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:

+(p1, p1) → p2
+(p1, +(p2, p2)) → p5
+(p5, p5) → p10
+(+(x, y), z) → +(x, +(y, z))
+(p1, +(p1, x)) → +(p2, x)
+(p1, +(p2, +(p2, x))) → +(p5, x)
+(p2, p1) → +(p1, p2)
+(p2, +(p1, x)) → +(p1, +(p2, x))
+(p2, +(p2, p2)) → +(p1, p5)
+(p2, +(p2, +(p2, x))) → +(p1, +(p5, x))
+(p5, p1) → +(p1, p5)
+(p5, +(p1, x)) → +(p1, +(p5, x))
+(p5, p2) → +(p2, p5)
+(p5, +(p2, x)) → +(p2, +(p5, x))
+(p5, +(p5, x)) → +(p10, x)
+(p10, p1) → +(p1, p10)
+(p10, +(p1, x)) → +(p1, +(p10, x))
+(p10, p2) → +(p2, p10)
+(p10, +(p2, x)) → +(p2, +(p10, x))
+(p10, p5) → +(p5, p10)
+(p10, +(p5, x)) → +(p5, +(p10, x))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

+(p1, p1) → p2
+(p1, +(p2, p2)) → p5
+(p5, p5) → p10
+(+(x, y), z) → +(x, +(y, z))
+(p1, +(p1, x)) → +(p2, x)
+(p1, +(p2, +(p2, x))) → +(p5, x)
+(p2, p1) → +(p1, p2)
+(p2, +(p1, x)) → +(p1, +(p2, x))
+(p2, +(p2, p2)) → +(p1, p5)
+(p2, +(p2, +(p2, x))) → +(p1, +(p5, x))
+(p5, p1) → +(p1, p5)
+(p5, +(p1, x)) → +(p1, +(p5, x))
+(p5, p2) → +(p2, p5)
+(p5, +(p2, x)) → +(p2, +(p5, x))
+(p5, +(p5, x)) → +(p10, x)
+(p10, p1) → +(p1, p10)
+(p10, +(p1, x)) → +(p1, +(p10, x))
+(p10, p2) → +(p2, p10)
+(p10, +(p2, x)) → +(p2, +(p10, x))
+(p10, p5) → +(p5, p10)
+(p10, +(p5, x)) → +(p5, +(p10, x))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

+1(p5, p2) → +1(p2, p5)
+1(p10, +(p5, x)) → +1(p10, x)
+1(p5, p1) → +1(p1, p5)
+1(p10, +(p1, x)) → +1(p1, +(p10, x))
+1(p2, +(p1, x)) → +1(p2, x)
+1(p10, +(p2, x)) → +1(p2, +(p10, x))
+1(p10, +(p2, x)) → +1(p10, x)
+1(p10, +(p5, x)) → +1(p5, +(p10, x))
+1(p10, p5) → +1(p5, p10)
+1(p2, p1) → +1(p1, p2)
+1(p1, +(p1, x)) → +1(p2, x)
+1(+(x, y), z) → +1(y, z)
+1(p2, +(p1, x)) → +1(p1, +(p2, x))
+1(p2, +(p2, +(p2, x))) → +1(p5, x)
+1(p10, +(p1, x)) → +1(p10, x)
+1(p2, +(p2, +(p2, x))) → +1(p1, +(p5, x))
+1(p5, +(p2, x)) → +1(p2, +(p5, x))
+1(p5, +(p1, x)) → +1(p5, x)
+1(p10, p2) → +1(p2, p10)
+1(p1, +(p2, +(p2, x))) → +1(p5, x)
+1(p5, +(p5, x)) → +1(p10, x)
+1(p10, p1) → +1(p1, p10)
+1(+(x, y), z) → +1(x, +(y, z))
+1(p5, +(p1, x)) → +1(p1, +(p5, x))
+1(p2, +(p2, p2)) → +1(p1, p5)
+1(p5, +(p2, x)) → +1(p5, x)

The TRS R consists of the following rules:

+(p1, p1) → p2
+(p1, +(p2, p2)) → p5
+(p5, p5) → p10
+(+(x, y), z) → +(x, +(y, z))
+(p1, +(p1, x)) → +(p2, x)
+(p1, +(p2, +(p2, x))) → +(p5, x)
+(p2, p1) → +(p1, p2)
+(p2, +(p1, x)) → +(p1, +(p2, x))
+(p2, +(p2, p2)) → +(p1, p5)
+(p2, +(p2, +(p2, x))) → +(p1, +(p5, x))
+(p5, p1) → +(p1, p5)
+(p5, +(p1, x)) → +(p1, +(p5, x))
+(p5, p2) → +(p2, p5)
+(p5, +(p2, x)) → +(p2, +(p5, x))
+(p5, +(p5, x)) → +(p10, x)
+(p10, p1) → +(p1, p10)
+(p10, +(p1, x)) → +(p1, +(p10, x))
+(p10, p2) → +(p2, p10)
+(p10, +(p2, x)) → +(p2, +(p10, x))
+(p10, p5) → +(p5, p10)
+(p10, +(p5, x)) → +(p5, +(p10, x))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof

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

+1(p5, p2) → +1(p2, p5)
+1(p10, +(p5, x)) → +1(p10, x)
+1(p5, p1) → +1(p1, p5)
+1(p10, +(p1, x)) → +1(p1, +(p10, x))
+1(p2, +(p1, x)) → +1(p2, x)
+1(p10, +(p2, x)) → +1(p2, +(p10, x))
+1(p10, +(p2, x)) → +1(p10, x)
+1(p10, +(p5, x)) → +1(p5, +(p10, x))
+1(p10, p5) → +1(p5, p10)
+1(p2, p1) → +1(p1, p2)
+1(p1, +(p1, x)) → +1(p2, x)
+1(+(x, y), z) → +1(y, z)
+1(p2, +(p1, x)) → +1(p1, +(p2, x))
+1(p2, +(p2, +(p2, x))) → +1(p5, x)
+1(p10, +(p1, x)) → +1(p10, x)
+1(p2, +(p2, +(p2, x))) → +1(p1, +(p5, x))
+1(p5, +(p2, x)) → +1(p2, +(p5, x))
+1(p5, +(p1, x)) → +1(p5, x)
+1(p10, p2) → +1(p2, p10)
+1(p1, +(p2, +(p2, x))) → +1(p5, x)
+1(p5, +(p5, x)) → +1(p10, x)
+1(p10, p1) → +1(p1, p10)
+1(+(x, y), z) → +1(x, +(y, z))
+1(p5, +(p1, x)) → +1(p1, +(p5, x))
+1(p2, +(p2, p2)) → +1(p1, p5)
+1(p5, +(p2, x)) → +1(p5, x)

The TRS R consists of the following rules:

+(p1, p1) → p2
+(p1, +(p2, p2)) → p5
+(p5, p5) → p10
+(+(x, y), z) → +(x, +(y, z))
+(p1, +(p1, x)) → +(p2, x)
+(p1, +(p2, +(p2, x))) → +(p5, x)
+(p2, p1) → +(p1, p2)
+(p2, +(p1, x)) → +(p1, +(p2, x))
+(p2, +(p2, p2)) → +(p1, p5)
+(p2, +(p2, +(p2, x))) → +(p1, +(p5, x))
+(p5, p1) → +(p1, p5)
+(p5, +(p1, x)) → +(p1, +(p5, x))
+(p5, p2) → +(p2, p5)
+(p5, +(p2, x)) → +(p2, +(p5, x))
+(p5, +(p5, x)) → +(p10, x)
+(p10, p1) → +(p1, p10)
+(p10, +(p1, x)) → +(p1, +(p10, x))
+(p10, p2) → +(p2, p10)
+(p10, +(p2, x)) → +(p2, +(p10, x))
+(p10, p5) → +(p5, p10)
+(p10, +(p5, x)) → +(p5, +(p10, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof

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

+1(p5, p2) → +1(p2, p5)
+1(p10, +(p5, x)) → +1(p10, x)
+1(p5, p1) → +1(p1, p5)
+1(p10, +(p1, x)) → +1(p1, +(p10, x))
+1(p2, +(p1, x)) → +1(p2, x)
+1(p10, +(p2, x)) → +1(p2, +(p10, x))
+1(p10, +(p2, x)) → +1(p10, x)
+1(p10, +(p5, x)) → +1(p5, +(p10, x))
+1(p10, p5) → +1(p5, p10)
+1(p2, p1) → +1(p1, p2)
+1(+(x, y), z) → +1(y, z)
+1(p1, +(p1, x)) → +1(p2, x)
+1(p2, +(p1, x)) → +1(p1, +(p2, x))
+1(p2, +(p2, +(p2, x))) → +1(p5, x)
+1(p10, +(p1, x)) → +1(p10, x)
+1(p5, +(p2, x)) → +1(p2, +(p5, x))
+1(p2, +(p2, +(p2, x))) → +1(p1, +(p5, x))
+1(p10, p2) → +1(p2, p10)
+1(p5, +(p1, x)) → +1(p5, x)
+1(p1, +(p2, +(p2, x))) → +1(p5, x)
+1(p10, p1) → +1(p1, p10)
+1(p5, +(p5, x)) → +1(p10, x)
+1(+(x, y), z) → +1(x, +(y, z))
+1(p5, +(p1, x)) → +1(p1, +(p5, x))
+1(p2, +(p2, p2)) → +1(p1, p5)
+1(p5, +(p2, x)) → +1(p5, x)

