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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
  ↳5 RisEmptyProof (⇔)
    ↳6 TRUE
  ↳7 RisEmptyProof (⇔)
    ↳8 TRUE

(0) Obligation:

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

+(0, y) → y
+(s(x), y) → s(+(x, y))
-(0, y) → 0
-(x, 0) → x
-(s(x), s(y)) → -(x, y)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

+(0, y) → y
-(0, y) → 0
-(x, 0) → x
-(s(x), s(y)) → -(x, y)


(2) Obligation:

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

+(s(x), y) → s(+(x, y))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

+(s(x), y) → s(+(x, y))


(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)

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

(6) TRUE

(7) RisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE