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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          isPal#_A(x1) = max{72, x1 + 20}
          isNePal#_A(x1) = max{19, x1 + 5}
          activate_A(x1) = max{52, x1 + 14}
          n_____A(x1,x2) = max{70, x1 + 46, x2}
          ___A(x1,x2) = max{84, x1 + 59, x2}
          U71#_A(x1,x2) = max{x1 + 31, x2 + 34}
          isQid_A(x1) = max{40, x1 + 6}
          tt_A = 41
          nil_A = 53
          n__nil_A = 52
          a_A = 53
          n__a_A = 53
          e_A = 51
          n__e_A = 36
          i_A = 55
          n__i_A = 42
          o_A = 54
          n__o_A = 41
          u_A = 53
          n__u_A = 40
    
      precedence:
      
        nil = n__nil = a > isNePal# = U71# > isPal# = n__i > activate = n__a = o > __ > n____ = n__o > isQid = tt = i = u > e = n__e = n__u
      
      partial status:
    
        pi(isPal#) = [1]
        pi(isNePal#) = [1]
        pi(activate) = [1]
        pi(n____) = [1, 2]
        pi(__) = [1, 2]
        pi(U71#) = [1, 2]
        pi(isQid) = [1]
        pi(tt) = []
        pi(nil) = []
        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) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          isPal#_A(x1) = x1 + 45
          isNePal#_A(x1) = x1 + 37
          activate_A(x1) = x1 + 7
          n_____A(x1,x2) = max{x1 - 8, x2 + 2}
          ___A(x1,x2) = max{x1 + 57, x2 + 56}
          U71#_A(x1,x2) = max{60, x2 + 53}
          isQid_A(x1) = x1 + 127
          tt_A = 107
          nil_A = 3
          n__nil_A = 4
          a_A = 3
          n__a_A = 4
          e_A = 3
          n__e_A = 4
          i_A = 3
          n__i_A = 4
          o_A = 3
          n__o_A = 4
          u_A = 3
          n__u_A = 4
    
      precedence:
      
        isPal# = isNePal# = activate = n____ = __ = isQid = nil = n__nil = a = n__a = e = n__e = i = n__i = o = n__o > U71# = u > tt = n__u
      
      partial status:
    
        pi(isPal#) = [1]
        pi(isNePal#) = [1]
        pi(activate) = [1]
        pi(n____) = [2]
        pi(__) = []
        pi(U71#) = []
        pi(isQid) = [1]
        pi(tt) = []
        pi(nil) = []
        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

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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          isNeList#_A(x1) = max{1, x1}
          n_____A(x1,x2) = max{x1 + 59, x2}
          U51#_A(x1,x2) = max{x1 + 36, x2}
          isNeList_A(x1) = max{22, x1 + 5}
          activate_A(x1) = max{14, x1}
          tt_A = 11
          isList#_A(x1) = max{14, x1}
          U21#_A(x1,x2) = max{x1 + 4, x2}
          isList_A(x1) = max{26, x1 + 23}
          U41#_A(x1,x2) = max{54, x1 + 16, x2}
          U22_A(x1) = max{27, x1 - 25}
          U42_A(x1) = max{12, x1 - 21}
          U52_A(x1) = max{12, x1 - 45}
          ___A(x1,x2) = max{x1 + 59, x2}
          nil_A = 13
          U11_A(x1) = max{22, x1 + 3}
          U21_A(x1,x2) = max{x1 + 16, x2 + 13}
          U31_A(x1) = max{6, x1 + 1}
          U41_A(x1,x2) = max{23, x1 - 33, x2 - 15}
          U51_A(x1,x2) = max{14, x2 + 1}
          isQid_A(x1) = max{3, x1 + 2}
          n__a_A = 10
          n__e_A = 10
          n__i_A = 12
          n__o_A = 12
          n__u_A = 12
          n__nil_A = 0
          a_A = 13
          e_A = 11
          i_A = 13
          o_A = 13
          u_A = 13
    
      precedence:
      
        U52 > isQid > a > n____ = __ > isNeList = U31 = U41 = U51 > U41# > U51# = U21# > isList# > isNeList# > o > activate > nil > isList > U21 > n__a > n__o > n__nil > U11 = i > e > U22 = n__e = n__i > u > tt = U42 = n__u
      
      partial status:
    
        pi(isNeList#) = [1]
        pi(n____) = [1, 2]
        pi(U51#) = [2]
        pi(isNeList) = [1]
        pi(activate) = [1]
        pi(tt) = []
        pi(isList#) = [1]
        pi(U21#) = [1, 2]
        pi(isList) = []
        pi(U41#) = [1, 2]
        pi(U22) = []
        pi(U42) = []
        pi(U52) = []
        pi(__) = [1, 2]
        pi(nil) = []
        pi(U11) = []
        pi(U21) = [2]
        pi(U31) = []
        pi(U41) = []
        pi(U51) = [2]
        pi(isQid) = [1]
        pi(n__a) = []
        pi(n__e) = []
        pi(n__i) = []
        pi(n__o) = []
        pi(n__u) = []
        pi(n__nil) = []
        pi(a) = []
        pi(e) = []
        pi(i) = []
        pi(o) = []
        pi(u) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          isNeList#_A(x1) = max{45, x1 + 6}
          n_____A(x1,x2) = max{118, x1 + 42, x2 + 110}
          U51#_A(x1,x2) = x2 + 82
          isNeList_A(x1) = 120
          activate_A(x1) = max{35, x1 + 2}
          tt_A = 122
          isList#_A(x1) = x1 + 46
          U21#_A(x1,x2) = x2 + 108
          isList_A(x1) = 135
          U41#_A(x1,x2) = max{x1 - 20, x2 + 82}
          U22_A(x1) = 107
          U42_A(x1) = 130
          U52_A(x1) = 123
          ___A(x1,x2) = max{119, x1 + 43, x2 + 110}
          nil_A = 36
          U11_A(x1) = 134
          U21_A(x1,x2) = x2 + 107
          U31_A(x1) = 1
          U41_A(x1,x2) = 119
          U51_A(x1,x2) = max{121, x2 + 1}
          isQid_A(x1) = max{121, x1 + 91}
          n__a_A = 123
          n__e_A = 35
          n__i_A = 31
          n__o_A = 35
          n__u_A = 32
          n__nil_A = 89
          a_A = 124
          e_A = 36
          i_A = 34
          o_A = 36
          u_A = 32
    
      precedence:
      
        __ > n____ = isNeList > U52 = U21 > U41# > isNeList# = U51# > isList# = U21# > nil = U41 > e > U31 > activate = a = i > isList > U11 > U22 = isQid > n__e > U51 > tt > n__nil > n__a = n__u = u > n__o > n__i > o > U42
      
      partial status:
    
        pi(isNeList#) = [1]
        pi(n____) = [1]
        pi(U51#) = []
        pi(isNeList) = []
        pi(activate) = [1]
        pi(tt) = []
        pi(isList#) = []
        pi(U21#) = [2]
        pi(isList) = []
        pi(U41#) = [2]
        pi(U22) = []
        pi(U42) = []
        pi(U52) = []
        pi(__) = [2]
        pi(nil) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(U51) = [2]
        pi(isQid) = []
        pi(n__a) = []
        pi(n__e) = []
        pi(n__i) = []
        pi(n__o) = []
        pi(n__u) = []
        pi(n__nil) = []
        pi(a) = []
        pi(e) = []
        pi(i) = []
        pi(o) = []
        pi(u) = []
    

The next rules are strictly ordered:

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

We remove them from the problem.

-- SCC decomposition.

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

p1: isNeList#(n____(V1,V2)) -> U51#(isNeList(activate(V1)),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 estimated dependency graph contains the following SCCs:

  (no SCCs)

-- 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          __#_A(x1,x2) = max{x1 + 4, x2 + 2}
          ___A(x1,x2) = max{x1 + 2, x2}
          nil_A = 0
          n_____A(x1,x2) = max{x1 + 2, x2}
    
      precedence:
      
        __# = __ > nil = n____
      
      partial status:
    
        pi(__#) = [1]
        pi(__) = [1, 2]
        pi(nil) = []
        pi(n____) = [1, 2]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          __#_A(x1,x2) = max{0, x1 - 3}
          ___A(x1,x2) = max{x1 + 5, x2 + 7}
          nil_A = 0
          n_____A(x1,x2) = max{x1 + 6, x2 + 8}
    
      precedence:
      
        __# = __ = nil = n____
      
      partial status:
    
        pi(__#) = []
        pi(__) = [2]
        pi(nil) = []
        pi(n____) = []
    

The next rules are strictly ordered:

  p1, p2

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