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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        isNePal#_A(x1) = ((1,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + (0,1,2,0)
        n_____A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,0,0,0),(0,1,0,0)) x1 + ((1,0,0,0),(0,0,0,0),(1,1,0,0),(0,0,1,0)) x2 + (10,3,0,0)
        ___A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,0,0,0),(0,1,0,0)) x1 + ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,0,0)) x2 + (11,2,1,0)
        isPal#_A(x1) = ((1,0,0,0),(0,0,0,0),(0,0,0,0),(0,1,0,0)) x1 + (8,0,1,21)
        activate_A(x1) = ((1,0,0,0),(0,0,0,0),(0,0,0,0),(0,1,0,0)) x1 + (3,1,3,0)
        U71#_A(x1,x2) = x2 + (4,2,0,0)
        isPalListKind_A(x1) = x1 + (2,2,0,4)
        tt_A() = (1,5,3,5)
        nil_A() = (1,2,4,2)
        and_A(x1,x2) = x1 + ((1,0,0,0),(0,0,0,0),(0,0,0,0),(0,1,0,0)) x2 + (3,0,0,1)
        n__nil_A() = (0,0,1,1)
        n__and_A(x1,x2) = x1 + ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,1,0)) x2 + (1,0,0,2)
        a_A() = (1,0,4,1)
        n__a_A() = (0,0,1,0)
        e_A() = (1,3,2,1)
        n__e_A() = (0,2,1,0)
        i_A() = (1,0,0,1)
        n__i_A() = (0,0,1,2)
        o_A() = (1,2,1,0)
        n__o_A() = (0,1,0,0)
        u_A() = (1,2,0,0)
        n__u_A() = (0,4,1,1)
        n__isPalListKind_A(x1) = ((1,0,0,0),(0,1,0,0),(0,1,0,0),(0,0,0,0)) x1 + (0,1,1,5)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        e > i > nil > n__u > and > n__isPalListKind > u > o > a > n__a > U71# > n__nil > isNePal# > isPal# > n____ > n__o > n__i > n__and > n__e > activate > isPalListKind > tt > __
      
      argument filter:
    
        pi(isNePal#) = []
        pi(n____) = []
        pi(__) = []
        pi(isPal#) = []
        pi(activate) = []
        pi(U71#) = []
        pi(isPalListKind) = []
        pi(tt) = []
        pi(nil) = []
        pi(and) = []
        pi(n__nil) = []
        pi(n__and) = [1]
        pi(a) = []
        pi(n__a) = []
        pi(e) = []
        pi(n__e) = []
        pi(i) = []
        pi(n__i) = []
        pi(o) = []
        pi(n__o) = []
        pi(u) = []
        pi(n__u) = []
        pi(n__isPalListKind) = []
    

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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        U52#_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,1,1,0),(1,0,1,0)) x1 + ((1,0,0,0),(0,1,0,0),(0,1,1,0),(0,1,1,1)) x2 + (22,0,0,0)
        tt_A() = (2,2,85,3)
        isList#_A(x1) = ((1,0,0,0),(0,1,0,0),(0,1,1,0),(1,1,0,1)) x1 + (16,45,121,47)
        activate_A(x1) = ((1,0,0,0),(1,1,0,0),(0,0,1,0),(0,0,0,1)) x1 + (7,4,32,30)
        n_____A(x1,x2) = ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,0,0)) x1 + x2 + (65,0,74,1)
        U21#_A(x1,x2,x3) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(1,0,0,0)) x1 + ((1,0,0,0),(0,0,0,0),(1,1,0,0),(0,0,0,0)) x2 + ((1,0,0,0),(0,1,0,0),(1,0,0,0),(1,0,0,0)) x3 + (65,53,156,85)
        and_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,1,1,0)) x1 + ((1,0,0,0),(1,0,0,0),(1,0,0,0),(1,1,0,0)) x2 + (6,3,48,31)
        isPalListKind_A(x1) = ((1,0,0,0),(1,0,0,0),(1,1,0,0),(1,0,1,0)) x1 + (2,3,1,0)
        n__isPalListKind_A(x1) = x1
        U11#_A(x1,x2) = ((0,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + ((1,0,0,0),(0,0,0,0),(1,0,0,0),(0,0,0,0)) x2 + (8,46,115,0)
        isNeList#_A(x1) = ((1,0,0,0),(1,0,0,0),(1,1,0,0),(0,1,1,0)) x1 + (0,0,105,14)
        U51#_A(x1,x2,x3) = ((0,0,0,0),(1,0,0,0),(1,1,0,0),(0,1,1,0)) x1 + ((1,0,0,0),(1,0,0,0),(1,0,0,0),(0,0,0,0)) x2 + x3 + (50,6,111,0)
        U41#_A(x1,x2,x3) = ((1,0,0,0),(1,1,0,0),(1,1,0,0),(0,0,0,0)) x2 + ((1,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0)) x3 + (49,48,156,57)
        U42#_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,1,1,0),(0,0,0,0)) x1 + ((1,0,0,0),(1,0,0,0),(0,0,0,0),(1,0,0,0)) x2 + (6,24,28,51)
        isList_A(x1) = ((1,0,0,0),(0,1,0,0),(1,0,1,0),(1,0,0,1)) x1 + (28,14,72,0)
        isNeList_A(x1) = ((1,0,0,0),(1,1,0,0),(0,1,1,0),(0,1,0,0)) x1 + (13,15,16,59)
        U22#_A(x1,x2) = ((1,0,0,0),(0,0,0,0),(1,0,0,0),(1,0,0,0)) x1 + ((1,0,0,0),(0,1,0,0),(0,1,1,0),(0,0,0,1)) x2 + (22,50,0,23)
        U23_A(x1) = ((1,0,0,0),(0,1,0,0),(0,1,1,0),(1,1,1,1)) x1 + (1,1,0,0)
        U43_A(x1) = ((0,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + (3,3,84,0)
        U53_A(x1) = ((1,0,0,0),(0,0,0,0),(1,0,0,0),(1,0,0,0)) x1 + (1,2,83,1)
        U12_A(x1) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(1,0,1,0)) x1 + (3,1,84,16)
        U22_A(x1,x2) = ((1,0,0,0),(1,1,0,0),(0,0,0,0),(0,0,0,0)) x1 + ((1,0,0,0),(1,1,0,0),(0,0,0,0),(0,0,0,0)) x2 + (35,16,130,0)
        U32_A(x1) = ((1,0,0,0),(1,1,0,0),(0,0,1,0),(0,0,0,1)) x1 + (1,0,1,1)
        U42_A(x1,x2) = ((0,0,0,0),(1,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + ((0,0,0,0),(1,0,0,0),(1,0,0,0),(1,0,0,0)) x2 + (4,2,84,19)
        U52_A(x1,x2) = ((1,0,0,0),(1,1,0,0),(0,0,1,0),(0,1,0,0)) x1 + ((1,0,0,0),(1,0,0,0),(1,0,0,0),(0,1,0,0)) x2 + (35,0,32,33)
        isQid_A(x1) = (3,3,86,4)
        n__a_A() = (85,0,0,1)
        n__e_A() = (0,1,0,0)
        n__i_A() = (4,0,0,1)
        n__o_A() = (0,1,0,0)
        n__u_A() = (0,85,4,1)
        ___A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,1,0,0),(0,0,1,0)) x1 + ((1,0,0,0),(0,1,0,0),(1,1,1,0),(1,1,0,1)) x2 + (71,70,107,32)
        nil_A() = (86,0,33,32)
        U11_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(1,1,1,0),(0,1,1,0)) x1 + ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,0,0)) x2 + (2,0,0,0)
        U21_A(x1,x2,x3) = ((0,0,0,0),(0,0,0,0),(1,0,0,0),(0,0,0,0)) x1 + ((1,0,0,0),(1,0,0,0),(0,0,0,0),(0,0,0,0)) x2 + ((1,0,0,0),(1,0,0,0),(1,0,0,0),(0,0,0,0)) x3 + (78,0,182,67)
        U31_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,1,1)) x1 + ((0,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x2 + (3,5,3,0)
        U41_A(x1,x2,x3) = ((1,0,0,0),(0,1,0,0),(0,0,0,0),(0,1,1,0)) x1 + ((0,0,0,0),(0,0,0,0),(0,0,0,0),(1,0,0,0)) x3 + (3,0,90,0)
        U51_A(x1,x2,x3) = x2 + x3 + (63,0,90,60)
        n__nil_A() = (85,0,0,1)
        a_A() = (86,1,33,32)
        e_A() = (1,6,1,1)
        i_A() = (5,9,1,2)
        o_A() = (1,1,0,0)
        u_A() = (1,90,37,32)
        n__and_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(0,0,0,0),(0,1,1,0)) x1 + ((1,0,0,0),(0,1,0,0),(0,0,0,0),(0,1,0,0)) x2
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        __ > U21 > isNeList > U42# > n____ > U51 > U52 > U22 > U51# > U21# > isNeList# > U41# > U22# > activate > isPalListKind > and > n__and > n__e > u > U31 > U41 > n__u > nil > o > n__o > U42 > U32 > U53 > isList > tt > U11 > a > U52# > isList# > U11# > n__isPalListKind > n__a > U23 > isQid > e > i > n__i > n__nil > U12 > U43
      
      argument filter:
    
