(0) Obligation:

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

f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

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(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

The set Q consists of the following terms:

f(empty, x0)
f(cons(x0, x1), x2)
g(x0, x1, x2)

(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(cons(x, k), l) → G(k, l, cons(x, k))
G(a, b, c) → F(a, cons(b, c))

The TRS R consists of the following rules:

f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

The set Q consists of the following terms:

f(empty, x0)
f(cons(x0, x1), x2)
g(x0, x1, x2)

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(cons(x, k), l) → G(k, l, cons(x, k))
G(a, b, c) → F(a, cons(b, c))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(x1, x2)  =  F(x1)
cons(x1, x2)  =  cons(x2)
G(x1, x2, x3)  =  G(x1, x3)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[F1, cons1, G2]

Status:
cons1: [1]
G2: [1,2]
F1: [1]


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

The set Q consists of the following terms:

f(empty, x0)
f(cons(x0, x1), x2)
g(x0, x1, x2)

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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE