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

0 QTRS
  ↳1 QTRS Reverse (⇔)
    ↳2 QTRS
  ↳3 QTRS Reverse (⇔)
    ↳4 QTRS
  ↳5 QTRSRRRProof (⇔)
    ↳6 QTRS
    ↳7 DependencyPairsProof (⇔)
    ↳8 QDP
    ↳9 DependencyGraphProof (⇔)
    ↳10 AND
      ↳11 QDP
        ↳12 UsableRulesProof (⇔)
          ↳13 QDP
          ↳14 UsableRulesReductionPairsProof (⇔)
          ↳15 QDP
          ↳16 PisEmptyProof (⇔)
          ↳17 TRUE
        ↳18 UsableRulesProof (⇔)
          ↳19 QDP
      ↳20 QDP
        ↳21 UsableRulesProof (⇔)
          ↳22 QDP
          ↳23 UsableRulesReductionPairsProof (⇔)
          ↳24 QDP
          ↳25 PisEmptyProof (⇔)
          ↳26 TRUE
        ↳27 UsableRulesProof (⇔)
          ↳28 QDP
      ↳29 QDP
        ↳30 UsableRulesProof (⇔)
          ↳31 QDP
          ↳32 UsableRulesReductionPairsProof (⇔)
          ↳33 QDP
          ↳34 PisEmptyProof (⇔)
          ↳35 TRUE
        ↳36 UsableRulesProof (⇔)
          ↳37 QDP
      ↳38 QDP
        ↳39 UsableRulesProof (⇔)
          ↳40 QDP
          ↳41 UsableRulesReductionPairsProof (⇔)
          ↳42 QDP
          ↳43 PisEmptyProof (⇔)
          ↳44 TRUE
        ↳45 UsableRulesProof (⇔)
          ↳46 QDP
      ↳47 QDP
        ↳48 UsableRulesProof (⇔)
          ↳49 QDP
          ↳50 UsableRulesReductionPairsProof (⇔)
          ↳51 QDP
          ↳52 PisEmptyProof (⇔)
          ↳53 TRUE
        ↳54 UsableRulesProof (⇔)
          ↳55 QDP
      ↳56 QDP
        ↳57 MRRProof (⇔)
        ↳58 QDP
        ↳59 QDPOrderProof (⇔)
        ↳60 QDP
        ↳61 RFCMatchBoundsDPProof (⇔)
        ↳62 TRUE

(0) Obligation:

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

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

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

f(f(active(x))) → f(g(f(c(mark(x)))))
c(active(x)) → d(mark(x))
h(active(x)) → d(c(mark(x)))
f(mark(x)) → mark(f(active(x)))
c(mark(x)) → c(active(x))
g(mark(x)) → g(active(x))
d(mark(x)) → d(active(x))
h(mark(x)) → mark(h(active(x)))
mark(f(x)) → f(x)
active(f(x)) → f(x)
mark(c(x)) → c(x)
active(c(x)) → c(x)
mark(g(x)) → g(x)
active(g(x)) → g(x)
mark(d(x)) → d(x)
active(d(x)) → d(x)
mark(h(x)) → h(x)
active(h(x)) → h(x)

Q is empty.

(3) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(4) Obligation:

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

f(f(active(x))) → f(g(f(c(mark(x)))))
c(active(x)) → d(mark(x))
h(active(x)) → d(c(mark(x)))
f(mark(x)) → mark(f(active(x)))
c(mark(x)) → c(active(x))
g(mark(x)) → g(active(x))
d(mark(x)) → d(active(x))
h(mark(x)) → mark(h(active(x)))
mark(f(x)) → f(x)
active(f(x)) → f(x)
mark(c(x)) → c(x)
active(c(x)) → c(x)
mark(g(x)) → g(x)
active(g(x)) → g(x)
mark(d(x)) → d(x)
active(d(x)) → d(x)
mark(h(x)) → h(x)
active(h(x)) → h(x)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(active(x1)) = x1   
POL(c(x1)) = x1   
POL(d(x1)) = x1   
POL(f(x1)) = x1   
POL(g(x1)) = x1   
POL(h(x1)) = 1 + x1   
POL(mark(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

active(h(X)) → mark(c(d(X)))


(6) Obligation:

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

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.

