YES

We show the termination of the TRS R:

  __(__(X,Y),Z) -> __(X,__(Y,Z))
  __(X,nil()) -> X
  __(nil(),X) -> X
  and(tt(),X) -> activate(X)
  isList(V) -> isNeList(activate(V))
  isList(n__nil()) -> tt()
  isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
  isNeList(V) -> isQid(activate(V))
  isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
  isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
  isNePal(V) -> isQid(activate(V))
  isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
  isPal(V) -> isNePal(activate(V))
  isPal(n__nil()) -> tt()
  isQid(n__a()) -> tt()
  isQid(n__e()) -> tt()
  isQid(n__i()) -> tt()
  isQid(n__o()) -> tt()
  isQid(n__u()) -> tt()
  nil() -> n__nil()
  __(X1,X2) -> n____(X1,X2)
  isList(X) -> n__isList(X)
  isNeList(X) -> n__isNeList(X)
  isPal(X) -> n__isPal(X)
  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__isList(X)) -> isList(X)
  activate(n__isNeList(X)) -> isNeList(X)
  activate(n__isPal(X)) -> isPal(X)
  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: and#(tt(),X) -> activate#(X)
p4: isList#(V) -> isNeList#(activate(V))
p5: isList#(V) -> activate#(V)
p6: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p7: isList#(n____(V1,V2)) -> isList#(activate(V1))
p8: isList#(n____(V1,V2)) -> activate#(V1)
p9: isList#(n____(V1,V2)) -> activate#(V2)
p10: isNeList#(V) -> isQid#(activate(V))
p11: isNeList#(V) -> activate#(V)
p12: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p13: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p14: isNeList#(n____(V1,V2)) -> activate#(V1)
p15: isNeList#(n____(V1,V2)) -> activate#(V2)
p16: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p17: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p18: isNeList#(n____(V1,V2)) -> activate#(V1)
p19: isNeList#(n____(V1,V2)) -> activate#(V2)
p20: isNePal#(V) -> isQid#(activate(V))
p21: isNePal#(V) -> activate#(V)
p22: isNePal#(n____(I,__(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P)))
p23: isNePal#(n____(I,__(P,I))) -> isQid#(activate(I))
p24: isNePal#(n____(I,__(P,I))) -> activate#(I)
p25: isNePal#(n____(I,__(P,I))) -> activate#(P)
p26: isPal#(V) -> isNePal#(activate(V))
p27: isPal#(V) -> activate#(V)
p28: activate#(n__nil()) -> nil#()
p29: activate#(n____(X1,X2)) -> __#(X1,X2)
p30: activate#(n__isList(X)) -> isList#(X)
p31: activate#(n__isNeList(X)) -> isNeList#(X)
p32: activate#(n__isPal(X)) -> isPal#(X)
p33: activate#(n__a()) -> a#()
p34: activate#(n__e()) -> e#()
p35: activate#(n__i()) -> i#()
p36: activate#(n__o()) -> o#()
p37: activate#(n__u()) -> u#()

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p3, p4, p5, p6, p7, p8, p9, p11, p12, p13, p14, p15, p16, p17, p18, p19, p21, p22, p24, p25, p26, p27, p30, p31, p32}
  {p1, p2}


-- Reduction pair.

