YES

We show the termination of the TRS R:

  __(__(X,Y),Z) -> __(X,__(Y,Z))
  __(X,nil()) -> X
  __(nil(),X) -> X
  U11(tt(),V) -> U12(isNeList(activate(V)))
  U12(tt()) -> tt()
  U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
  U22(tt(),V2) -> U23(isList(activate(V2)))
  U23(tt()) -> tt()
  U31(tt(),V) -> U32(isQid(activate(V)))
  U32(tt()) -> tt()
  U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
  U42(tt(),V2) -> U43(isNeList(activate(V2)))
  U43(tt()) -> tt()
  U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
  U52(tt(),V2) -> U53(isList(activate(V2)))
  U53(tt()) -> tt()
  U61(tt(),V) -> U62(isQid(activate(V)))
  U62(tt()) -> tt()
  U71(tt(),V) -> U72(isNePal(activate(V)))
  U72(tt()) -> tt()
  and(tt(),X) -> activate(X)
  isList(V) -> U11(isPalListKind(activate(V)),activate(V))
  isList(n__nil()) -> tt()
  isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
  isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
  isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
  isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
  isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
  isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))
  isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
  isPal(n__nil()) -> tt()
  isPalListKind(n__a()) -> tt()
  isPalListKind(n__e()) -> tt()
  isPalListKind(n__i()) -> tt()
  isPalListKind(n__nil()) -> tt()
  isPalListKind(n__o()) -> tt()
  isPalListKind(n__u()) -> tt()
  isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
  isQid(n__a()) -> tt()
  isQid(n__e()) -> tt()
  isQid(n__i()) -> tt()
  isQid(n__o()) -> tt()
  isQid(n__u()) -> tt()
  nil() -> n__nil()
  __(X1,X2) -> n____(X1,X2)
  isPalListKind(X) -> n__isPalListKind(X)
  and(X1,X2) -> n__and(X1,X2)
  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__isPalListKind(X)) -> isPalListKind(X)
  activate(n__and(X1,X2)) -> and(activate(X1),X2)
  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: U11#(tt(),V) -> U12#(isNeList(activate(V)))
p4: U11#(tt(),V) -> isNeList#(activate(V))
p5: U11#(tt(),V) -> activate#(V)
p6: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2))
p7: U21#(tt(),V1,V2) -> isList#(activate(V1))
p8: U21#(tt(),V1,V2) -> activate#(V1)
p9: U21#(tt(),V1,V2) -> activate#(V2)
p10: U22#(tt(),V2) -> U23#(isList(activate(V2)))
p11: U22#(tt(),V2) -> isList#(activate(V2))
p12: U22#(tt(),V2) -> activate#(V2)
p13: U31#(tt(),V) -> U32#(isQid(activate(V)))
p14: U31#(tt(),V) -> isQid#(activate(V))
p15: U31#(tt(),V) -> activate#(V)
p16: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2))
p17: U41#(tt(),V1,V2) -> isList#(activate(V1))
p18: U41#(tt(),V1,V2) -> activate#(V1)
p19: U41#(tt(),V1,V2) -> activate#(V2)
p20: U42#(tt(),V2) -> U43#(isNeList(activate(V2)))
p21: U42#(tt(),V2) -> isNeList#(activate(V2))
p22: U42#(tt(),V2) -> activate#(V2)
p23: U51#(tt(),V1,V2) -> U52#(isNeList(activate(V1)),activate(V2))
p24: U51#(tt(),V1,V2) -> isNeList#(activate(V1))
p25: U51#(tt(),V1,V2) -> activate#(V1)
p26: U51#(tt(),V1,V2) -> activate#(V2)
p27: U52#(tt(),V2) -> U53#(isList(activate(V2)))
p28: U52#(tt(),V2) -> isList#(activate(V2))
p29: U52#(tt(),V2) -> activate#(V2)
p30: U61#(tt(),V) -> U62#(isQid(activate(V)))
p31: U61#(tt(),V) -> isQid#(activate(V))
p32: U61#(tt(),V) -> activate#(V)
p33: U71#(tt(),V) -> U72#(isNePal(activate(V)))
p34: U71#(tt(),V) -> isNePal#(activate(V))
p35: U71#(tt(),V) -> activate#(V)
p36: and#(tt(),X) -> activate#(X)
p37: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V))
p38: isList#(V) -> isPalListKind#(activate(V))
p39: isList#(V) -> activate#(V)
p40: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p41: isList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p42: isList#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p43: isList#(n____(V1,V2)) -> activate#(V1)
p44: isList#(n____(V1,V2)) -> activate#(V2)
p45: isNeList#(V) -> U31#(isPalListKind(activate(V)),activate(V))
p46: isNeList#(V) -> isPalListKind#(activate(V))
p47: isNeList#(V) -> activate#(V)
p48: isNeList#(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p49: isNeList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p50: isNeList#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p51: isNeList#(n____(V1,V2)) -> activate#(V1)
p52: isNeList#(n____(V1,V2)) -> activate#(V2)
p53: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p54: isNeList#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p55: isNeList#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p56: isNeList#(n____(V1,V2)) -> activate#(V1)
p57: isNeList#(n____(V1,V2)) -> activate#(V2)
p58: isNePal#(V) -> U61#(isPalListKind(activate(V)),activate(V))
p59: isNePal#(V) -> isPalListKind#(activate(V))
p60: isNePal#(V) -> activate#(V)
p61: isNePal#(n____(I,n____(P,I))) -> and#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))
p62: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPalListKind(activate(I)))
p63: isNePal#(n____(I,n____(P,I))) -> isQid#(activate(I))
p64: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p65: isNePal#(n____(I,n____(P,I))) -> activate#(P)
p66: isPal#(V) -> U71#(isPalListKind(activate(V)),activate(V))
p67: isPal#(V) -> isPalListKind#(activate(V))
p68: isPal#(V) -> activate#(V)
p69: isPalListKind#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p70: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p71: isPalListKind#(n____(V1,V2)) -> activate#(V1)
p72: isPalListKind#(n____(V1,V2)) -> activate#(V2)
p73: activate#(n__nil()) -> nil#()
p74: activate#(n____(X1,X2)) -> __#(activate(X1),activate(X2))
p75: activate#(n____(X1,X2)) -> activate#(X1)
p76: activate#(n____(X1,X2)) -> activate#(X2)
p77: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p78: activate#(n__and(X1,X2)) -> and#(activate(X1),X2)
p79: activate#(n__and(X1,X2)) -> activate#(X1)
p80: activate#(n__isPal(X)) -> isPal#(X)
p81: activate#(n__a()) -> a#()
p82: activate#(n__e()) -> e#()
p83: activate#(n__i()) -> i#()
p84: activate#(n__o()) -> o#()
p85: activate#(n__u()) -> u#()

