(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(f(X)) → a__c(f(g(f(X))))
a__c(X) → d(X)
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(h(X)) → a__h(mark(X))
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
a__c(X) → c(X)
a__h(X) → h(X)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
[ah1, mark1, h1] > af1 > f1 > [ac1, d1, c1]
[ah1, mark1, h1] > af1 > g1 > [ac1, d1, c1]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__f(f(X)) → a__c(f(g(f(X))))
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__c(X) → d(X)
mark(h(X)) → a__h(mark(X))
a__c(X) → c(X)
a__h(X) → h(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
ac1 > d1 > [mark1, h1, ah1]
ac1 > c1 > [mark1, h1, ah1]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__c(X) → d(X)
a__c(X) → c(X)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(h(X)) → a__h(mark(X))
a__h(X) → h(X)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
mark1 > [h1, ah1]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(h(X)) → a__h(mark(X))
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__h(X) → h(X)
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
ah1 > h1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__h(X) → h(X)
(8) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) TRUE