Termination w.r.t. Q proof of /tmp/tmpZJQhtd/Ex1_Zan97

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__g(X) → a__h(X)
a__c → d
a__h(d) → a__g(c)
mark(g(X)) → a__g(X)
mark(h(X)) → a__h(X)
mark(c) → a__c
mark(d) → d
a__g(X) → g(X)
a__h(X) → h(X)
a__c → c

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > [a_{_c}, d] > a_{_g₁} > a_{_h₁} > c
mark₁ > [a_{_c}, d] > a_{_g₁} > a_{_h₁} > h₁
mark₁ > [a_{_c}, d] > a_{_g₁} > g₁

Status:

c: []
a_{_c}: []
a_{_h₁}: [1]
g₁: [1]
a_{_g₁}: [1]
h₁: [1]
mark₁: [1]
d: []

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

a__g(X) → a__h(X)
a__h(d) → a__g(c)
mark(g(X)) → a__g(X)
mark(h(X)) → a__h(X)
mark(c) → a__c
mark(d) → d
a__g(X) → g(X)
a__h(X) → h(X)
a__c → c

(2) Obligation:

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

a__c → d

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

a_{_c} > d

Status:

a_{_c}: []
d: []

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

a__c → d

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)