,0),(0,1,0,0),(1,1,0,0)) x1 + ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,1,0)) x2 + (0,7,0,0)
      tt_A() = (1,11,10,11)
      n__isPalListKind_A(x1) = ((1,0,0,0),(0,1,0,0),(1,0,0,0),(0,0,1,0)) x1 + (1,5,1,5)
      activate_A(x1) = x1 + (2,3,3,3)
      isPalListKind_A(x1) = ((1,0,0,0),(1,1,0,0),(0,0,1,0),(1,0,0,1)) x1 + (2,4,4,4)
      ___A(x1,x2) = ((1,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0)) x1 + x2 + (9,2,4,4)
      nil_A() = (2,1,4,4)
      and_A(x1,x2) = x1 + x2 + (2,0,2,0)
      n__nil_A() = (1,7,7,7)
      a_A() = (8,2,2,2)
      n__a_A() = (7,1,1,1)
      e_A() = (2,4,4,4)
      n__e_A() = (1,5,5,5)
      i_A() = (2,4,8,0)
      n__i_A() = (1,1,7,7)
      o_A() = (2,4,4,4)
      n__o_A() = (1,1,1,7)
      u_A() = (8,4,0,2)
      n__u_A() = (7,5,7,1)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7

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: __#(__(X,Y),Z) -> __#(X,__(Y,Z))
p2: __#(__(X,Y),Z) -> __#(Y,Z)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,__(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: a() -> n__a()
r49: e() -> n__e()
r50: i() -> n__i()
r51: o() -> n__o()
r52: u() -> n__u()
r53: activate(n__nil()) -> nil()
r54: activate(n____(X1,X2)) -> __(X1,X2)
r55: activate(n__isPalListKind(X)) -> isPalListKind(X)
r56: activate(n__and(X1,X2)) -> and(X1,X2)
r57: activate(n__a()) -> a()
r58: activate(n__e()) -> e()
r59: activate(n__i()) -> i()
r60: activate(n__o()) -> o()
r61: activate(n__u()) -> u()
r62: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r45

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^4
    order: lexicographic order
    interpretations:
      __#_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,1,1,1)) x1 + ((0,0,0,0),(1,0,0,0),(0,1,0,0),(1,0,1,0)) x2
      ___A(x1,x2) = ((1,0,0,0),(1,1,0,0),(1,0,1,0),(0,0,1,1)) x1 + ((1,0,0,0),(0,1,0,0),(1,0,0,0),(1,0,1,0)) x2 + (3,2,3,2)
      nil_A() = (1,1,1,1)
      n_____A(x1,x2) = (0,0,0,0)

The next rules are strictly ordered:

  p1, p2

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