Termination w.r.t. Q proof of /tmp/tmpN3k2D7/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 with status [RPO].
Quasi-Precedence:

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

Status:

f₁: multiset
a_{_h₁}: [1]
c₁: multiset
a_{_f₁}: multiset
a_{_c₁}: multiset
g₁: [1]
h₁: [1]
mark₁: [1]
d₁: multiset

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)

(2) Obligation:

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

a__c(X) → c(X)
a__h(X) → h(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

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

a_{_c₁} > [c₁, h₁]
a_{_h₁} > [c₁, h₁]

Status:

a_{_h₁}: multiset
c₁: multiset
a_{_c₁}: multiset
h₁: multiset

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)
a__h(X) → h(X)

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)