Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
    ↳6 QDPOrderProof (⇔)
    ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 QDPOrderProof (⇔)
    ↳16 QDP
    ↳17 QDPOrderProof (⇔)
    ↳18 QDP
    ↳19 QDPOrderProof (⇔)
    ↳20 QDP
    ↳21 PisEmptyProof (⇔)
    ↳22 TRUE
  ↳23 QDP
    ↳24 QDPOrderProof (⇔)
    ↳25 QDP
    ↳26 QDPOrderProof (⇔)
    ↳27 QDP
    ↳28 QDPOrderProof (⇔)
    ↳29 QDP
    ↳30 QDPOrderProof (⇔)
    ↳31 QDP
    ↳32 PisEmptyProof (⇔)
    ↳33 TRUE
  ↳34 QDP
    ↳35 QDPOrderProof (⇔)
    ↳36 QDP
    ↳37 QDPOrderProof (⇔)
    ↳38 QDP
    ↳39 QDPOrderProof (⇔)
    ↳40 QDP
    ↳41 QDPOrderProof (⇔)
    ↳42 QDP
    ↳43 PisEmptyProof (⇔)
    ↳44 TRUE
  ↳45 QDP
    ↳46 QDPOrderProof (⇔)
    ↳47 QDP
    ↳48 QDPOrderProof (⇔)
    ↳49 QDP
    ↳50 PisEmptyProof (⇔)
    ↳51 TRUE
  ↳52 QDP
    ↳53 QDPOrderProof (⇔)
    ↳54 QDP
    ↳55 QDPOrderProof (⇔)
    ↳56 QDP
    ↳57 QDPOrderProof (⇔)
    ↳58 QDP
    ↳59 QDPOrderProof (⇔)
    ↳60 QDP
    ↳61 PisEmptyProof (⇔)
    ↳62 TRUE
  ↳63 QDP
    ↳64 QDPOrderProof (⇔)
    ↳65 QDP
    ↳66 QDPOrderProof (⇔)
    ↳67 QDP
    ↳68 PisEmptyProof (⇔)
    ↳69 TRUE
  ↳70 QDP
    ↳71 QDPOrderProof (⇔)
    ↳72 QDP
    ↳73 QDPOrderProof (⇔)
    ↳74 QDP
    ↳75 QDPOrderProof (⇔)
    ↳76 QDP
    ↳77 QDPOrderProof (⇔)
    ↳78 QDP
    ↳79 PisEmptyProof (⇔)
    ↳80 TRUE
  ↳81 QDP
    ↳82 QDPOrderProof (⇔)
    ↳83 QDP
    ↳84 QDPOrderProof (⇔)
    ↳85 QDP
    ↳86 QDPOrderProof (⇔)
    ↳87 QDP
    ↳88 QDPOrderProof (⇔)
    ↳89 QDP
    ↳90 PisEmptyProof (⇔)
    ↳91 TRUE
  ↳92 QDP
    ↳93 QDPOrderProof (⇔)
    ↳94 QDP
    ↳95 QDPOrderProof (⇔)
    ↳96 QDP
    ↳97 PisEmptyProof (⇔)
    ↳98 TRUE
  ↳99 QDP
    ↳100 QDPOrderProof (⇔)
    ↳101 QDP
    ↳102 QDPOrderProof (⇔)
    ↳103 QDP
    ↳104 QDPOrderProof (⇔)
    ↳105 QDP
    ↳106 QDPOrderProof (⇔)
    ↳107 QDP
    ↳108 PisEmptyProof (⇔)
    ↳109 TRUE
  ↳110 QDP
    ↳111 QDPOrderProof (⇔)
    ↳112 QDP
    ↳113 QDPOrderProof (⇔)
    ↳114 QDP
    ↳115 QDPOrderProof (⇔)
    ↳116 QDP
    ↳117 QDPOrderProof (⇔)
    ↳118 QDP
    ↳119 PisEmptyProof (⇔)
    ↳120 TRUE
  ↳121 QDP
    ↳122 QDPOrderProof (⇔)
    ↳123 QDP
    ↳124 QDPOrderProof (⇔)
    ↳125 QDP
    ↳126 PisEmptyProof (⇔)
    ↳127 TRUE
  ↳128 QDP
    ↳129 QDPOrderProof (⇔)
    ↳130 QDP
    ↳131 QDPOrderProof (⇔)
    ↳132 QDP
    ↳133 QDPOrderProof (⇔)
    ↳134 QDP
    ↳135 QDPOrderProof (⇔)
    ↳136 QDP
    ↳137 PisEmptyProof (⇔)
    ↳138 TRUE
  ↳139 QDP
    ↳140 QDPOrderProof (⇔)
    ↳141 QDP
    ↳142 QDPOrderProof (⇔)
    ↳143 QDP
    ↳144 PisEmptyProof (⇔)
    ↳145 TRUE
  ↳146 QDP
    ↳147 QDPOrderProof (⇔)
    ↳148 QDP
    ↳149 QDPOrderProof (⇔)
    ↳150 QDP
    ↳151 PisEmptyProof (⇔)
    ↳152 TRUE
  ↳153 QDP
    ↳154 QDPOrderProof (⇔)
    ↳155 QDP
    ↳156 QDPOrderProof (⇔)
    ↳157 QDP
    ↳158 QDPOrderProof (⇔)
    ↳159 QDP
    ↳160 QDPOrderProof (⇔)
    ↳161 QDP
    ↳162 PisEmptyProof (⇔)
    ↳163 TRUE
  ↳164 QDP
    ↳165 QDPOrderProof (⇔)
    ↳166 QDP
    ↳167 QDPOrderProof (⇔)
    ↳168 QDP
    ↳169 QDPOrderProof (⇔)
    ↳170 QDP
    ↳171 QDPOrderProof (⇔)
    ↳172 QDP
    ↳173 PisEmptyProof (⇔)
    ↳174 TRUE
  ↳175 QDP
    ↳176 QDPOrderProof (⇔)
    ↳177 QDP
    ↳178 QDPOrderProof (⇔)
    ↳179 QDP
    ↳180 QDPOrderProof (⇔)
    ↳181 QDP
    ↳182 DependencyGraphProof (⇔)
    ↳183 QDP
    ↳184 QDPOrderProof (⇔)
    ↳185 QDP
    ↳186 QDPOrderProof (⇔)
    ↳187 QDP
    ↳188 QDPOrderProof (⇔)
    ↳189 QDP
    ↳190 PisEmptyProof (⇔)
    ↳191 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
ACTIVE(U11(tt, V1, V2)) → U121(isNat(V1), V2)
ACTIVE(U11(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
ACTIVE(U12(tt, V2)) → U131(isNat(V2))
ACTIVE(U12(tt, V2)) → ISNAT(V2)
ACTIVE(U13(tt)) → MARK(tt)
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ACTIVE(U21(tt, V1)) → U221(isNat(V1))
ACTIVE(U21(tt, V1)) → ISNAT(V1)
ACTIVE(U22(tt)) → MARK(tt)
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
ACTIVE(U31(tt, V1, V2)) → U321(isNat(V1), V2)
ACTIVE(U31(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
ACTIVE(U32(tt, V2)) → U331(isNat(V2))
ACTIVE(U32(tt, V2)) → ISNAT(V2)
ACTIVE(U33(tt)) → MARK(tt)
ACTIVE(U41(tt, N)) → MARK(N)
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
ACTIVE(U51(tt, M, N)) → S(plus(N, M))
ACTIVE(U51(tt, M, N)) → PLUS(N, M)
ACTIVE(U61(tt)) → MARK(0)
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
ACTIVE(U71(tt, M, N)) → PLUS(x(N, M), N)
ACTIVE(U71(tt, M, N)) → X(N, M)
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
ACTIVE(isNat(plus(V1, V2))) → U111(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ACTIVE(isNat(plus(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V2)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNat(s(V1))) → U211(isNatKind(V1), V1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
ACTIVE(isNat(x(V1, V2))) → U311(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ACTIVE(isNat(x(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
ACTIVE(isNat(x(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNat(x(V1, V2))) → ISNATKIND(V2)
ACTIVE(isNatKind(0)) → MARK(tt)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
ACTIVE(isNatKind(plus(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V2)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
ACTIVE(isNatKind(x(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
ACTIVE(isNatKind(x(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(x(V1, V2))) → ISNATKIND(V2)
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
ACTIVE(plus(N, 0)) → U411(and(isNat(N), isNatKind(N)), N)
ACTIVE(plus(N, 0)) → AND(isNat(N), isNatKind(N))
ACTIVE(plus(N, 0)) → ISNAT(N)
ACTIVE(plus(N, 0)) → ISNATKIND(N)
ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
ACTIVE(plus(N, s(M))) → U511(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ACTIVE(plus(N, s(M))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ACTIVE(plus(N, s(M))) → AND(isNat(M), isNatKind(M))
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(plus(N, s(M))) → ISNATKIND(M)
ACTIVE(plus(N, s(M))) → AND(isNat(N), isNatKind(N))
ACTIVE(plus(N, s(M))) → ISNAT(N)
ACTIVE(plus(N, s(M))) → ISNATKIND(N)
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
ACTIVE(x(N, 0)) → U611(and(isNat(N), isNatKind(N)))
ACTIVE(x(N, 0)) → AND(isNat(N), isNatKind(N))
ACTIVE(x(N, 0)) → ISNAT(N)
ACTIVE(x(N, 0)) → ISNATKIND(N)
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
ACTIVE(x(N, s(M))) → U711(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ACTIVE(x(N, s(M))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ACTIVE(x(N, s(M))) → AND(isNat(M), isNatKind(M))
ACTIVE(x(N, s(M))) → ISNAT(M)
ACTIVE(x(N, s(M))) → ISNATKIND(M)
ACTIVE(x(N, s(M))) → AND(isNat(N), isNatKind(N))
ACTIVE(x(N, s(M))) → ISNAT(N)
ACTIVE(x(N, s(M))) → ISNATKIND(N)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → U111(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(U12(X1, X2)) → U121(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U13(X)) → ACTIVE(U13(mark(X)))
MARK(U13(X)) → U131(mark(X))
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → U211(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
MARK(U22(X)) → U221(mark(X))
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
MARK(U31(X1, X2, X3)) → U311(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
MARK(U32(X1, X2)) → U321(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → ACTIVE(U33(mark(X)))
MARK(U33(X)) → U331(mark(X))
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U41(X1, X2)) → U411(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → U511(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
MARK(U61(X)) → U611(mark(X))
MARK(U61(X)) → MARK(X)
MARK(0) → ACTIVE(0)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → U711(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(x(X1, X2)) → X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(and(X1, X2)) → AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, mark(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
U111(active(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
U131(mark(X)) → U131(X)
U131(active(X)) → U131(X)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X)) → U221(X)
U221(active(X)) → U221(X)
U311(mark(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, mark(X2), X3) → U311(X1, X2, X3)
U311(X1, X2, mark(X3)) → U311(X1, X2, X3)
U311(active(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, active(X2), X3) → U311(X1, X2, X3)
U311(X1, X2, active(X3)) → U311(X1, X2, X3)
U321(mark(X1), X2) → U321(X1, X2)
U321(X1, mark(X2)) → U321(X1, X2)
U321(active(X1), X2) → U321(X1, X2)
U321(X1, active(X2)) → U321(X1, X2)
U331(mark(X)) → U331(X)
U331(active(X)) → U331(X)
U411(mark(X1), X2) → U411(X1, X2)
U411(X1, mark(X2)) → U411(X1, X2)
U411(active(X1), X2) → U411(X1, X2)
U411(X1, active(X2)) → U411(X1, X2)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(X1, mark(X2), X3) → U511(X1, X2, X3)
U511(X1, X2, mark(X3)) → U511(X1, X2, X3)
U511(active(X1), X2, X3) → U511(X1, X2, X3)
U511(X1, active(X2), X3) → U511(X1, X2, X3)
U511(X1, X2, active(X3)) → U511(X1, X2, X3)
S(mark(X)) → S(X)
S(active(X)) → S(X)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)
U611(mark(X)) → U611(X)
U611(active(X)) → U611(X)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, mark(X2), X3) → U711(X1, X2, X3)
U711(X1, X2, mark(X3)) → U711(X1, X2, X3)
U711(active(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, active(X2), X3) → U711(X1, X2, X3)
U711(X1, X2, active(X3)) → U711(X1, X2, X3)
X(mark(X1), X2) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)
ISNATKIND(mark(X)) → ISNATKIND(X)
ISNATKIND(active(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 19 SCCs with 79 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATKIND(active(X)) → ISNATKIND(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ISNATKIND1, active1]

Status:
active1: multiset
ISNATKIND1: multiset


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATKIND(mark(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATKIND(mark(X)) → ISNATKIND(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > ISNATKIND1

Status:
ISNATKIND1: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(X1, mark(X2)) → AND(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AND(X1, mark(X2)) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  AND(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[AND1, mark1]

Status:
AND1: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AND(mark(X1), X2) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  AND(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > AND2

Status:
AND2: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AND(X1, active(X2)) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  x2
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(active(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AND(active(X1), X2) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  AND(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[AND1, active1]

Status:
active1: [1]
AND1: multiset


The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(21) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(22) TRUE

(23) Obligation:

Q DP problem:
The TRS P consists of the following rules:

X(X1, mark(X2)) → X(X1, X2)
X(mark(X1), X2) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


X(X1, mark(X2)) → X(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  X(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[X1, mark1]

Status:
X1: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

X(mark(X1), X2) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


X(mark(X1), X2) → X(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  X(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > X2

Status:
X2: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


X(X1, active(X2)) → X(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  x2
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

X(active(X1), X2) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


X(active(X1), X2) → X(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  X(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[X1, active1]

Status:
X1: multiset
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(32) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(33) TRUE

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U711(X1, mark(X2), X3) → U711(X1, X2, X3)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, X2, mark(X3)) → U711(X1, X2, X3)
U711(active(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, active(X2), X3) → U711(X1, X2, X3)
U711(X1, X2, active(X3)) → U711(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(X1, mark(X2), X3) → U711(X1, X2, X3)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, X2, mark(X3)) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  U711(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U71^13: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U711(active(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, active(X2), X3) → U711(X1, X2, X3)
U711(X1, X2, active(X3)) → U711(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(X1, active(X2), X3) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  U711(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U71^11, active1]

Status:
active1: [1]
U71^11: multiset


The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U711(active(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, X2, active(X3)) → U711(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(X1, X2, active(X3)) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  x3
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(40) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U711(active(X1), X2, X3) → U711(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(active(X1), X2, X3) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  U711(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U71^11, active1]

Status:
active1: multiset
U71^11: multiset


The following usable rules [FROCOS05] were oriented: none

(42) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(43) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(44) TRUE

(45) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U611(active(X)) → U611(X)
U611(mark(X)) → U611(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(active(X)) → U611(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1)  =  U611(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U61^11, active1]

Status:
active1: multiset
U61^11: multiset


The following usable rules [FROCOS05] were oriented: none

(47) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U611(mark(X)) → U611(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(mark(X)) → U611(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U61^11

Status:
U61^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(49) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(50) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(51) TRUE

(52) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(X1, mark(X2)) → PLUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PLUS1, mark1]

Status:
mark1: multiset
PLUS1: multiset


The following usable rules [FROCOS05] were oriented: none

(54) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(mark(X1), X2) → PLUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > PLUS2

Status:
PLUS2: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(56) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(X1, active(X2)) → PLUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  x2
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(58) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS(active(X1), X2) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(active(X1), X2) → PLUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[PLUS1, active1]

Status:
active1: [1]
PLUS1: multiset


The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(61) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(62) TRUE

(63) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(64) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(active(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[S1, active1]

Status:
active1: multiset
S1: multiset


The following usable rules [FROCOS05] were oriented: none

(65) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(mark(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > S1

Status:
mark1: multiset
S1: multiset


The following usable rules [FROCOS05] were oriented: none

(67) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(68) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(69) TRUE

(70) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U511(X1, mark(X2), X3) → U511(X1, X2, X3)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(X1, X2, mark(X3)) → U511(X1, X2, X3)
U511(active(X1), X2, X3) → U511(X1, X2, X3)
U511(X1, active(X2), X3) → U511(X1, X2, X3)
U511(X1, X2, active(X3)) → U511(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(X1, mark(X2), X3) → U511(X1, X2, X3)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(X1, X2, mark(X3)) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2, x3)  =  U511(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U51^13: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(72) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U511(active(X1), X2, X3) → U511(X1, X2, X3)
U511(X1, active(X2), X3) → U511(X1, X2, X3)
U511(X1, X2, active(X3)) → U511(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(X1, active(X2), X3) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2, x3)  =  U511(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U51^11, active1]

Status:
active1: [1]
U51^11: multiset


The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U511(active(X1), X2, X3) → U511(X1, X2, X3)
U511(X1, X2, active(X3)) → U511(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(X1, X2, active(X3)) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2, x3)  =  x3
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(76) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U511(active(X1), X2, X3) → U511(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(active(X1), X2, X3) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2, x3)  =  U511(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U51^11, active1]

Status:
active1: multiset
U51^11: multiset


The following usable rules [FROCOS05] were oriented: none

(78) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(79) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(80) TRUE

(81) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U411(X1, mark(X2)) → U411(X1, X2)
U411(mark(X1), X2) → U411(X1, X2)
U411(active(X1), X2) → U411(X1, X2)
U411(X1, active(X2)) → U411(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(X1, mark(X2)) → U411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2)  =  U411(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U41^11, mark1]

Status:
U41^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U411(mark(X1), X2) → U411(X1, X2)
U411(active(X1), X2) → U411(X1, X2)
U411(X1, active(X2)) → U411(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(84) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(mark(X1), X2) → U411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2)  =  U411(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U41^12

Status:
U41^12: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(85) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U411(active(X1), X2) → U411(X1, X2)
U411(X1, active(X2)) → U411(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(X1, active(X2)) → U411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2)  =  x2
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U411(active(X1), X2) → U411(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(active(X1), X2) → U411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2)  =  U411(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U41^11, active1]

Status:
active1: [1]
U41^11: multiset


The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(90) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(91) TRUE

(92) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U331(active(X)) → U331(X)
U331(mark(X)) → U331(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U331(active(X)) → U331(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U331(x1)  =  U331(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U33^11, active1]

Status:
active1: multiset
U33^11: multiset


The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U331(mark(X)) → U331(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U331(mark(X)) → U331(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U33^11

Status:
U33^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(96) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(97) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(98) TRUE

(99) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U321(X1, mark(X2)) → U321(X1, X2)
U321(mark(X1), X2) → U321(X1, X2)
U321(active(X1), X2) → U321(X1, X2)
U321(X1, active(X2)) → U321(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(X1, mark(X2)) → U321(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1, x2)  =  U321(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U32^11, mark1]

Status:
mark1: multiset
U32^11: multiset


The following usable rules [FROCOS05] were oriented: none

(101) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U321(mark(X1), X2) → U321(X1, X2)
U321(active(X1), X2) → U321(X1, X2)
U321(X1, active(X2)) → U321(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(mark(X1), X2) → U321(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1, x2)  =  U321(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U32^12

Status:
U32^12: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(103) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U321(active(X1), X2) → U321(X1, X2)
U321(X1, active(X2)) → U321(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(X1, active(X2)) → U321(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1, x2)  =  x2
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U321(active(X1), X2) → U321(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(active(X1), X2) → U321(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1, x2)  =  U321(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U32^11, active1]

Status:
active1: [1]
U32^11: multiset


The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(108) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(109) TRUE

(110) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U311(X1, mark(X2), X3) → U311(X1, X2, X3)
U311(mark(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, X2, mark(X3)) → U311(X1, X2, X3)
U311(active(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, active(X2), X3) → U311(X1, X2, X3)
U311(X1, X2, active(X3)) → U311(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(X1, mark(X2), X3) → U311(X1, X2, X3)
U311(mark(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, X2, mark(X3)) → U311(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3)  =  U311(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U31^13: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(112) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U311(active(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, active(X2), X3) → U311(X1, X2, X3)
U311(X1, X2, active(X3)) → U311(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(X1, active(X2), X3) → U311(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3)  =  U311(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U31^11, active1]

Status:
active1: [1]
U31^11: multiset


The following usable rules [FROCOS05] were oriented: none

(114) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U311(active(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, X2, active(X3)) → U311(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(X1, X2, active(X3)) → U311(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3)  =  x3
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U311(active(X1), X2, X3) → U311(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(117) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(active(X1), X2, X3) → U311(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3)  =  U311(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U31^11, active1]

Status:
active1: multiset
U31^11: multiset


The following usable rules [FROCOS05] were oriented: none

(118) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(119) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(120) TRUE

(121) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U221(active(X)) → U221(X)
U221(mark(X)) → U221(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(active(X)) → U221(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U221(x1)  =  U221(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U22^11, active1]

Status:
active1: multiset
U22^11: multiset


The following usable rules [FROCOS05] were oriented: none

(123) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U221(mark(X)) → U221(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(mark(X)) → U221(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U22^11

Status:
U22^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(125) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(126) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(127) TRUE

(128) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U211(X1, mark(X2)) → U211(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(129) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(X1, mark(X2)) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  U211(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U21^11, mark1]

Status:
mark1: multiset
U21^11: multiset


The following usable rules [FROCOS05] were oriented: none

(130) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U211(mark(X1), X2) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(mark(X1), X2) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  U211(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U21^12

Status:
U21^12: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(132) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(133) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(X1, active(X2)) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  x2
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(134) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U211(active(X1), X2) → U211(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(135) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(active(X1), X2) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  U211(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U21^11, active1]

Status:
active1: [1]
U21^11: multiset


The following usable rules [FROCOS05] were oriented: none

(136) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(137) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(138) TRUE

(139) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U131(active(X)) → U131(X)
U131(mark(X)) → U131(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U131(active(X)) → U131(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131(x1)  =  U131(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U13^11, active1]

Status:
active1: multiset
U13^11: multiset


The following usable rules [FROCOS05] were oriented: none

(141) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U131(mark(X)) → U131(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(142) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U131(mark(X)) → U131(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U13^11

Status:
mark1: multiset
U13^11: multiset


The following usable rules [FROCOS05] were oriented: none

(143) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(144) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(145) TRUE

(146) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(147) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(active(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ISNAT1, active1]

Status:
active1: multiset
ISNAT1: multiset


The following usable rules [FROCOS05] were oriented: none

(148) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(mark(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > ISNAT1

Status:
mark1: multiset
ISNAT1: multiset


The following usable rules [FROCOS05] were oriented: none

(150) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(151) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(152) TRUE

(153) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U121(X1, mark(X2)) → U121(X1, X2)
U121(mark(X1), X2) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(154) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(X1, mark(X2)) → U121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2)  =  U121(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U12^11, mark1]

Status:
U12^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(155) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U121(mark(X1), X2) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(156) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(mark(X1), X2) → U121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2)  =  U121(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U12^12

Status:
U12^12: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(157) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(158) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(X1, active(X2)) → U121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2)  =  x2
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(159) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U121(active(X1), X2) → U121(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(160) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(active(X1), X2) → U121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2)  =  U121(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U12^11, active1]

Status:
active1: [1]
U12^11: multiset


The following usable rules [FROCOS05] were oriented: none

(161) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(162) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(163) TRUE

(164) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U111(X1, mark(X2), X3) → U111(X1, X2, X3)
U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
U111(active(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(165) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(X1, mark(X2), X3) → U111(X1, X2, X3)
U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2, x3)  =  U111(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U11^13: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(166) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U111(active(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(167) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(X1, active(X2), X3) → U111(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2, x3)  =  U111(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U11^11, active1]

Status:
active1: [1]
U11^11: multiset


The following usable rules [FROCOS05] were oriented: none

(168) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U111(active(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(169) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(X1, X2, active(X3)) → U111(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2, x3)  =  x3
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(170) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U111(active(X1), X2, X3) → U111(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(active(X1), X2, X3) → U111(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2, x3)  =  U111(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U11^11, active1]

Status:
active1: multiset
U11^11: multiset


The following usable rules [FROCOS05] were oriented: none

(172) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(173) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(174) TRUE

(175) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
MARK(U13(X)) → ACTIVE(U13(mark(X)))
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
ACTIVE(U41(tt, N)) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → ACTIVE(U33(mark(X)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(176) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U13(X)) → ACTIVE(U13(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark(x1)
tt  =  tt
U12(x1, x2)  =  U12
isNat(x1)  =  isNat
U13(x1)  =  U13
U21(x1, x2)  =  U21
U22(x1)  =  U22
U31(x1, x2, x3)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2)  =  U41
U51(x1, x2, x3)  =  U51
s(x1)  =  s
plus(x1, x2)  =  plus
U71(x1, x2, x3)  =  U71
x(x1, x2)  =  x
and(x1, x2)  =  and
isNatKind(x1)  =  isNatKind
U61(x1)  =  U61
0  =  0
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[MARK, U11, tt, U12, isNat, U21, U22, U31, U32, U33, U41, U51, s, plus, U71, x, and, isNatKind, U61] > mark1 > U13 > active1 > 0

Status:
U13: multiset
MARK: []
U21: []
U51: []
U22: []
U32: []
U31: []
plus: []
U12: []
U11: []
mark1: [1]
U33: []
s: []
U71: []
0: multiset
isNat: []
active1: [1]
U61: []
tt: multiset
x: []
U41: []
and: []
isNatKind: []


The following usable rules [FROCOS05] were oriented:

and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
x(X1, active(X2)) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)
U61(active(X)) → U61(X)
U61(mark(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U32(X1, mark(X2)) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U33(active(X)) → U33(X)
U33(mark(X)) → U33(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U13(active(X)) → U13(X)
U13(mark(X)) → U13(X)

(177) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
ACTIVE(U41(tt, N)) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → ACTIVE(U33(mark(X)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(178) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U61(X)) → ACTIVE(U61(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark(x1)
tt  =  tt
U12(x1, x2)  =  U12
isNat(x1)  =  isNat
U13(x1)  =  x1
U21(x1, x2)  =  U21
U22(x1)  =  U22
U31(x1, x2, x3)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2)  =  U41
U51(x1, x2, x3)  =  U51
s(x1)  =  s
plus(x1, x2)  =  plus
U71(x1, x2, x3)  =  U71
x(x1, x2)  =  x
and(x1, x2)  =  and
isNatKind(x1)  =  isNatKind
U61(x1)  =  U61
0  =  0
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > 0 > [MARK, U11, U12, isNat, U21, U22, U31, U32, U33, U41, U51, s, plus, U71, x, and, isNatKind] > tt > [U61, active1]

Status:
MARK: []
U21: []
U51: []
U22: []
U32: []
U31: []
plus: []
U12: []
U11: []
mark1: multiset
U33: []
s: []
U71: []
0: multiset
isNat: []
active1: multiset
U61: multiset
tt: multiset
x: []
U41: []
and: []
isNatKind: []


The following usable rules [FROCOS05] were oriented:

and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
x(X1, active(X2)) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)
U61(active(X)) → U61(X)
U61(mark(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U32(X1, mark(X2)) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U33(active(X)) → U33(X)
U33(mark(X)) → U33(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)

(179) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
ACTIVE(U41(tt, N)) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → ACTIVE(U33(mark(X)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(180) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U12(X1, X2)) → MARK(X1)
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(U41(tt, N)) → MARK(N)
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U31(X1, X2, X3)) → MARK(X1)
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(U32(X1, X2)) → MARK(X1)
ACTIVE(and(tt, X)) → MARK(X)
MARK(U33(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(U41(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(s(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U71(X1, X2, X3)) → MARK(X1)
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  U11(x1, x2, x3)
ACTIVE(x1)  =  x1
mark(x1)  =  x1
tt  =  tt
U12(x1, x2)  =  U12(x1, x2)
isNat(x1)  =  x1
U13(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2)  =  U32(x1, x2)
U33(x1)  =  U33(x1)
U41(x1, x2)  =  U41(x1, x2)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)
and(x1, x2)  =  and(x1, x2)
isNatKind(x1)  =  x1
U61(x1)  =  x1
0  =  0
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U713, x2] > [U313, U322] > [tt, U331, and2, 0]
[U713, x2] > [U513, plus2] > [U113, U122] > [tt, U331, and2, 0]
[U713, x2] > [U513, plus2] > U412 > [tt, U331, and2, 0]
[U713, x2] > [U513, plus2] > s1 > U212 > [tt, U331, and2, 0]

Status:
plus2: [2,1]
U322: [1,2]
U113: multiset
x2: [1,2]
U122: multiset
and2: [2,1]
0: multiset
U212: [1,2]
U313: [2,3,1]
tt: multiset
U412: multiset
s1: [1]
U331: [1]
U513: [2,3,1]
U713: [3,2,1]


The following usable rules [FROCOS05] were oriented:

and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
x(X1, active(X2)) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(U41(tt, N)) → mark(N)
mark(U33(X)) → active(U33(mark(X)))
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
mark(U13(X)) → active(U13(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U61(X)) → active(U61(mark(X)))
mark(isNat(X)) → active(isNat(X))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(and(tt, X)) → mark(X)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
mark(isNatKind(X)) → active(isNatKind(X))
mark(s(X)) → active(s(mark(X)))
mark(U22(X)) → active(U22(mark(X)))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
U61(active(X)) → U61(X)
U61(mark(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
mark(tt) → active(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(U22(tt)) → mark(tt)
U32(X1, mark(X2)) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
active(U13(tt)) → mark(tt)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
active(U33(tt)) → mark(tt)
active(isNat(0)) → mark(tt)
active(U61(tt)) → mark(0)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U33(active(X)) → U33(X)
U33(mark(X)) → U33(X)
active(isNatKind(0)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U13(active(X)) → U13(X)
U13(mark(X)) → U13(X)

(181) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U22(X)) → ACTIVE(U22(mark(X)))
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
MARK(U33(X)) → ACTIVE(U33(mark(X)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(182) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 16 less nodes.

(183) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U22(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(184) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U22(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U22(x1)  =  U22(x1)
U13(x1)  =  x1
U61(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[MARK1, U221]

Status:
U221: multiset
MARK1: multiset


The following usable rules [FROCOS05] were oriented: none

(185) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U13(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(186) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U13(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U13(x1)  =  U13(x1)
U61(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[MARK1, U131]

Status:
MARK1: multiset
U131: multiset


The following usable rules [FROCOS05] were oriented: none

(187) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U61(X)) → MARK(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(188) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U61(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
U611 > MARK1

Status:
MARK1: multiset
U611: multiset


The following usable rules [FROCOS05] were oriented: none

(189) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(190) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(191) TRUE