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

(0) Obligation:

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

i(0) → 0
+(0, y) → y
+(x, 0) → x
i(i(x)) → x
+(i(x), x) → 0
+(x, i(x)) → 0
i(+(x, y)) → +(i(x), i(y))
+(x, +(y, z)) → +(+(x, y), z)
+(+(x, i(y)), y) → x
+(+(x, y), i(y)) → x

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x₁, x₂)) = 1 + x₁ + x₂   
POL(0) = 1   
POL(i(x₁)) = x₁   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(0, y) → y
+(x, 0) → x
+(+(x, i(y)), y) → x
+(+(x, y), i(y)) → x

(2) Obligation:

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

i(0) → 0
i(i(x)) → x
+(i(x), x) → 0
+(x, i(x)) → 0
i(+(x, y)) → +(i(x), i(y))
+(x, +(y, z)) → +(+(x, y), z)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x₁, x₂)) = 1 + x₁ + x₂   
POL(0) = 0   
POL(i(x₁)) = x₁   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(i(x), x) → 0
+(x, i(x)) → 0

(4) Obligation:

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

i(0) → 0
i(i(x)) → x
i(+(x, y)) → +(i(x), i(y))
+(x, +(y, z)) → +(+(x, y), z)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(0) = 2   
POL(i(x₁)) = 2·x₁   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

i(0) → 0
i(+(x, y)) → +(i(x), i(y))
+(x, +(y, z)) → +(+(x, y), z)

(6) Obligation:

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

i(i(x)) → x

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(i(x₁)) = 1 + x₁   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

i(i(x)) → x

(8) Obligation:

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

(9) RisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE

(11) RisEmptyProof (EQUIVALENT transformation)

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

(12) TRUE

(13) RisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE