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,n____(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)) -> __(activate(X1),activate(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,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P)))
p23: isNePal#(n____(I,n____(P,I))) -> isQid#(activate(I))
p24: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p25: isNePal#(n____(I,n____(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)) -> __#(activate(X1),activate(X2))
p30: activate#(n____(X1,X2)) -> activate#(X1)
p31: activate#(n____(X1,X2)) -> activate#(X2)
p32: activate#(n__isList(X)) -> isList#(X)
p33: activate#(n__isNeList(X)) -> isNeList#(X)
p34: activate#(n__isPal(X)) -> isPal#(X)
p35: activate#(n__a()) -> a#()
p36: activate#(n__e()) -> e#()
p37: activate#(n__i()) -> i#()
p38: activate#(n__o()) -> o#()
p39: 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,n____(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)) -> __(activate(X1),activate(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, p33, p34}
  {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,n____(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: activate#(n____(X1,X2)) -> activate#(X2)
p10: activate#(n____(X1,X2)) -> activate#(X1)
p11: isList#(n____(V1,V2)) -> activate#(V1)
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p14: and#(tt(),X) -> activate#(X)
p15: isList#(V) -> activate#(V)
p16: isList#(V) -> isNeList#(activate(V))
p17: isNeList#(n____(V1,V2)) -> activate#(V1)
p18: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p19: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p20: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p21: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p22: isNeList#(V) -> activate#(V)
p23: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p24: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P)))
p25: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          isPal#_A(x1) = ((0,1),(0,0)) x1 + (10,13)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (9,13)
          n__isPal_A(x1) = ((0,0),(0,1)) x1 + (1,2)
          isNePal#_A(x1) = ((0,1),(0,0)) x1 + (9,13)
          activate_A(x1) = ((1,1),(0,1)) x1 + (6,0)
          n_____A(x1,x2) = x1 + x2 + (4,6)
          n__isNeList_A(x1) = ((0,0),(0,1)) x1
          isNeList#_A(x1) = ((0,1),(0,0)) x1 + (9,13)
          n__isList_A(x1) = ((0,0),(0,1)) x1
          isList#_A(x1) = ((0,1),(0,0)) x1 + (9,13)
          and#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (5,13)
          isList_A(x1) = ((0,1),(0,1)) x1 + (6,0)
          tt_A() = (10,7)
          isNeList_A(x1) = ((0,1),(0,1)) x1 + (5,0)
          isQid_A(x1) = ((0,1),(0,1)) x1 + (1,0)
          isNePal_A(x1) = ((0,1),(0,1)) x1 + (9,1)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,1),(0,1)) x2 + (0,6)
          ___A(x1,x2) = x1 + x2 + (4,6)
          nil_A() = (2,7)
          isPal_A(x1) = ((0,1),(0,1)) x1 + (9,2)
          n__nil_A() = (1,7)
          a_A() = (10,10)
          n__a_A() = (9,10)
          e_A() = (12,10)
          n__e_A() = (11,10)
          i_A() = (12,10)
          n__i_A() = (11,10)
          o_A() = (2,10)
          n__o_A() = (1,10)
          u_A() = (12,10)
          n__u_A() = (11,10)
  
    precedence:
    
      isPal# = activate# = isNePal# = activate = isNeList# = isList# = and# = isList = isNePal = isPal = n__nil = o = u > and = nil > e > a = n__a > n____ = n__isNeList = isNeList = isQid = __ > tt > n__e = i = n__i = n__u > n__o > n__isPal = n__isList
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n__isPal) = []
      pi(isNePal#) = []
      pi(activate) = [1]
      pi(n____) = []
      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(__) = [1, 2]
      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:

  p17

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: isPal#(V) -> isNePal#(activate(V))
p4: isNePal#(n____(I,n____(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: activate#(n____(X1,X2)) -> activate#(X2)
p10: activate#(n____(X1,X2)) -> activate#(X1)
p11: isList#(n____(V1,V2)) -> activate#(V1)
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p14: and#(tt(),X) -> activate#(X)
p15: isList#(V) -> activate#(V)
p16: isList#(V) -> isNeList#(activate(V))
p17: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p18: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p19: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p20: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p21: isNeList#(V) -> activate#(V)
p22: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p23: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P)))
p24: 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,n____(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)) -> __(activate(X1),activate(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, p16, p17, p18, p19, p20, p21, p22, p23, p24}


-- Reduction pair.

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

p1: isPal#(V) -> activate#(V)
p2: activate#(n____(X1,X2)) -> activate#(X1)
p3: activate#(n____(X1,X2)) -> activate#(X2)
p4: activate#(n__isList(X)) -> isList#(X)
p5: isList#(V) -> isNeList#(activate(V))
p6: isNeList#(V) -> activate#(V)
p7: activate#(n__isNeList(X)) -> isNeList#(X)
p8: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p9: and#(tt(),X) -> activate#(X)
p10: activate#(n__isPal(X)) -> isPal#(X)
p11: isPal#(V) -> isNePal#(activate(V))
p12: isNePal#(V) -> activate#(V)
p13: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPal(activate(P)))
p14: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p15: isNePal#(n____(I,n____(P,I))) -> activate#(P)
p16: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p17: isList#(V) -> activate#(V)
p18: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p19: isList#(n____(V1,V2)) -> isList#(activate(V1))
p20: isList#(n____(V1,V2)) -> activate#(V1)
p21: isList#(n____(V1,V2)) -> activate#(V2)
p22: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p23: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p24: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          isPal#_A(x1) = ((1,1),(0,1)) x1 + (15,14)
          activate#_A(x1) = ((0,1),(0,1)) x1 + (3,3)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,6)
          n__isList_A(x1) = ((0,0),(0,1)) x1 + (22,3)
          isList#_A(x1) = ((0,1),(0,1)) x1 + (5,3)
          isNeList#_A(x1) = ((0,1),(0,1)) x1 + (4,3)
          activate_A(x1) = x1
          n__isNeList_A(x1) = ((0,0),(0,1)) x1 + (22,2)
          and#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,1)) x2 + (4,5)
          isList_A(x1) = ((0,0),(0,1)) x1 + (22,3)
          tt_A() = (4,0)
          n__isPal_A(x1) = ((0,1),(1,1)) x1 + (23,13)
          isNePal#_A(x1) = ((1,1),(0,1)) x1 + (6,6)
          isQid_A(x1) = (21,2)
          isNeList_A(x1) = ((0,0),(0,1)) x1 + (22,2)
          isNePal_A(x1) = ((0,1),(0,1)) x1 + (22,6)
          and_A(x1,x2) = x2 + (0,5)
          ___A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,6)
          nil_A() = (1,0)
          isPal_A(x1) = ((0,1),(1,1)) x1 + (23,13)
          n__nil_A() = (1,0)
          a_A() = (5,1)
          n__a_A() = (5,1)
          e_A() = (5,1)
          n__e_A() = (5,1)
          i_A() = (5,1)
          n__i_A() = (5,1)
          o_A() = (5,1)
          n__o_A() = (5,1)
          u_A() = (5,1)
          n__u_A() = (5,1)
  
    precedence:
    
      isPal# = n__isList = activate = n__isNeList = isList = isQid = isNeList = isNePal = and = a = n__a > tt = nil > n__nil > isNePal# = e > activate# = isList# = isNeList# = and# > n__isPal = isPal > n____ = __ > n__e = i = o > n__i = u > n__o = n__u
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n____) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(isNeList#) = []
      pi(activate) = []
      pi(n__isNeList) = []
      pi(and#) = []
      pi(isList) = []
      pi(tt) = []
      pi(n__isPal) = []
      pi(isNePal#) = [1]
      pi(isQid) = []
      pi(isNeList) = []
      pi(isNePal) = []
      pi(and) = []
      pi(__) = []
      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:

  p13

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____(X1,X2)) -> activate#(X1)
p3: activate#(n____(X1,X2)) -> activate#(X2)
p4: activate#(n__isList(X)) -> isList#(X)
p5: isList#(V) -> isNeList#(activate(V))
p6: isNeList#(V) -> activate#(V)
p7: activate#(n__isNeList(X)) -> isNeList#(X)
p8: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p9: and#(tt(),X) -> activate#(X)
p10: activate#(n__isPal(X)) -> isPal#(X)
p11: isPal#(V) -> isNePal#(activate(V))
p12: isNePal#(V) -> activate#(V)
p13: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p14: isNePal#(n____(I,n____(P,I))) -> activate#(P)
p15: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p16: isList#(V) -> activate#(V)
p17: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p18: isList#(n____(V1,V2)) -> isList#(activate(V1))
p19: isList#(n____(V1,V2)) -> activate#(V1)
p20: isList#(n____(V1,V2)) -> activate#(V2)
p21: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p22: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p23: 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,n____(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)) -> __(activate(X1),activate(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, p16, p17, p18, p19, p20, p21, p22, p23}


-- 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,n____(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: activate#(n____(X1,X2)) -> activate#(X2)
p10: activate#(n____(X1,X2)) -> activate#(X1)
p11: isList#(n____(V1,V2)) -> activate#(V1)
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isList(activate(V2)))
p14: and#(tt(),X) -> activate#(X)
p15: isList#(V) -> activate#(V)
p16: isList#(V) -> isNeList#(activate(V))
p17: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p18: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p19: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p20: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p21: isNeList#(V) -> activate#(V)
p22: isNePal#(n____(I,n____(P,I))) -> activate#(I)
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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          isPal#_A(x1) = ((1,0),(0,0)) x1 + (18,1)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (7,1)
          n__isPal_A(x1) = ((1,1),(0,1)) x1 + (17,3)
          isNePal#_A(x1) = ((1,0),(0,0)) x1 + (8,1)
          activate_A(x1) = x1
          n_____A(x1,x2) = x1 + x2 + (8,1)
          n__isNeList_A(x1) = ((1,0),(1,0)) x1 + (10,3)
          isNeList#_A(x1) = ((1,0),(0,0)) x1 + (16,1)
          n__isList_A(x1) = ((1,0),(1,0)) x1 + (11,3)
          isList#_A(x1) = ((1,0),(0,0)) x1 + (17,1)
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
          isList_A(x1) = ((1,0),(1,0)) x1 + (11,3)
          tt_A() = (6,2)
          isNeList_A(x1) = ((1,0),(1,0)) x1 + (10,3)
          isNePal_A(x1) = ((1,1),(0,1)) x1 + (10,3)
          isQid_A(x1) = ((1,0),(0,0)) x1 + (9,3)
          and_A(x1,x2) = x2 + (1,0)
          ___A(x1,x2) = x1 + x2 + (8,1)
          nil_A() = (1,1)
          isPal_A(x1) = ((1,1),(0,1)) x1 + (17,3)
          n__nil_A() = (1,1)
          n__a_A() = (1,2)
          n__e_A() = (7,1)
          n__i_A() = (1,2)
          n__o_A() = (0,0)
          n__u_A() = (1,1)
          a_A() = (1,2)
          e_A() = (7,1)
          i_A() = (1,2)
          o_A() = (0,0)
          u_A() = (1,1)
  
    precedence:
    
      isNePal# > and > n__a = a > n__isPal = isPal > activate > isList = tt = isNeList = __ > n__isList > isPal# = activate# > isNePal = isQid > n__isNeList > isNeList# = isList# > and# > n____ > u > nil = n__nil = e = o > n__o > n__e = n__i = n__u = i
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n__isPal) = []
      pi(isNePal#) = []
      pi(activate) = [1]
      pi(n____) = [2]
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(and#) = []
      pi(isList) = []
      pi(tt) = []
      pi(isNeList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(and) = []
      pi(__) = [1, 2]
      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:

  p13

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: isPal#(V) -> isNePal#(activate(V))
p4: isNePal#(n____(I,n____(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: activate#(n____(X1,X2)) -> activate#(X2)
p10: activate#(n____(X1,X2)) -> activate#(X1)
p11: isList#(n____(V1,V2)) -> activate#(V1)
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: and#(tt(),X) -> activate#(X)
p14: isList#(V) -> activate#(V)
p15: isList#(V) -> isNeList#(activate(V))
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,n____(P,I))) -> activate#(I)
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,n____(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)) -> __(activate(X1),activate(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, p16, p17, p18, p19, p20, p21, p22}


-- Reduction pair.

