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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          isNePal#_A(x1) = x1 + 2
          n_____A(x1,x2) = x1 + x2 + 7
          ___A(x1,x2) = x1 + x2 + 8
          isPal#_A(x1) = x1 + 14
          activate_A(x1) = x1 + 2
          U71#_A(x1,x2) = x2 + 5
          isPalListKind_A(x1) = x1 + 2
          tt_A() = 5
          nil_A() = 7
          and_A(x1,x2) = x2 + 3
          n__nil_A() = 6
          n__and_A(x1,x2) = x2 + 2
          a_A() = 5
          n__a_A() = 4
          e_A() = 7
          n__e_A() = 6
          i_A() = 7
          n__i_A() = 6
          o_A() = 7
          n__o_A() = 6
          u_A() = 7
          n__u_A() = 6
          n__isPalListKind_A(x1) = x1 + 1
  
    precedence:
    
      and = n__and = o > n__a > a = e = n__e = n__o > __ = isPal# = activate = isPalListKind > isNePal# = U71# > tt = i = n__i > nil = n__nil = u > n__u = n__isPalListKind > n____
    
    partial status:
  
      pi(isNePal#) = []
      pi(n____) = []
      pi(__) = [1, 2]
      pi(isPal#) = [1]
      pi(activate) = [1]
      pi(U71#) = []
      pi(isPalListKind) = [1]
      pi(tt) = []
      pi(nil) = []
      pi(and) = [2]
      pi(n__nil) = []
      pi(n__and) = [2]
      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

We remove them from the problem.

-- SCC decomposition.

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

p1: isPal#(V) -> U71#(isPalListKind(activate(V)),activate(V))
p2: 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 estimated dependency graph contains the following SCCs:

  (no SCCs)

-- 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          U52#_A(x1,x2) = x1 + x2 + 15
          tt_A() = 21
          isList#_A(x1) = x1 + 35
          activate_A(x1) = x1
          n_____A(x1,x2) = x1 + x2 + 13
          U21#_A(x1,x2,x3) = x2 + x3 + 48
          and_A(x1,x2) = x2
          isPalListKind_A(x1) = 21
          n__isPalListKind_A(x1) = 21
          U11#_A(x1,x2) = x1 + x2 + 2
          isNeList#_A(x1) = x1 + 22
          U51#_A(x1,x2,x3) = x2 + x3 + 22
          U41#_A(x1,x2,x3) = x2 + x3 + 35
          U42#_A(x1,x2) = x2 + 23
          isList_A(x1) = 25
          isNeList_A(x1) = x1 + 5
          U22#_A(x1,x2) = x2 + 47
          U23_A(x1) = 22
          U43_A(x1) = x1 + 1
          U53_A(x1) = x1 + 1
          U12_A(x1) = 22
          U22_A(x1,x2) = 23
          U32_A(x1) = x1 + 1
          U42_A(x1,x2) = x2 + 7
          U52_A(x1,x2) = x1 + 6
          isQid_A(x1) = x1 + 2
          n__a_A() = 22
          n__e_A() = 20
          n__i_A() = 22
          n__o_A() = 19
          n__u_A() = 22
          ___A(x1,x2) = x1 + x2 + 13
          nil_A() = 1
          U11_A(x1,x2) = 23
          U21_A(x1,x2,x3) = 24
          U31_A(x1,x2) = x2 + 4
          U41_A(x1,x2,x3) = x3 + 8
          U51_A(x1,x2,x3) = x2 + 12
          n__nil_A() = 1
          a_A() = 22
          e_A() = 20
          i_A() = 22
          o_A() = 19
          u_A() = 22
          n__and_A(x1,x2) = x2
  
    precedence:
    
