YES

We show the termination of the TRS R:

  __(__(X,Y),Z) -> __(X,__(Y,Z))
  __(X,nil()) -> X
  __(nil(),X) -> X
  U11(tt()) -> tt()
  U21(tt(),V2) -> U22(isList(activate(V2)))
  U22(tt()) -> tt()
  U31(tt()) -> tt()
  U41(tt(),V2) -> U42(isNeList(activate(V2)))
  U42(tt()) -> tt()
  U51(tt(),V2) -> U52(isList(activate(V2)))
  U52(tt()) -> tt()
  U61(tt()) -> tt()
  U71(tt(),P) -> U72(isPal(activate(P)))
  U72(tt()) -> tt()
  U81(tt()) -> tt()
  isList(V) -> U11(isNeList(activate(V)))
  isList(n__nil()) -> tt()
  isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2))
  isNeList(V) -> U31(isQid(activate(V)))
  isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2))
  isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2))
  isNePal(V) -> U61(isQid(activate(V)))
  isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P))
  isPal(V) -> U81(isNePal(activate(V)))
  isPal(n__nil()) -> tt()
  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)
  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__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: U21#(tt(),V2) -> U22#(isList(activate(V2)))
p4: U21#(tt(),V2) -> isList#(activate(V2))
p5: U21#(tt(),V2) -> activate#(V2)
p6: U41#(tt(),V2) -> U42#(isNeList(activate(V2)))
p7: U41#(tt(),V2) -> isNeList#(activate(V2))
p8: U41#(tt(),V2) -> activate#(V2)
p9: U51#(tt(),V2) -> U52#(isList(activate(V2)))
p10: U51#(tt(),V2) -> isList#(activate(V2))
p11: U51#(tt(),V2) -> activate#(V2)
p12: U71#(tt(),P) -> U72#(isPal(activate(P)))
p13: U71#(tt(),P) -> isPal#(activate(P))
p14: U71#(tt(),P) -> activate#(P)
p15: isList#(V) -> U11#(isNeList(activate(V)))
p16: isList#(V) -> isNeList#(activate(V))
p17: isList#(V) -> activate#(V)
p18: isList#(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2))
p19: isList#(n____(V1,V2)) -> isList#(activate(V1))
p20: isList#(n____(V1,V2)) -> activate#(V1)
p21: isList#(n____(V1,V2)) -> activate#(V2)
p22: isNeList#(V) -> U31#(isQid(activate(V)))
p23: isNeList#(V) -> isQid#(activate(V))
p24: isNeList#(V) -> activate#(V)
p25: isNeList#(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2))
p26: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p27: isNeList#(n____(V1,V2)) -> activate#(V1)
p28: isNeList#(n____(V1,V2)) -> activate#(V2)
p29: isNeList#(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2))
p30: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p31: isNeList#(n____(V1,V2)) -> activate#(V1)
p32: isNeList#(n____(V1,V2)) -> activate#(V2)
p33: isNePal#(V) -> U61#(isQid(activate(V)))
p34: isNePal#(V) -> isQid#(activate(V))
p35: isNePal#(V) -> activate#(V)
p36: isNePal#(n____(I,__(P,I))) -> U71#(isQid(activate(I)),activate(P))
p37: isNePal#(n____(I,__(P,I))) -> isQid#(activate(I))
p38: isNePal#(n____(I,__(P,I))) -> activate#(I)
p39: isNePal#(n____(I,__(P,I))) -> activate#(P)
p40: isPal#(V) -> U81#(isNePal(activate(V)))
p41: isPal#(V) -> isNePal#(activate(V))
p42: isPal#(V) -> activate#(V)
p43: activate#(n__nil()) -> nil#()
p44: activate#(n____(X1,X2)) -> __#(X1,X2)
p45: activate#(n__a()) -> a#()
p46: activate#(n__e()) -> e#()
p47: activate#(n__i()) -> i#()
p48: activate#(n__o()) -> o#()
p49: 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()) -> tt()
r5: U21(tt(),V2) -> U22(isList(activate(V2)))
r6: U22(tt()) -> tt()
r7: U31(tt()) -> tt()
r8: U41(tt(),V2) -> U42(isNeList(activate(V2)))
r9: U42(tt()) -> tt()
r10: U51(tt(),V2) -> U52(isList(activate(V2)))
r11: U52(tt()) -> tt()
r12: U61(tt()) -> tt()
r13: U71(tt(),P) -> U72(isPal(activate(P)))
r14: U72(tt()) -> tt()
r15: U81(tt()) -> tt()
r16: isList(V) -> U11(isNeList(activate(V)))
r17: isList(n__nil()) -> tt()
r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2))
r19: isNeList(V) -> U31(isQid(activate(V)))
r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2))
r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2))
r22: isNePal(V) -> U61(isQid(activate(V)))
r23: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P))
r24: isPal(V) -> U81(isNePal(activate(V)))
r25: isPal(n__nil()) -> tt()
r26: isQid(n__a()) -> tt()
r27: isQid(n__e()) -> tt()
r28: isQid(n__i()) -> tt()
r29: isQid(n__o()) -> tt()
r30: isQid(n__u()) -> tt()
r31: nil() -> n__nil()
r32: __(X1,X2) -> n____(X1,X2)
r33: a() -> n__a()
r34: e() -> n__e()
r35: i() -> n__i()
r36: o() -> n__o()
r37: u() -> n__u()
r38: activate(n__nil()) -> nil()
r39: activate(n____(X1,X2)) -> __(X1,X2)
r40: activate(n__a()) -> a()
r41: activate(n__e()) -> e()
r42: activate(n__i()) -> i()
r43: activate(n__o()) -> o()
r44: activate(n__u()) -> u()
r45: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p13, p36, p41}
  {p4, p7, p10, p16, p18, p19, p25, p26, p29, p30}
  {p1, p2}


-- Reduction pair.

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

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

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt()) -> tt()
r5: U21(tt(),V2) -> U22(isList(activate(V2)))
r6: U22(tt()) -> tt()
r7: U31(tt()) -> tt()
r8: U41(tt(),V2) -> U42(isNeList(activate(V2)))
r9: U42(tt()) -> tt()
r10: U51(tt(),V2) -> U52(isList(activate(V2)))
r11: U52(tt()) -> tt()
r12: U61(tt()) -> tt()
r13: U71(tt(),P) -> U72(isPal(activate(P)))
r14: U72(tt()) -> tt()
r15: U81(tt()) -> tt()
r16: isList(V) -> U11(isNeList(activate(V)))
r17: isList(n__nil()) -> tt()
r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2))
r19: isNeList(V) -> U31(isQid(activate(V)))
r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2))
r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2))
r22: isNePal(V) -> U61(isQid(activate(V)))
r23: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P))
r24: isPal(V) -> U81(isNePal(activate(V)))
r25: isPal(n__nil()) -> tt()
r26: isQid(n__a()) -> tt()
r27: isQid(n__e()) -> tt()
r28: isQid(n__i()) -> tt()
r29: isQid(n__o()) -> tt()
r30: isQid(n__u()) -> tt()
r31: nil() -> n__nil()
r32: __(X1,X2) -> n____(X1,X2)
r33: a() -> n__a()
r34: e() -> n__e()
r35: i() -> n__i()
r36: o() -> n__o()
r37: u() -> n__u()
r38: activate(n__nil()) -> nil()
r39: activate(n____(X1,X2)) -> __(X1,X2)
r40: activate(n__a()) -> a()
r41: activate(n__e()) -> e()
r42: activate(n__i()) -> i()
r43: activate(n__o()) -> o()
r44: activate(n__u()) -> u()
r45: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: lexicographic order
    interpretations:
      isPal#_A(x1) = ((0,0),(1,0)) x1 + (0,8)
      isNePal#_A(x1) = ((0,0),(1,0)) x1 + (0,5)
      activate_A(x1) = x1 + (2,2)
      n_____A(x1,x2) = x1 + x2 + (0,1)
      ___A(x1,x2) = x1 + x2 + (0,3)
      U71#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2
      isQid_A(x1) = ((1,0),(1,0)) x1
      tt_A() = (11,13)
      nil_A() = (2,1)
      n__nil_A() = (1,2)
      a_A() = (13,3)
      n__a_A() = (12,1)
      e_A() = (15,3)
      n__e_A() = (14,4)
      i_A() = (15,3)
      n__i_A() = (14,1)
      o_A() = (13,3)
      n__o_A() = (12,4)
      u_A() = (13,3)
      n__u_A() = (12,4)

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: isNeList#(n____(V1,V2)) -> U51#(isNeList(activate(V1)),activate(V2))
p2: U51#(tt(),V2) -> isList#(activate(V2))
p3: isList#(n____(V1,V2)) -> isList#(activate(V1))
p4: isList#(n____(V1,V2)) -> U21#(isList(activate(V1)),activate(V2))
p5: U21#(tt(),V2) -> isList#(activate(V2))
p6: isList#(V) -> isNeList#(activate(V))
p7: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p8: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p9: isNeList#(n____(V1,V2)) -> U41#(isList(activate(V1)),activate(V2))
p10: U41#(tt(),V2) -> isNeList#(activate(V2))

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt()) -> tt()
r5: U21(tt(),V2) -> U22(isList(activate(V2)))
r6: U22(tt()) -> tt()
r7: U31(tt()) -> tt()
r8: U41(tt(),V2) -> U42(isNeList(activate(V2)))
r9: U42(tt()) -> tt()
r10: U51(tt(),V2) -> U52(isList(activate(V2)))
r11: U52(tt()) -> tt()
r12: U61(tt()) -> tt()
r13: U71(tt(),P) -> U72(isPal(activate(P)))
r14: U72(tt()) -> tt()
r15: U81(tt()) -> tt()
r16: isList(V) -> U11(isNeList(activate(V)))
r17: isList(n__nil()) -> tt()
r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2))
r19: isNeList(V) -> U31(isQid(activate(V)))
r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2))
r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2))
r22: isNePal(V) -> U61(isQid(activate(V)))
r23: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P))
r24: isPal(V) -> U81(isNePal(activate(V)))
r25: isPal(n__nil()) -> tt()
r26: isQid(n__a()) -> tt()
r27: isQid(n__e()) -> tt()
r28: isQid(n__i()) -> tt()
r29: isQid(n__o()) -> tt()
r30: isQid(n__u()) -> tt()
r31: nil() -> n__nil()
r32: __(X1,X2) -> n____(X1,X2)
r33: a() -> n__a()
r34: e() -> n__e()
r35: i() -> n__i()
r36: o() -> n__o()
r37: u() -> n__u()
r38: activate(n__nil()) -> nil()
r39: activate(n____(X1,X2)) -> __(X1,X2)
r40: activate(n__a()) -> a()
r41: activate(n__e()) -> e()
r42: activate(n__i()) -> i()
r43: activate(n__o()) -> o()
r44: activate(n__u()) -> u()
r45: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r16, r17, r18, r19, r20, r21, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: lexicographic order
    interpretations:
      isNeList#_A(x1) = x1 + (4,21)
      n_____A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (9,14)
      U51#_A(x1,x2) = ((1,0),(1,1)) x1 + x2
      isNeList_A(x1) = x1 + (4,16)
      activate_A(x1) = x1 + (2,12)
      tt_A() = (12,13)
      isList#_A(x1) = x1 + (7,0)
      U21#_A(x1,x2) = x1 + x2 + (0,13)
      isList_A(x1) = ((1,0),(1,0)) x1 + (8,0)
      U41#_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (0,21)
      U22_A(x1) = x1 + (1,11)
      U42_A(x1) = x1 + (5,12)
      U52_A(x1) = x1 + (1,14)
      ___A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (10,11)
      nil_A() = (6,13)
      U11_A(x1) = ((1,0),(1,0)) x1 + (1,0)
      U21_A(x1,x2) = x1 + x2 + (0,10)
      U31_A(x1) = ((1,0),(1,0)) x1 + (1,0)
      U41_A(x1,x2) = x1 + x2 + (0,11)
      U51_A(x1,x2) = x1 + x2 + (4,15)
      isQid_A(x1) = x1 + (0,1)
      n__a_A() = (13,1)
      n__e_A() = (13,1)
      n__i_A() = (13,1)
      n__o_A() = (13,14)
      n__u_A() = (13,14)
      n__nil_A() = (5,14)
      a_A() = (14,0)
      e_A() = (14,0)
      i_A() = (14,2)
      o_A() = (14,13)
      u_A() = (14,13)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9, p10

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

-- Reduction pair.

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

p1: __#(__(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()) -> tt()
r5: U21(tt(),V2) -> U22(isList(activate(V2)))
r6: U22(tt()) -> tt()
r7: U31(tt()) -> tt()
r8: U41(tt(),V2) -> U42(isNeList(activate(V2)))
r9: U42(tt()) -> tt()
r10: U51(tt(),V2) -> U52(isList(activate(V2)))
r11: U52(tt()) -> tt()
r12: U61(tt()) -> tt()
r13: U71(tt(),P) -> U72(isPal(activate(P)))
r14: U72(tt()) -> tt()
r15: U81(tt()) -> tt()
r16: isList(V) -> U11(isNeList(activate(V)))
r17: isList(n__nil()) -> tt()
r18: isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2))
r19: isNeList(V) -> U31(isQid(activate(V)))
r20: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2))
r21: isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2))
r22: isNePal(V) -> U61(isQid(activate(V)))
r23: isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P))
r24: isPal(V) -> U81(isNePal(activate(V)))
r25: isPal(n__nil()) -> tt()
r26: isQid(n__a()) -> tt()
r27: isQid(n__e()) -> tt()
r28: isQid(n__i()) -> tt()
r29: isQid(n__o()) -> tt()
r30: isQid(n__u()) -> tt()
r31: nil() -> n__nil()
r32: __(X1,X2) -> n____(X1,X2)
r33: a() -> n__a()
r34: e() -> n__e()
r35: i() -> n__i()
r36: o() -> n__o()
r37: u() -> n__u()
r38: activate(n__nil()) -> nil()
r39: activate(n____(X1,X2)) -> __(X1,X2)
r40: activate(n__a()) -> a()
r41: activate(n__e()) -> e()
r42: activate(n__i()) -> i()
r43: activate(n__o()) -> o()
r44: activate(n__u()) -> u()
r45: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r32

Take the reduction pair:

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

The next rules are strictly ordered:

  p1, p2

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