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: isPal(X) -> n__isPal(X)
r49: a() -> n__a()
r50: e() -> n__e()
r51: i() -> n__i()
r52: o() -> n__o()
r53: u() -> n__u()
r54: activate(n__nil()) -> nil()
r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2))
r56: activate(n__isPalListKind(X)) -> isPalListKind(X)
r57: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r58: activate(n__isPal(X)) -> isPal(X)
r59: activate(n__a()) -> a()
r60: activate(n__e()) -> e()
r61: activate(n__i()) -> i()
r62: activate(n__o()) -> o()
r63: activate(n__u()) -> u()
r64: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p4, p6, p7, p11, p16, p17, p21, p23, p24, p28, p37, p40, p48, p53}
  {p32, p34, p35, p36, p58, p59, p60, p61, p62, p64, p65, p66, p67, p68, p69, p70, p71, p72, p75, p76, p77, p78, p79, p80}
  {p1, p2}


-- Reduction pair.

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

p1: U52#(tt(),V2) -> isList#(activate(V2))
p2: isList#(n____(V1,V2)) -> U21#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p3: U21#(tt(),V1,V2) -> isList#(activate(V1))
p4: isList#(V) -> U11#(isPalListKind(activate(V)),activate(V))
p5: U11#(tt(),V) -> isNeList#(activate(V))
p6: isNeList#(n____(V1,V2)) -> U51#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p7: U51#(tt(),V1,V2) -> isNeList#(activate(V1))
p8: isNeList#(n____(V1,V2)) -> U41#(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
p9: U41#(tt(),V1,V2) -> isList#(activate(V1))
p10: U41#(tt(),V1,V2) -> U42#(isList(activate(V1)),activate(V2))
p11: U42#(tt(),V2) -> isNeList#(activate(V2))
p12: U51#(tt(),V1,V2) -> U52#(isNeList(activate(V1)),activate(V2))
p13: U21#(tt(),V1,V2) -> U22#(isList(activate(V1)),activate(V2))
p14: U22#(tt(),V2) -> isList#(activate(V2))

