Termination w.r.t. Q proof of /tmp/tmpng_PNG/Ex6_GM04

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 RisEmptyProof (⇔)

↳4 TRUE

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.

Used ordering:
Recursive path order with status [RPO].
Precedence:

mark₁ > a_{_c} > a_{_f₁} > f₁
mark₁ > a_{_c} > g₁ > f₁
mark₁ > a_{_c} > c > f₁

Status:

c: multiset
a_{_c}: multiset
f₁: multiset
a_{_f₁}: multiset
g₁: multiset
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
a__f(X) → f(X)

Q restricted rewrite system:
R is empty.
Q is empty.

The TRS R is empty. Hence, termination is trivially proven.