(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