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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
  ↳13 RisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 RisEmptyProof (⇔)
    ↳16 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) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = 1 + x1 + x2   
POL(p1) = 0   
POL(p10) = 0   
POL(p2) = 0   
POL(p5) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(2) Obligation:

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

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

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = 2 + 2·x1 + x2   
POL(p1) = 1   
POL(p10) = 2   
POL(p2) = 2   
POL(p5) = 2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(4) Obligation:

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

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

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = 2·x1 + 2·x2   
POL(p1) = 2   
POL(p10) = 0   
POL(p2) = 2   
POL(p5) = 2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(6) Obligation:

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

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

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = 2·x1 + x2   
POL(p1) = 0   
POL(p10) = 2   
POL(p2) = 0   
POL(p5) = 1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(8) Obligation:

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

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

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = 2·x1 + 2·x2   
POL(p1) = 2   
POL(p2) = 2   
POL(p5) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(10) Obligation:

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

+(p2, +(p1, x)) → +(p1, +(p2, x))

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = 2·x1 + 2·x2   
POL(p1) = 2   
POL(p2) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(p2, +(p1, x)) → +(p1, +(p2, x))


(12) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(13) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(14) TRUE

(15) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(16) TRUE