      tt = activate = and = isPalListKind = n__isPalListKind = U22# = U43 = U42 = isQid = n__u = nil = U41 = i = o = u > U12 > isList = U11 = U21 > U42# > e > U51 > U52 > U22 > n__nil > n__e > n__a = __ = a > U52# = isList# = n____ = U21# = U11# = isNeList# = U51# = U41# > isNeList = U23 = n__i = n__o = U31 > U53 = U32 > n__and
    
    partial status:
  
      pi(U52#) = []
      pi(tt) = []
      pi(isList#) = []
      pi(activate) = []
      pi(n____) = []
      pi(U21#) = []
      pi(and) = []
      pi(isPalListKind) = []
      pi(n__isPalListKind) = []
      pi(U11#) = [1]
      pi(isNeList#) = []
      pi(U51#) = []
      pi(U41#) = []
      pi(U42#) = []
      pi(isList) = []
      pi(isNeList) = [1]
      pi(U22#) = []
      pi(U23) = []
      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) = []
      pi(U41) = []
      pi(U51) = [2]
      pi(n__nil) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(u) = []
      pi(n__and) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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: U11#(tt(),V) -> isNeList#(activate(V))
p5: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p6: U51#(tt(),V1,V2) -> isNeList#(activate(V1))
p7: isNeList#(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p8: U41#(tt(),V1,V2) -> isList#(activate(V1))
p9: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2))
p10: U42#(tt(),V2) -> isNeList#(activate(V2))
p11: U51#(tt(),V1,V2) -> U52#(isNeList(activate(V1)),activate(V2))
p12: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2))
p13: 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 estimated dependency graph contains the following SCCs:

  {p5, p6, p7, p9, p10}
  {p2, p3, p12, p13}


-- Reduction pair.

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

p1: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p2: U51#(tt(),V1,V2) -> isNeList#(activate(V1))
p3: isNeList#(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p4: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2))
p5: U42#(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(),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          isNeList#_A(x1) = x1 + 2
          n_____A(x1,x2) = x1 + x2 + 52
          U51#_A(x1,x2,x3) = x2 + 45
          and_A(x1,x2) = x2 + 9
          isPalListKind_A(x1) = x1 + 2
          activate_A(x1) = x1 + 8
          n__isPalListKind_A(x1) = x1 + 1
          tt_A() = 2
          U41#_A(x1,x2,x3) = x3 + 24
          U42#_A(x1,x2) = x1 + x2 + 9
          isList_A(x1) = 6
          U43_A(x1) = 3
          U53_A(x1) = 3
          U32_A(x1) = 3
          U42_A(x1,x2) = x1 + 2
          isNeList_A(x1) = x1 + 13
          U52_A(x1,x2) = x1 + x2 + 5
          isQid_A(x1) = x1 + 2
          n__a_A() = 3
          n__e_A() = 1
          n__i_A() = 1
          n__o_A() = 1
          n__u_A() = 1
          U23_A(x1) = 3
          U31_A(x1,x2) = x1 + 2
          U41_A(x1,x2,x3) = x1 + x2 + 7
          U51_A(x1,x2,x3) = x2 + x3 + 35
          U12_A(x1) = 3
          U22_A(x1,x2) = 4
          ___A(x1,x2) = x1 + x2 + 52
          nil_A() = 2
          U11_A(x1,x2) = 4
          U21_A(x1,x2,x3) = 5
          n__nil_A() = 1
          a_A() = 10
          e_A() = 2
          i_A() = 2
          o_A() = 2
          u_A() = 2
          n__and_A(x1,x2) = x2 + 8
  
    precedence:
    
      nil > U41# > U53 = U51 = U11 > and = isList > isPalListKind = activate > isNeList > U32 = U31 = i = o > n__o = a > u > n____ = __ > U52 = U41 > U43 = U42 = n__nil = e > isNeList# = U51# = U42# > n__isPalListKind > tt = isQid > n__a = n__u > U12 > U21 > U23 = U22 > n__i > n__e = n__and
    
    partial status:
  
