Termination w.r.t. Q proof of /tmp/tmpHBBHP1/MYNAT_nokinds_noand

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 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 PisEmptyProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 PisEmptyProof (⇔)

    ↳25 TRUE

  ↳26 QDP

    ↳27 QDPOrderProof (⇔)

    ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 PisEmptyProof (⇔)

    ↳32 TRUE

  ↳33 QDP

    ↳34 QDPOrderProof (⇔)

    ↳35 QDP

    ↳36 PisEmptyProof (⇔)

    ↳37 TRUE

  ↳38 QDP

    ↳39 QDPOrderProof (⇔)

    ↳40 QDP

    ↳41 QDPOrderProof (⇔)

    ↳42 QDP

    ↳43 QDPOrderProof (⇔)

    ↳44 QDP

    ↳45 PisEmptyProof (⇔)

    ↳46 TRUE

  ↳47 QDP

    ↳48 QDPOrderProof (⇔)

    ↳49 QDP

    ↳50 PisEmptyProof (⇔)

    ↳51 TRUE

  ↳52 QDP

    ↳53 QDPOrderProof (⇔)

    ↳54 QDP

    ↳55 QDPOrderProof (⇔)

    ↳56 QDP

    ↳57 PisEmptyProof (⇔)

    ↳58 TRUE

  ↳59 QDP

    ↳60 QDPOrderProof (⇔)

    ↳61 QDP

    ↳62 QDPOrderProof (⇔)

    ↳63 QDP

    ↳64 PisEmptyProof (⇔)

    ↳65 TRUE

  ↳66 QDP

    ↳67 QDPOrderProof (⇔)

    ↳68 QDP

    ↳69 QDPOrderProof (⇔)

    ↳70 QDP

    ↳71 PisEmptyProof (⇔)

    ↳72 TRUE

  ↳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 PisEmptyProof (⇔)

    ↳89 TRUE

  ↳90 QDP

    ↳91 QDPOrderProof (⇔)

    ↳92 QDP

    ↳93 PisEmptyProof (⇔)

    ↳94 TRUE

  ↳95 QDP

    ↳96 QDPOrderProof (⇔)

    ↳97 QDP

    ↳98 QDPOrderProof (⇔)

    ↳99 QDP

    ↳100 PisEmptyProof (⇔)

    ↳101 TRUE

  ↳102 QDP

    ↳103 QDPOrderProof (⇔)

    ↳104 QDP

    ↳105 QDPOrderProof (⇔)

    ↳106 QDP

    ↳107 QDPOrderProof (⇔)

    ↳108 QDP

    ↳109 QDPOrderProof (⇔)

    ↳110 QDP

    ↳111 QDPOrderProof (⇔)

    ↳112 QDP

    ↳113 PisEmptyProof (⇔)

    ↳114 TRUE

  ↳115 QDP

    ↳116 QDPOrderProof (⇔)

    ↳117 QDP

    ↳118 QDPOrderProof (⇔)

    ↳119 QDP

    ↳120 QDPOrderProof (⇔)

    ↳121 QDP

    ↳122 QDPOrderProof (⇔)

    ↳123 QDP

    ↳124 QDPOrderProof (⇔)

    ↳125 QDP

    ↳126 QDPOrderProof (⇔)

    ↳127 QDP

    ↳128 QDPOrderProof (⇔)

    ↳129 QDP

    ↳130 PisEmptyProof (⇔)

    ↳131 TRUE

  ↳132 QDP

(0) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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, V2)) → U12¹(isNat(V2))
ACTIVE(U11(tt, V2)) → ISNAT(V2)
ACTIVE(U31(tt, V2)) → U32¹(isNat(V2))
ACTIVE(U31(tt, V2)) → ISNAT(V2)
ACTIVE(U51(tt, M, N)) → U52¹(isNat(N), M, N)
ACTIVE(U51(tt, M, N)) → ISNAT(N)
ACTIVE(U52(tt, M, N)) → S(plus(N, M))
ACTIVE(U52(tt, M, N)) → PLUS(N, M)
ACTIVE(U71(tt, M, N)) → U72¹(isNat(N), M, N)
ACTIVE(U71(tt, M, N)) → ISNAT(N)
ACTIVE(U72(tt, M, N)) → PLUS(x(N, M), N)
ACTIVE(U72(tt, M, N)) → X(N, M)
ACTIVE(isNat(plus(V1, V2))) → U11¹(isNat(V1), V2)
ACTIVE(isNat(plus(V1, V2))) → ISNAT(V1)
ACTIVE(isNat(s(V1))) → U21¹(isNat(V1))
ACTIVE(isNat(s(V1))) → ISNAT(V1)
ACTIVE(isNat(x(V1, V2))) → U31¹(isNat(V1), V2)
ACTIVE(isNat(x(V1, V2))) → ISNAT(V1)
ACTIVE(plus(N, 0)) → U41¹(isNat(N), N)
ACTIVE(plus(N, 0)) → ISNAT(N)
ACTIVE(plus(N, s(M))) → U51¹(isNat(M), M, N)
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(x(N, 0)) → U61¹(isNat(N))
ACTIVE(x(N, 0)) → ISNAT(N)
ACTIVE(x(N, s(M))) → U71¹(isNat(M), M, N)
ACTIVE(x(N, s(M))) → ISNAT(M)
ACTIVE(U11(X1, X2)) → U11¹(active(X1), X2)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X)) → U12¹(active(X))
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U21(X)) → U21¹(active(X))
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → U31¹(active(X1), X2)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → U32¹(active(X))
ACTIVE(U32(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(U52(X1, X2, X3)) → U52¹(active(X1), X2, X3)
ACTIVE(U52(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(U72(X1, X2, X3)) → U72¹(active(X1), X2, X3)
ACTIVE(U72(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)
U11¹(mark(X1), X2) → U11¹(X1, X2)
U12¹(mark(X)) → U12¹(X)
U21¹(mark(X)) → U21¹(X)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U32¹(mark(X)) → U32¹(X)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U52¹(mark(X1), X2, X3) → U52¹(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)
U72¹(mark(X1), X2, X3) → U72¹(X1, X2, X3)
X(mark(X1), X2) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
PROPER(U11(X1, X2)) → U11¹(proper(X1), proper(X2))
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X)) → U12¹(proper(X))
PROPER(U12(X)) → PROPER(X)
PROPER(isNat(X)) → ISNAT(proper(X))
PROPER(isNat(X)) → PROPER(X)
PROPER(U21(X)) → U21¹(proper(X))
PROPER(U21(X)) → PROPER(X)
PROPER(U31(X1, X2)) → U31¹(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U32(X)) → U32¹(proper(X))
PROPER(U32(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(U52(X1, X2, X3)) → U52¹(proper(X1), proper(X2), proper(X3))
PROPER(U52(X1, X2, X3)) → PROPER(X1)
PROPER(U52(X1, X2, X3)) → PROPER(X2)
PROPER(U52(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(U72(X1, X2, X3)) → U72¹(proper(X1), proper(X2), proper(X3))
PROPER(U72(X1, X2, X3)) → PROPER(X1)
PROPER(U72(X1, X2, X3)) → PROPER(X2)
PROPER(U72(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)
U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
U12¹(ok(X)) → U12¹(X)
ISNAT(ok(X)) → ISNAT(X)
U21¹(ok(X)) → U21¹(X)
U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)
U32¹(ok(X)) → U32¹(X)
U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(X1, X2, X3)
U52¹(ok(X1), ok(X2), ok(X3)) → U52¹(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)
U72¹(ok(X1), ok(X2), ok(X3)) → U72¹(X1, X2, X3)
X(ok(X1), ok(X2)) → X(X1, X2)
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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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 18 SCCs with 59 less nodes.

(4) Complex Obligation (AND)

(5) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

ISNAT(ok(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
U11(x1, x2) = x2
tt = tt
mark(x1) = mark
U12(x1) = x1
isNat(x1) = isNat(x1)
U21(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2) = x2
U51(x1, x2, x3) = x1
U52(x1, x2, x3) = x3
s(x1) = x1
plus(x1, x2) = plus(x1)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = x1
U72(x1, x2, x3) = x3
x(x1, x2) = x2
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[plus₁, proper₁] > [isNat₁, 0] > ok₁ > top > [tt, mark]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(7) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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:

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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

X(X1, mark(X2)) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2) = x2
mark(x1) = mark(x1)
ok(x1) = x1
active(x1) = active(x1)
U11(x1, x2) = x1
tt = tt
U12(x1) = U12(x1)
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1, x2) = U31(x1)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U12₁ > [tt, U21₁] > [mark₁, U32₁, s₁] > top
active₁ > U41₂ > [mark₁, U32₁, s₁] > top
active₁ > U51₃ > [isNat, plus₂] > [tt, U21₁] > [mark₁, U32₁, s₁] > top
active₁ > U51₃ > [isNat, plus₂] > U31₁ > [mark₁, U32₁, s₁] > top
active₁ > U51₃ > U52₃ > [mark₁, U32₁, s₁] > top
active₁ > 0 > [tt, U21₁] > [mark₁, U32₁, s₁] > top
active₁ > U72₃ > [isNat, plus₂] > [tt, U21₁] > [mark₁, U32₁, s₁] > top
active₁ > U72₃ > [isNat, plus₂] > U31₁ > [mark₁, U32₁, s₁] > top
active₁ > U72₃ > x₂ > U71₃ > [mark₁, U32₁, s₁] > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(12) Obligation:

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

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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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
ok(x1) = ok(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1)
tt = tt
U12(x1) = x1
isNat(x1) = isNat(x1)
U21(x1) = U21(x1)
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1)
U52(x1, x2, x3) = x1
s(x1) = x1
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U21₁ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > U21₁ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > plus₂ > U11₁ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > plus₂ > U11₁ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > plus₂ > U51₁ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > plus₂ > U51₁ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > U71₂ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > U71₂ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > x₁ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
active₁ > x₁ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > U21₁ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > U21₁ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > plus₂ > U11₁ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > plus₂ > U11₁ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > plus₂ > U51₁ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > plus₂ > U51₁ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > U71₂ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > U71₂ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > x₁ > [mark, tt, U31₂] > U41₂ > [ok₁, isNat₁, U32₁, U61₁] > 0
proper₁ > x₁ > [mark, tt, U31₂] > U72₃ > [ok₁, isNat₁, U32₁, U61₁] > 0

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(14) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(15) 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)
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = U12(x1)
isNat(x1) = x1
U21(x1) = U21(x1)
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, 0, U72₃] > tt > U12₁ > [mark₁, U21₁, U32₁, ok] > top
[active₁, 0, U72₃] > tt > s₁ > U71₃ > [mark₁, U21₁, U32₁, ok] > top
[active₁, 0, U72₃] > tt > x₂ > U31₂ > [mark₁, U21₁, U32₁, ok] > top
[active₁, 0, U72₃] > tt > x₂ > U71₃ > [mark₁, U21₁, U32₁, ok] > top
[active₁, 0, U72₃] > [U51₃, U52₃, plus₂] > U11₂ > U12₁ > [mark₁, U21₁, U32₁, ok] > top
[active₁, 0, U72₃] > [U51₃, U52₃, plus₂] > U41₂ > [mark₁, U21₁, U32₁, ok] > top
[active₁, 0, U72₃] > [U51₃, U52₃, plus₂] > s₁ > U71₃ > [mark₁, U21₁, U32₁, ok] > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(16) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) Obligation:

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

U72¹(ok(X1), ok(X2), ok(X3)) → U72¹(X1, X2, X3)
U72¹(mark(X1), X2, X3) → U72¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(ok(X1), ok(X2), ok(X3)) → U72¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2, x3) = U72¹(x2)
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = x1
U11(x1, x2) = U11(x2)
tt = tt
U12(x1) = x1
isNat(x1) = isNat(x1)
U21(x1) = x1
U31(x1, x2) = U31(x2)
U32(x1) = x1
U41(x1, x2) = U41(x1)
U51(x1, x2, x3) = U51(x3)
U52(x1, x2, x3) = x1
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = x1
U72(x1, x2, x3) = x1
x(x1, x2) = x(x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

0 > tt > [mark, U11₁, U31₁, U51₁, s₁, plus₂, x₁, proper₁] > isNat₁ > [ok₁, U41₁] > U72^1₁ > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(21) Obligation:

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

U72¹(mark(X1), X2, X3) → U72¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(mark(X1), X2, X3) → U72¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2, x3) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = U12(x1)
isNat(x1) = x1
U21(x1) = U21(x1)
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = x1
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U32₁, U41₂, plus₂] > U11₂ > [mark₁, tt, 0]
[active₁, U32₁, U41₂, plus₂] > U12₁ > [mark₁, tt, 0]
[active₁, U32₁, U41₂, plus₂] > U21₁ > [mark₁, tt, 0]
[active₁, U32₁, U41₂, plus₂] > U51₃ > U52₃ > [mark₁, tt, 0]
[active₁, U32₁, U41₂, plus₂] > U72₃ > [U31₂, U71₃, x₂] > U61₁ > [mark₁, tt, 0]
top > [mark₁, tt, 0]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(23) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(24) PisEmptyProof (EQUIVALENT transformation)

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

(25) TRUE

(26) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) 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)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1)
tt = tt
U12(x1) = x1
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = x1
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U41₂, U51₃, U61₁, 0] > U21₁ > [U71^1₁, mark₁] > top
[active₁, U41₂, U51₃, U61₁, 0] > U21₁ > tt > top
[active₁, U41₂, U51₃, U61₁, 0] > U52₃ > [U71^1₁, mark₁] > top
[active₁, U41₂, U51₃, U61₁, 0] > U72₃ > plus₂ > U11₁ > [U71^1₁, mark₁] > top
[active₁, U41₂, U51₃, U61₁, 0] > U72₃ > plus₂ > U11₁ > isNat > tt > top
[active₁, U41₂, U51₃, U61₁, 0] > U72₃ > x₂ > U71₃ > [U71^1₁, mark₁] > top
[active₁, U41₂, U51₃, U61₁, 0] > U72₃ > x₂ > U71₃ > isNat > tt > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(28) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(29) 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(x1)
ok(x1) = ok(x1)
active(x1) = x1
U11(x1, x2) = U11(x2)
tt = tt
mark(x1) = mark
U12(x1) = U12(x1)
isNat(x1) = isNat(x1)
U21(x1) = U21(x1)
U31(x1, x2) = x1
U32(x1) = U32(x1)
U41(x1, x2) = x1
U51(x1, x2, x3) = U51(x1)
U52(x1, x2, x3) = x2
s(x1) = x1
plus(x1, x2) = plus(x2)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = x1
U72(x1, x2, x3) = x1
x(x1, x2) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[ok₁, U11₁, U12₁, isNat₁, U21₁, U32₁, U51₁, plus₁, proper₁] > U71^1₁ > mark
[ok₁, U11₁, U12₁, isNat₁, U21₁, U32₁, U51₁, plus₁, proper₁] > 0 > tt > mark
[ok₁, U11₁, U12₁, isNat₁, U21₁, U32₁, U51₁, plus₁, proper₁] > top > mark

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(30) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(31) PisEmptyProof (EQUIVALENT transformation)

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

(32) TRUE

(33) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) 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)
U61¹(mark(X)) → U61¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1) = U61¹(x1)
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U11₂ > [U61^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U31₂ > [U61^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U51₃ > U52₃ > [U61^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U71₃ > [U61^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > plus₂ > U51₃ > U52₃ > [U61^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U61₁ > 0 > tt > U52₃ > [U61^1₁, ok₁, mark₁] > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(35) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(36) PisEmptyProof (EQUIVALENT transformation)

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

(37) TRUE

(38) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

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)
ok(x1) = x1
active(x1) = active(x1)
U11(x1, x2) = x1
tt = tt
U12(x1) = x1
isNat(x1) = isNat
U21(x1) = x1
U31(x1, x2) = x1
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = x1
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U51₃] > U32₁ > [tt, isNat, U52₃, U71₃, U72₃, x₂] > [mark₁, U61₁, top]
[active₁, U51₃] > plus₂ > [tt, isNat, U52₃, U71₃, U72₃, x₂] > [mark₁, U61₁, top]
[active₁, U51₃] > plus₂ > U41₂ > [mark₁, U61₁, top]
[active₁, U51₃] > 0 > [tt, isNat, U52₃, U71₃, U72₃, x₂] > [mark₁, U61₁, top]
[active₁, U51₃] > 0 > U41₂ > [mark₁, U61₁, top]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(40) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(X1, mark(X2)) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2) = PLUS(x1, x2)
mark(x1) = mark(x1)
ok(x1) = x1
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = U12(x1)
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