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

p1: isPal#(V) -> activate#(V)
p2: activate#(n____(X1,X2)) -> activate#(X1)
p3: activate#(n____(X1,X2)) -> activate#(X2)
p4: activate#(n__isList(X)) -> isList#(X)
p5: isList#(V) -> isNeList#(activate(V))
p6: isNeList#(V) -> activate#(V)
p7: activate#(n__isNeList(X)) -> isNeList#(X)
p8: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p9: and#(tt(),X) -> activate#(X)
p10: activate#(n__isPal(X)) -> isPal#(X)
p11: isPal#(V) -> isNePal#(activate(V))
p12: isNePal#(V) -> activate#(V)
p13: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p14: isNePal#(n____(I,n____(P,I))) -> activate#(P)
p15: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p16: isList#(V) -> activate#(V)
p17: isList#(n____(V1,V2)) -> isList#(activate(V1))
p18: isList#(n____(V1,V2)) -> activate#(V1)
p19: isList#(n____(V1,V2)) -> activate#(V2)
p20: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p21: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p22: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          isPal#_A(x1) = ((1,0),(0,0)) x1 + (13,11)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (11,11)
          n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (5,5)
          n__isList_A(x1) = ((1,1),(1,0)) x1 + (6,2)
          isList#_A(x1) = ((1,1),(0,0)) x1 + (16,11)
          isNeList#_A(x1) = ((1,1),(0,0)) x1 + (15,11)
          activate_A(x1) = x1
          n__isNeList_A(x1) = ((1,1),(1,0)) x1 + (4,2)
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (12,11)
          isList_A(x1) = ((1,1),(1,0)) x1 + (6,2)
          tt_A() = (2,2)
          n__isPal_A(x1) = ((1,1),(0,1)) x1 + (13,11)
          isNePal#_A(x1) = ((1,0),(0,0)) x1 + (12,11)
          isNeList_A(x1) = ((1,1),(1,0)) x1 + (4,2)
          isNePal_A(x1) = ((0,1),(0,1)) x1 + (12,5)
          isQid_A(x1) = ((0,1),(0,0)) x1 + (3,2)
          and_A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (1,2)
          ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (5,5)
          nil_A() = (3,1)
          isPal_A(x1) = ((1,1),(0,1)) x1 + (13,11)
          n__nil_A() = (3,1)
          n__a_A() = (3,2)
          n__e_A() = (3,2)
          n__i_A() = (3,2)
          n__o_A() = (1,0)
          n__u_A() = (1,1)
          a_A() = (3,2)
          e_A() = (3,2)
          i_A() = (3,2)
          o_A() = (1,0)
          u_A() = (1,1)
  
    precedence:
    
      activate > isList# = nil = n__nil = n__a = n__o = n__u = a = i = o = u > __ = n__e = e > isList = isNeList > n__isList = and > isNePal > tt = isQid = n__i > isPal# = n__isNeList = n__isPal = isNePal# = isPal > activate# = isNeList# = and# > n____
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n____) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(isNeList#) = []
      pi(activate) = []
      pi(n__isNeList) = []
      pi(and#) = []
      pi(isList) = []
      pi(tt) = []
      pi(n__isPal) = []
      pi(isNePal#) = []
      pi(isNeList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(and) = [1]
      pi(__) = []
      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: isPal#(V) -> activate#(V)
p2: activate#(n____(X1,X2)) -> activate#(X1)
p3: activate#(n____(X1,X2)) -> activate#(X2)
p4: activate#(n__isList(X)) -> isList#(X)
p5: isNeList#(V) -> activate#(V)
p6: activate#(n__isNeList(X)) -> isNeList#(X)
p7: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p8: and#(tt(),X) -> activate#(X)
p9: activate#(n__isPal(X)) -> isPal#(X)
p10: isPal#(V) -> isNePal#(activate(V))
p11: isNePal#(V) -> activate#(V)
p12: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p13: isNePal#(n____(I,n____(P,I))) -> activate#(P)
p14: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p15: isList#(V) -> activate#(V)
p16: isList#(n____(V1,V2)) -> isList#(activate(V1))
p17: isList#(n____(V1,V2)) -> activate#(V1)
p18: isList#(n____(V1,V2)) -> activate#(V2)
p19: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p20: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p21: 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,n____(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)) -> __(activate(X1),activate(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, p16, p17, p18, p19, p20, p21}


-- 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,n____(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: activate#(n____(X1,X2)) -> activate#(X2)
p10: activate#(n____(X1,X2)) -> activate#(X1)
p11: isList#(n____(V1,V2)) -> activate#(V1)
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isList#(V) -> activate#(V)
p14: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p15: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p16: and#(tt(),X) -> activate#(X)
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,n____(P,I))) -> activate#(I)
p21: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          isPal#_A(x1) = ((1,1),(0,0)) x1 + (14,0)
          activate#_A(x1) = ((1,1),(0,0)) x1 + (1,0)
          n__isPal_A(x1) = ((0,1),(1,1)) x1 + (13,17)
          isNePal#_A(x1) = ((1,1),(0,0)) x1 + (2,0)
          activate_A(x1) = x1
          n_____A(x1,x2) = x1 + x2 + (6,8)
          n__isNeList_A(x1) = ((1,1),(0,0)) x1 + (7,9)
          isNeList#_A(x1) = ((1,1),(0,0)) x1 + (5,0)
          n__isList_A(x1) = ((1,1),(0,0)) x1 + (7,9)
          isList#_A(x1) = ((1,1),(0,0)) x1 + (5,0)
          and#_A(x1,x2) = ((1,1),(0,0)) x2 + (2,0)
          isNeList_A(x1) = ((1,1),(0,0)) x1 + (7,9)
          tt_A() = (0,2)
          isList_A(x1) = ((1,1),(0,0)) x1 + (7,9)
          isNePal_A(x1) = ((0,1),(1,1)) x1 + (12,8)
          isQid_A(x1) = (0,8)
          and_A(x1,x2) = x2 + (14,0)
          ___A(x1,x2) = x1 + x2 + (6,8)
          nil_A() = (1,2)
          isPal_A(x1) = ((0,1),(1,1)) x1 + (13,17)
          n__nil_A() = (1,2)
          n__a_A() = (1,1)
          n__e_A() = (1,0)
          n__i_A() = (1,0)
          n__o_A() = (1,2)
          n__u_A() = (1,2)
          a_A() = (1,1)
          e_A() = (1,0)
          i_A() = (1,0)
          o_A() = (1,2)
          u_A() = (1,2)
  
    precedence:
    
      n__isPal = activate = n__isNeList = n__isList = isNeList = isList = isNePal = and = isPal > n____ = __ > nil > isQid > tt > n__nil > n__i = i > n__o = o > isPal# = activate# = isNePal# = isNeList# = isList# = and# = n__a = n__e = n__u = a = e = u
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n__isPal) = []
      pi(isNePal#) = []
      pi(activate) = []
      pi(n____) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(and#) = []
      pi(isNeList) = []
      pi(tt) = []
      pi(isList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(and) = []
      pi(__) = []
      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:

  p19

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: isPal#(V) -> isNePal#(activate(V))
p4: isNePal#(n____(I,n____(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: activate#(n____(X1,X2)) -> activate#(X2)
p10: activate#(n____(X1,X2)) -> activate#(X1)
p11: isList#(n____(V1,V2)) -> activate#(V1)
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isList#(V) -> activate#(V)
p14: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p15: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p16: and#(tt(),X) -> activate#(X)
p17: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p18: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p19: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p20: 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,n____(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)) -> __(activate(X1),activate(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, p16, p17, p18, p19, p20}


-- Reduction pair.

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

p1: isPal#(V) -> activate#(V)
p2: activate#(n____(X1,X2)) -> activate#(X1)
p3: activate#(n____(X1,X2)) -> activate#(X2)
p4: activate#(n__isList(X)) -> isList#(X)
p5: isList#(V) -> activate#(V)
p6: activate#(n__isNeList(X)) -> isNeList#(X)
p7: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p8: and#(tt(),X) -> activate#(X)
p9: activate#(n__isPal(X)) -> isPal#(X)
p10: isPal#(V) -> isNePal#(activate(V))
p11: isNePal#(V) -> activate#(V)
p12: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p13: isNePal#(n____(I,n____(P,I))) -> activate#(P)
p14: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p15: isList#(n____(V1,V2)) -> isList#(activate(V1))
p16: isList#(n____(V1,V2)) -> activate#(V1)
p17: isList#(n____(V1,V2)) -> activate#(V2)
p18: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p19: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p20: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          isPal#_A(x1) = ((1,1),(1,1)) x1 + (4,0)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (2,0)
          n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (0,3)
          n__isList_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          isList#_A(x1) = ((1,0),(1,0)) x1 + (3,0)
          n__isNeList_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          isNeList#_A(x1) = ((1,0),(1,0)) x1 + (4,2)
          and#_A(x1,x2) = ((1,0),(1,0)) x2 + (2,0)
          isList_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          activate_A(x1) = x1
          tt_A() = (0,2)
          n__isPal_A(x1) = ((1,1),(1,0)) x1 + (3,7)
          isNePal#_A(x1) = ((1,1),(1,0)) x1 + (3,0)
          isNeList_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          isNePal_A(x1) = ((0,1),(1,0)) x1 + (3,7)
          isQid_A(x1) = (0,2)
          and_A(x1,x2) = x2
          ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (0,3)
          nil_A() = (0,0)
          isPal_A(x1) = ((1,1),(1,0)) x1 + (3,7)
          n__nil_A() = (0,0)
          n__a_A() = (1,0)
          n__e_A() = (1,1)
          n__i_A() = (1,1)
          n__o_A() = (1,2)
          n__u_A() = (1,2)
          a_A() = (1,0)
          e_A() = (1,1)
          i_A() = (1,1)
          o_A() = (1,2)
          u_A() = (1,2)
  
    precedence:
    
      n____ = n__isList = n__isNeList = isList = activate = isNeList = isNePal = and = __ = isPal > nil > n__isPal = isQid > isPal# = activate# = isList# = isNeList# = and# = isNePal# > n__nil = n__e = a = e = o > tt = n__o = n__u = u > n__a = n__i = i
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n____) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(and#) = []
      pi(isList) = []
      pi(activate) = []
      pi(tt) = []
      pi(n__isPal) = []
      pi(isNePal#) = []
      pi(isNeList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(and) = []
      pi(__) = []
      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:

  p12

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____(X1,X2)) -> activate#(X1)
p3: activate#(n____(X1,X2)) -> activate#(X2)
p4: activate#(n__isList(X)) -> isList#(X)
p5: isList#(V) -> activate#(V)
p6: activate#(n__isNeList(X)) -> isNeList#(X)
p7: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p8: and#(tt(),X) -> activate#(X)
p9: activate#(n__isPal(X)) -> isPal#(X)
p10: isPal#(V) -> isNePal#(activate(V))
p11: isNePal#(V) -> activate#(V)
p12: isNePal#(n____(I,n____(P,I))) -> activate#(P)
p13: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p14: isList#(n____(V1,V2)) -> isList#(activate(V1))
p15: isList#(n____(V1,V2)) -> activate#(V1)
p16: isList#(n____(V1,V2)) -> activate#(V2)
p17: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p18: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p19: 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,n____(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)) -> __(activate(X1),activate(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, 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__isPal(X)) -> isPal#(X)
p3: isPal#(V) -> isNePal#(activate(V))
p4: isNePal#(n____(I,n____(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: activate#(n____(X1,X2)) -> activate#(X2)
p10: activate#(n____(X1,X2)) -> activate#(X1)
p11: isList#(n____(V1,V2)) -> activate#(V1)
p12: isList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isList#(V) -> activate#(V)
p14: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p15: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p16: and#(tt(),X) -> activate#(X)
p17: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p18: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p19: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          isPal#_A(x1) = ((1,1),(0,0)) x1 + (5,1)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (3,1)
          n__isPal_A(x1) = ((1,0),(1,1)) x1 + (7,3)
          isNePal#_A(x1) = ((1,1),(0,0)) x1 + (4,1)
          activate_A(x1) = x1
          n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (3,1)
          n__isNeList_A(x1) = ((0,0),(1,1)) x1 + (5,4)
          isNeList#_A(x1) = ((1,1),(0,0)) x1 + (6,1)
          n__isList_A(x1) = ((0,0),(1,1)) x1 + (5,4)
          isList#_A(x1) = ((0,1),(0,0)) x1 + (6,1)
          and#_A(x1,x2) = ((0,1),(0,0)) x2 + (4,1)
          isNeList_A(x1) = ((0,0),(1,1)) x1 + (5,4)
          tt_A() = (2,0)
          isList_A(x1) = ((0,0),(1,1)) x1 + (5,4)
          isNePal_A(x1) = ((1,0),(1,1)) x1 + (7,3)
          isQid_A(x1) = ((0,0),(1,1)) x1 + (4,3)
          and_A(x1,x2) = x2
          ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (3,1)
          nil_A() = (3,1)
          isPal_A(x1) = ((1,0),(1,1)) x1 + (7,3)
          n__nil_A() = (3,1)
          n__a_A() = (3,1)
          n__e_A() = (3,1)
          n__i_A() = (3,1)
          n__o_A() = (3,1)
          n__u_A() = (1,1)
          a_A() = (3,1)
          e_A() = (3,1)
          i_A() = (3,1)
          o_A() = (3,1)
          u_A() = (1,1)
  
    precedence:
    
      isPal# = isNePal# = activate = n__isList = isNeList = isList = and > i > __ > a > nil > n____ > activate# > isPal > tt > n__nil > n__isPal = n__isNeList = n__i > o > n__a = n__o > n__e = n__u = e = u > isNeList# > isList# = and# > isNePal = isQid
    
    partial status:
  
      pi(isPal#) = []
      pi(activate#) = []
      pi(n__isPal) = []
      pi(isNePal#) = []
      pi(activate) = []
      pi(n____) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(and#) = []
      pi(isNeList) = []
      pi(tt) = []
      pi(isList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(and) = []
      pi(__) = []
      pi(nil) = []
      pi(isPal) = [1]
      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:

  p2

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: isPal#(V) -> isNePal#(activate(V))
p3: isNePal#(n____(I,n____(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: activate#(n____(X1,X2)) -> activate#(X2)
p9: activate#(n____(X1,X2)) -> activate#(X1)
p10: isList#(n____(V1,V2)) -> activate#(V1)
p11: isList#(n____(V1,V2)) -> isList#(activate(V1))
p12: isList#(V) -> activate#(V)
p13: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p14: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p15: and#(tt(),X) -> activate#(X)
p16: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p17: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p18: 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,n____(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)) -> __(activate(X1),activate(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, p7, 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__isNeList(X)) -> isNeList#(X)
p2: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p3: and#(tt(),X) -> activate#(X)
p4: activate#(n____(X1,X2)) -> activate#(X1)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n__isList(X)) -> isList#(X)
p7: isList#(V) -> activate#(V)
p8: isList#(n____(V1,V2)) -> isList#(activate(V1))
p9: isList#(n____(V1,V2)) -> activate#(V1)
p10: isList#(n____(V1,V2)) -> activate#(V2)
p11: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p12: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p13: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p14: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (5,1)
          n__isNeList_A(x1) = ((0,1),(1,1)) x1 + (5,0)
          isNeList#_A(x1) = ((1,1),(0,0)) x1 + (4,1)
          n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (3,0)
          and#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (4,1)
          isList_A(x1) = ((0,1),(1,1)) x1 + (5,2)
          activate_A(x1) = x1
          tt_A() = (2,2)
          n__isList_A(x1) = ((0,1),(1,1)) x1 + (5,2)
          isList#_A(x1) = ((0,1),(0,0)) x1 + (6,1)
          isNeList_A(x1) = ((0,1),(1,1)) x1 + (5,0)
          isNePal_A(x1) = ((1,1),(0,1)) x1 + (2,1)
          isQid_A(x1) = ((0,1),(0,1)) x1 + (1,0)
          and_A(x1,x2) = x2
          n__isPal_A(x1) = ((1,1),(0,1)) x1 + (7,1)
          ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (3,0)
          nil_A() = (1,1)
          isPal_A(x1) = ((1,1),(0,1)) x1 + (7,1)
          n__nil_A() = (1,1)
          n__a_A() = (1,2)
          n__e_A() = (2,2)
          n__i_A() = (1,2)
          n__o_A() = (3,2)
          n__u_A() = (1,2)
          a_A() = (1,2)
          e_A() = (2,2)
          i_A() = (1,2)
          o_A() = (3,2)
          u_A() = (1,2)
  
    precedence:
    
      activate# = isNeList# = n____ = and# = isList = activate = n__isList = isList# = isNeList = isNePal = and = n__isPal = __ = isPal = n__e = n__i = n__o = e = i = o > n__isNeList > isQid > tt = nil = n__nil > n__a = n__u = a = u
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n____) = []
      pi(and#) = []
      pi(isList) = []
      pi(activate) = []
      pi(tt) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(isNeList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(and) = []
      pi(n__isPal) = []
      pi(__) = []
      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:

  p9

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)) -> and#(isList(activate(V1)),n__isNeList(activate(V2)))
p3: and#(tt(),X) -> activate#(X)
p4: activate#(n____(X1,X2)) -> activate#(X1)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n__isList(X)) -> isList#(X)
p7: isList#(V) -> activate#(V)
p8: isList#(n____(V1,V2)) -> isList#(activate(V1))
p9: isList#(n____(V1,V2)) -> activate#(V2)
p10: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p11: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p12: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p13: 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,n____(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)) -> __(activate(X1),activate(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#(n____(V1,V2)) -> activate#(V2)
p3: activate#(n__isList(X)) -> isList#(X)
p4: isList#(n____(V1,V2)) -> activate#(V2)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n____(X1,X2)) -> activate#(X1)
p7: isList#(n____(V1,V2)) -> isList#(activate(V1))
p8: isList#(V) -> activate#(V)
p9: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p10: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p11: and#(tt(),X) -> activate#(X)
p12: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p13: isNeList#(n____(V1,V2)) -> and#(isList(activate(V1)),n__isNeList(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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(0,0)) x1 + (15,4)
          n__isNeList_A(x1) = ((1,0),(0,0)) x1 + (7,2)
          isNeList#_A(x1) = ((1,0),(0,0)) x1 + (17,4)
          n_____A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (14,3)
          n__isList_A(x1) = ((1,0),(0,0)) x1 + (16,1)
          isList#_A(x1) = ((1,0),(0,0)) x1 + (15,4)
          activate_A(x1) = ((1,0),(1,1)) x1
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (15,4)
          isNeList_A(x1) = ((1,0),(1,0)) x1 + (7,3)
          tt_A() = (7,2)
          isList_A(x1) = ((1,0),(1,0)) x1 + (16,5)
          isNePal_A(x1) = ((1,0),(1,0)) x1 + (28,8)
          isQid_A(x1) = ((1,0),(1,0)) x1 + (6,3)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(1,1)) x2 + (1,0)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (29,7)
          ___A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (14,4)
          nil_A() = (0,1)
          isPal_A(x1) = ((1,0),(1,0)) x1 + (29,8)
          n__nil_A() = (0,1)
          n__a_A() = (7,1)
          n__e_A() = (7,1)
          n__i_A() = (7,1)
          n__o_A() = (7,1)
          n__u_A() = (1,1)
          a_A() = (7,2)
          e_A() = (7,2)
          i_A() = (7,2)
          o_A() = (7,2)
          u_A() = (1,1)
  
    precedence:
    
      activate = isNePal = and = n__a = n__e = a = e = i > n__u = u > n__isNeList = isNeList = isList = n__isPal = isPal > isNeList# > n__isList = and# > isQid > isList# > activate# > n____ = tt = __ > nil = n__nil = n__i > n__o = o
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n____) = []
      pi(n__isList) = []
      pi(isList#) = []
      pi(activate) = []
      pi(and#) = []
      pi(isNeList) = []
      pi(tt) = []
      pi(isList) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(and) = []
      pi(n__isPal) = []
      pi(__) = [1, 2]
      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:

  p13

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: activate#(n__isList(X)) -> isList#(X)
p4: isList#(n____(V1,V2)) -> activate#(V2)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n____(X1,X2)) -> activate#(X1)
p7: isList#(n____(V1,V2)) -> isList#(activate(V1))
p8: isList#(V) -> activate#(V)
p9: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p10: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p11: and#(tt(),X) -> activate#(X)
p12: isNeList#(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,n____(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)) -> __(activate(X1),activate(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}


-- 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)) -> isList#(activate(V1))
p3: isList#(V) -> activate#(V)
p4: activate#(n____(X1,X2)) -> activate#(X1)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n__isList(X)) -> isList#(X)
p7: isList#(n____(V1,V2)) -> isList#(activate(V1))
p8: isList#(n____(V1,V2)) -> activate#(V2)
p9: isNeList#(n____(V1,V2)) -> and#(isNeList(activate(V1)),n__isList(activate(V2)))
p10: and#(tt(),X) -> activate#(X)
p11: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p12: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (9,1)
          n__isNeList_A(x1) = ((0,0),(0,1)) x1 + (0,2)
          isNeList#_A(x1) = ((0,1),(0,0)) x1 + (5,1)
          n_____A(x1,x2) = x1 + x2 + (8,8)
          isList#_A(x1) = ((0,1),(0,0)) x1 + (10,1)
          activate_A(x1) = ((1,1),(0,1)) x1 + (3,0)
          n__isList_A(x1) = ((0,0),(0,1)) x1 + (1,2)
          and#_A(x1,x2) = ((0,1),(0,0)) x2 + (10,1)
          isNeList_A(x1) = ((0,1),(0,1)) x1 + (0,2)
          tt_A() = (0,0)
          isNePal_A(x1) = ((0,1),(0,0)) x1 + (2,12)
          isQid_A(x1) = (0,0)
          and_A(x1,x2) = ((1,1),(0,1)) x2 + (4,0)
          n__isPal_A(x1) = ((0,1),(0,0)) x1 + (2,12)
          ___A(x1,x2) = x1 + x2 + (9,8)
          nil_A() = (2,1)
          isList_A(x1) = ((0,1),(0,1)) x1 + (2,2)
          n__nil_A() = (1,1)
          isPal_A(x1) = ((0,1),(0,0)) x1 + (2,12)
          n__a_A() = (1,1)
          n__e_A() = (1,1)
          n__i_A() = (1,1)
          n__o_A() = (1,1)
          n__u_A() = (1,1)
          a_A() = (2,1)
          e_A() = (1,1)
          i_A() = (4,1)
          o_A() = (2,1)
          u_A() = (1,1)
  
    precedence:
    
      n__a = a > n__nil > isNeList# > n__isPal = isPal > n__e = e > and# > activate# > isNePal > n__u = u > n____ = activate = __ = i > isList > isNeList > and > n__o = o > tt = isQid = nil > isList# > n__isNeList = n__isList = n__i
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n____) = []
      pi(isList#) = []
      pi(activate) = [1]
      pi(n__isList) = []
      pi(and#) = []
      pi(isNeList) = []
      pi(tt) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(and) = []
      pi(n__isPal) = []
      pi(__) = []
      pi(nil) = []
      pi(isList) = []
      pi(n__nil) = []
      pi(isPal) = []
      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:

  p9

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)) -> isList#(activate(V1))
p3: isList#(V) -> activate#(V)
p4: activate#(n____(X1,X2)) -> activate#(X1)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n__isList(X)) -> isList#(X)
p7: isList#(n____(V1,V2)) -> isList#(activate(V1))
p8: isList#(n____(V1,V2)) -> activate#(V2)
p9: and#(tt(),X) -> activate#(X)
p10: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p11: 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,n____(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)) -> __(activate(X1),activate(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, p10, p11}


-- 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: activate#(n__isList(X)) -> isList#(X)
p4: isList#(n____(V1,V2)) -> activate#(V2)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n____(X1,X2)) -> activate#(X1)
p7: isList#(n____(V1,V2)) -> isList#(activate(V1))
p8: isList#(V) -> activate#(V)
p9: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p10: isNeList#(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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(0,0)) x1
          n__isNeList_A(x1) = ((1,0),(0,0)) x1 + (4,4)
          isNeList#_A(x1) = ((1,0),(0,0)) x1 + (3,0)
          n_____A(x1,x2) = x1 + x2 + (14,15)
          n__isList_A(x1) = ((1,0),(0,0)) x1 + (5,5)
          isList#_A(x1) = ((1,0),(0,0)) x1 + (1,0)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,5)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (8,5)
          tt_A() = (2,4)
          isNePal_A(x1) = ((1,0),(1,0)) x1 + (7,4)
          isQid_A(x1) = (3,4)
          n__isPal_A(x1) = ((1,0),(0,0)) x1 + (23,4)
          n__a_A() = (3,2)
          n__e_A() = (3,1)
          n__i_A() = (1,1)
          n__o_A() = (1,0)
          n__u_A() = (3,1)
          ___A(x1,x2) = x1 + x2 + (14,15)
          nil_A() = (1,1)
          isList_A(x1) = ((1,0),(1,0)) x1 + (5,5)
          isNeList_A(x1) = ((1,0),(1,0)) x1 + (4,4)
          n__nil_A() = (1,1)
          isPal_A(x1) = ((1,0),(1,0)) x1 + (23,4)
          a_A() = (3,2)
          e_A() = (3,1)
          i_A() = (1,1)
          o_A() = (1,0)
          u_A() = (3,2)
  
    precedence:
    
      n__isList = activate = and = tt = isNePal = isQid = n__isPal = nil = isList = isNeList = n__nil = isPal = a = e = i = o = u > n__isNeList = n____ = n__i = n__o = __ > activate# = isNeList# = isList# = n__a = n__e = n__u
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n____) = [1, 2]
      pi(n__isList) = []
      pi(isList#) = []
      pi(activate) = []
      pi(and) = []
      pi(tt) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(n__isPal) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(__) = [1, 2]
      pi(nil) = []
      pi(isList) = []
      pi(isNeList) = []
      pi(n__nil) = []
      pi(isPal) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(u) = []

The next rules are strictly ordered:

  p8

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: activate#(n__isList(X)) -> isList#(X)
p4: isList#(n____(V1,V2)) -> activate#(V2)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n____(X1,X2)) -> activate#(X1)
p7: isList#(n____(V1,V2)) -> isList#(activate(V1))
p8: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p9: isNeList#(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,n____(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)) -> __(activate(X1),activate(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}


-- 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)) -> isList#(activate(V1))
p3: isList#(n____(V1,V2)) -> isList#(activate(V1))
p4: isList#(n____(V1,V2)) -> activate#(V2)
p5: activate#(n____(X1,X2)) -> activate#(X1)
p6: activate#(n____(X1,X2)) -> activate#(X2)
p7: activate#(n__isList(X)) -> isList#(X)
p8: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p9: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (9,0)
          n__isNeList_A(x1) = ((0,0),(0,1)) x1
          isNeList#_A(x1) = ((0,1),(0,0)) x1 + (8,0)
          n_____A(x1,x2) = x1 + x2 + (7,6)
          isList#_A(x1) = ((0,1),(0,0)) x1 + (4,0)
          activate_A(x1) = ((1,1),(0,1)) x1 + (6,0)
          n__isList_A(x1) = ((0,0),(0,1)) x1 + (5,0)
          and_A(x1,x2) = ((1,1),(0,1)) x2 + (6,0)
          tt_A() = (2,0)
          isNePal_A(x1) = (34,13)
          isQid_A(x1) = (3,0)
          n__isPal_A(x1) = (15,13)
          n__a_A() = (3,1)
          n__e_A() = (1,1)
          n__i_A() = (3,1)
          n__o_A() = (3,1)
          n__u_A() = (1,1)
          ___A(x1,x2) = x1 + x2 + (7,6)
          nil_A() = (2,0)
          isList_A(x1) = ((0,1),(0,1)) x1 + (11,0)
          isNeList_A(x1) = ((0,1),(0,1)) x1 + (6,0)
          n__nil_A() = (1,0)
          isPal_A(x1) = (34,13)
          a_A() = (4,1)
          e_A() = (2,1)
          i_A() = (4,1)
          o_A() = (4,1)
          u_A() = (2,1)
  
    precedence:
    
      n__nil > activate = n__isList = and = tt = isNePal = n__e = n__i = n__u = nil = isList = isNeList = isPal = e > n__isNeList > activate# = isNeList# = isList# > n__isPal > n__a = n__o > isQid = __ > n____ = a = i = u > o
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNeList) = []
      pi(isNeList#) = []
      pi(n____) = [2]
      pi(isList#) = []
      pi(activate) = []
      pi(n__isList) = []
      pi(and) = []
      pi(tt) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(n__isPal) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(__) = [1, 2]
      pi(nil) = []
      pi(isList) = []
      pi(isNeList) = []
      pi(n__nil) = []
      pi(isPal) = []
      pi(a) = []
      pi(e) = []
      pi(i) = []
      pi(o) = []
      pi(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: activate#(n__isNeList(X)) -> isNeList#(X)
p2: isNeList#(n____(V1,V2)) -> isList#(activate(V1))
p3: isList#(n____(V1,V2)) -> activate#(V2)
p4: activate#(n____(X1,X2)) -> activate#(X1)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n__isList(X)) -> isList#(X)
p7: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p8: 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,n____(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)) -> __(activate(X1),activate(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}


-- 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: activate#(n__isList(X)) -> isList#(X)
p4: isList#(n____(V1,V2)) -> activate#(V2)
p5: activate#(n____(X1,X2)) -> activate#(X2)
p6: activate#(n____(X1,X2)) -> activate#(X1)
p7: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p8: isNeList#(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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (1,1)
          n__isNeList_A(x1) = ((0,1),(1,1)) x1 + (3,2)
          isNeList#_A(x1) = ((1,1),(0,1)) x1 + (1,0)
          n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,10)
          n__isList_A(x1) = ((0,1),(1,1)) x1 + (4,3)
          isList#_A(x1) = ((0,1),(0,1)) x1 + (1,4)
          activate_A(x1) = x1
          and_A(x1,x2) = ((0,1),(1,1)) x1 + x2 + (7,0)
          tt_A() = (2,2)
          isNePal_A(x1) = ((1,1),(1,1)) x1 + (4,4)
          isQid_A(x1) = (3,2)
          n__isPal_A(x1) = ((1,1),(1,1)) x1 + (5,21)
          n__a_A() = (1,1)
          n__e_A() = (1,1)
          n__i_A() = (1,1)
          n__o_A() = (1,1)
          n__u_A() = (1,1)
          ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,10)
          nil_A() = (1,0)
          isList_A(x1) = ((0,1),(1,1)) x1 + (4,3)
          isNeList_A(x1) = ((0,1),(1,1)) x1 + (3,2)
          n__nil_A() = (1,0)
          isPal_A(x1) = ((1,1),(1,1)) x1 + (5,21)
          a_A() = (1,1)
          e_A() = (1,1)
          i_A() = (1,1)
          o_A() = (1,1)
          u_A() = (1,1)
  