The TRS R consists of the following rules:

+(p1, p1) → p2
+(p1, +(p2, p2)) → p5
+(p5, p5) → p10
+(+(x, y), z) → +(x, +(y, z))
+(p1, +(p1, x)) → +(p2, x)
+(p1, +(p2, +(p2, x))) → +(p5, x)
+(p2, p1) → +(p1, p2)
+(p2, +(p1, x)) → +(p1, +(p2, x))
+(p2, +(p2, p2)) → +(p1, p5)
+(p2, +(p2, +(p2, x))) → +(p1, +(p5, x))
+(p5, p1) → +(p1, p5)
+(p5, +(p1, x)) → +(p1, +(p5, x))
+(p5, p2) → +(p2, p5)
+(p5, +(p2, x)) → +(p2, +(p5, x))
+(p5, +(p5, x)) → +(p10, x)
+(p10, p1) → +(p1, p10)
+(p10, +(p1, x)) → +(p1, +(p10, x))
+(p10, p2) → +(p2, p10)
+(p10, +(p2, x)) → +(p2, +(p10, x))
+(p10, p5) → +(p5, p10)
+(p10, +(p5, x)) → +(p5, +(p10, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 9 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ QDPOrderProof

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

+1(p2, +(p2, +(p2, x))) → +1(p5, x)
+1(p10, +(p5, x)) → +1(p10, x)
+1(p10, +(p1, x)) → +1(p1, +(p10, x))
+1(p10, +(p1, x)) → +1(p10, x)
+1(p2, +(p1, x)) → +1(p2, x)
+1(p2, +(p2, +(p2, x))) → +1(p1, +(p5, x))
+1(p5, +(p2, x)) → +1(p2, +(p5, x))
+1(p10, +(p2, x)) → +1(p2, +(p10, x))
+1(p5, +(p1, x)) → +1(p5, x)
+1(p1, +(p2, +(p2, x))) → +1(p5, x)
+1(p10, +(p2, x)) → +1(p10, x)
+1(p10, +(p5, x)) → +1(p5, +(p10, x))
+1(p5, +(p5, x)) → +1(p10, x)
+1(p1, +(p1, x)) → +1(p2, x)
+1(p5, +(p1, x)) → +1(p1, +(p5, x))
+1(p5, +(p2, x)) → +1(p5, x)
+1(p2, +(p1, x)) → +1(p1, +(p2, x))

The TRS R consists of the following rules:

+(p1, p1) → p2
+(p1, +(p2, p2)) → p5
+(p5, p5) → p10
+(+(x, y), z) → +(x, +(y, z))
+(p1, +(p1, x)) → +(p2, x)
+(p1, +(p2, +(p2, x))) → +(p5, x)
+(p2, p1) → +(p1, p2)
+(p2, +(p1, x)) → +(p1, +(p2, x))
+(p2, +(p2, p2)) → +(p1, p5)
+(p2, +(p2, +(p2, x))) → +(p1, +(p5, x))
+(p5, p1) → +(p1, p5)
+(p5, +(p1, x)) → +(p1, +(p5, x))
+(p5, p2) → +(p2, p5)
+(p5, +(p2, x)) → +(p2, +(p5, x))
+(p5, +(p5, x)) → +(p10, x)
+(p10, p1) → +(p1, p10)
+(p10, +(p1, x)) → +(p1, +(p10, x))
+(p10, p2) → +(p2, p10)
+(p10, +(p2, x)) → +(p2, +(p10, x))
+(p10, p5) → +(p5, p10)
+(p10, +(p5, x)) → +(p5, +(p10, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


+1(p2, +(p2, +(p2, x))) → +1(p5, x)
+1(p10, +(p5, x)) → +1(p10, x)
+1(p10, +(p1, x)) → +1(p10, x)
+1(p2, +(p1, x)) → +1(p2, x)
+1(p2, +(p2, +(p2, x))) → +1(p1, +(p5, x))
+1(p5, +(p1, x)) → +1(p5, x)
+1(p1, +(p2, +(p2, x))) → +1(p5, x)
+1(p10, +(p2, x)) → +1(p10, x)
+1(p5, +(p5, x)) → +1(p10, x)
+1(p1, +(p1, x)) → +1(p2, x)
+1(p5, +(p2, x)) → +1(p5, x)
The remaining pairs can at least be oriented weakly.

+1(p10, +(p1, x)) → +1(p1, +(p10, x))
+1(p5, +(p2, x)) → +1(p2, +(p5, x))
+1(p10, +(p2, x)) → +1(p2, +(p10, x))
+1(p10, +(p5, x)) → +1(p5, +(p10, x))
+1(p5, +(p1, x)) → +1(p1, +(p5, x))
+1(p2, +(p1, x)) → +1(p1, +(p2, x))
Used ordering: Combined order from the following AFS and order.
+1(x1, x2)  =  +1(x2)
p2  =  p2
+(x1, x2)  =  +(x2)
p5  =  p5
p10  =  p10
p1  =  p1

Lexicographic path order with status [19].
Quasi-Precedence:
[p2, p5, p10, p1] > +^11 > +1

Status:
p1: multiset
+^11: [1]
p10: multiset
p2: multiset
+1: [1]
p5: multiset


The following usable rules [14] were oriented:

+(p5, p1) → +(p1, p5)
+(p1, p1) → p2
+(p10, p2) → +(p2, p10)
+(p2, +(p2, p2)) → +(p1, p5)
+(p5, p2) → +(p2, p5)
+(p5, p5) → p10
+(p10, p1) → +(p1, p10)
+(p10, p5) → +(p5, p10)
+(p2, p1) → +(p1, p2)
+(p5, +(p1, x)) → +(p1, +(p5, x))
+(p2, +(p1, x)) → +(p1, +(p2, x))
+(p10, +(p1, x)) → +(p1, +(p10, x))
+(p5, +(p2, x)) → +(p2, +(p5, x))
+(p1, +(p2, +(p2, x))) → +(p5, x)
+(p1, +(p1, x)) → +(p2, x)
+(p2, +(p2, +(p2, x))) → +(p1, +(p5, x))
+(p10, +(p5, x)) → +(p5, +(p10, x))
+(p10, +(p2, x)) → +(p2, +(p10, x))
+(p5, +(p5, x)) → +(p10, x)
+(p1, +(p2, p2)) → p5



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ QDPOrderProof
                ↳ QDP
                  ↳ DependencyGraphProof

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

+1(p10, +(p5, x)) → +1(p5, +(p10, x))
+1(p10, +(p1, x)) → +1(p1, +(p10, x))
+1(p5, +(p1, x)) → +1(p1, +(p5, x))
+1(p5, +(p2, x)) → +1(p2, +(p5, x))
+1(p10, +(p2, x)) → +1(p2, +(p10, x))
+1(p2, +(p1, x)) → +1(p1, +(p2, x))

The TRS R consists of the following rules:

+(p1, p1) → p2
+(p1, +(p2, p2)) → p5
+(p5, p5) → p10
+(+(x, y), z) → +(x, +(y, z))
+(p1, +(p1, x)) → +(p2, x)
+(p1, +(p2, +(p2, x))) → +(p5, x)
+(p2, p1) → +(p1, p2)
+(p2, +(p1, x)) → +(p1, +(p2, x))
+(p2, +(p2, p2)) → +(p1, p5)
+(p2, +(p2, +(p2, x))) → +(p1, +(p5, x))
+(p5, p1) → +(p1, p5)
+(p5, +(p1, x)) → +(p1, +(p5, x))
+(p5, p2) → +(p2, p5)
+(p5, +(p2, x)) → +(p2, +(p5, x))
+(p5, +(p5, x)) → +(p10, x)
+(p10, p1) → +(p1, p10)
+(p10, +(p1, x)) → +(p1, +(p10, x))
+(p10, p2) → +(p2, p10)
+(p10, +(p2, x)) → +(p2, +(p10, x))
+(p10, p5) → +(p5, p10)
+(p10, +(p5, x)) → +(p5, +(p10, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 6 less nodes.