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

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MARK(U12(X1, X2, X3)) → MARK(X1)
A__U22(tt, M, N) → MARK(N)
A__U22(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(s(X)) → MARK(X)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U11(tt, M, N)
MARK(plus(X1, X2)) → MARK(X2)
MARK(x(X1, X2)) → MARK(X1)
A__U11(tt, M, N) → A__U12(tt, M, N)
A__U22(tt, M, N) → MARK(M)
A__PLUS(N, 0) → MARK(N)
A__U12(tt, M, N) → MARK(N)
MARK(U22(X1, X2, X3)) → A__U22(mark(X1), X2, X3)
A__U21(tt, M, N) → A__U22(tt, M, N)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__U22(tt, M, N) → A__X(mark(N), mark(M))
MARK(plus(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(x(X1, X2)) → MARK(X2)
A__X(N, s(M)) → A__U21(tt, M, N)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U12(tt, M, N) → MARK(M)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)

The TRS R consists of the following rules:

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
A__U22(tt, M, N) → MARK(N)
A__U22(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(s(X)) → MARK(X)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U11(tt, M, N)
MARK(plus(X1, X2)) → MARK(X2)
MARK(x(X1, X2)) → MARK(X1)
A__U11(tt, M, N) → A__U12(tt, M, N)
A__U22(tt, M, N) → MARK(M)
A__PLUS(N, 0) → MARK(N)
A__U12(tt, M, N) → MARK(N)
MARK(U22(X1, X2, X3)) → A__U22(mark(X1), X2, X3)
A__U21(tt, M, N) → A__U22(tt, M, N)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__U22(tt, M, N) → A__X(mark(N), mark(M))
MARK(plus(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(x(X1, X2)) → MARK(X2)
A__X(N, s(M)) → A__U21(tt, M, N)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U12(tt, M, N) → MARK(M)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)

The TRS R consists of the following rules:

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U12(X1, X2, X3)) → MARK(X1)
A__U22(tt, M, N) → MARK(N)
A__U22(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(s(X)) → MARK(X)
A__PLUS(N, s(M)) → A__U11(tt, M, N)
MARK(plus(X1, X2)) → MARK(X2)
MARK(x(X1, X2)) → MARK(X1)
A__U22(tt, M, N) → MARK(M)
A__PLUS(N, 0) → MARK(N)
A__U12(tt, M, N) → MARK(N)
MARK(U22(X1, X2, X3)) → A__U22(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__U22(tt, M, N) → A__X(mark(N), mark(M))
MARK(plus(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(x(X1, X2)) → MARK(X2)
A__X(N, s(M)) → A__U21(tt, M, N)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U12(tt, M, N) → MARK(M)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)

The remaining pairs can at least be oriented weakly.

A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U11(tt, M, N) → A__U12(tt, M, N)
A__U21(tt, M, N) → A__U22(tt, M, N)

Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U12(x1, x2, x3) = U12(x1, x2, x3)
A__U22(x1, x2, x3) = A__U22(x1, x2, x3)
tt = tt
A__PLUS(x1, x2) = A__PLUS(x1, x2)
a__x(x1, x2) = a__x(x1, x2)
mark(x1) = x1
s(x1) = s(x1)
A__U12(x1, x2, x3) = A__U12(x2, x3)
A__U11(x1, x2, x3) = A__U11(x2, x3)
plus(x1, x2) = plus(x1, x2)
x(x1, x2) = x(x1, x2)
0 = 0
U22(x1, x2, x3) = U22(x1, x2, x3)
A__U21(x1, x2, x3) = A__U21(x1, x2, x3)
U11(x1, x2, x3) = U11(x1, x2, x3)
A__X(x1, x2) = A__X(x1, x2)
U21(x1, x2, x3) = U21(x1, x2, x3)
a__U11(x1, x2, x3) = a__U11(x1, x2, x3)
a__U12(x1, x2, x3) = a__U12(x1, x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)
a__U21(x1, x2, x3) = a__U21(x1, x2, x3)
a__U22(x1, x2, x3) = a__U22(x1, x2, x3)

Recursive path order with status [2].
Quasi-Precedence:

[A_{_U22₃}, a_{_x₂}, x₂, U22₃, A_{_U21₃}, A_{_X₂}, U21₃, a_{_U21₃}, a_{_U22₃}] > [MARK₁, A_{_PLUS₂}, A_{_U12₂}, A_{_U11₂}] > tt
[A_{_U22₃}, a_{_x₂}, x₂, U22₃, A_{_U21₃}, A_{_X₂}, U21₃, a_{_U21₃}, a_{_U22₃}] > [U12₃, plus₂, U11₃, a_{_U11₃}, a_{_U12₃}, a_{_plus₂}] > s₁ > tt
[A_{_U22₃}, a_{_x₂}, x₂, U22₃, A_{_U21₃}, A_{_X₂}, U21₃, a_{_U21₃}, a_{_U22₃}] > 0

Status:

plus₂: multiset
A_{_PLUS₂}: multiset
A_{_U12₂}: multiset
A_{_U11₂}: multiset
A_{_U21₃}: multiset
U11₃: multiset
a_{_U21₃}: multiset
x₂: multiset
a_{_U12₃}: multiset
0: multiset
A_{_U22₃}: multiset
a_{_x₂}: multiset
a_{_plus₂}: multiset
a_{_U22₃}: multiset
MARK₁: multiset
tt: multiset
U22₃: multiset
a_{_U11₃}: multiset
A_{_X₂}: multiset
s₁: multiset
U12₃: multiset
U21₃: multiset

The following usable rules [17] were oriented:

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
a__U21(tt, M, N) → a__U22(tt, M, N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
a__plus(N, 0) → mark(N)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, 0) → 0
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(tt) → tt
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

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

A__U11(tt, M, N) → A__U12(tt, M, N)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U21(tt, M, N) → A__U22(tt, M, N)

The TRS R consists of the following rules:

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.