Termination w.r.t. Q proof of /tmp/tmpuy0V9X/Ex25_Luc06

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

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 QTRSRRRProof (⇔)

↳4 QTRS

↳5 RisEmptyProof (⇔)

↳6 TRUE

(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:

mark₁ > a_{_f₁} > [f₁, a_{_c₁}, g₁, c₁] > d₁
mark₁ > a_{_h₁} > [f₁, a_{_c₁}, g₁, c₁] > d₁
mark₁ > a_{_h₁} > h₁ > d₁

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__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__h(X) → h(X)

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__c(X) → c(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive Path Order [RPO].
Precedence:

a_{_c₁} > c₁

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__c(X) → c(X)

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)