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

p1: isPal#(V) -> activate#(V)
p2: activate#(n__isPal(X)) -> isPal#(X)
p3: isPal#(V) -> isNePal#(activate(V))
p4: isNePal#(n____(I,__(P,I))) -> activate#(P)
p5: activate#(n__isNeList(X)) -> isNeList#(X)
p6: isNeList#(n____(V1,V2)) -> activate#(V2)
p7: activate#(n__isList(X)) -> isList#(X)
p8: isList#(n____(V1,V2)) -> activate#(V2)
p9: isList#(n____(V1,V2)) -> activate#(V1)
p10: isList#(n____(V1,V2)) -> isList#(activate(V1))
p11: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p12: and#(tt(),X) -> activate#(X)
p13: isList#(V) -> activate#(V)
p14: isList#(V) -> isNeList#(activate(V))
p15: isNeList#(n____(V1,V2)) -> activate#(V1)
p16: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p17: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p18: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p19: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p20: isNeList#(V) -> activate#(V)
p21: isNePal#(n____(I,__(P,I))) -> activate#(I)
p22: isNePal#(n____(I,__(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P)))
p23: isNePal#(V) -> activate#(V)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isPal#_A(x1) = ((1,0),(1,1)) x1 + (12,33)
          activate#_A(x1) = ((1,0),(1,1)) x1 + (5,33)
          n__isPal_A(x1) = x1 + (44,34)
          isNePal#_A(x1) = x1 + (8,34)
          activate_A(x1) = ((1,0),(1,1)) x1 + (3,67)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (22,7)
          ___A(x1,x2) = x1 + x2 + (25,0)
          n__isNeList_A(x1) = ((1,0),(1,0)) x1 + (5,1)
          isNeList#_A(x1) = x1 + (6,122)
          n__isList_A(x1) = ((1,0),(0,0)) x1 + (11,1)
          isList#_A(x1) = x1 + (10,61)
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,34)
          isList_A(x1) = ((1,0),(1,0)) x1 + (11,78)
          tt_A() = (4,1)
          isNeList_A(x1) = ((1,0),(0,0)) x1 + (8,6)
          isQid_A(x1) = ((0,0),(1,0)) x1 + (5,64)
          isNePal_A(x1) = ((1,0),(1,1)) x1 + (7,68)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
          nil_A() = (6,3)
          isPal_A(x1) = ((1,0),(0,0)) x1 + (45,68)
          n__nil_A() = (5,2)
          a_A() = (6,0)
          n__a_A() = (5,2)
          e_A() = (6,3)
          n__e_A() = (5,2)
          i_A() = (6,3)
          n__i_A() = (5,2)
          o_A() = (8,3)
          n__o_A() = (5,2)
          u_A() = (6,3)
          n__u_A() = (5,2)
  
    precedence:
    
      n__isPal = activate = n__isList = isList = tt = isNeList = isQid = isNePal = and = isPal > isNePal# = n____ = n__isNeList = nil = n__nil = a = n__a = e = n__e = i = n__i = o = n__o = u > isPal# = activate# = __ = isNeList# = isList# = and# = n__u
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n__isPal) = []
      pi(isNePal#) = [1]
      pi(activate) = []
      pi(n____) = []
      pi(__) = [1, 2]
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(and#) = []
      pi(isList) = []
      pi(tt) = []
      pi(isNeList) = []
      pi(isQid) = []
      pi(isNePal) = []
      pi(and) = []
      pi(nil) = []
      pi(isPal) = []
      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:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: isPal#(V) -> activate#(V)
p2: activate#(n__isPal(X)) -> isPal#(X)
p3: isNePal#(n____(I,__(P,I))) -> activate#(P)
p4: activate#(n__isNeList(X)) -> isNeList#(X)
p5: isNeList#(n____(V1,V2)) -> activate#(V2)
p6: activate#(n__isList(X)) -> isList#(X)
p7: isList#(n____(V1,V2)) -> activate#(V2)
p8: isList#(n____(V1,V2)) -> activate#(V1)
p9: isList#(n____(V1,V2)) -> isList#(activate(V1))
p10: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p11: and#(tt(),X) -> activate#(X)
p12: isList#(V) -> activate#(V)
p13: isList#(V) -> isNeList#(activate(V))
p14: isNeList#(n____(V1,V2)) -> activate#(V1)
p15: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p16: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p17: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p18: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p19: isNeList#(V) -> activate#(V)
p20: isNePal#(n____(I,__(P,I))) -> activate#(I)
p21: isNePal#(n____(I,__(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P)))
p22: isNePal#(V) -> activate#(V)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19}


-- Reduction pair.

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

p1: isPal#(V) -> activate#(V)
p2: activate#(n__isList(X)) -> isList#(X)
p3: isList#(V) -> isNeList#(activate(V))
p4: isNeList#(V) -> activate#(V)
p5: activate#(n__isNeList(X)) -> isNeList#(X)
p6: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p7: and#(tt(),X) -> activate#(X)
p8: activate#(n__isPal(X)) -> isPal#(X)
p9: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p10: isList#(V) -> activate#(V)
p11: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isList#(n____(V1,V2)) -> activate#(V1)
p14: isList#(n____(V1,V2)) -> activate#(V2)
p15: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p16: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p17: isNeList#(n____(V1,V2)) -> activate#(V1)
p18: isNeList#(n____(V1,V2)) -> activate#(V2)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isPal#_A(x1) = ((1,0),(0,0)) x1 + (39,1)
          activate#_A(x1) = x1 + (25,3)
          n__isList_A(x1) = ((1,0),(1,1)) x1 + (8,8)
          isList#_A(x1) = ((1,0),(1,1)) x1 + (32,9)
          isNeList#_A(x1) = x1 + (26,49)
          activate_A(x1) = x1 + (5,36)
          n__isNeList_A(x1) = ((1,0),(1,1)) x1 + (2,47)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (24,2)
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (24,4)
          isList_A(x1) = ((1,0),(0,0)) x1 + (13,7)
          tt_A() = (2,2)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (40,5)
          isNeList_A(x1) = ((1,0),(0,0)) x1 + (6,6)
          isNePal_A(x1) = x1 + (4,1)
          isQid_A(x1) = (3,5)
          ___A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (28,1)
          and_A(x1,x2) = ((1,0),(0,0)) x2 + (6,5)
          nil_A() = (4,4)
          isPal_A(x1) = ((1,0),(0,0)) x1 + (41,38)
          n__nil_A() = (3,3)
          n__a_A() = (3,3)
          n__e_A() = (3,3)
          n__i_A() = (3,3)
          n__o_A() = (1,3)
          n__u_A() = (3,3)
          a_A() = (4,4)
          e_A() = (4,4)
          i_A() = (4,4)
          o_A() = (2,4)
          u_A() = (4,4)
  
    precedence:
    
      isPal# = __ > n__isPal = isPal > activate = n__isNeList = isList = isNeList = and > isQid > isNePal > n__isList = n____ = and# > activate# = isList# = isNeList# = tt = nil = n__nil = n__a = n__e = n__i = n__o = n__u = a = e = i = o = u
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(isNeList#) = []
      pi(activate) = []
      pi(n__isNeList) = []
      pi(n____) = [2]
      pi(and#) = []
      pi(isList) = []
      pi(tt) = []
      pi(n__isPal) = []
      pi(isNeList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(__) = []
      pi(and) = []
      pi(nil) = []
      pi(isPal) = []
      pi(n__nil) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(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__isList(X)) -> isList#(X)
p2: isList#(V) -> isNeList#(activate(V))
p3: isNeList#(V) -> activate#(V)
p4: activate#(n__isNeList(X)) -> isNeList#(X)
p5: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p6: and#(tt(),X) -> activate#(X)
p7: activate#(n__isPal(X)) -> isPal#(X)
p8: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p9: isList#(V) -> activate#(V)
p10: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p11: isList#(n____(V1,V2)) -> isList#(activate(V1))
p12: isList#(n____(V1,V2)) -> activate#(V1)
p13: isList#(n____(V1,V2)) -> activate#(V2)
p14: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p15: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p16: isNeList#(n____(V1,V2)) -> activate#(V1)
p17: isNeList#(n____(V1,V2)) -> activate#(V2)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17}


-- Reduction pair.