      pi(isNeList#) = []
      pi(n____) = []
      pi(U51#) = []
      pi(and) = [2]
      pi(isPalListKind) = [1]
      pi(activate) = [1]
      pi(n__isPalListKind) = []
      pi(tt) = []
      pi(U41#) = []
      pi(U42#) = []
      pi(isList) = []
      pi(U43) = []
      pi(U53) = []
      pi(U32) = []
      pi(U42) = []
      pi(isNeList) = []
      pi(U52) = []
      pi(isQid) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(U23) = []
      pi(U31) = []
      pi(U41) = []
      pi(U51) = [2]
      pi(U12) = []
      pi(U22) = []
      pi(__) = []
      pi(nil) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__nil) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(u) = []
      pi(n__and) = []

The next rules are strictly ordered:

  p3

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#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p2: U51#(tt(),V1,V2) -> isNeList#(activate(V1))
p3: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2))
p4: U42#(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(),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:

  {p1, p2}


-- Reduction pair.

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

p1: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p2: U51#(tt(),V1,V2) -> isNeList#(activate(V1))

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          isNeList#_A(x1) = x1 + 2
          n_____A(x1,x2) = x1 + x2 + 8
          U51#_A(x1,x2,x3) = x2 + x3 + 5
          and_A(x1,x2) = x2 + 3
          isPalListKind_A(x1) = x1 + 2
          activate_A(x1) = x1 + 2
          n__isPalListKind_A(x1) = x1 + 1
          tt_A() = 5
          ___A(x1,x2) = x1 + x2 + 9
          nil_A() = 7
          n__nil_A() = 6
          a_A() = 7
          n__a_A() = 6
          e_A() = 7
          n__e_A() = 6
          i_A() = 5
          n__i_A() = 4
          o_A() = 5
          n__o_A() = 4
          u_A() = 7
          n__u_A() = 6
          n__and_A(x1,x2) = x2 + 2
  
    precedence:
    
      and = activate > n____ = U51# > isNeList# = isPalListKind > n__isPalListKind > tt = nil = n__nil = a = n__a = e = n__e = i = n__i = o = n__o = u > __ = n__u = n__and
    
    partial status:
  
      pi(isNeList#) = [1]
      pi(n____) = [1, 2]
      pi(U51#) = [2, 3]
      pi(and) = []
      pi(isPalListKind) = [1]
      pi(activate) = []
      pi(n__isPalListKind) = [1]
      pi(tt) = []
      pi(__) = [1, 2]
      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) = []
      pi(n__and) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: U51#(tt(),V1,V2) -> isNeList#(activate(V1))

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:

  (no SCCs)

-- Reduction pair.

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

p1: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p2: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2))
p3: U22#(tt(),V2) -> isList#(activate(V2))
p4: U21#(tt(),V1,V2) -> isList#(activate(V1))

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          isList#_A(x1) = x1 + 6
          n_____A(x1,x2) = x1 + x2 + 9
          U21#_A(x1,x2,x3) = x2 + x3 + 8
          and_A(x1,x2) = x1 + x2 + 1
          isPalListKind_A(x1) = x1 + 1
          activate_A(x1) = x1
          n__isPalListKind_A(x1) = x1 + 1
          tt_A() = 5
          U22#_A(x1,x2) = x2 + 7
          isList_A(x1) = x1 + 4
          U43_A(x1) = 6
          U53_A(x1) = x1 + 1
          U32_A(x1) = x1
          U42_A(x1,x2) = 7
          isNeList_A(x1) = x1
          U52_A(x1,x2) = x1 + x2
          isQid_A(x1) = x1
          n__a_A() = 6
          n__e_A() = 6
          n__i_A() = 6
          n__o_A() = 6
          n__u_A() = 6
          U23_A(x1) = 6
          U31_A(x1,x2) = x2
          U41_A(x1,x2,x3) = x1 + 3
          U51_A(x1,x2,x3) = x2 + x3
          U12_A(x1) = 6
          U22_A(x1,x2) = 7
          ___A(x1,x2) = x1 + x2 + 9
          nil_A() = 6
          U11_A(x1,x2) = x1 + 2
          U21_A(x1,x2,x3) = 8
          n__nil_A() = 6
          a_A() = 6
          e_A() = 6
          i_A() = 6
          o_A() = 6
          u_A() = 6
          n__and_A(x1,x2) = x1 + x2 + 1
  
    precedence:
    
      and = n__and > isNeList = U31 > U53 = U52 = U51 = U21 > isQid > U42 = U41 = U22 > U12 > U23 > activate = tt = isList = U32 = nil > isPalListKind > n__isPalListKind > U43 > n____ = U21# = __ > n__nil > U11 > U22# > isList# = a > n__a = e > n__e = i > n__i = o > n__o = u > n__u
    
    partial status:
  
      pi(isList#) = [1]
      pi(n____) = [1, 2]
      pi(U21#) = []
      pi(and) = []
      pi(isPalListKind) = [1]
      pi(activate) = [1]
      pi(n__isPalListKind) = [1]
      pi(tt) = []
      pi(U22#) = []
      pi(isList) = [1]
      pi(U43) = []
      pi(U53) = []
      pi(U32) = []
      pi(U42) = []
      pi(isNeList) = [1]
      pi(U52) = []
      pi(isQid) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(U23) = []
      pi(U31) = []
      pi(U41) = []
      pi(U51) = []
      pi(U12) = []
      pi(U22) = []
      pi(__) = [1, 2]
      pi(nil) = []
      pi(U11) = [1]
      pi(U21) = []
      pi(n__nil) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(u) = []
      pi(n__and) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p2: U22#(tt(),V2) -> isList#(activate(V2))
p3: U21#(tt(),V1,V2) -> isList#(activate(V1))

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:

  {p1, p3}


-- Reduction pair.

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

p1: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p2: U21#(tt(),V1,V2) -> isList#(activate(V1))

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          isList#_A(x1) = x1 + 2
          n_____A(x1,x2) = x1 + x2 + 8
          U21#_A(x1,x2,x3) = x2 + x3 + 5
          and_A(x1,x2) = x2 + 3
          isPalListKind_A(x1) = x1 + 2
          activate_A(x1) = x1 + 2
          n__isPalListKind_A(x1) = x1 + 1
          tt_A() = 5
          ___A(x1,x2) = x1 + x2 + 9
          nil_A() = 7
          n__nil_A() = 6
          a_A() = 7
          n__a_A() = 6
          e_A() = 7
          n__e_A() = 6
          i_A() = 5
          n__i_A() = 4
          o_A() = 7
          n__o_A() = 6
          u_A() = 5
          n__u_A() = 4
          n__and_A(x1,x2) = x2 + 2
  
    precedence:
    
      U21# > isList# = n____ = and = isPalListKind = activate = n__isPalListKind = tt = __ = nil = n__nil = a = n__a = n__e > e = i = n__i = o = n__o = u = n__u = n__and
    
    partial status:
  
      pi(isList#) = [1]
      pi(n____) = [2]
      pi(U21#) = []
      pi(and) = [2]
      pi(isPalListKind) = [1]
      pi(activate) = [1]
      pi(n__isPalListKind) = []
      pi(tt) = []
      pi(__) = [1, 2]
      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) = []
      pi(n__and) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: U21#(tt(),V1,V2) -> isList#(activate(V1))

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:

  (no SCCs)

