(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → 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(x)) → n__f(n__g(n__f(c(x))))
c(x) → activate(d(x))
h(x) → n__d(c(x))
f(x) → n__f(x)
g(x) → n__g(x)
d(x) → n__d(x)
n__f(activate(x)) → activate(f(x))
n__g(activate(x)) → g(x)
n__d(activate(x)) → d(x)
activate(x) → 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(x)) → n__f(n__g(n__f(c(x))))
c(x) → activate(d(x))
h(x) → n__d(c(x))
f(x) → n__f(x)
g(x) → n__g(x)
d(x) → n__d(x)
n__f(activate(x)) → activate(f(x))
n__g(activate(x)) → g(x)
n__d(activate(x)) → d(x)
activate(x) → x
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(activate(x1)) = x1
POL(c(x1)) = x1
POL(d(x1)) = x1
POL(f(x1)) = x1
POL(g(x1)) = x1
POL(h(x1)) = 1 + x1
POL(n__d(x1)) = x1
POL(n__f(x1)) = x1
POL(n__g(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
h(X) → c(n__d(X))
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → 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:
F(f(X)) → C(n__f(n__g(n__f(X))))
C(X) → D(activate(X))
C(X) → ACTIVATE(X)
ACTIVATE(n__f(X)) → F(activate(X))
ACTIVATE(n__f(X)) → ACTIVATE(X)
ACTIVATE(n__g(X)) → G(X)
ACTIVATE(n__d(X)) → D(X)
The TRS R consists of the following rules:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → 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 1 SCC with 3 less nodes.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(X) → ACTIVATE(X)
ACTIVATE(n__f(X)) → F(activate(X))
F(f(X)) → C(n__f(n__g(n__f(X))))
ACTIVATE(n__f(X)) → ACTIVATE(X)
The TRS R consists of the following rules:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) 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:
ACTIVATE(n__f(X)) → ACTIVATE(X)
Used ordering: Polynomial interpretation [POLO]:
POL(ACTIVATE(x1)) = x1
POL(C(x1)) = x1
POL(F(x1)) = 1 + x1
POL(activate(x1)) = x1
POL(c(x1)) = x1
POL(d(x1)) = x1
POL(f(x1)) = 1 + x1
POL(g(x1)) = x1
POL(n__d(x1)) = x1
POL(n__f(x1)) = 1 + x1
POL(n__g(x1)) = x1
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(X) → ACTIVATE(X)
ACTIVATE(n__f(X)) → F(activate(X))
F(f(X)) → C(n__f(n__g(n__f(X))))
The TRS R consists of the following rules:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) RFCMatchBoundsDPProof (EQUIVALENT transformation)
Finiteness of the DP problem can be shown by a matchbound of 4.
As the DP problem is minimal we only have to initialize the certificate graph by the rules of P:
C(X) → ACTIVATE(X)
ACTIVATE(n__f(X)) → F(activate(X))
F(f(X)) → C(n__f(n__g(n__f(X))))
To find matches we regarded all rules of R and P:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → X
C(X) → ACTIVATE(X)
ACTIVATE(n__f(X)) → F(activate(X))
F(f(X)) → C(n__f(n__g(n__f(X))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1750852, 1750853, 1750854, 1750857, 1750855, 1750856, 1750858, 1750859, 1750860, 1750862, 1750861, 1750864, 1750865, 1750863, 1750866, 1750867, 1750868
Node 1750852 is start node and node 1750853 is final node.
Those nodes are connect through the following edges:
- 1750852 to 1750854 labelled F_1(0)
- 1750852 to 1750855 labelled C_1(0), ACTIVATE_1(1)
- 1750852 to 1750853 labelled ACTIVATE_1(0)
- 1750852 to 1750858 labelled F_1(1)
- 1750852 to 1750860 labelled C_1(1), ACTIVATE_1(2)
- 1750852 to 1750863 labelled C_1(2), ACTIVATE_1(3)
- 1750852 to 1750867 labelled F_1(2)
- 1750852 to 1750868 labelled F_1(3)
- 1750853 to 1750853 labelled #_1(0)
- 1750854 to 1750853 labelled activate_1(0), g_1(1), ACTIVATE_1(1), n__d_1(1), f_1(1), c_1(1), n__f_1(1), n__g_1(1), C_1(1), d_1(1), F_1(1), activate_1(1), ACTIVATE_1(2), n__g_1(2), n__f_1(2), n__d_1(2)
- 1750854 to 1750858 labelled f_1(1), n__f_1(2)
- 1750854 to 1750859 labelled d_1(2), n__d_1(3)
- 1750854 to 1750854 labelled F_1(1), f_1(1), n__f_1(2)
- 1750854 to 1750860 labelled C_1(1), c_1(1), ACTIVATE_1(2)
- 1750854 to 1750863 labelled C_1(2), c_1(2), ACTIVATE_1(3)
- 1750854 to 1750866 labelled d_1(3), n__d_1(4)
- 1750854 to 1750867 labelled F_1(2)
- 1750854 to 1750868 labelled F_1(3)
- 1750857 to 1750853 labelled n__f_1(0)
- 1750855 to 1750856 labelled n__f_1(0)
- 1750856 to 1750857 labelled n__g_1(0)
- 1750858 to 1750853 labelled activate_1(1), g_1(1), ACTIVATE_1(1), n__d_1(1), f_1(1), c_1(1), n__f_1(1), n__g_1(1), C_1(1), d_1(1), F_1(1), n__g_1(2), ACTIVATE_1(2), n__f_1(2), n__d_1(2)
- 1750858 to 1750854 labelled f_1(1), n__f_1(2), F_1(1)
- 1750858 to 1750856 labelled activate_1(1)
- 1750858 to 1750859 labelled d_1(2), n__d_1(3)
- 1750858 to 1750863 labelled c_1(2), C_1(2), ACTIVATE_1(3)
- 1750858 to 1750860 labelled C_1(1), c_1(1), ACTIVATE_1(2)
- 1750858 to 1750857 labelled g_1(1), n__g_1(1), n__g_1(2)
- 1750858 to 1750866 labelled d_1(3), n__d_1(4)
- 1750858 to 1750867 labelled F_1(2)
- 1750858 to 1750868 labelled F_1(3)
- 1750859 to 1750853 labelled activate_1(2), g_1(1), ACTIVATE_1(1), n__d_1(1), f_1(1), c_1(1), n__f_1(1), n__g_1(1), C_1(1), d_1(1), F_1(1), activate_1(1), n__g_1(2), ACTIVATE_1(2), n__f_1(2), n__d_1(2)
- 1750859 to 1750860 labelled activate_1(2), C_1(1), c_1(1), ACTIVATE_1(2)
- 1750859 to 1750854 labelled f_1(1), n__f_1(2), F_1(1)
- 1750859 to 1750859 labelled d_1(2), n__d_1(3)
- 1750859 to 1750863 labelled c_1(2), C_1(2), ACTIVATE_1(3)
- 1750859 to 1750861 labelled n__f_1(2)
- 1750859 to 1750867 labelled f_1(2), n__f_1(3), F_1(2)
- 1750859 to 1750866 labelled d_1(3), n__d_1(4)
- 1750859 to 1750868 labelled F_1(3)
- 1750860 to 1750861 labelled n__f_1(1)
- 1750862 to 1750853 labelled n__f_1(1)
- 1750862 to 1750858 labelled n__f_1(1)
- 1750862 to 1750854 labelled n__f_1(1)
- 1750861 to 1750862 labelled n__g_1(1)
- 1750864 to 1750865 labelled n__g_1(2)
- 1750865 to 1750853 labelled n__f_1(2)
- 1750865 to 1750858 labelled n__f_1(2)
- 1750865 to 1750854 labelled n__f_1(2)
- 1750863 to 1750864 labelled n__f_1(2)
- 1750866 to 1750863 labelled activate_1(3)
- 1750866 to 1750864 labelled n__f_1(3)
- 1750866 to 1750868 labelled f_1(3), n__f_1(4)
- 1750867 to 1750861 labelled activate_1(2)
- 1750867 to 1750862 labelled g_1(2), n__g_1(2), n__g_1(3)
- 1750868 to 1750864 labelled activate_1(3)
- 1750868 to 1750865 labelled g_1(3), n__g_1(3), n__g_1(4)
(14) TRUE