Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

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

f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

The TRS R 2 is

f(t, x, y) → f(g(x, y), x, s(y))

The signature Sigma is {f}

(2) Obligation:

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

f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

The set Q consists of the following terms:

f(t, x0, x1)
g(s(x0), 0)
g(s(x0), s(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(t, x, y) → F(g(x, y), x, s(y))
F(t, x, y) → G(x, y)
G(s(x), s(y)) → G(x, y)

The TRS R consists of the following rules:

f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

The set Q consists of the following terms:

f(t, x0, x1)
g(s(x0), 0)
g(s(x0), s(x1))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(6) Complex Obligation (AND)

(7) Obligation:

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

G(s(x), s(y)) → G(x, y)

The TRS R consists of the following rules:

f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

The set Q consists of the following terms:

f(t, x0, x1)
g(s(x0), 0)
g(s(x0), s(x1))

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

G(s(x), s(y)) → G(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
G(x1, x2)  =  G(x2)
s(x1)  =  s(x1)
f(x1, x2, x3)  =  f(x2)
t  =  t
g(x1, x2)  =  g(x1, x2)
0  =  0

Recursive path order with status [RPO].
Precedence:

G₁ > f₁
0 > t > s₁ > g₂ > f₁

Status:

G₁: multiset
s₁: multiset
f₁: [1]
t: multiset
g₂: [2,1]
0: multiset

The following usable rules [FROCOS05] were oriented:

f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

The set Q consists of the following terms:

f(t, x0, x1)
g(s(x0), 0)
g(s(x0), s(x1))

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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

F(t, x, y) → F(g(x, y), x, s(y))

The TRS R consists of the following rules:

f(t, x, y) → f(g(x, y), x, s(y))
g(s(x), 0) → t
g(s(x), s(y)) → g(x, y)

The set Q consists of the following terms:

f(t, x0, x1)
g(s(x0), 0)
g(s(x0), s(x1))

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