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:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

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

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(f(0, x), 1) → F(g(f(x, x)), x)
F(f(0, x), 1) → F(x, x)
F(g(x), y) → F(x, y)

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

We have to consider all minimal (P,Q,R)-chains.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

F(f(0, x), 1) → F(x, x)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(F(x1, x2)) =

1A

+

1A

·

x1

+

-I

·

x2

POL(f(x1, x2)) =

0A

+

1A

·

x1

+

4A

·

x2

POL(0) =

4A

POL(1) =

0A

POL(g(x1)) =

-I

+

0A

·

x1

The following usable rules [FROCOS05] were oriented:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(f(0, x), 1) → F(g(f(x, x)), x)
F(g(x), y) → F(x, y)

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

We have to consider all minimal (P,Q,R)-chains.

(7) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule F(g(x), y) → F(x, y) we obtained the following new rules [LPAR04]:

F(g(f(0, y_0)), 1) → F(f(0, y_0), 1)
F(g(g(y_0)), x1) → F(g(y_0), x1)

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(f(0, x), 1) → F(g(f(x, x)), x)
F(g(f(0, y_0)), 1) → F(f(0, y_0), 1)
F(g(g(y_0)), x1) → F(g(y_0), x1)

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

We have to consider all minimal (P,Q,R)-chains.

(9) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule F(f(0, x), 1) → F(g(f(x, x)), x) at position [] we obtained the following new rules [LPAR04]:

F(f(0, g(x0)), 1) → F(g(g(f(x0, g(x0)))), g(x0))

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(g(f(0, y_0)), 1) → F(f(0, y_0), 1)
F(g(g(y_0)), x1) → F(g(y_0), x1)
F(f(0, g(x0)), 1) → F(g(g(f(x0, g(x0)))), g(x0))

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

F(f(0, g(x0)), 1) → F(g(g(f(x0, g(x0)))), g(x0))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(1) = 1   
POL(F(x₁, x₂)) = x₂   
POL(f(x₁, x₂)) = 0   
POL(g(x₁)) = 0   

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(g(f(0, y_0)), 1) → F(f(0, y_0), 1)
F(g(g(y_0)), x1) → F(g(y_0), x1)

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

We have to consider all minimal (P,Q,R)-chains.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(g(g(y_0)), x1) → F(g(y_0), x1)

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

We have to consider all minimal (P,Q,R)-chains.

(15) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(g(g(y_0)), x1) → F(g(y_0), x1)

R is empty.
The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

We have to consider all minimal (P,Q,R)-chains.

(17) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

f(f(0, x0), 1)
f(g(x0), x1)

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(g(g(y_0)), x1) → F(g(y_0), x1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• F(g(g(y_0)), x1) → F(g(y_0), x1)
  The graph contains the following edges 1 > 1, 2 >= 2

(20) TRUE