Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
    ↳6 QDPOrderProof (⇔)
    ↳7 QDP
    ↳8 PisEmptyProof (⇔)
    ↳9 TRUE
  ↳10 QDP
    ↳11 QDPOrderProof (⇔)
    ↳12 QDP
    ↳13 PisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 QDP
    ↳16 QDPOrderProof (⇔)
    ↳17 QDP
    ↳18 QDPOrderProof (⇔)
    ↳19 QDP
    ↳20 PisEmptyProof (⇔)
    ↳21 TRUE
  ↳22 QDP
    ↳23 QDPOrderProof (⇔)
    ↳24 QDP
    ↳25 QDPOrderProof (⇔)
    ↳26 QDP
    ↳27 PisEmptyProof (⇔)
    ↳28 TRUE
  ↳29 QDP
    ↳30 QDPOrderProof (⇔)
    ↳31 QDP
    ↳32 QDPOrderProof (⇔)
    ↳33 QDP
    ↳34 PisEmptyProof (⇔)
    ↳35 TRUE
  ↳36 QDP
    ↳37 QDPOrderProof (⇔)
    ↳38 QDP
    ↳39 QDPOrderProof (⇔)
    ↳40 QDP
    ↳41 PisEmptyProof (⇔)
    ↳42 TRUE
  ↳43 QDP
    ↳44 QDPOrderProof (⇔)
    ↳45 QDP
    ↳46 QDPOrderProof (⇔)
    ↳47 QDP
    ↳48 PisEmptyProof (⇔)
    ↳49 TRUE
  ↳50 QDP
    ↳51 QDPOrderProof (⇔)
    ↳52 QDP
    ↳53 QDPOrderProof (⇔)
    ↳54 QDP
    ↳55 PisEmptyProof (⇔)
    ↳56 TRUE
  ↳57 QDP
    ↳58 QDPOrderProof (⇔)
    ↳59 QDP
    ↳60 QDPOrderProof (⇔)
    ↳61 QDP
    ↳62 PisEmptyProof (⇔)
    ↳63 TRUE
  ↳64 QDP
    ↳65 QDPOrderProof (⇔)
    ↳66 QDP
    ↳67 QDPOrderProof (⇔)
    ↳68 QDP
    ↳69 PisEmptyProof (⇔)
    ↳70 TRUE
  ↳71 QDP
    ↳72 QDPOrderProof (⇔)
    ↳73 QDP
    ↳74 QDPOrderProof (⇔)
    ↳75 QDP
    ↳76 PisEmptyProof (⇔)
    ↳77 TRUE
  ↳78 QDP
    ↳79 QDPOrderProof (⇔)
    ↳80 QDP
    ↳81 QDPOrderProof (⇔)
    ↳82 QDP
    ↳83 PisEmptyProof (⇔)
    ↳84 TRUE
  ↳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 PisEmptyProof (⇔)
    ↳105 TRUE
  ↳106 QDP
    ↳107 QDPOrderProof (⇔)
    ↳108 QDP
    ↳109 QDPOrderProof (⇔)
    ↳110 QDP
    ↳111 PisEmptyProof (⇔)
    ↳112 TRUE
  ↳113 QDP
    ↳114 QDPOrderProof (⇔)
    ↳115 QDP
    ↳116 QDPOrderProof (⇔)
    ↳117 QDP
    ↳118 PisEmptyProof (⇔)
    ↳119 TRUE
  ↳120 QDP
    ↳121 QDPOrderProof (⇔)
    ↳122 QDP
    ↳123 QDPOrderProof (⇔)
    ↳124 QDP
    ↳125 PisEmptyProof (⇔)
    ↳126 TRUE
  ↳127 QDP
    ↳128 QDPOrderProof (⇔)
    ↳129 QDP
    ↳130 PisEmptyProof (⇔)
    ↳131 TRUE
  ↳132 QDP
    ↳133 QDPOrderProof (⇔)
    ↳134 QDP
    ↳135 QDPOrderProof (⇔)
    ↳136 QDP
    ↳137 QDPOrderProof (⇔)
    ↳138 QDP
    ↳139 QDPOrderProof (⇔)
    ↳140 QDP
    ↳141 QDPOrderProof (⇔)
    ↳142 QDP
    ↳143 QDPOrderProof (⇔)
    ↳144 QDP
    ↳145 QDPOrderProof (⇔)
    ↳146 QDP
    ↳147 PisEmptyProof (⇔)
    ↳148 TRUE
  ↳149 QDP

