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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
↳13 QTRSRRRProof (⇔)
↳14 QTRS
↳15 QTRSRRRProof (⇔)
↳16 QTRS
  ↳17 RisEmptyProof (⇔)
    ↳18 TRUE
  ↳19 RisEmptyProof (⇔)
    ↳20 TRUE

(0) Obligation:

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

a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__f(x1)) = x1   
POL(a__if(x1, x2, x3)) = x1 + x2 + x3   
POL(c) = 0   
POL(f(x1)) = x1   
POL(false) = 1   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(mark(x1)) = x1   
POL(true) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__if(false, X, Y) → mark(Y)


(2) Obligation:

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

a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__f(x1)) = 2 + 2·x1   
POL(a__if(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(c) = 0   
POL(f(x1)) = 1 + 2·x1   
POL(false) = 2   
POL(if(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(mark(x1)) = 2·x1   
POL(true) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__f(X) → a__if(mark(X), c, f(true))
mark(false) → false
a__f(X) → f(X)


(4) Obligation:

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

a__if(true, X, Y) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
a__if(X1, X2, X3) → if(X1, X2, X3)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__f(x1)) = x1   
POL(a__if(x1, x2, x3)) = x1 + x2 + x3   
POL(c) = 0   
POL(f(x1)) = x1   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(mark(x1)) = x1   
POL(true) = 1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__if(true, X, Y) → mark(X)


(6) Obligation:

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

mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
a__if(X1, X2, X3) → if(X1, X2, X3)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__f(x1)) = x1   
POL(a__if(x1, x2, x3)) = x1 + x2 + x3   
POL(c) = 0   
POL(f(x1)) = 1 + x1   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(mark(x1)) = x1   
POL(true) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(f(X)) → a__f(mark(X))


(8) Obligation:

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

mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
a__if(X1, X2, X3) → if(X1, X2, X3)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__if(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3   
POL(c) = 2   
POL(if(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(mark(x1)) = 2·x1   
POL(true) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(c) → c


(10) Obligation:

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

mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(true) → true
a__if(X1, X2, X3) → if(X1, X2, X3)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__if(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3   
POL(if(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(mark(x1)) = 2·x1   
POL(true) = 2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(true) → true


(12) Obligation:

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

mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
a__if(X1, X2, X3) → if(X1, X2, X3)

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__if(x1, x2, x3)) = 2 + x1 + 2·x2 + x3   
POL(if(x1, x2, x3)) = 1 + x1 + 2·x2 + x3   
POL(mark(x1)) = 2·x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__if(X1, X2, X3) → if(X1, X2, X3)


(14) Obligation:

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

mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__if(x1, x2, x3)) = x1 + x2 + x3   
POL(if(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(mark(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)


(16) Obligation:

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

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE