(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))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
x1
cons(
x1,
x2) =
cons(
x2)
G(
x1,
x2,
x3) =
x1
Recursive path order with status [RPO].
Quasi-Precedence:
trivial
Status:
trivial
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(8) TRUE