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: isPal(X) -> n__isPal(X)
r49: a() -> n__a()
r50: e() -> n__e()
r51: i() -> n__i()
r52: o() -> n__o()
r53: u() -> n__u()
r54: activate(n__nil()) -> nil()
r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2))
r56: activate(n__isPalListKind(X)) -> isPalListKind(X)
r57: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r58: activate(n__isPal(X)) -> isPal(X)
r59: activate(n__a()) -> a()
r60: activate(n__e()) -> e()
r61: activate(n__i()) -> i()
r62: activate(n__o()) -> o()
r63: activate(n__u()) -> u()
r64: 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, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        U52#_A(x1,x2) = x2 + (3,0)
        tt_A() = (2,1)
        isList#_A(x1) = x1 + (2,0)
        activate_A(x1) = x1
        n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (5,3)
        U21#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + x3 + (7,0)
        and_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (1,1)
        isPalListKind_A(x1) = ((0,1),(1,1)) x1 + (1,1)
        n__isPalListKind_A(x1) = ((0,1),(1,1)) x1 + (1,1)
        U11#_A(x1,x2) = x2 + (1,0)
        isNeList#_A(x1) = x1
        U51#_A(x1,x2,x3) = x2 + x3 + (4,0)
        U41#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + x3 + (3,0)
        U42#_A(x1,x2) = x2 + (1,0)
        isList_A(x1) = ((1,1),(1,1)) x1 + (5,1)
        isNeList_A(x1) = ((0,1),(1,0)) x1 + (5,1)
        U22#_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (2,0)
        U62_A(x1) = (3,1)
        U61_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,1),(1,0)) x2 + (4,0)
        isQid_A(x1) = x1 + (3,1)
        U23_A(x1) = (3,1)
        U43_A(x1) = (3,1)
        U53_A(x1) = (3,1)
        U72_A(x1) = (3,1)
        isNePal_A(x1) = ((0,1),(1,1)) x1 + (5,1)
        n__and_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (1,1)
        n__isPal_A(x1) = ((1,1),(1,0)) x1 + (4,1)
        U12_A(x1) = (3,1)
        U22_A(x1,x2) = (4,1)
        U32_A(x1) = (3,1)
        U42_A(x1,x2) = (4,1)
        U52_A(x1,x2) = (4,1)
        U71_A(x1,x2) = x1 + (2,1)
        n__a_A() = (1,2)
        n__e_A() = (1,2)
        n__i_A() = (1,2)
        n__o_A() = (0,1)
        n__u_A() = (1,2)
        ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (5,3)
        nil_A() = (1,1)
        U11_A(x1,x2) = ((0,1),(0,0)) x1 + (3,1)
        U21_A(x1,x2,x3) = x2 + (5,1)
        U31_A(x1,x2) = (4,1)
        U41_A(x1,x2,x3) = x2 + (5,1)
        U51_A(x1,x2,x3) = ((1,1),(0,0)) x2 + (5,1)
        isPal_A(x1) = ((1,1),(1,0)) x1 + (4,1)
        n__nil_A() = (1,1)
        a_A() = (1,2)
        e_A() = (1,2)
        i_A() = (1,2)
        o_A() = (0,1)
        u_A() = (1,2)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        U52#_A(x1,x2) = (0,4)
        tt_A() = (5,4)
        isList#_A(x1) = (0,1)
        activate_A(x1) = x1 + (2,3)
        n_____A(x1,x2) = x2
        U21#_A(x1,x2,x3) = (0,1)
        and_A(x1,x2) = (7,2)
        isPalListKind_A(x1) = (6,4)
        n__isPalListKind_A(x1) = (5,1)
        U11#_A(x1,x2) = (1,2)
        isNeList#_A(x1) = x1 + (0,3)
        U51#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + (2,3)
        U41#_A(x1,x2,x3) = (3,4)
        U42#_A(x1,x2) = (3,5)
        isList_A(x1) = ((1,0),(1,0)) x1 + (0,3)
        isNeList_A(x1) = (1,1)
        U22#_A(x1,x2) = (1,0)
        U62_A(x1) = (0,4)
        U61_A(x1,x2) = (1,2)
        isQid_A(x1) = x1 + (6,4)
        U23_A(x1) = (6,6)
        U43_A(x1) = (1,4)
        U53_A(x1) = (6,4)
        U72_A(x1) = (4,6)
        isNePal_A(x1) = (8,1)
        n__and_A(x1,x2) = (6,0)
        n__isPal_A(x1) = (1,1)
        U12_A(x1) = (2,5)
        U22_A(x1,x2) = (1,5)
        U32_A(x1) = (6,3)
        U42_A(x1,x2) = (0,3)
        U52_A(x1,x2) = (0,3)
        U71_A(x1,x2) = (3,5)
        n__a_A() = (0,0)
        n__e_A() = (0,1)
        n__i_A() = (1,0)
        n__o_A() = (0,1)
        n__u_A() = (0,1)
        ___A(x1,x2) = x2 + (1,0)
        nil_A() = (2,1)
        U11_A(x1,x2) = (1,4)
        U21_A(x1,x2,x3) = ((0,1),(0,0)) x2 + (0,4)
        U31_A(x1,x2) = (7,2)
        U41_A(x1,x2,x3) = x2 + (1,2)
        U51_A(x1,x2,x3) = (2,2)
        isPal_A(x1) = (2,4)
        n__nil_A() = (1,1)
        a_A() = (1,3)
        e_A() = (1,1)
        i_A() = (2,3)
        o_A() = (1,1)
        u_A() = (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        U52#_A(x1,x2) = (0,0)
        tt_A() = (6,1)
        isList#_A(x1) = (4,0)
        activate_A(x1) = ((1,1),(1,1)) x1 + (2,3)
        n_____A(x1,x2) = (2,3)
        U21#_A(x1,x2,x3) = (3,0)
        and_A(x1,x2) = (3,2)
        isPalListKind_A(x1) = (7,1)
        n__isPalListKind_A(x1) = (8,2)
        U11#_A(x1,x2) = (2,0)
        isNeList#_A(x1) = ((0,1),(1,0)) x1
        U51#_A(x1,x2,x3) = (4,2)
        U41#_A(x1,x2,x3) = (3,0)
        U42#_A(x1,x2) = (3,1)
        isList_A(x1) = x1 + (1,1)
        isNeList_A(x1) = (2,1)
        U22#_A(x1,x2) = (5,1)
        U62_A(x1) = (7,1)
        U61_A(x1,x2) = (6,1)
        isQid_A(x1) = ((0,0),(1,0)) x1 + (7,1)
        U23_A(x1) = (7,7)
        U43_A(x1) = (1,1)
        U53_A(x1) = (1,1)
        U72_A(x1) = (5,6)
        isNePal_A(x1) = (1,1)
        n__and_A(x1,x2) = (3,3)
        n__isPal_A(x1) = (0,0)
        U12_A(x1) = (1,3)
        U22_A(x1,x2) = (3,6)
        U32_A(x1) = (7,1)
        U42_A(x1,x2) = (0,0)
        U52_A(x1,x2) = (0,0)
        U71_A(x1,x2) = (4,5)
        n__a_A() = (1,1)
        n__e_A() = (0,1)
        n__i_A() = (0,0)
        n__o_A() = (1,0)
        n__u_A() = (1,1)
        ___A(x1,x2) = (3,3)
        nil_A() = (0,0)
        U11_A(x1,x2) = (0,2)
        U21_A(x1,x2,x3) = (2,5)
        U31_A(x1,x2) = (8,2)
        U41_A(x1,x2,x3) = ((0,0),(1,1)) x2 + (1,0)
        U51_A(x1,x2,x3) = (3,2)
        isPal_A(x1) = (3,4)
        n__nil_A() = (1,0)
        a_A() = (5,6)
        e_A() = (1,0)
        i_A() = (1,0)
        o_A() = (4,5)
        u_A() = (5,6)
    

The next rules are strictly ordered:

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

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: isPal#(V) -> activate#(V)
p2: activate#(n__isPal(X)) -> isPal#(X)
p3: isPal#(V) -> isPalListKind#(activate(V))
p4: isPalListKind#(n____(V1,V2)) -> activate#(V2)
p5: activate#(n__and(X1,X2)) -> activate#(X1)
p6: activate#(n__and(X1,X2)) -> and#(activate(X1),X2)
p7: and#(tt(),X) -> activate#(X)
p8: activate#(n__isPalListKind(X)) -> isPalListKind#(X)
p9: isPalListKind#(n____(V1,V2)) -> activate#(V1)
p10: activate#(n____(X1,X2)) -> activate#(X2)
p11: activate#(n____(X1,X2)) -> activate#(X1)
p12: isPalListKind#(n____(V1,V2)) -> isPalListKind#(activate(V1))
p13: isPalListKind#(n____(V1,V2)) -> and#(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
p14: isPal#(V) -> U71#(isPalListKind(activate(V)),activate(V))
p15: U71#(tt(),V) -> activate#(V)
p16: U71#(tt(),V) -> isNePal#(activate(V))
p17: isNePal#(n____(I,n____(P,I))) -> activate#(P)
p18: isNePal#(n____(I,n____(P,I))) -> activate#(I)
p19: isNePal#(n____(I,n____(P,I))) -> and#(isQid(activate(I)),n__isPalListKind(activate(I)))
p20: isNePal#(n____(I,n____(P,I))) -> and#(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))
p21: isNePal#(V) -> activate#(V)
p22: isNePal#(V) -> isPalListKind#(activate(V))
p23: isNePal#(V) -> U61#(isPalListKind(activate(V)),activate(V))
p24: U61#(tt(),V) -> activate#(V)

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: isPal(X) -> n__isPal(X)
r49: a() -> n__a()
r50: e() -> n__e()
r51: i() -> n__i()
r52: o() -> n__o()
r53: u() -> n__u()
r54: activate(n__nil()) -> nil()
r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2))
r56: activate(n__isPalListKind(X)) -> isPalListKind(X)
r57: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r58: activate(n__isPal(X)) -> isPal(X)
r59: activate(n__a()) -> a()
r60: activate(n__e()) -> e()
r61: activate(n__i()) -> i()
r62: activate(n__o()) -> o()
r63: activate(n__u()) -> u()
r64: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r17, r18, r19, r20, r21, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        isPal#_A(x1) = ((1,1),(0,0)) x1 + (2,0)
        activate#_A(x1) = ((0,1),(0,0)) x1
        n__isPal_A(x1) = ((0,0),(1,1)) x1 + (4,3)
        isPalListKind#_A(x1) = ((1,1),(0,0)) x1
        activate_A(x1) = x1
        n_____A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (4,1)
        n__and_A(x1,x2) = ((0,0),(1,1)) x1 + x2 + (1,1)
        and#_A(x1,x2) = x1 + ((0,1),(0,0)) x2
        tt_A() = (1,1)
        n__isPalListKind_A(x1) = ((0,1),(1,1)) x1 + (1,1)
        isPalListKind_A(x1) = ((0,1),(1,1)) x1 + (1,1)
        U71#_A(x1,x2) = ((1,1),(0,0)) x2 + (1,0)
        isNePal#_A(x1) = ((1,1),(0,0)) x1 + (1,0)
        isQid_A(x1) = ((1,0),(1,1)) x1 + (2,1)
        and_A(x1,x2) = ((0,0),(1,1)) x1 + x2 + (1,1)
        U61#_A(x1,x2) = ((0,1),(0,0)) x2 + (1,0)
        U62_A(x1) = (2,1)
        U61_A(x1,x2) = (3,1)
        U72_A(x1) = (2,1)
        isNePal_A(x1) = ((1,0),(1,1)) x1 + (4,7)
        U71_A(x1,x2) = (3,1)
        ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (4,1)
        nil_A() = (0,0)
        isPal_A(x1) = ((0,0),(1,1)) x1 + (4,3)
        n__nil_A() = (0,0)
        a_A() = (0,0)
        n__a_A() = (0,0)
        e_A() = (0,0)
        n__e_A() = (0,0)
        i_A() = (0,1)
        n__i_A() = (0,1)
        o_A() = (0,0)
        n__o_A() = (0,0)
        u_A() = (0,0)
        n__u_A() = (0,0)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        isPal#_A(x1) = (12,1)
        activate#_A(x1) = (12,0)
        n__isPal_A(x1) = (1,1)
        isPalListKind#_A(x1) = ((1,1),(0,1)) x1 + (13,1)
        activate_A(x1) = ((1,1),(0,1)) x1 + (12,1)
        n_____A(x1,x2) = ((1,1),(0,1)) x1 + (1,3)
        n__and_A(x1,x2) = x2 + (1,1)
        and#_A(x1,x2) = ((1,1),(0,0)) x1 + (0,5)
        tt_A() = (8,5)
        n__isPalListKind_A(x1) = (0,4)
        isPalListKind_A(x1) = (9,5)
        U71#_A(x1,x2) = (12,5)
        isNePal#_A(x1) = (12,5)
        isQid_A(x1) = ((1,1),(1,1)) x1 + (5,0)
        and_A(x1,x2) = ((0,1),(0,1)) x2 + (2,1)
        U61#_A(x1,x2) = (11,0)
        U62_A(x1) = (9,5)
        U61_A(x1,x2) = (0,2)
        U72_A(x1) = (4,3)
        isNePal_A(x1) = (8,1)
        U71_A(x1,x2) = (3,2)
        ___A(x1,x2) = ((1,1),(0,1)) x1 + (2,3)
        nil_A() = (0,0)
        isPal_A(x1) = (2,1)
        n__nil_A() = (0,0)
        a_A() = (3,1)
        n__a_A() = (2,0)
        e_A() = (5,0)
        n__e_A() = (4,0)
        i_A() = (4,1)
        n__i_A() = (3,1)
        o_A() = (2,1)
        n__o_A() = (1,1)
        u_A() = (2,0)
        n__u_A() = (1,0)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        isPal#_A(x1) = (3,5)
        activate#_A(x1) = (2,1)
        n__isPal_A(x1) = (1,1)
        isPalListKind#_A(x1) = ((1,1),(0,1)) x1 + (1,2)
        activate_A(x1) = ((0,1),(1,0)) x1 + (4,3)
        n_____A(x1,x2) = ((0,1),(0,0)) x1 + (5,1)
        n__and_A(x1,x2) = (0,0)
        and#_A(x1,x2) = ((0,1),(0,0)) x1 + (0,4)
        tt_A() = (9,7)
        n__isPalListKind_A(x1) = (11,6)
        isPalListKind_A(x1) = (10,6)
        U71#_A(x1,x2) = (2,5)
        isNePal#_A(x1) = (2,5)
        isQid_A(x1) = ((0,1),(1,0)) x1 + (10,7)
        and_A(x1,x2) = (5,3)
        U61#_A(x1,x2) = (1,0)
        U62_A(x1) = (8,7)
        U61_A(x1,x2) = (7,0)
        U72_A(x1) = (4,3)
        isNePal_A(x1) = (6,3)
        U71_A(x1,x2) = (3,2)
        ___A(x1,x2) = ((0,1),(0,1)) x1 + (6,6)
        nil_A() = (0,4)
        isPal_A(x1) = (2,1)
        n__nil_A() = (0,0)
        a_A() = (2,0)
        n__a_A() = (1,0)
        e_A() = (0,0)
        n__e_A() = (0,1)
        i_A() = (0,0)
        n__i_A() = (1,0)
        o_A() = (2,0)
        n__o_A() = (1,0)
        u_A() = (5,5)
        n__u_A() = (1,0)
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p17, p18, p19, p20, p21, p22, p23, p24

We remove them from the problem.

-- SCC decomposition.

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

p1: U71#(tt(),V) -> isNePal#(activate(V))

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: isPal(X) -> n__isPal(X)
r49: a() -> n__a()
r50: e() -> n__e()
r51: i() -> n__i()
r52: o() -> n__o()
r53: u() -> n__u()
r54: activate(n__nil()) -> nil()
r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2))
r56: activate(n__isPalListKind(X)) -> isPalListKind(X)
r57: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r58: activate(n__isPal(X)) -> isPal(X)
r59: activate(n__a()) -> a()
r60: activate(n__e()) -> e()
r61: activate(n__i()) -> i()
r62: activate(n__o()) -> o()
r63: activate(n__u()) -> u()
r64: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

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

and R consists of:

r1: __(__(X,Y),Z) -> __(X,__(Y,Z))
r2: __(X,nil()) -> X
r3: __(nil(),X) -> X
r4: U11(tt(),V) -> U12(isNeList(activate(V)))
r5: U12(tt()) -> tt()
r6: U21(tt(),V1,V2) -> U22(isList(activate(V1)),activate(V2))
r7: U22(tt(),V2) -> U23(isList(activate(V2)))
r8: U23(tt()) -> tt()
r9: U31(tt(),V) -> U32(isQid(activate(V)))
r10: U32(tt()) -> tt()
r11: U41(tt(),V1,V2) -> U42(isList(activate(V1)),activate(V2))
r12: U42(tt(),V2) -> U43(isNeList(activate(V2)))
r13: U43(tt()) -> tt()
r14: U51(tt(),V1,V2) -> U52(isNeList(activate(V1)),activate(V2))
r15: U52(tt(),V2) -> U53(isList(activate(V2)))
r16: U53(tt()) -> tt()
r17: U61(tt(),V) -> U62(isQid(activate(V)))
r18: U62(tt()) -> tt()
r19: U71(tt(),V) -> U72(isNePal(activate(V)))
r20: U72(tt()) -> tt()
r21: and(tt(),X) -> activate(X)
r22: isList(V) -> U11(isPalListKind(activate(V)),activate(V))
r23: isList(n__nil()) -> tt()
r24: isList(n____(V1,V2)) -> U21(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r25: isNeList(V) -> U31(isPalListKind(activate(V)),activate(V))
r26: isNeList(n____(V1,V2)) -> U41(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r27: isNeList(n____(V1,V2)) -> U51(and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2))),activate(V1),activate(V2))
r28: isNePal(V) -> U61(isPalListKind(activate(V)),activate(V))
r29: isNePal(n____(I,n____(P,I))) -> and(and(isQid(activate(I)),n__isPalListKind(activate(I))),n__and(n__isPal(activate(P)),n__isPalListKind(activate(P))))
r30: isPal(V) -> U71(isPalListKind(activate(V)),activate(V))
r31: isPal(n__nil()) -> tt()
r32: isPalListKind(n__a()) -> tt()
r33: isPalListKind(n__e()) -> tt()
r34: isPalListKind(n__i()) -> tt()
r35: isPalListKind(n__nil()) -> tt()
r36: isPalListKind(n__o()) -> tt()
r37: isPalListKind(n__u()) -> tt()
r38: isPalListKind(n____(V1,V2)) -> and(isPalListKind(activate(V1)),n__isPalListKind(activate(V2)))
r39: isQid(n__a()) -> tt()
r40: isQid(n__e()) -> tt()
r41: isQid(n__i()) -> tt()
r42: isQid(n__o()) -> tt()
r43: isQid(n__u()) -> tt()
r44: nil() -> n__nil()
r45: __(X1,X2) -> n____(X1,X2)
r46: isPalListKind(X) -> n__isPalListKind(X)
r47: and(X1,X2) -> n__and(X1,X2)
r48: isPal(X) -> n__isPal(X)
r49: a() -> n__a()
r50: e() -> n__e()
r51: i() -> n__i()
r52: o() -> n__o()
r53: u() -> n__u()
r54: activate(n__nil()) -> nil()
r55: activate(n____(X1,X2)) -> __(activate(X1),activate(X2))
r56: activate(n__isPalListKind(X)) -> isPalListKind(X)
r57: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r58: activate(n__isPal(X)) -> isPal(X)
r59: activate(n__a()) -> a()
r60: activate(n__e()) -> e()
r61: activate(n__i()) -> i()
r62: activate(n__o()) -> o()
r63: activate(n__u()) -> u()
r64: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r45

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        __#_A(x1,x2) = ((1,1),(1,1)) x1 + x2
        ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,1)
        nil_A() = (1,1)
        n_____A(x1,x2) = x2
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        __#_A(x1,x2) = x1 + ((1,0),(1,0)) x2
        ___A(x1,x2) = x1 + x2 + (1,1)
        nil_A() = (1,1)
        n_____A(x1,x2) = ((0,0),(1,0)) x2 + (2,2)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        __#_A(x1,x2) = x1 + ((1,0),(1,0)) x2
        ___A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (1,1)
        nil_A() = (1,1)
        n_____A(x1,x2) = (2,2)
    

The next rules are strictly ordered:

  p1, p2

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