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 RFCMatchBoundsDPProof (⇔)
        ↳60 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) RFCMatchBoundsDPProof (EQUIVALENT transformation)

Finiteness of the DP problem can be shown by a matchbound of 8.
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(g(X)) → ACTIVE(g(X))
MARK(d(X)) → ACTIVE(d(X))
MARK(h(X)) → ACTIVE(h(mark(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(g(X)) → ACTIVE(g(X))
MARK(d(X)) → ACTIVE(d(X))
MARK(h(X)) → ACTIVE(h(mark(X)))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

9986123, 9986124, 9986125, 9986126, 9986127, 9986130, 9986131, 9986129, 9986128, 9986133, 9986132, 9986134, 9986135, 9986136, 9986137, 9986138, 9986139, 9986140, 9986141, 9986142, 9986143, 9986144, 9986145, 9986146, 9986147, 9986149, 9986150, 9986151, 9986148, 9986152, 9986153, 9986154, 9986155, 9986156

Node 9986123 is start node and node 9986124 is final node.

Those nodes are connect through the following edges:

  • 9986123 to 9986125 labelled ACTIVE_1(0)
  • 9986123 to 9986127 labelled MARK_1(0), ACTIVE_1(0), ACTIVE_1(1), MARK_1(1)
  • 9986123 to 9986128 labelled MARK_1(0)
  • 9986123 to 9986132 labelled ACTIVE_1(0)
  • 9986123 to 9986139 labelled ACTIVE_1(2), MARK_1(2)
  • 9986123 to 9986140 labelled MARK_1(1)
  • 9986123 to 9986144 labelled ACTIVE_1(3), MARK_1(3)
  • 9986123 to 9986145 labelled ACTIVE_1(3)
  • 9986123 to 9986146 labelled MARK_1(4), ACTIVE_1(4)
  • 9986123 to 9986147 labelled ACTIVE_1(4)
  • 9986123 to 9986152 labelled ACTIVE_1(5), MARK_1(5)
  • 9986123 to 9986153 labelled ACTIVE_1(6)
  • 9986123 to 9986154 labelled MARK_1(6), ACTIVE_1(6)
  • 9986123 to 9986155 labelled ACTIVE_1(7), MARK_1(7)
  • 9986123 to 9986156 labelled ACTIVE_1(8)
  • 9986124 to 9986124 labelled #_1(0)
  • 9986125 to 9986126 labelled h_1(0)
  • 9986125 to 9986124 labelled g_1(0), c_1(0), g_1(1), c_1(1), h_1(1)
  • 9986125 to 9986134 labelled h_1(1)
  • 9986125 to 9986135 labelled h_1(1)
  • 9986125 to 9986127 labelled h_1(1)
  • 9986125 to 9986137 labelled h_1(1)
  • 9986125 to 9986139 labelled h_1(1)
  • 9986125 to 9986140 labelled h_1(1)
  • 9986125 to 9986144 labelled h_1(1)
  • 9986125 to 9986146 labelled h_1(1)
  • 9986125 to 9986148 labelled h_1(1)
  • 9986125 to 9986145 labelled h_1(1)
  • 9986125 to 9986152 labelled h_1(1)
  • 9986125 to 9986147 labelled h_1(1)
  • 9986125 to 9986153 labelled h_1(1)
  • 9986125 to 9986154 labelled h_1(1)
  • 9986125 to 9986155 labelled h_1(1)
  • 9986125 to 9986156 labelled h_1(1)
  • 9986126 to 9986124 labelled mark_1(0)
  • 9986126 to 9986134 labelled active_1(1)
  • 9986126 to 9986127 labelled active_1(1)
  • 9986126 to 9986135 labelled active_1(1)
  • 9986126 to 9986137 labelled active_1(1)
  • 9986126 to 9986139 labelled mark_1(2), active_1(2)
  • 9986126 to 9986144 labelled active_1(3), mark_1(3)
  • 9986126 to 9986140 labelled mark_1(1)
  • 9986126 to 9986145 labelled active_1(3)
  • 9986126 to 9986146 labelled mark_1(4), active_1(4)
  • 9986126 to 9986148 labelled mark_1(2)
  • 9986126 to 9986152 labelled active_1(5), mark_1(5)
  • 9986126 to 9986147 labelled active_1(4)
  • 9986126 to 9986153 labelled active_1(6)
  • 9986126 to 9986154 labelled mark_1(6), active_1(6)
  • 9986126 to 9986155 labelled active_1(7), mark_1(7)
  • 9986126 to 9986156 labelled active_1(8)
  • 9986127 to 9986124 labelled d_1(0), d_1(1)
  • 9986127 to 9986129 labelled c_1(1), d_1(1)
  • 9986130 to 9986131 labelled g_1(0)
  • 9986131 to 9986124 labelled f_1(0), f_1(1)
  • 9986129 to 9986130 labelled f_1(0)
  • 9986128 to 9986129 labelled c_1(0)
  • 9986133 to 9986124 labelled mark_1(0)
  • 9986133 to 9986134 labelled active_1(1)
  • 9986133 to 9986127 labelled active_1(1)
  • 9986133 to 9986135 labelled active_1(1)
  • 9986133 to 9986137 labelled active_1(1)
  • 9986133 to 9986139 labelled mark_1(2), active_1(2)
  • 9986133 to 9986144 labelled active_1(3), mark_1(3)
  • 9986133 to 9986140 labelled mark_1(1)
  • 9986133 to 9986145 labelled active_1(3)
  • 9986133 to 9986146 labelled mark_1(4), active_1(4)
  • 9986133 to 9986148 labelled mark_1(2)
  • 9986133 to 9986152 labelled active_1(5), mark_1(5)
  • 9986133 to 9986147 labelled active_1(4)
  • 9986133 to 9986153 labelled active_1(6)
  • 9986133 to 9986154 labelled mark_1(6), active_1(6)
  • 9986133 to 9986155 labelled active_1(7), mark_1(7)
  • 9986133 to 9986156 labelled active_1(8)
  • 9986132 to 9986133 labelled f_1(0)
  • 9986132 to 9986124 labelled f_1(1)
  • 9986132 to 9986135 labelled f_1(1)
  • 9986132 to 9986127 labelled f_1(1)
  • 9986132 to 9986134 labelled f_1(1)
  • 9986132 to 9986137 labelled f_1(1)
  • 9986132 to 9986139 labelled f_1(1)
  • 9986132 to 9986140 labelled f_1(1)
  • 9986132 to 9986144 labelled f_1(1)
  • 9986132 to 9986148 labelled f_1(1)
  • 9986132 to 9986146 labelled f_1(1)
  • 9986132 to 9986145 labelled f_1(1)
  • 9986132 to 9986152 labelled f_1(1)
  • 9986132 to 9986147 labelled f_1(1)
  • 9986132 to 9986153 labelled f_1(1)
  • 9986132 to 9986154 labelled f_1(1)
  • 9986132 to 9986155 labelled f_1(1)
  • 9986132 to 9986156 labelled f_1(1)
  • 9986134 to 9986124 labelled g_1(1), c_1(1)
  • 9986135 to 9986136 labelled f_1(1)
  • 9986135 to 9986124 labelled f_1(2), f_1(1)
  • 9986135 to 9986135 labelled f_1(2)
  • 9986135 to 9986125 labelled f_1(2)
  • 9986135 to 9986137 labelled f_1(2)
  • 9986135 to 9986127 labelled f_1(2)
  • 9986135 to 9986139 labelled f_1(2)
  • 9986135 to 9986140 labelled f_1(2)
  • 9986135 to 9986148 labelled f_1(2)
  • 9986135 to 9986145 labelled f_1(2)
  • 9986135 to 9986144 labelled f_1(2)
  • 9986135 to 9986146 labelled f_1(2)
  • 9986135 to 9986147 labelled f_1(2)
  • 9986135 to 9986152 labelled f_1(2)
  • 9986135 to 9986153 labelled f_1(2)
  • 9986135 to 9986154 labelled f_1(2)
  • 9986135 to 9986155 labelled f_1(2)
  • 9986135 to 9986156 labelled f_1(2)
  • 9986136 to 9986124 labelled mark_1(1)
  • 9986136 to 9986125 labelled active_1(1)
  • 9986136 to 9986127 labelled active_1(1), mark_1(1)
  • 9986136 to 9986135 labelled active_1(1)
  • 9986136 to 9986137 labelled active_1(1)
  • 9986136 to 9986139 labelled mark_1(2), active_1(2)
  • 9986136 to 9986140 labelled mark_1(1)
  • 9986136 to 9986145 labelled active_1(3)
  • 9986136 to 9986144 labelled active_1(3), mark_1(3)
  • 9986136 to 9986148 labelled mark_1(2)
  • 9986136 to 9986146 labelled mark_1(4), active_1(4)
  • 9986136 to 9986152 labelled active_1(5), mark_1(5)
  • 9986136 to 9986147 labelled active_1(4)
  • 9986136 to 9986153 labelled active_1(6)
  • 9986136 to 9986154 labelled mark_1(6), active_1(6)
  • 9986136 to 9986155 labelled active_1(7), mark_1(7)
  • 9986136 to 9986156 labelled active_1(8)
  • 9986137 to 9986138 labelled h_1(1)
  • 9986137 to 9986124 labelled h_1(2), h_1(1)
  • 9986137 to 9986137 labelled h_1(2)
  • 9986137 to 9986135 labelled h_1(2)
  • 9986137 to 9986125 labelled h_1(2)
  • 9986137 to 9986127 labelled h_1(2)
  • 9986137 to 9986139 labelled h_1(2)
  • 9986137 to 9986140 labelled h_1(2)
  • 9986137 to 9986148 labelled h_1(2)
  • 9986137 to 9986145 labelled h_1(2)
  • 9986137 to 9986144 labelled h_1(2)
  • 9986137 to 9986146 labelled h_1(2)
  • 9986137 to 9986147 labelled h_1(2)
  • 9986137 to 9986152 labelled h_1(2)
  • 9986137 to 9986153 labelled h_1(2)
  • 9986137 to 9986154 labelled h_1(2)
  • 9986137 to 9986155 labelled h_1(2)
  • 9986137 to 9986156 labelled h_1(2)
  • 9986138 to 9986124 labelled mark_1(1)
  • 9986138 to 9986125 labelled active_1(1)
  • 9986138 to 9986127 labelled active_1(1), mark_1(1)
  • 9986138 to 9986135 labelled active_1(1)
  • 9986138 to 9986137 labelled active_1(1)
  • 9986138 to 9986139 labelled mark_1(2), active_1(2)
  • 9986138 to 9986140 labelled mark_1(1)
  • 9986138 to 9986145 labelled active_1(3)
  • 9986138 to 9986144 labelled active_1(3), mark_1(3)
  • 9986138 to 9986148 labelled mark_1(2)
  • 9986138 to 9986146 labelled mark_1(4), active_1(4)
  • 9986138 to 9986152 labelled active_1(5), mark_1(5)
  • 9986138 to 9986147 labelled active_1(4)
  • 9986138 to 9986153 labelled active_1(6)
  • 9986138 to 9986154 labelled mark_1(6), active_1(6)
  • 9986138 to 9986155 labelled active_1(7), mark_1(7)
  • 9986138 to 9986156 labelled active_1(8)
  • 9986139 to 9986124 labelled d_1(2), d_1(1)
  • 9986139 to 9986129 labelled d_1(2), c_1(2)
  • 9986139 to 9986141 labelled c_1(2)
  • 9986140 to 9986141 labelled c_1(1)
  • 9986141 to 9986142 labelled f_1(1)
  • 9986142 to 9986143 labelled g_1(1)
  • 9986143 to 9986124 labelled f_1(1), f_1(2)
  • 9986143 to 9986136 labelled f_1(1)
  • 9986143 to 9986135 labelled f_1(2), f_1(1)
  • 9986143 to 9986137 labelled f_1(2), f_1(1)
  • 9986143 to 9986139 labelled f_1(2), f_1(1)
  • 9986143 to 9986125 labelled f_1(2), f_1(1)
  • 9986143 to 9986140 labelled f_1(2), f_1(1)
  • 9986143 to 9986127 labelled f_1(2), f_1(1)
  • 9986143 to 9986148 labelled f_1(2), f_1(1)
  • 9986143 to 9986145 labelled f_1(2), f_1(1)
  • 9986143 to 9986144 labelled f_1(2), f_1(1)
  • 9986143 to 9986146 labelled f_1(2), f_1(1)
  • 9986143 to 9986147 labelled f_1(2), f_1(1)
  • 9986143 to 9986152 labelled f_1(2), f_1(1)
  • 9986143 to 9986153 labelled f_1(2), f_1(1)
  • 9986143 to 9986154 labelled f_1(2), f_1(1)
  • 9986143 to 9986155 labelled f_1(2), f_1(1)
  • 9986143 to 9986156 labelled f_1(2), f_1(1)
  • 9986144 to 9986124 labelled d_1(3), d_1(1)
  • 9986144 to 9986129 labelled d_1(3), c_1(3)
  • 9986144 to 9986141 labelled d_1(3)
  • 9986144 to 9986149 labelled c_1(3)
  • 9986145 to 9986141 labelled c_1(3)
  • 9986146 to 9986129 labelled d_1(4)
  • 9986146 to 9986124 labelled d_1(4), d_1(1)
  • 9986146 to 9986141 labelled d_1(4)
  • 9986146 to 9986149 labelled d_1(4), c_1(4)
  • 9986147 to 9986129 labelled c_1(4)
  • 9986149 to 9986150 labelled f_1(2)
  • 9986150 to 9986151 labelled g_1(2)
  • 9986151 to 9986124 labelled f_1(2), f_1(1)
  • 9986151 to 9986127 labelled f_1(2)
  • 9986151 to 9986136 labelled f_1(2)
  • 9986151 to 9986135 labelled f_1(2)
  • 9986151 to 9986137 labelled f_1(2)
  • 9986151 to 9986125 labelled f_1(2)
  • 9986151 to 9986148 labelled f_1(3), f_1(2)
  • 9986151 to 9986139 labelled f_1(3), f_1(2)
  • 9986151 to 9986140 labelled f_1(2)
  • 9986151 to 9986145 labelled f_1(3), f_1(2)
  • 9986151 to 9986144 labelled f_1(3), f_1(2)
  • 9986151 to 9986146 labelled f_1(3), f_1(2)
  • 9986151 to 9986147 labelled f_1(3), f_1(2)
  • 9986151 to 9986152 labelled f_1(3), f_1(2)
  • 9986151 to 9986153 labelled f_1(3), f_1(2)
  • 9986151 to 9986154 labelled f_1(3), f_1(2)
  • 9986151 to 9986155 labelled f_1(3), f_1(2)
  • 9986151 to 9986156 labelled f_1(3), f_1(2)
  • 9986148 to 9986149 labelled c_1(2)
  • 9986152 to 9986124 labelled d_1(5), d_1(1)
  • 9986152 to 9986129 labelled d_1(5)
  • 9986152 to 9986141 labelled d_1(5)
  • 9986152 to 9986149 labelled c_1(5), d_1(5)
  • 9986153 to 9986124 labelled d_1(6), d_1(1)
  • 9986153 to 9986129 labelled d_1(6)
  • 9986153 to 9986141 labelled d_1(6)
  • 9986153 to 9986149 labelled c_1(6)
  • 9986154 to 9986149 labelled d_1(6)
  • 9986155 to 9986149 labelled d_1(7)
  • 9986156 to 9986149 labelled d_1(8)

(60) TRUE