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

p1: activate#(n__isList(X)) -> isList#(X)
p2: isList#(n____(V1,V2)) -> activate#(V2)
p3: activate#(n__isNeList(X)) -> isNeList#(X)
p4: isNeList#(n____(V1,V2)) -> activate#(V2)
p5: isNeList#(n____(V1,V2)) -> activate#(V1)
p6: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p7: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p8: and#(tt(),X) -> activate#(X)
p9: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p10: isList#(n____(V1,V2)) -> activate#(V1)
p11: isList#(n____(V1,V2)) -> isList#(activate(V1))
p12: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p13: isList#(V) -> activate#(V)
p14: isList#(V) -> isNeList#(activate(V))
p15: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p16: isNeList#(V) -> activate#(V)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (3,2)
          n__isList_A(x1) = ((1,0),(0,0)) x1 + (13,52)
          isList#_A(x1) = ((1,0),(0,0)) x1 + (12,50)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (48,3)
          n__isNeList_A(x1) = ((1,0),(0,0)) x1 + (5,50)
          isNeList#_A(x1) = ((1,0),(1,0)) x1 + (4,1)
          activate_A(x1) = x1 + (7,54)
          and#_A(x1,x2) = ((1,0),(1,0)) x2 + (27,2)
          isNeList_A(x1) = ((1,0),(0,0)) x1 + (6,51)
          tt_A() = (2,1)
          isList_A(x1) = ((1,0),(0,0)) x1 + (14,53)
          isNePal_A(x1) = ((1,0),(0,0)) x1 + (11,5)
          isQid_A(x1) = (3,6)
          ___A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (49,2)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,4)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (91,6)
          nil_A() = (4,3)
          isPal_A(x1) = ((1,0),(0,0)) x1 + (92,7)
          n__nil_A() = (3,2)
          n__a_A() = (3,2)
          n__e_A() = (3,2)
          n__i_A() = (1,2)
          n__o_A() = (3,2)
          n__u_A() = (3,2)
          a_A() = (4,3)
          e_A() = (4,3)
          i_A() = (2,3)
          o_A() = (4,3)
          u_A() = (4,3)
  
    precedence:
    
      activate# = isNeList# = activate = and# = isNePal = __ = and = n__isPal = isPal > isList# > n__nil = n__e = e > tt = isQid = n__o = o > n__isList = n__isNeList = isNeList = isList > n____ > nil > n__a > n__i = n__u = a = i = u
    
    partial status:
  
      pi(activate#) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(n____) = [2]
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(activate) = []
      pi(and#) = []
      pi(isNeList) = []
      pi(tt) = []
      pi(isList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(__) = []
      pi(and) = []
      pi(n__isPal) = []
      pi(nil) = []
      pi(isPal) = []
      pi(n__nil) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(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: isList#(n____(V1,V2)) -> activate#(V2)
p2: activate#(n__isNeList(X)) -> isNeList#(X)
p3: isNeList#(n____(V1,V2)) -> activate#(V2)
p4: isNeList#(n____(V1,V2)) -> activate#(V1)
p5: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p6: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p7: and#(tt(),X) -> activate#(X)
p8: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p9: isList#(n____(V1,V2)) -> activate#(V1)
p10: isList#(n____(V1,V2)) -> isList#(activate(V1))
p11: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p12: isList#(V) -> activate#(V)
p13: isList#(V) -> isNeList#(activate(V))
p14: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p15: isNeList#(V) -> activate#(V)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15}


-- Reduction pair.

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

p1: isList#(n____(V1,V2)) -> activate#(V2)
p2: activate#(n__isNeList(X)) -> isNeList#(X)
p3: isNeList#(V) -> activate#(V)
p4: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p5: and#(tt(),X) -> activate#(X)
p6: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p7: isList#(V) -> isNeList#(activate(V))
p8: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p9: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p10: isNeList#(n____(V1,V2)) -> activate#(V1)
p11: isNeList#(n____(V1,V2)) -> activate#(V2)
p12: isList#(V) -> activate#(V)
p13: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p14: isList#(n____(V1,V2)) -> isList#(activate(V1))
p15: isList#(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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isList#_A(x1) = x1 + (8,40)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (54,61)
          activate#_A(x1) = x1 + (1,3)
          n__isNeList_A(x1) = x1 + (3,42)
          isNeList#_A(x1) = x1 + (2,19)
          and#_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (27,2)
          isList_A(x1) = x1 + (10,82)
          activate_A(x1) = x1 + (5,20)
          tt_A() = (2,2)
          isNeList_A(x1) = x1 + (4,61)
          n__isList_A(x1) = x1 + (6,63)
          isNePal_A(x1) = ((1,0),(1,0)) x1 + (6,13)
          isQid_A(x1) = (3,3)
          ___A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (58,62)
          and_A(x1,x2) = x2 + (6,101)
          n__isPal_A(x1) = ((1,0),(1,0)) x1 + (46,18)
          nil_A() = (4,1)
          isPal_A(x1) = ((1,0),(1,0)) x1 + (47,19)
          n__nil_A() = (3,3)
          n__a_A() = (1,1)
          n__e_A() = (3,3)
          n__i_A() = (3,3)
          n__o_A() = (3,3)
          n__u_A() = (3,3)
          a_A() = (5,2)
          e_A() = (4,4)
          i_A() = (4,4)
          o_A() = (4,4)
          u_A() = (4,4)
  
    precedence:
    
      n____ = isNeList# = activate = n__isList = isNePal = __ = and = isPal > isList# = activate# = n__isPal > isList = tt = isNeList = isQid > n__isNeList = and# > n__nil = n__i > nil = n__a = a > n__e = n__o = n__u = e = i = o = u
    
    partial status:
  
      pi(isList#) = []
      pi(n____) = []
      pi(activate#) = [1]
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(and#) = []
      pi(isList) = []
      pi(activate) = []
      pi(tt) = []
      pi(isNeList) = []
      pi(n__isList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(__) = []
      pi(and) = []
      pi(n__isPal) = []
      pi(nil) = []
      pi(isPal) = []
      pi(n__nil) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(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__isNeList(X)) -> isNeList#(X)
p2: isNeList#(V) -> activate#(V)
p3: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p4: and#(tt(),X) -> activate#(X)
p5: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p6: isList#(V) -> isNeList#(activate(V))
p7: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p8: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p9: isNeList#(n____(V1,V2)) -> activate#(V1)
p10: isNeList#(n____(V1,V2)) -> activate#(V2)
p11: isList#(V) -> activate#(V)
p12: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p13: isList#(n____(V1,V2)) -> isList#(activate(V1))
p14: isList#(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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: activate#(n__isNeList(X)) -> isNeList#(X)
p2: isNeList#(n____(V1,V2)) -> activate#(V2)
p3: isNeList#(n____(V1,V2)) -> activate#(V1)
p4: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p5: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p6: and#(tt(),X) -> activate#(X)
p7: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p8: isList#(n____(V1,V2)) -> activate#(V1)
p9: isList#(n____(V1,V2)) -> isList#(activate(V1))
p10: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p11: isList#(V) -> activate#(V)
p12: isList#(V) -> isNeList#(activate(V))
p13: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p14: isNeList#(V) -> activate#(V)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (2,1)
          n__isNeList_A(x1) = x1 + (2,18)
          isNeList#_A(x1) = ((1,0),(1,0)) x1 + (3,1)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (21,32)
          activate_A(x1) = x1 + (6,25)
          and#_A(x1,x2) = ((1,0),(1,0)) x2 + (3,2)
          isNeList_A(x1) = ((1,0),(1,0)) x1 + (5,17)
          n__isList_A(x1) = ((1,0),(1,0)) x1 + (7,5)
          tt_A() = (3,0)
          isList#_A(x1) = ((1,0),(1,1)) x1 + (10,0)
          isList_A(x1) = ((1,0),(1,0)) x1 + (12,24)
          isNePal_A(x1) = x1 + (9,2)
          isQid_A(x1) = (4,66)
          ___A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (22,33)
          and_A(x1,x2) = x2 + (7,26)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (38,27)
          nil_A() = (5,2)
          isPal_A(x1) = ((1,0),(0,0)) x1 + (39,28)
          n__nil_A() = (4,1)
          n__a_A() = (4,1)
          n__e_A() = (1,1)
          n__i_A() = (4,1)
          n__o_A() = (4,1)
          n__u_A() = (4,1)
          a_A() = (5,0)
          e_A() = (2,2)
          i_A() = (5,2)
          o_A() = (5,2)
          u_A() = (5,2)
  
    precedence:
    
      activate = isNeList = isList = and > and# > activate# = isNeList# > n____ = tt = __ > n__isList = isList# = isQid > nil > isNePal = n__isPal = isPal > n__isNeList = n__nil = n__a > n__e = n__i = n__o = n__u = a = e = i = o = u
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n____) = []
      pi(activate) = []
      pi(and#) = []
      pi(isNeList) = []
      pi(n__isList) = []
      pi(tt) = []
      pi(isList#) = [1]
      pi(isList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(__) = [2]
      pi(and) = []
      pi(n__isPal) = []
      pi(nil) = []
      pi(isPal) = []
      pi(n__nil) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(u) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__isNeList(X)) -> isNeList#(X)
p2: isNeList#(n____(V1,V2)) -> activate#(V2)
p3: isNeList#(n____(V1,V2)) -> activate#(V1)
p4: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p5: and#(tt(),X) -> activate#(X)
p6: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p7: isList#(n____(V1,V2)) -> activate#(V1)
p8: isList#(n____(V1,V2)) -> isList#(activate(V1))
p9: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p10: isList#(V) -> activate#(V)
p11: isList#(V) -> isNeList#(activate(V))
p12: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p13: isNeList#(V) -> activate#(V)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: activate#(n__isNeList(X)) -> isNeList#(X)
p2: isNeList#(V) -> activate#(V)
p3: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p4: and#(tt(),X) -> activate#(X)
p5: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p6: isList#(V) -> isNeList#(activate(V))
p7: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p8: isNeList#(n____(V1,V2)) -> activate#(V1)
p9: isNeList#(n____(V1,V2)) -> activate#(V2)
p10: isList#(V) -> activate#(V)
p11: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isList#(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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,1)) x1 + (1,25)
          n__isNeList_A(x1) = ((1,0),(0,0)) x1 + (3,1)
          isNeList#_A(x1) = x1 + (2,30)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (11,24)
          and#_A(x1,x2) = ((1,0),(1,1)) x2 + (1,26)
          isList_A(x1) = ((1,0),(0,0)) x1 + (7,4)
          activate_A(x1) = ((1,0),(0,0)) x1 + (2,5)
          tt_A() = (0,1)
          isList#_A(x1) = ((1,0),(1,1)) x1 + (5,48)
          n__isList_A(x1) = ((1,0),(1,0)) x1 + (6,1)
          isNePal_A(x1) = ((1,0),(1,0)) x1 + (9,15)
          isQid_A(x1) = (3,38)
          ___A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (11,25)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (3,6)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (25,1)
          n__a_A() = (1,2)
          n__e_A() = (1,2)
          n__i_A() = (1,2)
          n__o_A() = (1,2)
          n__u_A() = (0,2)
          nil_A() = (1,4)
          isNeList_A(x1) = ((1,0),(0,0)) x1 + (4,23)
          isPal_A(x1) = ((1,0),(0,0)) x1 + (27,2)
          n__nil_A() = (1,3)
          a_A() = (3,3)
          e_A() = (3,3)
          i_A() = (2,3)
          o_A() = (1,3)
          u_A() = (2,3)
  
    precedence:
    
      isList = activate = tt = isList# = n__isList = isNePal = __ = and = isNeList = isPal > isQid > activate# = isNeList# = and# > n__isNeList = n____ > n__isPal = n__a = n__u = a > n__e = e > n__i = i = o > n__o = u > nil = n__nil
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__isNeList) = []
      pi(isNeList#) = [1]
      pi(n____) = []
      pi(and#) = [2]
      pi(isList) = []
      pi(activate) = []
      pi(tt) = []
      pi(isList#) = [1]
      pi(n__isList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(__) = []
      pi(and) = []
      pi(n__isPal) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(nil) = []
      pi(isNeList) = []
      pi(isPal) = []
      pi(n__nil) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(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: isNeList#(V) -> activate#(V)
p2: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p3: and#(tt(),X) -> activate#(X)
p4: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p5: isList#(V) -> isNeList#(activate(V))
p6: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p7: isNeList#(n____(V1,V2)) -> activate#(V1)
p8: isNeList#(n____(V1,V2)) -> activate#(V2)
p9: isList#(V) -> activate#(V)
p10: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p11: isList#(n____(V1,V2)) -> isList#(activate(V1))
p12: isList#(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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p4, p5, p6, p11}


-- Reduction pair.

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

p1: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p2: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p3: isList#(n____(V1,V2)) -> isList#(activate(V1))
p4: isList#(V) -> isNeList#(activate(V))

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNeList#_A(x1) = x1 + (1,2)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (20,12)
          activate_A(x1) = x1 + (2,5)
          isList#_A(x1) = x1 + (4,8)
          and_A(x1,x2) = x2 + (3,13)
          tt_A() = (1,6)
          isNePal_A(x1) = ((1,0),(0,0)) x1 + (4,3)
          isQid_A(x1) = ((1,0),(0,0)) x1 + (1,2)
          ___A(x1,x2) = x1 + x2 + (21,1)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (6,0)
          n__a_A() = (2,7)
          n__e_A() = (2,7)
          n__i_A() = (2,1)
          n__o_A() = (2,7)
          n__u_A() = (2,0)
          nil_A() = (3,1)
          isList_A(x1) = ((1,0),(1,0)) x1 + (15,15)
          isNeList_A(x1) = ((1,0),(1,0)) x1 + (12,12)
          n__nil_A() = (2,1)
          n__isList_A(x1) = ((1,0),(1,0)) x1 + (14,16)
          n__isNeList_A(x1) = ((1,0),(1,0)) x1 + (11,11)
          isPal_A(x1) = ((1,0),(0,0)) x1 + (7,4)
          a_A() = (3,0)
          e_A() = (3,13)
          i_A() = (3,0)
          o_A() = (3,6)
          u_A() = (3,1)
  
    precedence:
    
      activate = and = n__a = n__e = n__i = isList = isNeList = e = i > isList# > isNeList# = n__isPal = n__o > n__u > nil = n__nil = n__isList > n__isNeList = a = o > u > n____ = tt = isNePal = isQid = __ = isPal
    
    partial status:
  
      pi(isNeList#) = [1]
      pi(n____) = []
      pi(activate) = []
      pi(isList#) = []
      pi(and) = []
      pi(tt) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(__) = [1, 2]
      pi(n__isPal) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(nil) = []
      pi(isList) = []
      pi(isNeList) = []
      pi(n__nil) = []
      pi(n__isList) = []
      pi(n__isNeList) = []
      pi(isPal) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(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: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p2: isList#(n____(V1,V2)) -> isList#(activate(V1))
p3: isList#(V) -> isNeList#(activate(V))

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p2: isList#(V) -> isNeList#(activate(V))
p3: isList#(n____(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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNeList#_A(x1) = ((1,0),(1,0)) x1 + (1,5)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (18,32)
          isList#_A(x1) = x1 + (7,9)
          activate_A(x1) = ((1,0),(1,1)) x1 + (5,24)
          and_A(x1,x2) = ((1,0),(0,0)) x2 + (6,1)
          tt_A() = (2,31)
          isNePal_A(x1) = ((1,0),(0,0)) x1 + (6,23)
          isQid_A(x1) = ((0,0),(1,0)) x1 + (3,27)
          ___A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (19,33)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (25,85)
          n__a_A() = (1,1)
          n__e_A() = (3,30)
          n__i_A() = (5,30)
          n__o_A() = (1,1)
          n__u_A() = (3,32)
          nil_A() = (4,33)
          isList_A(x1) = ((1,0),(1,1)) x1 + (10,32)
          isNeList_A(x1) = x1 + (4,25)
          n__nil_A() = (3,32)
          n__isList_A(x1) = ((1,0),(1,0)) x1 + (6,1)
          n__isNeList_A(x1) = ((1,0),(1,0)) x1 + (1,26)
          isPal_A(x1) = ((1,0),(0,0)) x1 + (25,85)
          a_A() = (2,2)
          e_A() = (4,0)
          i_A() = (6,0)
          o_A() = (2,2)
          u_A() = (4,0)
  
    precedence:
    
      isList# = n__a = n__e > isNeList# = n____ = __ = n__i = n__u = nil = isNeList = n__nil > n__isPal = n__o = n__isNeList = isPal = e = i = o > and = tt = isNePal = isQid > n__isList > activate > isList = a = u
    
    partial status:
  
      pi(isNeList#) = []
      pi(n____) = []
      pi(isList#) = [1]
      pi(activate) = [1]
      pi(and) = []
      pi(tt) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(__) = [1, 2]
      pi(n__isPal) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(nil) = []
      pi(isList) = [1]
      pi(isNeList) = [1]
      pi(n__nil) = []
      pi(n__isList) = []
      pi(n__isNeList) = []
      pi(isPal) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(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: isList#(V) -> isNeList#(activate(V))
p2: isList#(n____(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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

