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 RisEmptyProof (⇔)
    ↳10 TRUE
  ↳11 RisEmptyProof (⇔)
    ↳12 TRUE

(0) Obligation:

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

f(x, y) → g1(x, x, y)
f(x, y) → g1(y, x, x)
f(x, y) → g2(x, y, y)
f(x, y) → g2(y, y, x)
g1(x, x, y) → h(x, y)
g1(y, x, x) → h(x, y)
g2(x, y, y) → h(x, y)
g2(y, y, x) → h(x, y)
h(x, x) → x

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(f(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(g1(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(g2(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(h(x1, x2)) = 1 + x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

h(x, x) → x


(2) Obligation:

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

f(x, y) → g1(x, x, y)
f(x, y) → g1(y, x, x)
f(x, y) → g2(x, y, y)
f(x, y) → g2(y, y, x)
g1(x, x, y) → h(x, y)
g1(y, x, x) → h(x, y)
g2(x, y, y) → h(x, y)
g2(y, y, x) → h(x, y)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(f(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(g1(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(g2(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(h(x1, x2)) = 1 + x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f(x, y) → g1(x, x, y)
f(x, y) → g1(y, x, x)
f(x, y) → g2(x, y, y)
f(x, y) → g2(y, y, x)


(4) Obligation:

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

g1(x, x, y) → h(x, y)
g1(y, x, x) → h(x, y)
g2(x, y, y) → h(x, y)
g2(y, y, x) → h(x, y)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(g1(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(g2(x1, x2, x3)) = x1 + x2 + x3   
POL(h(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

g1(x, x, y) → h(x, y)
g1(y, x, x) → h(x, y)


(6) Obligation:

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

g2(x, y, y) → h(x, y)
g2(y, y, x) → h(x, y)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(g2(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3   
POL(h(x1, x2)) = 1 + x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

g2(x, y, y) → h(x, y)
g2(y, y, x) → h(x, y)


(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