(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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)) → U121(isNat(V1), V2)
ACTIVE(U11(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U12(tt, V2)) → U131(isNat(V2))
ACTIVE(U12(tt, V2)) → ISNAT(V2)
ACTIVE(U21(tt, V1)) → U221(isNat(V1))
ACTIVE(U21(tt, V1)) → ISNAT(V1)
ACTIVE(U31(tt, V1, V2)) → U321(isNat(V1), V2)
ACTIVE(U31(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U32(tt, V2)) → U331(isNat(V2))
ACTIVE(U32(tt, V2)) → ISNAT(V2)
ACTIVE(U51(tt, M, N)) → S(plus(N, M))
ACTIVE(U51(tt, M, N)) → PLUS(N, M)
ACTIVE(U71(tt, M, N)) → PLUS(x(N, M), N)
ACTIVE(U71(tt, M, N)) → X(N, M)
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))) → U211(isNatKind(V1), V1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
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(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))) → ISNATKIND(V1)
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)) → 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))) → 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)) → 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))) → 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)
ACTIVE(U11(X1, X2, X3)) → U111(active(X1), X2, X3)
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → U121(active(X1), X2)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → U131(active(X))
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → U211(active(X1), X2)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → U221(active(X))
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3)) → U311(active(X1), X2, X3)
ACTIVE(U31(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → U321(active(X1), X2)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → U331(active(X))
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → U411(active(X1), X2)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(X1, X2, X3)) → U511(active(X1), X2, X3)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → PLUS(active(X1), X2)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → PLUS(X1, active(X2))
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(U61(X)) → U611(active(X))
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2, X3)) → U711(active(X1), X2, X3)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → X(active(X1), X2)
ACTIVE(x(X1, X2)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → X(X1, active(X2))
ACTIVE(x(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U121(mark(X1), X2) → U121(X1, X2)
U131(mark(X)) → U131(X)
U211(mark(X1), X2) → U211(X1, X2)
U221(mark(X)) → U221(X)
U311(mark(X1), X2, X3) → U311(X1, X2, X3)
U321(mark(X1), X2) → U321(X1, X2)
U331(mark(X)) → U331(X)
U411(mark(X1), X2) → U411(X1, X2)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
S(mark(X)) → S(X)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
U611(mark(X)) → U611(X)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
X(mark(X1), X2) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
PROPER(U11(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
PROPER(U11(X1, X2, X3)) → PROPER(X1)
PROPER(U11(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X3)
PROPER(U12(X1, X2)) → U121(proper(X1), proper(X2))
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → ISNAT(proper(X))
PROPER(isNat(X)) → PROPER(X)
PROPER(U13(X)) → U131(proper(X))
PROPER(U13(X)) → PROPER(X)
PROPER(U21(X1, X2)) → U211(proper(X1), proper(X2))
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X)) → U221(proper(X))
PROPER(U22(X)) → PROPER(X)
PROPER(U31(X1, X2, X3)) → U311(proper(X1), proper(X2), proper(X3))
PROPER(U31(X1, X2, X3)) → PROPER(X1)
PROPER(U31(X1, X2, X3)) → PROPER(X2)
PROPER(U31(X1, X2, X3)) → PROPER(X3)
PROPER(U32(X1, X2)) → U321(proper(X1), proper(X2))
PROPER(U32(X1, X2)) → PROPER(X1)
PROPER(U32(X1, X2)) → PROPER(X2)
PROPER(U33(X)) → U331(proper(X))
PROPER(U33(X)) → PROPER(X)
PROPER(U41(X1, X2)) → U411(proper(X1), proper(X2))
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → U511(proper(X1), proper(X2), proper(X3))
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PLUS(proper(X1), proper(X2))
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U61(X)) → U611(proper(X))
PROPER(U61(X)) → PROPER(X)
PROPER(U71(X1, X2, X3)) → U711(proper(X1), proper(X2), proper(X3))
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(x(X1, X2)) → X(proper(X1), proper(X2))
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatKind(X)) → ISNATKIND(proper(X))
PROPER(isNatKind(X)) → PROPER(X)
U111(ok(X1), ok(X2), ok(X3)) → U111(X1, X2, X3)
U121(ok(X1), ok(X2)) → U121(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
U131(ok(X)) → U131(X)
U211(ok(X1), ok(X2)) → U211(X1, X2)
U221(ok(X)) → U221(X)
U311(ok(X1), ok(X2), ok(X3)) → U311(X1, X2, X3)
U321(ok(X1), ok(X2)) → U321(X1, X2)
U331(ok(X)) → U331(X)
U411(ok(X1), ok(X2)) → U411(X1, X2)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
S(ok(X)) → S(X)
PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
U611(ok(X)) → U611(X)
U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)
X(ok(X1), ok(X2)) → X(X1, X2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ISNATKIND(ok(X)) → ISNATKIND(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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 21 SCCs with 93 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNATKIND(ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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(ok(X)) → ISNATKIND(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
ok1 > ISNATKIND1

The following usable rules [FROCOS05] were oriented: none

(7) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

ISNAT(ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(ok(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
ok1 > ISNAT1

The following usable rules [FROCOS05] were oriented: none

(12) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

AND(ok(X1), ok(X2)) → AND(X1, X2)
AND(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AND(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > AND1
mark1 > AND1

The following usable rules [FROCOS05] were oriented: none

(17) Obligation:

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

AND(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(18) 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)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(19) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(20) PisEmptyProof (EQUIVALENT transformation)

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

(21) TRUE

(22) 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(ok(X1), ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(23) 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)
X(ok(X1), ok(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
mark(x1)  =  mark(x1)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

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

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

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

(25) 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)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(26) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(27) PisEmptyProof (EQUIVALENT transformation)

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

(28) TRUE

(29) Obligation:

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

U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)
U711(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


U711(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > U71^11
mark1 > U71^11

The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

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

U711(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(mark(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)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
mark1 > U71^11

The following usable rules [FROCOS05] were oriented: none

(33) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(34) PisEmptyProof (EQUIVALENT transformation)

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

(35) TRUE

(36) Obligation:

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

U611(ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


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

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(38) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


U611(mark(X)) → U611(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
mark1 > U61^11

The following usable rules [FROCOS05] were oriented: none

(40) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(41) PisEmptyProof (EQUIVALENT transformation)

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

(42) TRUE

(43) 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(ok(X1), ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(44) 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)
PLUS(ok(X1), ok(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
mark(x1)  =  mark(x1)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

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

PLUS(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


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)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(47) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(48) PisEmptyProof (EQUIVALENT transformation)

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

(49) TRUE

(50) Obligation:

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

S(ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(52) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


S(mark(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
mark1 > S1

The following usable rules [FROCOS05] were oriented: none

(54) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(55) PisEmptyProof (EQUIVALENT transformation)

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

(56) TRUE

(57) Obligation:

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

U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
U511(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > U51^11
mark1 > U51^11

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

U511(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(mark(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)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
mark1 > U51^11

The following usable rules [FROCOS05] were oriented: none

(61) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(62) PisEmptyProof (EQUIVALENT transformation)

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

(63) TRUE

(64) Obligation:

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

U411(ok(X1), ok(X2)) → U411(X1, X2)
U411(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > U41^11
mark1 > U41^11

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

U411(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(67) 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)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(68) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(69) PisEmptyProof (EQUIVALENT transformation)

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

(70) TRUE

(71) Obligation:

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

U331(ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(73) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(74) 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 [RPO].
Precedence:
mark1 > U33^11

The following usable rules [FROCOS05] were oriented: none

(75) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE

(78) Obligation:

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

U321(ok(X1), ok(X2)) → U321(X1, X2)
U321(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > U32^11
mark1 > U32^11

The following usable rules [FROCOS05] were oriented: none

(80) Obligation:

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

U321(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(81) 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)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(82) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(83) PisEmptyProof (EQUIVALENT transformation)

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

(84) TRUE

(85) Obligation:

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

U311(ok(X1), ok(X2), ok(X3)) → U311(X1, X2, X3)
U311(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


U311(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > U31^11
mark1 > U31^11

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

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

U311(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


U311(mark(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)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
mark1 > U31^11

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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

U221(ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


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

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(94) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


U221(mark(X)) → U221(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
mark1 > U22^11

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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

U211(ok(X1), ok(X2)) → U211(X1, X2)
U211(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


U211(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > U21^11
mark1 > U21^11

The following usable rules [FROCOS05] were oriented: none

(101) Obligation:

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

U211(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


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)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(103) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(104) PisEmptyProof (EQUIVALENT transformation)

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

(105) TRUE

(106) Obligation:

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

U131(ok(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(108) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(109) 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 [RPO].
Precedence:
mark1 > U13^11

The following usable rules [FROCOS05] were oriented: none

(110) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(111) PisEmptyProof (EQUIVALENT transformation)

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

(112) TRUE

(113) Obligation:

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

U121(ok(X1), ok(X2)) → U121(X1, X2)
U121(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > U12^11
mark1 > U12^11

The following usable rules [FROCOS05] were oriented: none

(115) Obligation:

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

U121(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(116) 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)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(117) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(118) PisEmptyProof (EQUIVALENT transformation)

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

(119) TRUE

(120) Obligation:

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

U111(ok(X1), ok(X2), ok(X3)) → U111(X1, X2, X3)
U111(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(121) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > U11^11
mark1 > U11^11

The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

U111(mark(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(mark(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)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
mark1 > U11^11

The following usable rules [FROCOS05] were oriented: none

(124) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(125) PisEmptyProof (EQUIVALENT transformation)

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

(126) TRUE

(127) Obligation:

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

PROPER(U11(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X1)
PROPER(U11(X1, X2, X3)) → PROPER(X3)
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → PROPER(X)
PROPER(U13(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X)) → PROPER(X)
PROPER(U31(X1, X2, X3)) → PROPER(X1)
PROPER(U31(X1, X2, X3)) → PROPER(X2)
PROPER(U31(X1, X2, X3)) → PROPER(X3)
PROPER(U32(X1, X2)) → PROPER(X1)
PROPER(U32(X1, X2)) → PROPER(X2)
PROPER(U33(X)) → PROPER(X)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U61(X)) → PROPER(X)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatKind(X)) → PROPER(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(128) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U11(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X1)
PROPER(U11(X1, X2, X3)) → PROPER(X3)
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → PROPER(X)
PROPER(U13(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X)) → PROPER(X)
PROPER(U31(X1, X2, X3)) → PROPER(X1)
PROPER(U31(X1, X2, X3)) → PROPER(X2)
PROPER(U31(X1, X2, X3)) → PROPER(X3)
PROPER(U32(X1, X2)) → PROPER(X1)
PROPER(U32(X1, X2)) → PROPER(X2)
PROPER(U33(X)) → PROPER(X)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U61(X)) → PROPER(X)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatKind(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
U113 > PROPER1
U122 > PROPER1
isNat1 > PROPER1
U131 > PROPER1
U212 > PROPER1
U221 > PROPER1
U313 > PROPER1
U322 > PROPER1
U331 > PROPER1
U412 > PROPER1
U513 > PROPER1
s1 > PROPER1
plus2 > PROPER1
U611 > PROPER1
U713 > PROPER1
x2 > PROPER1
and2 > PROPER1
isNatKind1 > PROPER1

The following usable rules [FROCOS05] were oriented: none

(129) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(130) PisEmptyProof (EQUIVALENT transformation)

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

(131) TRUE

(132) Obligation:

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

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

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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(x(X1, X2)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → ACTIVE(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U12(x1, x2)  =  x1
U11(x1, x2, x3)  =  U11(x1)
U13(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1)  =  x1
U31(x1, x2, x3)  =  x1
U32(x1, x2)  =  x1
U33(x1)  =  x1
U41(x1, x2)  =  x1
U51(x1, x2, x3)  =  U51(x1)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
U61(x1)  =  x1
U71(x1, x2, x3)  =  x1
x(x1, x2)  =  x(x1, x2)
and(x1, x2)  =  x1

Recursive Path Order [RPO].
Precedence:
U111 > ACTIVE1
U511 > ACTIVE1
plus2 > ACTIVE1
x2 > ACTIVE1

The following usable rules [FROCOS05] were oriented: none

(134) Obligation:

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

ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U12(x1, x2)  =  x1
U13(x1)  =  U13(x1)
U21(x1, x2)  =  U21(x1)
U22(x1)  =  U22(x1)
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2)  =  x1
U33(x1)  =  U33(x1)
U41(x1, x2)  =  x1
s(x1)  =  s(x1)
U61(x1)  =  U61(x1)
U71(x1, x2, x3)  =  x1
and(x1, x2)  =  x1

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(136) Obligation:

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

ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

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

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

(137) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U12(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U12(x1, x2)  =  U12(x1)
U32(x1, x2)  =  x1
U41(x1, x2)  =  x1
U71(x1, x2, x3)  =  x1
and(x1, x2)  =  x1

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(138) Obligation:

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

ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

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

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

(139) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U32(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U32(x1, x2)  =  U32(x1)
U41(x1, x2)  =  x1
U71(x1, x2, x3)  =  x1
and(x1, x2)  =  x1

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(140) Obligation:

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

ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

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

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

(141) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U41(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U71(x1, x2, x3)  =  x1
and(x1, x2)  =  x1

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(142) Obligation:

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

ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

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

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

(143) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U71(x1, x2, x3)  =  U71(x1)
and(x1, x2)  =  x1

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(144) Obligation:

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

ACTIVE(and(X1, X2)) → ACTIVE(X1)

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

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

(145) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(and(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
and2 > ACTIVE1

The following usable rules [FROCOS05] were oriented: none

(146) 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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(147) PisEmptyProof (EQUIVALENT transformation)

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

(148) TRUE

(149) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(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))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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