PLUS₂ > [mark₁, top]
[active₁, U11₂, U12₁] > 0 > [tt, U71₃, U72₃] > [U31₂, U32₁] > [mark₁, top]
[active₁, U11₂, U12₁] > 0 > [tt, U71₃, U72₃] > plus₂ > [U51₃, s₁] > U52₃ > [mark₁, top]
[active₁, U11₂, U12₁] > 0 > [tt, U71₃, U72₃] > x₂ > [mark₁, top]
[active₁, U11₂, U12₁] > 0 > U41₂ > [mark₁, top]
[active₁, U11₂, U12₁] > 0 > U61₁ > [mark₁, top]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(42) Obligation:

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

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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
ok(x1) = ok(x1)
active(x1) = x1
U11(x1, x2) = x2
tt = tt
mark(x1) = mark
U12(x1) = U12(x1)
isNat(x1) = x1
U21(x1) = U21(x1)
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2) = x2
U51(x1, x2, x3) = U51(x1)
U52(x1, x2, x3) = x2
s(x1) = s(x1)
plus(x1, x2) = x1
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2)
U72(x1, x2, x3) = U72(x1)
x(x1, x2) = x(x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

proper₁ > U12₁ > [tt, 0] > U61₁ > [ok₁, U51₁, U72₁] > mark
proper₁ > x₁ > [U21₁, s₁, U71₂] > [tt, 0] > U61₁ > [ok₁, U51₁, U72₁] > mark
top > mark

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(44) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(45) PisEmptyProof (EQUIVALENT transformation)

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

(46) TRUE

(47) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) 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)
S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U11₂ > [S₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U31₂ > [S₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U51₃ > U52₃ > [S₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U71₃ > [S₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > plus₂ > U51₃ > U52₃ > [S₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U61₁ > 0 > tt > U52₃ > [S₁, ok₁, mark₁] > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(49) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(50) PisEmptyProof (EQUIVALENT transformation)

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

(51) TRUE

(52) Obligation:

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

U52¹(ok(X1), ok(X2), ok(X3)) → U52¹(X1, X2, X3)
U52¹(mark(X1), X2, X3) → U52¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

U52¹(ok(X1), ok(X2), ok(X3)) → U52¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2, x3) = U52¹(x3)
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = x1
U11(x1, x2) = x2
tt = tt
U12(x1) = x1
isNat(x1) = isNat(x1)
U21(x1) = x1
U31(x1, x2) = x2
U32(x1) = x1
U41(x1, x2) = x1
U51(x1, x2, x3) = U51(x1, x3)
U52(x1, x2, x3) = x3
s(x1) = s(x1)
plus(x1, x2) = x1
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = U71(x1, x2)
U72(x1, x2, x3) = U72(x1, x2)
x(x1, x2) = x(x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

0 > [isNat₁, U71₂] > [U52^1₁, ok₁, s₁] > [mark, top]
0 > [isNat₁, U71₂] > tt > [mark, top]
[U72₂, proper₁] > U51₂ > [isNat₁, U71₂] > [U52^1₁, ok₁, s₁] > [mark, top]
[U72₂, proper₁] > U51₂ > [isNat₁, U71₂] > tt > [mark, top]
[U72₂, proper₁] > x₁ > [isNat₁, U71₂] > [U52^1₁, ok₁, s₁] > [mark, top]
[U72₂, proper₁] > x₁ > [isNat₁, U71₂] > tt > [mark, top]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(54) Obligation:

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

U52¹(mark(X1), X2, X3) → U52¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(mark(X1), X2, X3) → U52¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2, x3) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = U12(x1)
isNat(x1) = isNat(x1)
U21(x1) = U21(x1)
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = x1
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

proper₁ > [active₁, U32₁] > U11₂ > ok > U12₁ > tt > [isNat₁, U61₁, U71₃, U72₃, x₂] > [mark₁, U21₁, 0]
proper₁ > [active₁, U32₁] > U11₂ > ok > U52₃ > plus₂ > U41₂ > [mark₁, U21₁, 0]
proper₁ > [active₁, U32₁] > U11₂ > ok > U52₃ > plus₂ > U51₃ > [isNat₁, U61₁, U71₃, U72₃, x₂] > [mark₁, U21₁, 0]
proper₁ > [active₁, U32₁] > U11₂ > ok > top
proper₁ > [active₁, U32₁] > U31₂ > ok > U12₁ > tt > [isNat₁, U61₁, U71₃, U72₃, x₂] > [mark₁, U21₁, 0]
proper₁ > [active₁, U32₁] > U31₂ > ok > U52₃ > plus₂ > U41₂ > [mark₁, U21₁, 0]
proper₁ > [active₁, U32₁] > U31₂ > ok > U52₃ > plus₂ > U51₃ > [isNat₁, U61₁, U71₃, U72₃, x₂] > [mark₁, U21₁, 0]
proper₁ > [active₁, U32₁] > U31₂ > ok > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(56) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(57) PisEmptyProof (EQUIVALENT transformation)

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

(58) TRUE

(59) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1)
tt = tt
U12(x1) = x1
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = x1
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U41₂, U51₃, U61₁, 0] > U21₁ > [U51^1₁, mark₁] > top
[active₁, U41₂, U51₃, U61₁, 0] > U21₁ > tt > top
[active₁, U41₂, U51₃, U61₁, 0] > U52₃ > [U51^1₁, mark₁] > top
[active₁, U41₂, U51₃, U61₁, 0] > U72₃ > plus₂ > U11₁ > [U51^1₁, mark₁] > top
[active₁, U41₂, U51₃, U61₁, 0] > U72₃ > plus₂ > U11₁ > isNat > tt > top
[active₁, U41₂, U51₃, U61₁, 0] > U72₃ > x₂ > U71₃ > [U51^1₁, mark₁] > top
[active₁, U41₂, U51₃, U61₁, 0] > U72₃ > x₂ > U71₃ > isNat > tt > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(61) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) 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(x1)
ok(x1) = ok(x1)
active(x1) = x1
U11(x1, x2) = U11(x2)
tt = tt
mark(x1) = mark
U12(x1) = U12(x1)
isNat(x1) = isNat(x1)
U21(x1) = U21(x1)
U31(x1, x2) = x1
U32(x1) = U32(x1)
U41(x1, x2) = x1
U51(x1, x2, x3) = U51(x1)
U52(x1, x2, x3) = x2
s(x1) = x1
plus(x1, x2) = plus(x2)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = x1
U72(x1, x2, x3) = x1
x(x1, x2) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[ok₁, U11₁, U12₁, isNat₁, U21₁, U32₁, U51₁, plus₁, proper₁] > U51^1₁ > mark
[ok₁, U11₁, U12₁, isNat₁, U21₁, U32₁, U51₁, plus₁, proper₁] > 0 > tt > mark
[ok₁, U11₁, U12₁, isNat₁, U21₁, U32₁, U51₁, plus₁, proper₁] > top > mark

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(63) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(64) PisEmptyProof (EQUIVALENT transformation)

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

(65) TRUE

(66) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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(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
active(x1) = active(x1)
U11(x1, x2) = U11(x2)
tt = tt
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x2)
U32(x1) = x1
U41(x1, x2) = x2
U51(x1, x2, x3) = x2
U52(x1, x2, x3) = U52(x3)
s(x1) = x1
plus(x1, x2) = x2
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1)
U72(x1, x2, x3) = U72(x1)
x(x1, x2) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

U41^1₁ > [ok₁, mark, tt, U71₁]
0 > [active₁, U11₁, U52₁, U61₁, U72₁, top] > [ok₁, mark, tt, U71₁]
proper₁ > U31₁ > [active₁, U11₁, U52₁, U61₁, U72₁, top] > [ok₁, mark, tt, U71₁]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(68) 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2) = U41¹(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = x1
tt = tt
U12(x1) = U12(x1)
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1, x2) = U31(x1)
U32(x1) = x1
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > [U31₁, proper₁] > U41₂ > [U41^1₁, mark₁, U12₁, U21₁, 0] > tt > [isNat, ok]
top > active₁ > [U31₁, proper₁] > s₁ > [U71₃, U72₃, x₂] > [U51₃, U52₃, plus₂] > [U41^1₁, mark₁, U12₁, U21₁, 0] > tt > [isNat, ok]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(70) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(71) PisEmptyProof (EQUIVALENT transformation)

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

(72) TRUE

(73) Obligation:

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

U32¹(ok(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

U32¹(ok(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1) = U32¹(x1)
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U11₂ > [U32^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U31₂ > [U32^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U51₃ > U52₃ > [U32^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U71₃ > [U32^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > plus₂ > U51₃ > U52₃ > [U32^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U61₁ > 0 > tt > U52₃ > [U32^1₁, ok₁, mark₁] > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(75) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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:

U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2) = U31¹(x2)
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = active(x1)
U11(x1, x2) = U11(x2)
tt = tt
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x2)
U32(x1) = x1
U41(x1, x2) = x2
U51(x1, x2, x3) = x2
U52(x1, x2, x3) = U52(x3)
s(x1) = x1
plus(x1, x2) = x2
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1)
U72(x1, x2, x3) = U72(x1)
x(x1, x2) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

U31^1₁ > [ok₁, mark, tt, U71₁]
0 > [active₁, U11₁, U52₁, U61₁, U72₁, top] > [ok₁, mark, tt, U71₁]
proper₁ > U31₁ > [active₁, U11₁, U52₁, U61₁, U72₁, top] > [ok₁, mark, tt, U71₁]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(80) Obligation:

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

U31¹(mark(X1), X2) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

U31¹(mark(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2) = U31¹(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = x1
tt = tt
U12(x1) = U12(x1)
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1, x2) = U31(x1)
U32(x1) = x1
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > [U31₁, proper₁] > U41₂ > [U31^1₁, mark₁, U12₁, U21₁, 0] > tt > [isNat, ok]
top > active₁ > [U31₁, proper₁] > s₁ > [U71₃, U72₃, x₂] > [U51₃, U52₃, plus₂] > [U31^1₁, mark₁, U12₁, U21₁, 0] > tt > [isNat, ok]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(82) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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:

U21¹(ok(X)) → U21¹(X)
U21¹(mark(X)) → U21¹(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

U21¹(ok(X)) → U21¹(X)
U21¹(mark(X)) → U21¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1) = U21¹(x1)
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U11₂ > [U21^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U31₂ > [U21^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U51₃ > U52₃ > [U21^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U71₃ > [U21^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > plus₂ > U51₃ > U52₃ > [U21^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U61₁ > 0 > tt > U52₃ > [U21^1₁, ok₁, mark₁] > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(87) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) PisEmptyProof (EQUIVALENT transformation)

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

(89) TRUE

(90) Obligation:

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

U12¹(ok(X)) → U12¹(X)
U12¹(mark(X)) → U12¹(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(91) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(ok(X)) → U12¹(X)
U12¹(mark(X)) → U12¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1) = U12¹(x1)
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U11₂ > [U12^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U31₂ > [U12^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U51₃ > U52₃ > [U12^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > s₁ > U71₃ > [U12^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > plus₂ > U51₃ > U52₃ > [U12^1₁, ok₁, mark₁] > top
[active₁, U32₁, U41₂, U72₃, x₂, proper₁] > U61₁ > 0 > tt > U52₃ > [U12^1₁, ok₁, mark₁] > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(92) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) PisEmptyProof (EQUIVALENT transformation)

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

(94) TRUE

(95) Obligation:

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

U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(96) 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)) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = U11¹(x2)
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = active(x1)
U11(x1, x2) = U11(x2)
tt = tt
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x2)
U32(x1) = x1
U41(x1, x2) = x2
U51(x1, x2, x3) = x2
U52(x1, x2, x3) = U52(x3)
s(x1) = x1
plus(x1, x2) = x2
U61(x1) = U61(x1)
0 = 0
U71(x1, x2, x3) = U71(x1)
U72(x1, x2, x3) = U72(x1)
x(x1, x2) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

U11^1₁ > [ok₁, mark, tt, U71₁]
0 > [active₁, U11₁, U52₁, U61₁, U72₁, top] > [ok₁, mark, tt, U71₁]
proper₁ > U31₁ > [active₁, U11₁, U52₁, U61₁, U72₁, top] > [ok₁, mark, tt, U71₁]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(97) Obligation:

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

U11¹(mark(X1), X2) → U11¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X1), X2) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = U11¹(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
U11(x1, x2) = x1
tt = tt
U12(x1) = U12(x1)
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1, x2) = U31(x1)
U32(x1) = x1
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
0 = 0
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > [U31₁, proper₁] > U41₂ > [U11^1₁, mark₁, U12₁, U21₁, 0] > tt > [isNat, ok]
top > active₁ > [U31₁, proper₁] > s₁ > [U71₃, U72₃, x₂] > [U51₃, U52₃, plus₂] > [U11^1₁, mark₁, U12₁, U21₁, 0] > tt > [isNat, ok]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(99) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) PisEmptyProof (EQUIVALENT transformation)

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

(101) TRUE

(102) Obligation:

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

PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U12(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U32(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(U52(X1, X2, X3)) → PROPER(X1)
PROPER(U52(X1, X2, X3)) → PROPER(X2)
PROPER(U52(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(U72(X1, X2, X3)) → PROPER(X1)
PROPER(U72(X1, X2, X3)) → PROPER(X2)
PROPER(U72(X1, X2, X3)) → PROPER(X3)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(103) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
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(U52(X1, X2, X3)) → PROPER(X1)
PROPER(U52(X1, X2, X3)) → PROPER(X2)
PROPER(U52(X1, X2, X3)) → PROPER(X3)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U72(X1, X2, X3)) → PROPER(X1)
PROPER(U72(X1, X2, X3)) → PROPER(X2)
PROPER(U72(X1, X2, X3)) → PROPER(X3)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U11(x1, x2) = U11(x1, x2)
U12(x1) = x1
isNat(x1) = x1
U21(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = x1
U41(x1, x2) = U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = x1
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
active(x1) = active(x1)
tt = tt
mark(x1) = mark
0 = 0
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

PROPER₁ > top
[U11₂, U41₂, active₁] > U31₂ > [x₂, mark, 0] > U72₃ > top
[U11₂, U41₂, active₁] > U51₃ > [U52₃, plus₂] > [x₂, mark, 0] > U72₃ > top
[U11₂, U41₂, active₁] > U71₃ > [x₂, mark, 0] > U72₃ > top
[U11₂, U41₂, active₁] > tt > [U52₃, plus₂] > [x₂, mark, 0] > U72₃ > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(104) Obligation:

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

PROPER(U12(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(105) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNat(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U12(x1) = x1
isNat(x1) = isNat(x1)
U21(x1) = x1
U32(x1) = x1
s(x1) = x1
U61(x1) = x1
active(x1) = active
U11(x1, x2) = x1
tt = tt
mark(x1) = mark
U31(x1, x2) = x1
U41(x1, x2) = x1
U51(x1, x2, x3) = x1
U52(x1, x2, x3) = x1
plus(x1, x2) = plus
0 = 0
U71(x1, x2, x3) = U71
U72(x1, x2, x3) = x1
x(x1, x2) = x
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

0 > [active, mark, plus, U71, x] > [tt, ok] > isNat₁
0 > [active, mark, plus, U71, x] > [tt, ok] > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(106) Obligation:

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

PROPER(U12(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

PROPER(U21(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U12(x1) = x1
U21(x1) = U21(x1)
U32(x1) = U32(x1)
s(x1) = s(x1)
U61(x1) = x1
active(x1) = active(x1)
U11(x1, x2) = x1
tt = tt
mark(x1) = mark
isNat(x1) = isNat(x1)
U31(x1, x2) = U31(x1)
U41(x1, x2) = U41
U51(x1, x2, x3) = U51(x1, x2)
U52(x1, x2, x3) = U52
plus(x1, x2) = plus(x1)
0 = 0
U71(x1, x2, x3) = U71
U72(x1, x2, x3) = U72(x2, x3)
x(x1, x2) = x(x1)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

