Termination w.r.t. Q proof of /tmp/tmpJl5Fiv/21.xml

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 DependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 QDP

        ↳11 QDPOrderProof (⇔)

        ↳12 QDP

        ↳13 PisEmptyProof (⇔)

        ↳14 TRUE

(0) Obligation:

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.

(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:

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(+(x₁, x₂)) = x₁ + x₂
POL(+¹(x₁, x₂)) = x₂
POL(p1) = 1
POL(p10) = 0
POL(p2) = 1
POL(p5) = 1

The following usable rules [FROCOS05] were oriented:

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

(7) Obligation:

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

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

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(9) TRUE

(10) Obligation:

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

+¹(+(x, y), z) → +¹(y, z)
+¹(+(x, y), z) → +¹(x, +(y, z))

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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

+¹(+(x, y), z) → +¹(y, z)
+¹(+(x, y), z) → +¹(x, +(y, z))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(+(x₁, x₂)) = 1 + x₁ + x₂
POL(+¹(x₁, x₂)) = x₁
POL(p1) = 0
POL(p10) = 0
POL(p2) = 0
POL(p5) = 0

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

Q DP problem:
P is empty.
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.

(13) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) DependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) PisEmptyProof (EQUIVALENT transformation)

(14) TRUE