Termination w.r.t. Q proof of /tmp/tmpMrue8w/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:
Lexicographic Path Order [LPO].
Precedence:

[a_{_h₁}, mark₁, h₁] > a_{_f₁} > f₁ > [a_{_c₁}, d₁, c₁]
[a_{_h₁}, mark₁, h₁] > a_{_f₁} > g₁ > [a_{_c₁}, 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:
Lexicographic Path Order [LPO].
Precedence:

a_{_c₁} > d₁
a_{_c₁} > c₁
mark₁ > a_{_h₁} > h₁

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)
mark(h(X)) → a__h(mark(X))
a__c(X) → c(X)
a__h(X) → h(X)

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)