    precedence:
    
      n__u = u > n__i = i > n__isList = isList > n__isNeList = activate = isNeList > and > n__e = e > n____ = __ > activate# = isNeList# = isList# > isNePal = n__isPal = isPal > n__a = a > tt = isQid = n__o = o > nil = n__nil
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNeList) = []
      pi(isNeList#) = [1]
      pi(n____) = [1, 2]
      pi(n__isList) = []
      pi(isList#) = []
      pi(activate) = [1]
      pi(and) = []
      pi(tt) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(n__isPal) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(__) = [1, 2]
      pi(nil) = []
      pi(isList) = []
      pi(isNeList) = []
      pi(n__nil) = []
      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)) -> activate#(V2)
p2: activate#(n__isList(X)) -> isList#(X)
p3: isList#(n____(V1,V2)) -> activate#(V2)
p4: activate#(n____(X1,X2)) -> activate#(X2)
p5: activate#(n____(X1,X2)) -> activate#(X1)
p6: isNeList#(n____(V1,V2)) -> isNeList#(activate(V1))
p7: isNeList#(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,n____(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)) -> __(activate(X1),activate(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:

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


-- Reduction pair.

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

p1: isNeList#(n____(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: 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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          isNeList#_A(x1) = ((1,1),(1,1)) x1 + (5,2)
          n_____A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (4,1)
          activate_A(x1) = x1
          and_A(x1,x2) = ((1,1),(0,1)) x2 + (1,0)
          tt_A() = (2,0)
          isNePal_A(x1) = ((1,1),(0,1)) x1 + (7,1)
          isQid_A(x1) = (3,0)
          n__isPal_A(x1) = ((1,1),(0,1)) x1 + (7,1)
          n__a_A() = (1,1)
          n__e_A() = (1,1)
          n__i_A() = (1,0)
          n__o_A() = (3,1)
          n__u_A() = (1,1)
          ___A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (4,1)
          nil_A() = (1,1)
          isList_A(x1) = ((0,1),(0,0)) x1 + (5,0)
          isNeList_A(x1) = ((0,1),(0,0)) x1 + (5,0)
          n__nil_A() = (1,1)
          n__isList_A(x1) = ((0,1),(0,0)) x1 + (5,0)
          n__isNeList_A(x1) = ((0,1),(0,0)) x1 + (5,0)
          isPal_A(x1) = ((1,1),(0,1)) x1 + (7,1)
          a_A() = (1,1)
          e_A() = (1,1)
          i_A() = (1,0)
          o_A() = (3,1)
          u_A() = (1,1)
  
    precedence:
    
      isNeList# = n____ = activate = and = tt = isNePal = isQid = n__isPal = n__a = n__e = n__i = n__o = n__u = __ = nil = isList = isNeList = n__nil = n__isList = n__isNeList = isPal = a = e = i = o = u
    
    partial status:
  
      pi(isNeList#) = []
      pi(n____) = []
      pi(activate) = []
      pi(and) = []
      pi(tt) = []
      pi(isNePal) = []
      pi(isQid) = []
      pi(n__isPal) = []
      pi(n__a) = []
      pi(n__e) = []
      pi(n__i) = []
      pi(n__o) = []
      pi(n__u) = []
      pi(__) = []
      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: activate#(n__isList(X)) -> isList#(X)
p2: isList#(n____(V1,V2)) -> activate#(V2)
p3: activate#(n____(X1,X2)) -> activate#(X1)
p4: activate#(n____(X1,X2)) -> activate#(X2)

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,n____(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)) -> __(activate(X1),activate(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = x1 + (1,1)
          n__isList_A(x1) = ((1,1),(1,1)) x1 + (2,1)
          isList#_A(x1) = ((1,1),(0,1)) x1 + (1,1)
          n_____A(x1,x2) = x1 + x2 + (2,1)
  
    precedence:
    
      n____ > n__isList = isList# > activate#
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__isList) = []
      pi(isList#) = []
      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: isList#(n____(V1,V2)) -> activate#(V2)
p2: activate#(n____(X1,X2)) -> activate#(X1)
p3: activate#(n____(X1,X2)) -> activate#(X2)

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,n____(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)) -> __(activate(X1),activate(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, p3}


-- Reduction pair.

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

p1: activate#(n____(X1,X2)) -> activate#(X1)
p2: activate#(n____(X1,X2)) -> activate#(X2)

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,n____(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)) -> __(activate(X1),activate(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          n_____A(x1,x2) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + (1,1)
  
    precedence:
    
      activate# = n____
    
    partial status:
  
      pi(activate#) = []
      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: activate#(n____(X1,X2)) -> activate#(X1)

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,n____(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)) -> __(activate(X1),activate(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: activate#(n____(X1,X2)) -> activate#(X1)

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,n____(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)) -> __(activate(X1),activate(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

  (no rules)

Take the monotone reduction pair:

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

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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          __#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (3,2)
          ___A(x1,x2) = x1 + x2 + (2,1)
          nil_A() = (1,1)
          n_____A(x1,x2) = (1,1)
  
    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,n____(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)) -> __(activate(X1),activate(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,n____(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)) -> __(activate(X1),activate(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: standard order
        interpretations:
          __#_A(x1,x2) = ((1,1),(0,1)) x1 + (2,2)
          ___A(x1,x2) = x1 + x2 + (1,1)
          nil_A() = (1,1)
          n_____A(x1,x2) = (0,0)
  
    precedence:
    
      __ > __# = nil = n____
    
    partial status:
  
      pi(__#) = [1]
      pi(__) = []
      pi(nil) = []
      pi(n____) = []

The next rules are strictly ordered:

  p1

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