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 QDP
    ↳11 MRRProof (⇔)
    ↳12 QDP
    ↳13 RFCMatchBoundsDPProof (⇔)
    ↳14 TRUE

(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:

8291310, 8291311, 8291312, 8291313, 8291314, 8291315, 8291316, 8291317, 8291319, 8291320, 8291318, 8291321, 8291322, 8291323, 8291326, 8291324, 8291325, 8291327, 8291328, 8291329

Node 8291310 is start node and node 8291311 is final node.

Those nodes are connect through the following edges:

  • 8291310 to 8291312 labelled F_1(0)
  • 8291310 to 8291313 labelled C_1(0), ACTIVATE_1(1)
  • 8291310 to 8291311 labelled ACTIVATE_1(0)
  • 8291310 to 8291316 labelled F_1(1)
  • 8291310 to 8291321 labelled C_1(1), ACTIVATE_1(2)
  • 8291310 to 8291318 labelled C_1(1), ACTIVATE_1(2)
  • 8291310 to 8291324 labelled C_1(2), ACTIVATE_1(3)
  • 8291310 to 8291328 labelled F_1(2)
  • 8291310 to 8291329 labelled F_1(3)
  • 8291311 to 8291311 labelled #_1(0)
  • 8291312 to 8291311 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), n__f_1(2), n__g_1(2), ACTIVATE_1(2), n__d_1(2)
  • 8291312 to 8291316 labelled f_1(1), n__f_1(2)
  • 8291312 to 8291317 labelled d_1(2), n__d_1(3)
  • 8291312 to 8291312 labelled F_1(1), f_1(1), n__f_1(2)
  • 8291312 to 8291318 labelled c_1(1), C_1(1), ACTIVATE_1(2)
  • 8291312 to 8291324 labelled c_1(2), C_1(2), ACTIVATE_1(3)
  • 8291312 to 8291327 labelled d_1(3), n__d_1(4)
  • 8291312 to 8291328 labelled F_1(2)
  • 8291312 to 8291329 labelled F_1(3)
  • 8291313 to 8291314 labelled n__f_1(0)
  • 8291314 to 8291315 labelled n__g_1(0)
  • 8291315 to 8291311 labelled n__f_1(0)
  • 8291316 to 8291311 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), n__d_1(2), ACTIVATE_1(2), n__f_1(2)
  • 8291316 to 8291314 labelled activate_1(1)
  • 8291316 to 8291312 labelled f_1(1), n__f_1(2), F_1(1)
  • 8291316 to 8291317 labelled d_1(2), n__d_1(3)
  • 8291316 to 8291318 labelled C_1(1), c_1(1), ACTIVATE_1(2)
  • 8291316 to 8291324 labelled c_1(2), C_1(2), ACTIVATE_1(3)
  • 8291316 to 8291315 labelled g_1(1), n__g_1(1), n__g_1(2)
  • 8291316 to 8291327 labelled d_1(3), n__d_1(4)
  • 8291316 to 8291328 labelled F_1(2)
  • 8291316 to 8291329 labelled F_1(3)
  • 8291317 to 8291311 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)
  • 8291317 to 8291318 labelled activate_1(2), c_1(1), C_1(1), ACTIVATE_1(2)
  • 8291317 to 8291312 labelled f_1(1), n__f_1(2), F_1(1)
  • 8291317 to 8291317 labelled d_1(2), n__d_1(3)
  • 8291317 to 8291319 labelled n__f_1(2)
  • 8291317 to 8291328 labelled f_1(2), n__f_1(3), F_1(2)
  • 8291317 to 8291324 labelled c_1(2), C_1(2), ACTIVATE_1(3)
  • 8291317 to 8291327 labelled d_1(3), n__d_1(4)
  • 8291317 to 8291329 labelled F_1(3)
  • 8291319 to 8291320 labelled n__g_1(1)
  • 8291320 to 8291311 labelled n__f_1(1)
  • 8291320 to 8291312 labelled n__f_1(1)
  • 8291318 to 8291319 labelled n__f_1(1)
  • 8291321 to 8291322 labelled n__f_1(1)
  • 8291322 to 8291323 labelled n__g_1(1)
  • 8291323 to 8291316 labelled n__f_1(1)
  • 8291326 to 8291311 labelled n__f_1(2)
  • 8291326 to 8291312 labelled n__f_1(2)
  • 8291326 to 8291316 labelled n__f_1(2)
  • 8291324 to 8291325 labelled n__f_1(2)
  • 8291325 to 8291326 labelled n__g_1(2)
  • 8291327 to 8291324 labelled activate_1(3)
  • 8291327 to 8291325 labelled n__f_1(3)
  • 8291327 to 8291329 labelled f_1(3), n__f_1(4)
  • 8291328 to 8291319 labelled activate_1(2)
  • 8291328 to 8291322 labelled activate_1(2)
  • 8291328 to 8291320 labelled g_1(2), n__g_1(2), n__g_1(3)
  • 8291328 to 8291323 labelled g_1(2), n__g_1(2), n__g_1(3)
  • 8291329 to 8291325 labelled activate_1(3)
  • 8291329 to 8291326 labelled g_1(3), n__g_1(3), n__g_1(4)

(14) TRUE