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:

s(a) → a
s(s(x)) → x
s(f(x, y)) → f(s(y), s(x))
s(g(x, y)) → g(s(x), s(y))
f(x, a) → x
f(a, y) → y
f(g(x, y), g(u, v)) → g(f(x, u), f(y, v))
g(a, a) → a

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a) = 0   
POL(f(x₁, x₂)) = x₁ + x₂   
POL(g(x₁, x₂)) = 1 + x₁ + x₂   
POL(s(x₁)) = x₁   

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

f(g(x, y), g(u, v)) → g(f(x, u), f(y, v))
g(a, a) → a

(2) Obligation:

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

s(a) → a
s(s(x)) → x
s(f(x, y)) → f(s(y), s(x))
s(g(x, y)) → g(s(x), s(y))
f(x, a) → x
f(a, y) → y

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a) = 1   
POL(f(x₁, x₂)) = x₁ + x₂   
POL(g(x₁, x₂)) = x₁ + x₂   
POL(s(x₁)) = x₁   

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

f(x, a) → x
f(a, y) → y

(4) Obligation:

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

s(a) → a
s(s(x)) → x
s(f(x, y)) → f(s(y), s(x))
s(g(x, y)) → g(s(x), s(y))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a) = 0   
POL(f(x₁, x₂)) = x₁ + x₂   
POL(g(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(s(x₁)) = 2·x₁   

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

s(g(x, y)) → g(s(x), s(y))

(6) Obligation:

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

s(a) → a
s(s(x)) → x
s(f(x, y)) → f(s(y), s(x))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a) = 1   
POL(f(x₁, x₂)) = 1 + x₁ + x₂   
POL(s(x₁)) = 2·x₁   

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

s(a) → a
s(f(x, y)) → f(s(y), s(x))

(8) Obligation:

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

s(s(x)) → x

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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(s(x)) → x

(10) Obligation:

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

(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