(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(x)) → s(s(f(p(s(x)))))
f(0(x)) → 0(x)
p(s(x)) → x
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
s(f(x)) → s(p(f(s(s(x)))))
0(f(x)) → 0(x)
s(p(x)) → x
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0(x1)) = x1
POL(f(x1)) = 1 + x1
POL(p(x1)) = x1
POL(s(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
0(f(x)) → 0(x)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
s(f(x)) → s(p(f(s(s(x)))))
s(p(x)) → x
Q is empty.
(5) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
s(f(x)) → s(p(f(s(s(x)))))
s(p(x)) → x
The set Q consists of the following terms:
s(f(x0))
s(p(x0))
(7) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S(f(x)) → S(p(f(s(s(x)))))
S(f(x)) → S(s(x))
S(f(x)) → S(x)
The TRS R consists of the following rules:
s(f(x)) → s(p(f(s(s(x)))))
s(p(x)) → x
The set Q consists of the following terms:
s(f(x0))
s(p(x0))
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S(f(x)) → S(x)
S(f(x)) → S(s(x))
The TRS R consists of the following rules:
s(f(x)) → s(p(f(s(s(x)))))
s(p(x)) → x
The set Q consists of the following terms:
s(f(x0))
s(p(x0))
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
S(f(x)) → S(x)
S(f(x)) → S(s(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(S(x1)) = x1
POL(f(x1)) = 1 + x1
POL(p(x1)) = x1
POL(s(x1)) = x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
s(f(x)) → s(p(f(s(s(x)))))
s(p(x)) → x
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
s(f(x)) → s(p(f(s(s(x)))))
s(p(x)) → x
The set Q consists of the following terms:
s(f(x0))
s(p(x0))
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) YES