PROPER₁ > [U32₁, mark, U41, 0]
isNat₁ > U21₁ > tt > [U32₁, mark, U41, 0]
isNat₁ > U31₁ > [U32₁, mark, U41, 0]
U52 > [s₁, active₁, U51₂, plus₁] > U21₁ > tt > [U32₁, mark, U41, 0]
U52 > [s₁, active₁, U51₂, plus₁] > U71 > [U32₁, mark, U41, 0]
U52 > [s₁, active₁, U51₂, plus₁] > x₁ > U31₁ > [U32₁, mark, U41, 0]
U72₂ > [s₁, active₁, U51₂, plus₁] > U21₁ > tt > [U32₁, mark, U41, 0]
U72₂ > [s₁, active₁, U51₂, plus₁] > U71 > [U32₁, mark, U41, 0]
U72₂ > [s₁, active₁, U51₂, plus₁] > x₁ > U31₁ > [U32₁, mark, U41, 0]
top > [U32₁, mark, U41, 0]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(108) Obligation:

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

PROPER(U12(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

PROPER(U12(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U12(x1) = U12(x1)
U61(x1) = x1
active(x1) = active(x1)
U11(x1, x2) = U11
tt = tt
mark(x1) = mark
isNat(x1) = x1
U21(x1) = U21
U31(x1, x2) = U31
U32(x1) = U32
U41(x1, x2) = x1
U51(x1, x2, x3) = U51(x2)
U52(x1, x2, x3) = U52(x1, x2, x3)
s(x1) = s
plus(x1, x2) = plus
0 = 0
U71(x1, x2, x3) = U71
U72(x1, x2, x3) = x1
x(x1, x2) = x2
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

proper₁ > [PROPER₁, U12₁, active₁] > U52₃ > [U11, U21, U31, U32, s, plus, ok] > tt > [mark, U51₁]
proper₁ > [PROPER₁, U12₁, active₁] > 0 > [U11, U21, U31, U32, s, plus, ok] > tt > [mark, U51₁]
proper₁ > [PROPER₁, U12₁, active₁] > U71 > [U11, U21, U31, U32, s, plus, ok] > tt > [mark, U51₁]
top > [mark, U51₁]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(110) Obligation:

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

PROPER(U61(X)) → PROPER(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U61(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U61(x1) = U61(x1)
active(x1) = x1
U11(x1, x2) = U11(x1)
tt = tt
mark(x1) = mark
U12(x1) = x1
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1, x2) = U31(x2)
U32(x1) = U32(x1)
U41(x1, x2) = U41
U51(x1, x2, x3) = U51(x1, x3)
U52(x1, x2, x3) = U52(x1, x2)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
0 = 0
U71(x1, x2, x3) = U71(x1)
U72(x1, x2, x3) = U72
x(x1, x2) = x(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[PROPER₁, U61₁] > [mark, isNat, U31₁, U52₂, 0, x₂] > U21₁ > top
[PROPER₁, U61₁] > [mark, isNat, U31₁, U52₂, 0, x₂] > [U51₂, plus₂] > U11₁ > top
[PROPER₁, U61₁] > [mark, isNat, U31₁, U52₂, 0, x₂] > U71₁ > top
tt > U32₁ > [mark, isNat, U31₁, U52₂, 0, x₂] > U21₁ > top
tt > U32₁ > [mark, isNat, U31₁, U52₂, 0, x₂] > [U51₂, plus₂] > U11₁ > top
tt > U32₁ > [mark, isNat, U31₁, U52₂, 0, x₂] > U71₁ > top
U41 > [mark, isNat, U31₁, U52₂, 0, x₂] > U21₁ > top
U41 > [mark, isNat, U31₁, U52₂, 0, x₂] > [U51₂, plus₂] > U11₁ > top
U41 > [mark, isNat, U31₁, U52₂, 0, x₂] > U71₁ > top
s₁ > [mark, isNat, U31₁, U52₂, 0, x₂] > U21₁ > top
s₁ > [mark, isNat, U31₁, U52₂, 0, x₂] > [U51₂, plus₂] > U11₁ > top
s₁ > [mark, isNat, U31₁, U52₂, 0, x₂] > U71₁ > top
U72 > [mark, isNat, U31₁, U52₂, 0, x₂] > U21₁ > top
U72 > [mark, isNat, U31₁, U52₂, 0, x₂] > [U51₂, plus₂] > U11₁ > top
U72 > [mark, isNat, U31₁, U52₂, 0, x₂] > U71₁ > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(112) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(113) PisEmptyProof (EQUIVALENT transformation)

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

(114) TRUE

(115) Obligation:

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

ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(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(U72(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → ACTIVE(X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

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) = x1
U11(x1, x2) = x1
U21(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2) = x1
U51(x1, x2, x3) = x1
U52(x1, x2, x3) = x1
s(x1) = x1
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
U71(x1, x2, x3) = x1
U72(x1, x2, x3) = x1
x(x1, x2) = x(x1, x2)
active(x1) = active(x1)
tt = tt
mark(x1) = mark
isNat(x1) = isNat(x1)
0 = 0
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

active₁ > mark > plus₂ > ok
active₁ > mark > x₂ > ok
active₁ > isNat₁ > ok
active₁ > 0
tt > mark > plus₂ > ok
tt > mark > x₂ > ok
tt > isNat₁ > ok
tt > 0
proper₁ > plus₂ > ok
proper₁ > x₂ > ok
proper₁ > isNat₁ > ok
proper₁ > 0

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(117) Obligation:

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

ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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) 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
U12(x1) = x1
U11(x1, x2) = x1
U21(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2) = x1
U51(x1, x2, x3) = x1
U52(x1, x2, x3) = x1
s(x1) = x1
U61(x1) = x1
U71(x1, x2, x3) = U71(x1)
U72(x1, x2, x3) = x1
active(x1) = active(x1)
tt = tt
mark(x1) = mark
isNat(x1) = x1
plus(x1, x2) = x1
0 = 0
x(x1, x2) = x
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

active₁ > [U71₁, tt, proper₁, top] > 0 > [mark, x]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(119) Obligation:

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

ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(120) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U12(X)) → ACTIVE(X)
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) = ACTIVE(x1)
U12(x1) = U12(x1)
U11(x1, x2) = x1
U21(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2) = U41(x1)
U51(x1, x2, x3) = x1
U52(x1, x2, x3) = x1
s(x1) = x1
U61(x1) = x1
U72(x1, x2, x3) = x1
active(x1) = active(x1)
tt = tt
mark(x1) = mark
isNat(x1) = isNat(x1)
plus(x1, x2) = plus
0 = 0
U71(x1, x2, x3) = U71(x1)
x(x1, x2) = x
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

ACTIVE₁ > [U12₁, U41₁, U71₁]
isNat₁ > [tt, mark, plus] > [0, ok] > active₁ > [U12₁, U41₁, U71₁]
isNat₁ > [tt, mark, plus] > top > active₁ > [U12₁, U41₁, U71₁]
x > [tt, mark, plus] > [0, ok] > active₁ > [U12₁, U41₁, U71₁]
x > [tt, mark, plus] > top > active₁ > [U12₁, U41₁, U71₁]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(121) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
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) = ACTIVE(x1)
U11(x1, x2) = x1
U21(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U51(x1, x2, x3) = U51(x1, x3)
U52(x1, x2, x3) = U52(x1, x3)
s(x1) = x1
U61(x1) = U61(x1)
U72(x1, x2, x3) = x1
active(x1) = active(x1)
tt = tt
mark(x1) = x1
U12(x1) = U12
isNat(x1) = isNat
U41(x1, x2) = U41(x1, x2)
plus(x1, x2) = plus(x1)
0 = 0
U71(x1, x2, x3) = x1
x(x1, x2) = x
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

[U61₁, active₁, U41₂, plus₁] > U51₂
[U61₁, active₁, U41₂, plus₁] > U52₂
[U61₁, active₁, U41₂, plus₁] > [tt, x] > U12
[U61₁, active₁, U41₂, plus₁] > [tt, x] > [isNat, 0]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(123) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(s(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
U11(x1, x2) = x1
U21(x1) = x1
U31(x1, x2) = U31(x1)
U32(x1) = U32(x1)
s(x1) = s(x1)
U72(x1, x2, x3) = x1
active(x1) = active(x1)
tt = tt
mark(x1) = mark
U12(x1) = U12
isNat(x1) = isNat
U41(x1, x2) = x2
U51(x1, x2, x3) = x1
U52(x1, x2, x3) = U52
plus(x1, x2) = x2
U61(x1) = U61
0 = 0
U71(x1, x2, x3) = U71(x2)
x(x1, x2) = x(x1)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U12 > tt > [x₁, proper₁] > U61 > [U32₁, s₁, mark, U52, 0, ok, top]
active₁ > U12 > tt > [x₁, proper₁] > U71₁ > [U31₁, isNat] > [U32₁, s₁, mark, U52, 0, ok, top]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(125) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

(126) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U72(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) = ACTIVE(x1)
U11(x1, x2) = x1
U21(x1) = x1
U72(x1, x2, x3) = U72(x1, x2, x3)
active(x1) = active(x1)
tt = tt
mark(x1) = mark
U12(x1) = U12
isNat(x1) = isNat(x1)
U31(x1, x2) = U31
U32(x1) = U32
U41(x1, x2) = x1
U51(x1, x2, x3) = x3
U52(x1, x2, x3) = U52
s(x1) = s
plus(x1, x2) = plus
U61(x1) = U61
0 = 0
U71(x1, x2, x3) = x3
x(x1, x2) = x1
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

ACTIVE₁ > [tt, mark, U12]
U32 > [U52, proper₁] > plus > [active₁, isNat₁] > U72₃ > [tt, mark, U12]
U32 > [U52, proper₁] > plus > [active₁, isNat₁] > U31 > [tt, mark, U12]
U32 > [U52, proper₁] > plus > [active₁, isNat₁] > s > [tt, mark, U12]
U32 > [U52, proper₁] > plus > [active₁, isNat₁] > U61 > [tt, mark, U12]
0 > [active₁, isNat₁] > U72₃ > [tt, mark, U12]
0 > [active₁, isNat₁] > U31 > [tt, mark, U12]
0 > [active₁, isNat₁] > s > [tt, mark, U12]
0 > [active₁, isNat₁] > U61 > [tt, mark, U12]
top > [tt, mark, U12]

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(127) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(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
U11(x1, x2) = U11(x1, x2)
U21(x1) = U21(x1)
active(x1) = x1
tt = tt
mark(x1) = mark
U12(x1) = U12
isNat(x1) = isNat
U31(x1, x2) = U31
U32(x1) = U32
U41(x1, x2) = U41
U51(x1, x2, x3) = U51
U52(x1, x2, x3) = U52
s(x1) = s
plus(x1, x2) = plus
U61(x1) = U61
0 = 0
U71(x1, x2, x3) = U71
U72(x1, x2, x3) = U72
x(x1, x2) = x(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic Path Order [LPO].
Precedence:

proper₁ > [U12, isNat, U31, U32, U41, U51, U52, s, plus, U61, U71, U72, x₂, ok] > U21₁ > [U11₂, mark] > top
proper₁ > [U12, isNat, U31, U32, U41, U51, U52, s, plus, U61, U71, U72, x₂, ok] > tt > [U11₂, mark] > top
proper₁ > [U12, isNat, U31, U32, U41, U51, U52, s, plus, U61, U71, U72, x₂, ok] > tt > 0 > top

The following usable rules [FROCOS05] were oriented:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(129) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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:

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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.