(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__c → a__f(g(c))
a__f(g(X)) → g(X)
mark(c) → a__c
mark(f(X)) → a__f(X)
mark(g(X)) → g(X)
a__c → c
a__f(X) → f(X)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > ac > g1 > [af1, f1]
mark1 > ac > c > [af1, f1]
Status:
c: multiset
ac: multiset
f1: multiset
af1: multiset
g1: [1]
mark1: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__c → a__f(g(c))
a__f(g(X)) → g(X)
mark(c) → a__c
mark(f(X)) → a__f(X)
mark(g(X)) → g(X)
a__c → c
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X) → f(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
af1 > f1
Status:
f1: multiset
af1: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__f(X) → f(X)
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) TRUE