(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(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(active(x)) → f(h(g(mark(x))))
f(mark(x)) → mark(f(active(x)))
g(mark(x)) → g(active(x))
h(mark(x)) → mark(h(active(x)))
mark(f(x)) → f(x)
active(f(x)) → f(x)
mark(g(x)) → g(x)
active(g(x)) → g(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(active(x)) → f(h(g(mark(x))))
f(mark(x)) → mark(f(active(x)))
g(mark(x)) → g(active(x))
h(mark(x)) → mark(h(active(x)))
mark(f(x)) → f(x)
active(f(x)) → f(x)
mark(g(x)) → g(x)
active(g(x)) → g(x)
mark(h(x)) → h(x)
active(h(x)) → h(x)
Q is empty.
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(f(X)) → MARK(g(h(f(X))))
ACTIVE(f(X)) → G(h(f(X)))
ACTIVE(f(X)) → H(f(X))
MARK(f(X)) → ACTIVE(f(mark(X)))
MARK(f(X)) → F(mark(X))
MARK(f(X)) → MARK(X)
MARK(g(X)) → ACTIVE(g(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)
G(mark(X)) → G(X)
G(active(X)) → G(X)
H(mark(X)) → H(X)
H(active(X)) → H(X)
The TRS R consists of the following rules:
active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 4 less nodes.
(8) Complex Obligation (AND)
(9) 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(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(10) 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.
(11) 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.
(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) 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(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) 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.
(20) 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.
(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:
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.
(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:
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
(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) 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(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(28) 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.
(29) 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.
(30) 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
(31) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(32) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(33) TRUE
(34) 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.
(35) 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.
(36) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(X)) → MARK(g(h(f(X))))
MARK(f(X)) → MARK(X)
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → MARK(X)
The TRS R consists of the following rules:
active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(37) 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)
Used ordering: Polynomial interpretation [POLO]:
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(active(x1)) = x1
POL(f(x1)) = 1 + x1
POL(g(x1)) = x1
POL(h(x1)) = x1
POL(mark(x1)) = x1
(38) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(X)) → MARK(g(h(f(X))))
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → MARK(X)
The TRS R consists of the following rules:
active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(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) RFCMatchBoundsDPProof (EQUIVALENT transformation)
Finiteness of the DP problem can be shown by a matchbound of 3.
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(X)) → MARK(g(h(f(X))))
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → MARK(X)
To find matches we regarded all rules of R and P:
active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(X)) → MARK(g(h(f(X))))
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → MARK(X)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
9776865, 9776866, 9776867, 9776868, 9776869, 9776870, 9776871, 9776872, 9776873, 9776874, 9776875, 9776876, 9776877, 9776878, 9776879, 9776880, 9776881, 9776882, 9776883, 9776884, 9776885, 9776886
Node 9776865 is start node and node 9776866 is final node.
Those nodes are connect through the following edges:
- 9776865 to 9776867 labelled ACTIVE_1(0), ACTIVE_1(1)
- 9776865 to 9776866 labelled MARK_1(0), MARK_1(1)
- 9776865 to 9776869 labelled MARK_1(0)
- 9776865 to 9776872 labelled ACTIVE_1(0)
- 9776865 to 9776874 labelled ACTIVE_1(1)
- 9776865 to 9776877 labelled ACTIVE_1(1)
- 9776865 to 9776875 labelled ACTIVE_1(1)
- 9776865 to 9776879 labelled MARK_1(1)
- 9776865 to 9776882 labelled ACTIVE_1(2)
- 9776865 to 9776883 labelled MARK_1(2)
- 9776865 to 9776886 labelled ACTIVE_1(3)
- 9776866 to 9776866 labelled #_1(0)
- 9776867 to 9776868 labelled h_1(0)
- 9776867 to 9776866 labelled g_1(0), g_1(1), h_1(1)
- 9776867 to 9776867 labelled h_1(1)
- 9776867 to 9776875 labelled h_1(1)
- 9776867 to 9776877 labelled h_1(1)
- 9776867 to 9776883 labelled h_1(1)
- 9776867 to 9776886 labelled h_1(1)
- 9776868 to 9776866 labelled mark_1(0)
- 9776868 to 9776867 labelled active_1(1)
- 9776868 to 9776875 labelled active_1(1)
- 9776868 to 9776877 labelled active_1(1)
- 9776868 to 9776883 labelled mark_1(2)
- 9776868 to 9776886 labelled active_1(3)
- 9776869 to 9776870 labelled g_1(0)
- 9776870 to 9776871 labelled h_1(0)
- 9776871 to 9776866 labelled f_1(0), f_1(1)
- 9776872 to 9776873 labelled f_1(0)
- 9776872 to 9776866 labelled f_1(1)
- 9776872 to 9776877 labelled f_1(1)
- 9776872 to 9776867 labelled f_1(1)
- 9776872 to 9776875 labelled f_1(1)
- 9776872 to 9776883 labelled f_1(1)
- 9776872 to 9776886 labelled f_1(1)
- 9776873 to 9776866 labelled mark_1(0)
- 9776873 to 9776867 labelled active_1(1)
- 9776873 to 9776875 labelled active_1(1)
- 9776873 to 9776877 labelled active_1(1)
- 9776873 to 9776883 labelled mark_1(2)
- 9776873 to 9776886 labelled active_1(3)
- 9776874 to 9776870 labelled g_1(1)
- 9776875 to 9776876 labelled f_1(1)
- 9776875 to 9776866 labelled f_1(2), f_1(1)
- 9776875 to 9776875 labelled f_1(2)
- 9776875 to 9776867 labelled f_1(2)
- 9776875 to 9776877 labelled f_1(2)
- 9776875 to 9776883 labelled f_1(2)
- 9776875 to 9776886 labelled f_1(2)
- 9776876 to 9776866 labelled mark_1(1)
- 9776876 to 9776867 labelled active_1(1)
- 9776876 to 9776875 labelled active_1(1)
- 9776876 to 9776877 labelled active_1(1)
- 9776876 to 9776883 labelled mark_1(2)
- 9776876 to 9776886 labelled active_1(3)
- 9776877 to 9776878 labelled h_1(1)
- 9776877 to 9776866 labelled h_1(2), h_1(1)
- 9776877 to 9776875 labelled h_1(2)
- 9776877 to 9776877 labelled h_1(2)
- 9776877 to 9776867 labelled h_1(2)
- 9776877 to 9776883 labelled h_1(2)
- 9776877 to 9776886 labelled h_1(2)
- 9776878 to 9776866 labelled mark_1(1)
- 9776878 to 9776867 labelled active_1(1)
- 9776878 to 9776875 labelled active_1(1)
- 9776878 to 9776877 labelled active_1(1)
- 9776878 to 9776883 labelled mark_1(2)
- 9776878 to 9776886 labelled active_1(3)
- 9776879 to 9776880 labelled g_1(1)
- 9776880 to 9776881 labelled h_1(1)
- 9776881 to 9776873 labelled f_1(1)
- 9776881 to 9776866 labelled f_1(1)
- 9776881 to 9776877 labelled f_1(2), f_1(1)
- 9776881 to 9776867 labelled f_1(2), f_1(1)
- 9776881 to 9776875 labelled f_1(2), f_1(1)
- 9776881 to 9776883 labelled f_1(2), f_1(1)
- 9776881 to 9776886 labelled f_1(2), f_1(1)
- 9776882 to 9776880 labelled g_1(2)
- 9776883 to 9776884 labelled g_1(2)
- 9776884 to 9776885 labelled h_1(2)
- 9776885 to 9776876 labelled f_1(2)
- 9776885 to 9776866 labelled f_1(2), f_1(1)
- 9776885 to 9776875 labelled f_1(2)
- 9776885 to 9776867 labelled f_1(2)
- 9776885 to 9776877 labelled f_1(2)
- 9776885 to 9776883 labelled f_1(3), f_1(2)
- 9776885 to 9776886 labelled f_1(3), f_1(2)
- 9776886 to 9776884 labelled g_1(3)
(40) TRUE