(7) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(f(f(X))) → MARK(c(f(g(f(X)))))
ACTIVE(f(f(X))) → C(f(g(f(X))))
ACTIVE(f(f(X))) → F(g(f(X)))
ACTIVE(f(f(X))) → G(f(X))
ACTIVE(c(X)) → MARK(d(X))
ACTIVE(c(X)) → D(X)
MARK(f(X)) → ACTIVE(f(mark(X)))
MARK(f(X)) → F(mark(X))
MARK(f(X)) → MARK(X)
MARK(c(X)) → ACTIVE(c(X))
MARK(g(X)) → ACTIVE(g(X))
MARK(d(X)) → ACTIVE(d(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → H(mark(X))
MARK(h(X)) → MARK(X)
F(mark(X)) → F(X)
F(active(X)) → F(X)
C(mark(X)) → C(X)
C(active(X)) → C(X)
G(mark(X)) → G(X)
G(active(X)) → G(X)
D(mark(X)) → D(X)
D(active(X)) → D(X)
H(mark(X)) → H(X)
H(active(X)) → H(X)

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 6 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(active(X)) → H(X)
H(mark(X)) → H(X)

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(active(X)) → H(X)
H(mark(X)) → H(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(14) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

H(active(X)) → H(X)
H(mark(X)) → H(X)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(H(x1)) = 2·x1   
POL(active(x1)) = 2·x1   
POL(mark(x1)) = 2·x1   

(15) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(16) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(17) TRUE

(18) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(active(X)) → H(X)
H(mark(X)) → H(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

D(active(X)) → D(X)
D(mark(X)) → D(X)

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(21) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

D(active(X)) → D(X)
D(mark(X)) → D(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

D(active(X)) → D(X)
D(mark(X)) → D(X)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(D(x1)) = 2·x1   
POL(active(x1)) = 2·x1   
POL(mark(x1)) = 2·x1   

(24) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(26) TRUE

(27) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

D(active(X)) → D(X)
D(mark(X)) → D(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

G(active(X)) → G(X)
G(mark(X)) → G(X)

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(30) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(31) Obligation:

Q DP problem:
The TRS P consists of the following rules:

G(active(X)) → G(X)
G(mark(X)) → G(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(32) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

G(active(X)) → G(X)
G(mark(X)) → G(X)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(G(x1)) = 2·x1   
POL(active(x1)) = 2·x1   
POL(mark(x1)) = 2·x1   

(33) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(34) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(35) TRUE

(36) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(37) Obligation:

Q DP problem:
The TRS P consists of the following rules:

G(active(X)) → G(X)
G(mark(X)) → G(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(active(X)) → C(X)
C(mark(X)) → C(X)

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(39) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(40) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(active(X)) → C(X)
C(mark(X)) → C(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(41) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

C(active(X)) → C(X)
C(mark(X)) → C(X)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(C(x1)) = 2·x1   
POL(active(x1)) = 2·x1   
POL(mark(x1)) = 2·x1   

(42) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(43) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(44) TRUE

(45) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(46) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C(active(X)) → C(X)
C(mark(X)) → C(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(47) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(active(X)) → F(X)
F(mark(X)) → F(X)

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(48) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(49) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(active(X)) → F(X)
F(mark(X)) → F(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(50) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

F(active(X)) → F(X)
F(mark(X)) → F(X)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(F(x1)) = 2·x1   
POL(active(x1)) = 2·x1   
POL(mark(x1)) = 2·x1   

(51) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(52) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(53) TRUE

(54) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(55) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(active(X)) → F(X)
F(mark(X)) → F(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(56) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(f(X))) → MARK(c(f(g(f(X)))))
MARK(f(X)) → MARK(X)
MARK(c(X)) → ACTIVE(c(X))
ACTIVE(c(X)) → MARK(d(X))
MARK(g(X)) → ACTIVE(g(X))
MARK(d(X)) → ACTIVE(d(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → MARK(X)

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(57) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MARK(f(X)) → MARK(X)
MARK(h(X)) → MARK(X)


Used ordering: Polynomial interpretation [POLO]:

POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(active(x1)) = x1   
POL(c(x1)) = x1   
POL(d(x1)) = x1   
POL(f(x1)) = 1 + x1   
POL(g(x1)) = x1   
POL(h(x1)) = 1 + x1   
POL(mark(x1)) = x1   

(58) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(f(X))) → MARK(c(f(g(f(X)))))
MARK(c(X)) → ACTIVE(c(X))
ACTIVE(c(X)) → MARK(d(X))
MARK(g(X)) → ACTIVE(g(X))
MARK(d(X)) → ACTIVE(d(X))
MARK(h(X)) → ACTIVE(h(mark(X)))

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE(x1) ) = 1


POL( f(x1) ) = max{0, -1}


POL( h(x1) ) = x1 + 1


POL( mark(x1) ) = 1


POL( g(x1) ) = 1


POL( active(x1) ) = 1


POL( c(x1) ) = 0


POL( d(x1) ) = max{0, -1}


POL( MARK(x1) ) = x1 + 1



The following usable rules [FROCOS05] were oriented:

d(active(X)) → d(X)
d(mark(X)) → d(X)
c(active(X)) → c(X)
c(mark(X)) → c(X)

(60) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(f(X))) → MARK(c(f(g(f(X)))))
MARK(c(X)) → ACTIVE(c(X))
ACTIVE(c(X)) → MARK(d(X))
MARK(d(X)) → ACTIVE(d(X))

The TRS R consists of the following rules:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(61) RFCMatchBoundsDPProof (EQUIVALENT transformation)

Finiteness of the DP problem can be shown by a matchbound of 6.
As the DP problem is minimal we only have to initialize the certificate graph by the rules of P:

MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(f(X))) → MARK(c(f(g(f(X)))))
MARK(c(X)) → ACTIVE(c(X))
ACTIVE(c(X)) → MARK(d(X))
MARK(d(X)) → ACTIVE(d(X))

To find matches we regarded all rules of R and P:

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(f(X))) → MARK(c(f(g(f(X)))))
MARK(c(X)) → ACTIVE(c(X))
ACTIVE(c(X)) → MARK(d(X))
MARK(d(X)) → ACTIVE(d(X))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

8126749, 8126750, 8126751, 8126755, 8126752, 8126753, 8126754, 8126756, 8126757, 8126758, 8126759, 8126760, 8126761, 8126762, 8126763, 8126764, 8126765, 8126766, 8126767, 8126768, 8126769, 8126770, 8126771, 8126775, 8126772, 8126773, 8126774, 8126776, 8126777, 8126778, 8126779, 8126780

Node 8126749 is start node and node 8126750 is final node.

Those nodes are connect through the following edges:

  • 8126749 to 8126751 labelled MARK_1(0), ACTIVE_1(0), MARK_1(1), ACTIVE_1(1)
  • 8126749 to 8126752 labelled MARK_1(0)
  • 8126749 to 8126756 labelled ACTIVE_1(0)
  • 8126749 to 8126758 labelled ACTIVE_1(1), MARK_1(1)
  • 8126749 to 8126765 labelled ACTIVE_1(2)
  • 8126749 to 8126764 labelled MARK_1(2), ACTIVE_1(2)
  • 8126749 to 8126766 labelled MARK_1(1)
  • 8126749 to 8126770 labelled ACTIVE_1(3), MARK_1(3)
  • 8126749 to 8126771 labelled ACTIVE_1(4)
  • 8126749 to 8126776 labelled ACTIVE_1(4)
  • 8126749 to 8126777 labelled ACTIVE_1(4)
  • 8126749 to 8126778 labelled MARK_1(4)
  • 8126749 to 8126779 labelled ACTIVE_1(5), MARK_1(5)
  • 8126749 to 8126780 labelled ACTIVE_1(6)
  • 8126750 to 8126750 labelled #_1(0)
  • 8126751 to 8126750 labelled d_1(0), c_1(0), d_1(1), c_1(1)
  • 8126755 to 8126750 labelled f_1(0), f_1(1)
  • 8126752 to 8126753 labelled c_1(0)
  • 8126753 to 8126754 labelled f_1(0)
  • 8126754 to 8126755 labelled g_1(0)
  • 8126756 to 8126757 labelled f_1(0)
  • 8126756 to 8126750 labelled f_1(1)
  • 8126756 to 8126760 labelled f_1(1)
  • 8126756 to 8126759 labelled f_1(1)
  • 8126756 to 8126751 labelled f_1(1)
  • 8126756 to 8126762 labelled f_1(1)
  • 8126756 to 8126764 labelled f_1(1)
  • 8126756 to 8126766 labelled f_1(1)
  • 8126756 to 8126770 labelled f_1(1)
  • 8126756 to 8126765 labelled f_1(1)
  • 8126756 to 8126772 labelled f_1(1)
  • 8126756 to 8126771 labelled f_1(1)
  • 8126756 to 8126776 labelled f_1(1)
  • 8126756 to 8126778 labelled f_1(1)
  • 8126756 to 8126777 labelled f_1(1)
  • 8126756 to 8126779 labelled f_1(1)
  • 8126756 to 8126780 labelled f_1(1)
  • 8126757 to 8126750 labelled mark_1(0)
  • 8126757 to 8126759 labelled active_1(1)
  • 8126757 to 8126751 labelled active_1(1), mark_1(1)
  • 8126757 to 8126760 labelled active_1(1)
  • 8126757 to 8126762 labelled active_1(1)
  • 8126757 to 8126764 labelled mark_1(2), active_1(2)
  • 8126757 to 8126770 labelled active_1(3), mark_1(3)
  • 8126757 to 8126765 labelled active_1(2)
  • 8126757 to 8126766 labelled mark_1(1)
  • 8126757 to 8126772 labelled mark_1(2)
  • 8126757 to 8126776 labelled active_1(4)
  • 8126757 to 8126771 labelled active_1(4)
  • 8126757 to 8126777 labelled active_1(4)
  • 8126757 to 8126778 labelled mark_1(4)
  • 8126757 to 8126779 labelled active_1(5), mark_1(5)
  • 8126757 to 8126780 labelled active_1(6)
  • 8126758 to 8126750 labelled d_1(1), c_1(1)
  • 8126758 to 8126753 labelled c_1(1)
  • 8126759 to 8126750 labelled g_1(1)
  • 8126760 to 8126761 labelled f_1(1)
  • 8126760 to 8126750 labelled f_1(2), f_1(1)
  • 8126760 to 8126759 labelled f_1(2)
  • 8126760 to 8126762 labelled f_1(2)
  • 8126760 to 8126760 labelled f_1(2)
  • 8126760 to 8126751 labelled f_1(2)
  • 8126760 to 8126764 labelled f_1(2)
  • 8126760 to 8126766 labelled f_1(2)
  • 8126760 to 8126772 labelled f_1(2)
  • 8126760 to 8126770 labelled f_1(2)
  • 8126760 to 8126765 labelled f_1(2)
  • 8126760 to 8126778 labelled f_1(2)
  • 8126760 to 8126771 labelled f_1(2)
  • 8126760 to 8126777 labelled f_1(2)
  • 8126760 to 8126776 labelled f_1(2)
  • 8126760 to 8126779 labelled f_1(2)
  • 8126760 to 8126780 labelled f_1(2)
  • 8126761 to 8126750 labelled mark_1(1)
  • 8126761 to 8126759 labelled active_1(1)
  • 8126761 to 8126751 labelled active_1(1), mark_1(1)
  • 8126761 to 8126760 labelled active_1(1)
  • 8126761 to 8126762 labelled active_1(1)
  • 8126761 to 8126764 labelled mark_1(2), active_1(2)
  • 8126761 to 8126766 labelled mark_1(1)
  • 8126761 to 8126765 labelled active_1(2)
  • 8126761 to 8126770 labelled active_1(3), mark_1(3)
  • 8126761 to 8126772 labelled mark_1(2)
  • 8126761 to 8126777 labelled active_1(4)
  • 8126761 to 8126778 labelled mark_1(4)
  • 8126761 to 8126776 labelled active_1(4)
  • 8126761 to 8126771 labelled active_1(4)
  • 8126761 to 8126779 labelled active_1(5), mark_1(5)
  • 8126761 to 8126780 labelled active_1(6)
  • 8126762 to 8126763 labelled h_1(1)
  • 8126762 to 8126750 labelled h_1(2), h_1(1)
  • 8126762 to 8126759 labelled h_1(2)
  • 8126762 to 8126762 labelled h_1(2)
  • 8126762 to 8126760 labelled h_1(2)
  • 8126762 to 8126751 labelled h_1(2)
  • 8126762 to 8126764 labelled h_1(2)
  • 8126762 to 8126766 labelled h_1(2)
  • 8126762 to 8126772 labelled h_1(2)
  • 8126762 to 8126770 labelled h_1(2)
  • 8126762 to 8126765 labelled h_1(2)
  • 8126762 to 8126778 labelled h_1(2)
  • 8126762 to 8126771 labelled h_1(2)
  • 8126762 to 8126777 labelled h_1(2)
  • 8126762 to 8126776 labelled h_1(2)
  • 8126762 to 8126779 labelled h_1(2)
  • 8126762 to 8126780 labelled h_1(2)
  • 8126763 to 8126750 labelled mark_1(1)
  • 8126763 to 8126759 labelled active_1(1)
  • 8126763 to 8126751 labelled active_1(1), mark_1(1)
  • 8126763 to 8126760 labelled active_1(1)
  • 8126763 to 8126762 labelled active_1(1)
  • 8126763 to 8126764 labelled mark_1(2), active_1(2)
  • 8126763 to 8126766 labelled mark_1(1)
  • 8126763 to 8126765 labelled active_1(2)
  • 8126763 to 8126770 labelled active_1(3), mark_1(3)
  • 8126763 to 8126772 labelled mark_1(2)
  • 8126763 to 8126777 labelled active_1(4)
  • 8126763 to 8126778 labelled mark_1(4)
  • 8126763 to 8126776 labelled active_1(4)
  • 8126763 to 8126771 labelled active_1(4)
  • 8126763 to 8126779 labelled active_1(5), mark_1(5)
  • 8126763 to 8126780 labelled active_1(6)
  • 8126764 to 8126750 labelled d_1(2), d_1(1)
  • 8126764 to 8126753 labelled d_1(2)
  • 8126765 to 8126750 labelled c_1(2), c_1(1)
  • 8126765 to 8126753 labelled c_1(2)
  • 8126765 to 8126767 labelled c_1(2)
  • 8126766 to 8126767 labelled c_1(1)
  • 8126767 to 8126768 labelled f_1(1)
  • 8126768 to 8126769 labelled g_1(1)
  • 8126769 to 8126750 labelled f_1(1), f_1(2)
  • 8126769 to 8126761 labelled f_1(1)
  • 8126769 to 8126759 labelled f_1(2), f_1(1)
  • 8126769 to 8126762 labelled f_1(2), f_1(1)
  • 8126769 to 8126760 labelled f_1(2), f_1(1)
  • 8126769 to 8126751 labelled f_1(2), f_1(1)
  • 8126769 to 8126764 labelled f_1(2), f_1(1)
  • 8126769 to 8126766 labelled f_1(2), f_1(1)
  • 8126769 to 8126772 labelled f_1(2), f_1(1)
  • 8126769 to 8126770 labelled f_1(2), f_1(1)
  • 8126769 to 8126765 labelled f_1(2), f_1(1)
  • 8126769 to 8126778 labelled f_1(2), f_1(1)
  • 8126769 to 8126771 labelled f_1(2), f_1(1)
  • 8126769 to 8126777 labelled f_1(2), f_1(1)
  • 8126769 to 8126776 labelled f_1(2), f_1(1)
  • 8126769 to 8126779 labelled f_1(2), f_1(1)
  • 8126769 to 8126780 labelled f_1(2), f_1(1)
  • 8126770 to 8126750 labelled d_1(3), d_1(1)
  • 8126770 to 8126753 labelled d_1(3)
  • 8126770 to 8126767 labelled d_1(3)
  • 8126770 to 8126773 labelled c_1(3)
  • 8126771 to 8126750 labelled d_1(4), d_1(1)
  • 8126771 to 8126753 labelled d_1(4)
  • 8126775 to 8126762 labelled f_1(2)
  • 8126775 to 8126751 labelled f_1(2)
  • 8126775 to 8126750 labelled f_1(2), f_1(1)
  • 8126775 to 8126760 labelled f_1(2)
  • 8126775 to 8126761 labelled f_1(2)
  • 8126775 to 8126759 labelled f_1(2)
  • 8126775 to 8126764 labelled f_1(3), f_1(2)
  • 8126775 to 8126766 labelled f_1(2)
  • 8126775 to 8126772 labelled f_1(3), f_1(2)
  • 8126775 to 8126770 labelled f_1(3), f_1(2)
  • 8126775 to 8126765 labelled f_1(3), f_1(2)
  • 8126775 to 8126778 labelled f_1(3), f_1(2)
  • 8126775 to 8126771 labelled f_1(3), f_1(2)
  • 8126775 to 8126777 labelled f_1(3), f_1(2)
  • 8126775 to 8126776 labelled f_1(3), f_1(2)
  • 8126775 to 8126779 labelled f_1(3), f_1(2)
  • 8126775 to 8126780 labelled f_1(3), f_1(2)
  • 8126772 to 8126773 labelled c_1(2)
  • 8126773 to 8126774 labelled f_1(2)
  • 8126774 to 8126775 labelled g_1(2)
  • 8126776 to 8126767 labelled d_1(4)
  • 8126777 to 8126773 labelled c_1(4)
  • 8126778 to 8126773 labelled d_1(4)
  • 8126779 to 8126773 labelled d_1(5)
  • 8126780 to 8126773 labelled d_1(6)

(62) TRUE