-- 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          isPalListKind#_A(x1) = x1 + 2
          n_____A(x1,x2) = x1 + x2 + 12
          activate#_A(x1) = x1 + 4
          n__and_A(x1,x2) = x1 + x2 + 3
          and#_A(x1,x2) = x1 + x2 + 2
          tt_A() = 3
          n__isPalListKind_A(x1) = x1 + 1
          activate_A(x1) = x1 + 2
          isPalListKind_A(x1) = x1 + 2
          ___A(x1,x2) = x1 + x2 + 13
          nil_A() = 5
          and_A(x1,x2) = x1 + x2 + 4
          n__nil_A() = 4
          a_A() = 3
          n__a_A() = 2
          e_A() = 5
          n__e_A() = 4
          i_A() = 3
          n__i_A() = 2
          o_A() = 5
          n__o_A() = 4
          u_A() = 3
          n__u_A() = 2
  
    precedence:
    
      isPalListKind# = n____ = activate# = n__and = and# = tt = n__isPalListKind = activate = isPalListKind = __ = nil = and = n__nil = a = n__a = e = i > n__e = n__i = o = n__o = u = n__u
    
    partial status:
  
      pi(isPalListKind#) = [1]
      pi(n____) = [2]
      pi(activate#) = []
      pi(n__and) = [2]
      pi(and#) = []
      pi(tt) = []
      pi(n__isPalListKind) = [1]
      pi(activate) = [1]
      pi(isPalListKind) = [1]
      pi(__) = []
      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

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p4: isPalListKind#(n____(V1,V2)) -> activate#(V1)
p5: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p6: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p4: isPalListKind#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p5: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p6: isPalListKind#(n____(V1,V2)) -> activate#(V1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1
          n__and_A(x1,x2) = x2 + 2
          and#_A(x1,x2) = x2 + 2
          tt_A() = 3
          n__isPalListKind_A(x1) = x1 + 2
          isPalListKind#_A(x1) = x1 + 1
          n_____A(x1,x2) = x1 + x2 + 8
          isPalListKind_A(x1) = x1 + 3
          activate_A(x1) = x1 + 2
          ___A(x1,x2) = x1 + x2 + 9
          nil_A() = 5
          and_A(x1,x2) = x2 + 3
          n__nil_A() = 4
          a_A() = 5
          n__a_A() = 4
          e_A() = 5
          n__e_A() = 4
          i_A() = 2
          n__i_A() = 1
          o_A() = 5
          n__o_A() = 4
          u_A() = 2
          n__u_A() = 1
  
    precedence:
    
      n__and = tt = activate = __ = and = a = e = n__e = i = n__i = o = n__o > isPalListKind# > activate# = and# > n__isPalListKind = n____ = isPalListKind = n__nil > nil = n__a > u > n__u
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__and) = []
      pi(and#) = [2]
      pi(tt) = []
      pi(n__isPalListKind) = [1]
      pi(isPalListKind#) = []
      pi(n____) = [1]
      pi(isPalListKind) = [1]
      pi(activate) = []
      pi(__) = []
      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

We remove them from the problem.

-- SCC decomposition.

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

p1: and#(tt(),X) -> activate#(X)
p2: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p3: isPalListKind#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p4: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p5: isPalListKind#(n____(V1,V2)) -> activate#(V1)

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:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: and#(tt(),X) -> activate#(X)
p2: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p3: isPalListKind#(n____(V1,V2)) -> activate#(V1)
p4: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p5: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = x1 + x2
          tt_A() = 3
          activate#_A(x1) = x1 + 2
          n__isPalListKind_A(x1) = x1 + 1
          isPalListKind#_A(x1) = x1 + 2
          n_____A(x1,x2) = x1 + x2 + 12
          activate_A(x1) = x1 + 4
          isPalListKind_A(x1) = x1 + 2
          ___A(x1,x2) = x1 + x2 + 13
          nil_A() = 5
          and_A(x1,x2) = x2 + 5
          n__nil_A() = 4
          n__and_A(x1,x2) = x2 + 4
          a_A() = 5
          n__a_A() = 4
          e_A() = 5
          n__e_A() = 4
          i_A() = 5
          n__i_A() = 4
          o_A() = 3
          n__o_A() = 2
          u_A() = 7
          n__u_A() = 4
  