p1: isList#(n____(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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isList#_A(x1) = ((0,0),(1,0)) x1 + (26,5)
          n_____A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (25,7)
          activate_A(x1) = ((1,0),(1,1)) x1 + (3,35)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (3,36)
          tt_A() = (2,34)
          isNePal_A(x1) = ((1,0),(1,0)) x1 + (4,1)
          isQid_A(x1) = (3,53)
          ___A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (26,68)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (6,6)
          n__a_A() = (3,35)
          n__e_A() = (3,35)
          n__i_A() = (3,35)
          n__o_A() = (3,35)
          n__u_A() = (1,1)
          nil_A() = (31,1)
          isList_A(x1) = ((0,0),(1,0)) x1 + (27,58)
          isNeList_A(x1) = ((0,0),(1,0)) x1 + (27,54)
          n__nil_A() = (30,0)
          n__isList_A(x1) = ((0,0),(1,0)) x1 + (24,0)
          n__isNeList_A(x1) = ((0,0),(1,0)) x1 + (24,0)
          isPal_A(x1) = ((1,0),(1,0)) x1 + (8,5)
          a_A() = (4,36)
          e_A() = (4,36)
          i_A() = (4,36)
          o_A() = (4,0)
          u_A() = (2,2)
  
    precedence:
    
      isList# = activate = __ = n__isPal = n__e = n__i = isPal = e = i > n____ > n__o > nil > isList = n__nil = n__isList > tt > isNeList = n__isNeList > and = isNePal = n__a > n__u = a = o = u > isQid
    
    partial status:
  
      pi(isList#) = []
      pi(n____) = []
      pi(activate) = []
      pi(and) = []
      pi(tt) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(__) = []
      pi(n__isPal) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(nil) = []
      pi(isList) = []
      pi(isNeList) = []
      pi(n__nil) = []
      pi(n__isList) = []
      pi(n__isNeList) = []
      pi(isPal) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(u) = []

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: __#(__(X,Y),Z) -> __#(X,__(Y,Z))
p2: __#(__(X,Y),Z) -> __#(Y,Z)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r21

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          __#_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (1,1)
          ___A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,0)
          nil_A() = (1,1)
          n_____A(x1,x2) = (1,0)
  
    precedence:
    
      __ > __# = nil = n____
    
    partial status:
  
      pi(__#) = [1]
      pi(__) = []
      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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: 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: and(tt(),X) -> activate(X)
r5: isList(V) -> isNeList(activate(V))
r6: isList(n__nil()) -> tt()
r7: isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2)))
r8: isNeList(V) -> isQid(activate(V))
r9: isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2)))
r10: isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2)))
r11: isNePal(V) -> isQid(activate(V))
r12: isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P)))
r13: isPal(V) -> isNePal(activate(V))
r14: isPal(n__nil()) -> tt()
r15: isQid(n__a()) -> tt()
r16: isQid(n__e()) -> tt()
r17: isQid(n__i()) -> tt()
r18: isQid(n__o()) -> tt()
r19: isQid(n__u()) -> tt()
r20: nil() -> n__nil()
r21: __(X1,X2) -> n____(X1,X2)
r22: isList(X) -> n__isList(X)
r23: isNeList(X) -> n__isNeList(X)
r24: isPal(X) -> n__isPal(X)
r25: a() -> n__a()
r26: e() -> n__e()
r27: i() -> n__i()
r28: o() -> n__o()
r29: u() -> n__u()
r30: activate(n__nil()) -> nil()
r31: activate(n____(X1,X2)) -> __(X1,X2)
r32: activate(n__isList(X)) -> isList(X)
r33: activate(n__isNeList(X)) -> isNeList(X)
r34: activate(n__isPal(X)) -> isPal(X)
r35: activate(n__a()) -> a()
r36: activate(n__e()) -> e()
r37: activate(n__i()) -> i()
r38: activate(n__o()) -> o()
r39: activate(n__u()) -> u()
r40: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r21

Take the reduction pair:

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

The next rules are strictly ordered:

  p1

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