Termination w.r.t. Q proof of /tmp/tmp52NH7R/MYNAT_nosorts_noand

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

    ↳5 QDP

        ↳6 QDPOrderProof (⇔)

        ↳7 QDP

        ↳8 QDPOrderProof (⇔)

        ↳9 QDP

        ↳10 QDPOrderProof (⇔)

        ↳11 QDP

        ↳12 QDPOrderProof (⇔)

        ↳13 QDP

        ↳14 PisEmptyProof (⇔)

        ↳15 TRUE

    ↳16 QDP

        ↳17 QDPOrderProof (⇔)

        ↳18 QDP

        ↳19 QDPOrderProof (⇔)

        ↳20 QDP

        ↳21 QDPOrderProof (⇔)

        ↳22 QDP

        ↳23 QDPOrderProof (⇔)

        ↳24 QDP

        ↳25 QDPOrderProof (⇔)

        ↳26 QDP

        ↳27 QDPOrderProof (⇔)

        ↳28 QDP

        ↳29 PisEmptyProof (⇔)

        ↳30 TRUE

    ↳31 QDP

        ↳32 QDPOrderProof (⇔)

        ↳33 QDP

        ↳34 QDPOrderProof (⇔)

        ↳35 QDP

        ↳36 QDPOrderProof (⇔)

        ↳37 QDP

        ↳38 QDPOrderProof (⇔)

        ↳39 QDP

        ↳40 QDPOrderProof (⇔)

        ↳41 QDP

        ↳42 QDPOrderProof (⇔)

        ↳43 QDP

        ↳44 PisEmptyProof (⇔)

        ↳45 TRUE

    ↳46 QDP

        ↳47 QDPOrderProof (⇔)

        ↳48 QDP

        ↳49 QDPOrderProof (⇔)

        ↳50 QDP

        ↳51 QDPOrderProof (⇔)

        ↳52 QDP

        ↳53 QDPOrderProof (⇔)

        ↳54 QDP

        ↳55 PisEmptyProof (⇔)

        ↳56 TRUE

    ↳57 QDP

        ↳58 QDPOrderProof (⇔)

        ↳59 QDP

        ↳60 QDPOrderProof (⇔)

        ↳61 QDP

        ↳62 PisEmptyProof (⇔)

        ↳63 TRUE

    ↳64 QDP

        ↳65 QDPOrderProof (⇔)

        ↳66 QDP

        ↳67 QDPOrderProof (⇔)

        ↳68 QDP

        ↳69 QDPOrderProof (⇔)

        ↳70 QDP

        ↳71 QDPOrderProof (⇔)

        ↳72 QDP

        ↳73 QDPOrderProof (⇔)

        ↳74 QDP

        ↳75 QDPOrderProof (⇔)

        ↳76 QDP

        ↳77 PisEmptyProof (⇔)

        ↳78 TRUE

    ↳79 QDP

        ↳80 QDPOrderProof (⇔)

        ↳81 QDP

        ↳82 QDPOrderProof (⇔)

        ↳83 QDP

        ↳84 QDPOrderProof (⇔)

        ↳85 QDP

        ↳86 QDPOrderProof (⇔)

        ↳87 QDP

        ↳88 QDPOrderProof (⇔)

        ↳89 QDP

        ↳90 QDPOrderProof (⇔)

        ↳91 QDP

        ↳92 PisEmptyProof (⇔)

        ↳93 TRUE

    ↳94 QDP

        ↳95 QDPOrderProof (⇔)

        ↳96 QDP

        ↳97 QDPOrderProof (⇔)

        ↳98 QDP

        ↳99 QDPOrderProof (⇔)

        ↳100 QDP

        ↳101 DependencyGraphProof (⇔)

        ↳102 TRUE

(0) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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, M, N)) → MARK(U12(tt, M, N))
ACTIVE(U11(tt, M, N)) → U12¹(tt, M, N)
ACTIVE(U12(tt, M, N)) → MARK(s(plus(N, M)))
ACTIVE(U12(tt, M, N)) → S(plus(N, M))
ACTIVE(U12(tt, M, N)) → PLUS(N, M)
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
ACTIVE(U21(tt, M, N)) → U22¹(tt, M, N)
ACTIVE(U22(tt, M, N)) → MARK(plus(x(N, M), N))
ACTIVE(U22(tt, M, N)) → PLUS(x(N, M), N)
ACTIVE(U22(tt, M, N)) → X(N, M)
ACTIVE(plus(N, 0)) → MARK(N)
ACTIVE(plus(N, s(M))) → MARK(U11(tt, M, N))
ACTIVE(plus(N, s(M))) → U11¹(tt, M, N)
ACTIVE(x(N, 0)) → MARK(0)
ACTIVE(x(N, s(M))) → MARK(U21(tt, M, N))
ACTIVE(x(N, s(M))) → U21¹(tt, M, N)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → U11¹(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U12(X1, X2, X3)) → U12¹(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U21(X1, X2, X3)) → U21¹(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
MARK(U22(X1, X2, X3)) → U22¹(mark(X1), X2, X3)
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(x(X1, X2)) → X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(0) → ACTIVE(0)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)
S(mark(X)) → S(X)
S(active(X)) → S(X)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
U21¹(X1, X2, mark(X3)) → U21¹(X1, X2, X3)
U21¹(active(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)
U22¹(mark(X1), X2, X3) → U22¹(X1, X2, X3)
U22¹(X1, mark(X2), X3) → U22¹(X1, X2, X3)
U22¹(X1, X2, mark(X3)) → U22¹(X1, X2, X3)
U22¹(active(X1), X2, X3) → U22¹(X1, X2, X3)
U22¹(X1, active(X2), X3) → U22¹(X1, X2, X3)
U22¹(X1, X2, active(X3)) → U22¹(X1, X2, X3)
X(mark(X1), X2) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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 8 SCCs with 18 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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.

X(mark(X1), X2) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
X(x1, x2) = X(x1)

Tags:
X has tags [1,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

X(active(X1), X2) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
X(x1, x2) = X(x1)

Tags:
X has tags [1,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark
active(x1) = active(x1)

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

trivial

Status:

mark: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(10) 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: SCNP Order with the following components:
Level mapping:
Top level AFS:
X(x1, x2) = X(x2)

Tags:
X has tags [0,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

X(X1, active(X2)) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
X(x1, x2) = X(x2)

Tags:
X has tags [1,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

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

U22¹(X1, mark(X2), X3) → U22¹(X1, X2, X3)
U22¹(mark(X1), X2, X3) → U22¹(X1, X2, X3)
U22¹(X1, X2, mark(X3)) → U22¹(X1, X2, X3)
U22¹(active(X1), X2, X3) → U22¹(X1, X2, X3)
U22¹(X1, active(X2), X3) → U22¹(X1, X2, X3)
U22¹(X1, X2, active(X3)) → U22¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, X2, mark(X3)) → U22¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U22¹(x1, x2, x3) = U22¹(x3)

Tags:
U22¹ has tags [2,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

U22¹(X1, mark(X2), X3) → U22¹(X1, X2, X3)
U22¹(mark(X1), X2, X3) → U22¹(X1, X2, X3)
U22¹(active(X1), X2, X3) → U22¹(X1, X2, X3)
U22¹(X1, active(X2), X3) → U22¹(X1, X2, X3)
U22¹(X1, X2, active(X3)) → U22¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(active(X1), X2, X3) → U22¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U22¹(x1, x2, x3) = U22¹(x1)

Tags:
U22¹ has tags [2,3,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = x1
active(x1) = active(x1)

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

U22¹(X1, mark(X2), X3) → U22¹(X1, X2, X3)
U22¹(mark(X1), X2, X3) → U22¹(X1, X2, X3)
U22¹(X1, active(X2), X3) → U22¹(X1, X2, X3)
U22¹(X1, X2, active(X3)) → U22¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, active(X2), X3) → U22¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U22¹(x1, x2, x3) = U22¹(x2)

Tags:
U22¹ has tags [2,3,2]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = x1
active(x1) = active(x1)

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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

U22¹(X1, mark(X2), X3) → U22¹(X1, X2, X3)
U22¹(mark(X1), X2, X3) → U22¹(X1, X2, X3)
U22¹(X1, X2, active(X3)) → U22¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, X2, active(X3)) → U22¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U22¹(x1, x2, x3) = U22¹(x3)

Tags:
U22¹ has tags [2,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark
active(x1) = active(x1)

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

trivial

Status:

mark: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

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

U22¹(X1, mark(X2), X3) → U22¹(X1, X2, X3)
U22¹(mark(X1), X2, X3) → U22¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(mark(X1), X2, X3) → U22¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U22¹(x1, x2, x3) = U22¹(x1)

Tags:
U22¹ has tags [3,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(26) Obligation:

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

U22¹(X1, mark(X2), X3) → U22¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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.

U22¹(X1, mark(X2), X3) → U22¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U22¹(x1, x2, x3) = U22¹(x2)

Tags:
U22¹ has tags [0,2,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(29) PisEmptyProof (EQUIVALENT transformation)

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

(30) TRUE

(31) Obligation:

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

U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(X1, X2, mark(X3)) → U21¹(X1, X2, X3)
U21¹(active(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, X2, mark(X3)) → U21¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U21¹(x1, x2, x3) = U21¹(x3)

Tags:
U21¹ has tags [2,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(33) Obligation:

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

U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(active(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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.

U21¹(active(X1), X2, X3) → U21¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U21¹(x1, x2, x3) = U21¹(x1)

Tags:
U21¹ has tags [2,3,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = x1
active(x1) = active(x1)

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(35) Obligation:

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

U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, active(X2), X3) → U21¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U21¹(x1, x2, x3) = U21¹(x2)

Tags:
U21¹ has tags [2,3,2]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = x1
active(x1) = active(x1)

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

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

U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)
U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, X2, active(X3)) → U21¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U21¹(x1, x2, x3) = U21¹(x3)

Tags:
U21¹ has tags [2,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark
active(x1) = active(x1)

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

trivial

Status:

mark: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)
U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2, X3) → U21¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U21¹(x1, x2, x3) = U21¹(x1)

Tags:
U21¹ has tags [3,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, mark(X2), X3) → U21¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U21¹(x1, x2, x3) = U21¹(x2)

Tags:
U21¹ has tags [0,2,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(44) PisEmptyProof (EQUIVALENT transformation)

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

(45) TRUE

(46) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(47) 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: SCNP Order with the following components:
Level mapping:
Top level AFS:
PLUS(x1, x2) = PLUS(x1)

Tags:
PLUS has tags [1,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(active(X1), X2) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
PLUS(x1, x2) = PLUS(x1)

Tags:
PLUS has tags [1,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark
active(x1) = active(x1)

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

trivial

Status:

mark: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(X1, mark(X2)) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
PLUS(x1, x2) = PLUS(x2)

Tags:
PLUS has tags [0,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

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

PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(X1, active(X2)) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
PLUS(x1, x2) = PLUS(x2)

Tags:
PLUS has tags [1,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(54) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(55) PisEmptyProof (EQUIVALENT transformation)

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

(56) TRUE

(57) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
S(x1) = S(x1)

Tags:
S has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
active(x1) = active(x1)
mark(x1) = x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
S(x1) = S(x1)

Tags:
S has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(62) PisEmptyProof (EQUIVALENT transformation)

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

(63) TRUE

(64) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U12¹(x1, x2, x3) = U12¹(x3)

Tags:
U12¹ has tags [2,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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.

U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U12¹(x1, x2, x3) = U12¹(x1)

Tags:
U12¹ has tags [2,3,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = x1
active(x1) = active(x1)

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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.

U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U12¹(x1, x2, x3) = U12¹(x2)

Tags:
U12¹ has tags [2,3,2]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = x1
active(x1) = active(x1)

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(70) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U12¹(x1, x2, x3) = U12¹(x3)

Tags:
U12¹ has tags [2,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark
active(x1) = active(x1)

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

trivial

Status:

mark: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(72) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U12¹(x1, x2, x3) = U12¹(x1)

Tags:
U12¹ has tags [3,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U12¹(x1, x2, x3) = U12¹(x2)

Tags:
U12¹ has tags [0,2,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(76) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(77) PisEmptyProof (EQUIVALENT transformation)

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

(78) TRUE

(79) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U11¹(x1, x2, x3) = U11¹(x3)

Tags:
U11¹ has tags [2,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U11¹(x1, x2, x3) = U11¹(x1)

Tags:
U11¹ has tags [2,3,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = x1
active(x1) = active(x1)

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(84) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U11¹(x1, x2, x3) = U11¹(x2)

Tags:
U11¹ has tags [2,3,2]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = x1
active(x1) = active(x1)

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(85) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U11¹(x1, x2, x3) = U11¹(x3)

Tags:
U11¹ has tags [2,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
mark(x1) = mark
active(x1) = active(x1)

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

trivial

Status:

mark: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U11¹(x1, x2, x3) = U11¹(x1)

Tags:
U11¹ has tags [3,1,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(90) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U11¹(x1, x2, x3) = U11¹(x2)

Tags:
U11¹ has tags [0,2,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(91) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(92) PisEmptyProof (EQUIVALENT transformation)

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

(93) TRUE

(94) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U22(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
ACTIVE(plus(N, 0)) → MARK(N)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
ACTIVE(plus(N, s(M))) → MARK(U11(tt, M, N))
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(x(N, s(M))) → MARK(U21(tt, M, N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(U12(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
ACTIVE(U22(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(plus(N, 0)) → MARK(N)
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(plus(N, s(M))) → MARK(U11(tt, M, N))
MARK(U22(X1, X2, X3)) → MARK(X1)
ACTIVE(x(N, s(M))) → MARK(U21(tt, M, N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x1) = MARK(x1)
ACTIVE(x1) = ACTIVE(x1)

Tags:
MARK has tags [0]
ACTIVE has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U11(x1, x2, x3) = U11(x1, x2, x3)
mark(x1) = x1
tt = tt
U12(x1, x2, x3) = U12(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2, x3) = U22(x1, x2, x3)
x(x1, x2) = x(x1, x2)
0 = 0
active(x1) = x1

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

[U21₃, U22₃, x₂] > [U11₃, U12₃, plus₂] > [tt, s₁]

Status:

U11₃: [2,3,1]
tt: multiset
U12₃: [2,3,1]
s₁: multiset
plus₂: [2,1]
U21₃: [3,2,1]
U22₃: [3,2,1]
x₂: [1,2]
0: multiset

The following usable rules [FROCOS05] were oriented:

mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
active(U11(tt, M, N)) → mark(U12(tt, M, N))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
mark(s(X)) → active(s(mark(X)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
active(plus(N, 0)) → mark(N)
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
active(plus(N, s(M))) → mark(U11(tt, M, N))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(tt) → active(tt)
mark(0) → active(0)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
plus(X1, mark(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(X1, mark(X2)) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
active(x(N, 0)) → mark(0)

(96) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x1) = MARK(x1)
ACTIVE(x1) = ACTIVE(x1)

Tags:
MARK has tags [1]
ACTIVE has tags [1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U11(x1, x2, x3) = U11(x3)
mark(x1) = x1
tt = tt
U12(x1, x2, x3) = x3
s(x1) = s
U21(x1, x2, x3) = U21(x2, x3)
U22(x1, x2, x3) = U22(x2, x3)
plus(x1, x2) = plus
x(x1, x2) = x
active(x1) = x1
0 = 0

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

tt > s > U11₁
tt > [U21₂, U22₂, x] > U11₁
tt > plus > U11₁
0 > U11₁

Status:

U11₁: multiset
tt: multiset
s: []
U21₂: [1,2]
U22₂: [1,2]
plus: []
x: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
plus(X1, mark(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
x(X1, mark(X2)) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

(98) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(99) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x1) = MARK(x1)
ACTIVE(x1) = ACTIVE(x1)

Tags:
MARK has tags [0]
ACTIVE has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U11(x1, x2, x3) = U11
mark(x1) = mark
U12(x1, x2, x3) = U12
s(x1) = s
U21(x1, x2, x3) = U21
tt = tt
U22(x1, x2, x3) = U22
plus(x1, x2) = plus
x(x1, x2) = x
active(x1) = active
0 = 0

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

s > [plus, active] > mark > U11 > [tt, x]
s > [plus, active] > mark > U12 > [tt, x]
s > [plus, active] > mark > U22 > [tt, x]
U21 > [plus, active] > mark > U11 > [tt, x]
U21 > [plus, active] > mark > U12 > [tt, x]
U21 > [plus, active] > mark > U22 > [tt, x]
0 > [plus, active] > mark > U11 > [tt, x]
0 > [plus, active] > mark > U12 > [tt, x]
0 > [plus, active] > mark > U22 > [tt, x]

Status:

U11: []
mark: multiset
U12: multiset
s: []
U21: multiset
tt: multiset
U22: multiset
plus: []
x: []
active: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
plus(X1, mark(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
x(X1, mark(X2)) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

(100) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(101) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 7 less nodes.