    precedence:
    
      n____ = activate = and > and# > tt = activate# = isPalListKind# = isPalListKind > n__isPalListKind = __ = n__a = e = n__e = i = n__i = o > nil = n__nil = n__and = a = n__o = u = n__u
    
    partial status:
  
      pi(and#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__isPalListKind) = [1]
      pi(isPalListKind#) = []
      pi(n____) = []
      pi(activate) = []
      pi(isPalListKind) = []
      pi(__) = [1, 2]
      pi(nil) = []
      pi(and) = []
      pi(n__nil) = []
      pi(n__and) = [2]
      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

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p2: isPalListKind#(n____(V1,V2)) -> activate#(V1)
p3: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p4: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p2: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p3: isPalListKind#(n____(V1,V2)) -> activate#(V1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + 1
          n__isPalListKind_A(x1) = x1 + 1
          isPalListKind#_A(x1) = x1
          n_____A(x1,x2) = x1 + x2 + 7
          activate_A(x1) = x1 + 2
          ___A(x1,x2) = x1 + x2 + 8
          nil_A() = 5
          and_A(x1,x2) = x2 + 3
          tt_A() = 3
          isPalListKind_A(x1) = x1 + 2
          n__a_A() = 2
          n__e_A() = 2
          n__i_A() = 4
          n__nil_A() = 4
          n__o_A() = 4
          n__u_A() = 4
          n__and_A(x1,x2) = x2 + 2
          a_A() = 3
          e_A() = 3
          i_A() = 5
          o_A() = 5
          u_A() = 5
  
    precedence:
    
      activate# = isPalListKind# = n____ = activate = __ = nil = and = tt = isPalListKind = n__a = n__e > n__isPalListKind = n__i = n__nil = n__o = n__u = n__and = a = e = i = o = u
    
    partial status:
  
      pi(activate#) = []
      pi(n__isPalListKind) = [1]
      pi(isPalListKind#) = []
      pi(n____) = [1, 2]
      pi(activate) = [1]
      pi(__) = []
      pi(nil) = []
      pi(and) = [2]
      pi(tt) = []
      pi(isPalListKind) = [1]
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__nil) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(n__and) = [2]
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(u) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p2: isPalListKind#(n____(V1,V2)) -> activate#(V1)

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:

  {p1, p2}


-- Reduction pair.

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

p1: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p2: isPalListKind#(n____(V1,V2)) -> activate#(V1)

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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1
          n__isPalListKind_A(x1) = x1 + 1
          isPalListKind#_A(x1) = x1
          n_____A(x1,x2) = x1 + x2 + 1
  
    precedence:
    
      activate# = isPalListKind# > n__isPalListKind = n____
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__isPalListKind) = []
      pi(isPalListKind#) = [1]
      pi(n____) = [1, 2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: isPalListKind#(n____(V1,V2)) -> activate#(V1)

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:

  (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(),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          __#_A(x1,x2) = x1 + 1
          ___A(x1,x2) = x1 + x2 + 2
          nil_A() = 1
          n_____A(x1,x2) = x2
  
    precedence:
    
      __# = __ = nil = n____
    
    partial status:
  
      pi(__#) = [1]
      pi(__) = [1, 2]
      pi(nil) = []
      pi(n____) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: __#(__(X,Y),Z) -> __#(X,__(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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: __#(__(X,Y),Z) -> __#(X,__(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          __#_A(x1,x2) = x1
          ___A(x1,x2) = x1 + x2 + 1
          nil_A() = 1
          n_____A(x1,x2) = 0
  
    precedence:
    
      __# = __ = nil = n____
    
    partial status:
  
      pi(__#) = [1]
      pi(__) = [1, 2]
      pi(nil) = []
      pi(n____) = []

The next rules are strictly ordered:

  p1

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