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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 QDPOrderProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 QDP
↳7 QDPOrderProof (⇔)
↳8 QDP
↳9 DependencyGraphProof (⇔)
↳10 TRUE

(0) Obligation:

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.

(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:

A__U11(tt, M, N) → A__U12(tt, M, N)
A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U12(tt, M, N) → MARK(N)
A__U12(tt, M, 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__U22(tt, M, N) → A__X(mark(N), mark(M))
A__U22(tt, M, N) → MARK(N)
A__U22(tt, M, N) → MARK(M)
A__PLUS(N, 0) → MARK(N)
A__PLUS(N, s(M)) → A__U11(tt, M, N)
A__X(N, s(M)) → A__U21(tt, M, N)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2, X3)) → A__U22(mark(X1), X2, X3)
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

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.

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__U12(tt, M, N) → MARK(N)
A__U12(tt, M, N) → MARK(M)
A__U21(tt, M, N) → A__U22(tt, M, N)
A__U22(tt, M, N) → A__X(mark(N), mark(M))
A__U22(tt, M, N) → MARK(N)
A__U22(tt, M, N) → MARK(M)
A__PLUS(N, 0) → MARK(N)
A__X(N, s(M)) → A__U21(tt, M, N)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2, X3)) → A__U22(mark(X1), X2, X3)
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__U11(x0, x1, x2, x3)  =  A__U11(x0, x2)
A__U12(x0, x1, x2, x3)  =  A__U12(x0, x2)
A__PLUS(x0, x1, x2)  =  A__PLUS(x0, x2)
MARK(x0, x1)  =  MARK(x1)
A__U21(x0, x1, x2, x3)  =  A__U21(x0, x1, x2, x3)
A__U22(x0, x1, x2, x3)  =  A__U22(x0)
A__X(x0, x1, x2)  =  A__X(x0)

Tags:
A__U11 has argument tags [0,24,24,25] and root tag 4
A__U12 has argument tags [0,7,4,8] and root tag 4
A__PLUS has argument tags [0,16,4] and root tag 4
MARK has argument tags [0,0] and root tag 0
A__U21 has argument tags [18,1,0,0] and root tag 6
A__U22 has argument tags [0,0,28,0] and root tag 4
A__X has argument tags [0,0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__U11(x1, x2, x3)  =  x3
tt  =  tt
A__U12(x1, x2, x3)  =  x3
A__PLUS(x1, x2)  =  x1
mark(x1)  =  x1
MARK(x1)  =  MARK
A__U21(x1, x2, x3)  =  A__U21(x2, x3)
A__U22(x1, x2, x3)  =  A__U22(x2, x3)
a__x(x1, x2)  =  a__x(x1, x2)
A__X(x1, x2)  =  A__X(x1, x2)
0  =  0
s(x1)  =  s(x1)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
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)
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)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[AU212, AU222, ax2, AX2, U213, U223, x2, aU213, aU223] > [U113, U123, plus2, aU113, aU123, aplus2] > tt > s1 > 0
[AU212, AU222, ax2, AX2, U213, U223, x2, aU213, aU223] > [U113, U123, plus2, aU113, aU123, aplus2] > MARK > 0

Status:
tt: []
MARK: []
AU212: [2,1]
AU222: [2,1]
ax2: [1,2]
AX2: [1,2]
0: []
s1: [1]
U113: [3,2,1]
U123: [3,2,1]
plus2: [1,2]
U213: [3,2,1]
U223: [3,2,1]
x2: [1,2]
aU113: [3,2,1]
aU123: [3,2,1]
aplus2: [1,2]
aU213: [3,2,1]
aU223: [3,2,1]


The following usable rules [FROCOS05] were oriented:

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

(4) Obligation:

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__U22(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__PLUS(N, s(M)) → A__U11(tt, M, N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))

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.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(6) Obligation:

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

A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U11(tt, M, N)
A__U11(tt, M, N) → A__U12(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.

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__U12(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U11(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:
A__U12(x0, x1, x2, x3)  =  A__U12(x2)
A__PLUS(x0, x1, x2)  =  A__PLUS(x2)
A__U11(x0, x1, x2, x3)  =  A__U11(x0, x1, x2)

Tags:
A__U12 has argument tags [4,9,8,1] and root tag 0
A__PLUS has argument tags [0,0,2] and root tag 2
A__U11 has argument tags [2,15,8,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__U12(x1, x2, x3)  =  A__U12(x1, x2)
tt  =  tt
A__PLUS(x1, x2)  =  x1
mark(x1)  =  x1
s(x1)  =  s(x1)
A__U11(x1, x2, x3)  =  A__U11(x1)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
a__U11(x1, x2, x3)  =  a__U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
a__U12(x1, x2, x3)  =  a__U12(x1, x2, x3)
a__plus(x1, x2)  =  a__plus(x1, x2)
0  =  0
plus(x1, x2)  =  plus(x1, x2)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
a__U21(x1, x2, x3)  =  a__U21(x1, x2, x3)
a__U22(x1, x2, x3)  =  a__U22(x1, x2, x3)
a__x(x1, x2)  =  a__x(x1, x2)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:
AU122 > [tt, s1, AU111]
[U213, aU213, aU223, ax2, U223, x2] > [U113, aU113, U123, aU123, aplus2, plus2] > [tt, s1, AU111]
[U213, aU213, aU223, ax2, U223, x2] > 0 > [tt, s1, AU111]

Status:
AU122: [2,1]
tt: []
s1: [1]
AU111: [1]
U113: [3,2,1]
aU113: [3,2,1]
U123: [3,2,1]
aU123: [3,2,1]
aplus2: [1,2]
0: []
plus2: [1,2]
U213: [2,3,1]
aU213: [2,3,1]
aU223: [2,3,1]
ax2: [2,1]
U223: [2,3,1]
x2: [2,1]


The following usable rules [FROCOS05] were oriented:

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

(8) Obligation:

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

A__U11(tt, M, N) → A__U12(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.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(10) TRUE