        pi(U52#) = []
        pi(tt) = []
        pi(isList#) = 1
        pi(activate) = []
        pi(n____) = []
        pi(U21#) = []
        pi(and) = []
        pi(isPalListKind) = []
        pi(n__isPalListKind) = []
        pi(U11#) = []
        pi(isNeList#) = []
        pi(U51#) = []
        pi(U41#) = []
        pi(U42#) = []
        pi(isList) = []
        pi(isNeList) = []
        pi(U22#) = []
        pi(U23) = 1
        pi(U43) = []
        pi(U53) = []
        pi(U12) = []
        pi(U22) = []
        pi(U32) = 1
        pi(U42) = []
        pi(U52) = []
        pi(isQid) = []
        pi(n__a) = []
        pi(n__e) = []
        pi(n__i) = []
        pi(n__o) = []
        pi(n__u) = []
        pi(__) = []
        pi(nil) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = 1
        pi(U41) = []
        pi(U51) = []
        pi(n__nil) = []
        pi(a) = []
        pi(e) = []
        pi(i) = []
        pi(o) = []
        pi(u) = []
        pi(n__and) = []
    

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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        isPalListKind#_A(x1) = ((1,0,0,0),(0,0,0,0),(1,1,0,0),(0,1,1,0)) x1 + (8,2,27,10)
        n_____A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,0,0,0),(1,1,0,0)) x1 + ((1,0,0,0),(0,0,0,0),(0,0,0,0),(1,1,0,0)) x2 + (14,1,1,2)
        activate#_A(x1) = ((1,0,0,0),(0,1,0,0),(0,0,0,0),(1,0,1,0)) x1 + (1,1,26,1)
        n__and_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(1,1,1,0),(0,1,1,1)) x1 + ((1,0,0,0),(1,1,0,0),(0,0,1,0),(0,0,0,0)) x2 + (0,1,1,1)
        and#_A(x1,x2) = ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,0,0)) x1 + x2
        tt_A() = (2,25,7,9)
        n__isPalListKind_A(x1) = ((1,0,0,0),(0,1,0,0),(0,0,0,0),(1,0,0,0)) x1 + (8,24,0,1)
        activate_A(x1) = ((1,0,0,0),(1,0,0,0),(0,0,0,0),(0,0,0,0)) x1 + (2,14,3,3)
        isPalListKind_A(x1) = ((1,0,0,0),(0,1,0,0),(1,0,1,0),(1,0,0,1)) x1 + (9,23,4,4)
        ___A(x1,x2) = ((1,0,0,0),(1,1,0,0),(1,0,1,0),(0,0,0,1)) x1 + ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,1,0)) x2 + (15,1,4,1)
        nil_A() = (2,4,2,4)
        and_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),(1,0,0,0),(1,0,0,0)) x2 + (1,2,0,0)
        n__nil_A() = (1,3,1,1)
        a_A() = (1,15,5,7)
        n__a_A() = (0,16,4,6)
        e_A() = (1,15,4,0)
        n__e_A() = (0,3,1,1)
        i_A() = (1,0,2,2)
        n__i_A() = (0,1,1,1)
        o_A() = (1,0,0,0)
        n__o_A() = (0,3,4,6)
        u_A() = (4,4,2,4)
        n__u_A() = (3,3,1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        i > n__nil > __ > n____ > n__a > tt > isPalListKind# > activate > n__i > o > isPalListKind > n__e > a > and > e > n__o > nil > n__and > and# > u > n__u > n__isPalListKind > activate#
      
      argument filter:
    
        pi(isPalListKind#) = []
        pi(n____) = []
        pi(activate#) = []
        pi(n__and) = [1]
        pi(and#) = []
        pi(tt) = []
        pi(n__isPalListKind) = []
        pi(activate) = []
        pi(isPalListKind) = 1
        pi(__) = 1
        pi(nil) = []
        pi(and) = []
        pi(n__nil) = []
        pi(a) = []
        pi(n__a) = []
        pi(e) = []
        pi(n__e) = []
        pi(i) = []
        pi(n__i) = []
        pi(o) = []
        pi(n__o) = []
        pi(u) = []
        pi(n__u) = []
    

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:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        __#_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,0,0,0),(0,1,0,0)) x1 + ((0,0,0,0),(0,0,0,0),(1,0,0,0),(1,1,0,0)) x2
        ___A(x1,x2) = x1 + x2 + (1,0,1,1)
        nil_A() = (1,1,1,1)
        n_____A(x1,x2) = (0,0,0,0)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        __ > __# > n____ > nil
      
      argument filter:
    
        pi(__#) = []
        pi(__) = []
        pi(nil) = []
        pi(n____) = []
    

The next rules are strictly ordered:

  p1, p2

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