(0) Obligation:

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

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none

The TRS R 2 is

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

The signature Sigma is {f, g}

(2) Obligation:

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

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

The set Q consists of the following terms:

f(c(x0, s(x1)))
g(c(s(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(c(X, s(Y))) → F(c(s(X), Y))
G(c(s(X), Y)) → F(c(X, s(Y)))

The TRS R consists of the following rules:

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

The set Q consists of the following terms:

f(c(x0, s(x1)))
g(c(s(x0), 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 1 SCC with 1 less node.

(6) Obligation:

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

F(c(X, s(Y))) → F(c(s(X), Y))

The TRS R consists of the following rules:

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

The set Q consists of the following terms:

f(c(x0, s(x1)))
g(c(s(x0), x1))

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(c(X, s(Y))) → F(c(s(X), Y))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(x1)  =  F(x1)
c(x1, x2)  =  c(x2)
s(x1)  =  s(x1)
f(x1)  =  f(x1)
g(x1)  =  g(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
g1 > f1 > [F1, c1, s1]

Status:
F1: multiset
c1: multiset
s1: multiset
f1: multiset
g1: multiset


The following usable rules [FROCOS05] were oriented:

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

(8) Obligation:

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

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

The set Q consists of the following terms:

f(c(x0, s(x1)))
g(c(s(x0), x1))

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

(9) PisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE