YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex25_Luc06_Z.ari

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 RFCMatchBoundsTRSProof (⇔, 0 ms)
↳4 YES

(0) Obligation:

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

f(f(X)) → c(n__f(g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
d(X) → n__d(X)
activate(n__f(X)) → f(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(g(n__f(c(X))))
c(X) → activate(d(X))
h(X) → n__d(c(X))
f(X) → n__f(X)
d(X) → n__d(X)
n__f(activate(X)) → f(X)
n__d(activate(X)) → d(X)
activate(X) → X

Q is empty.

(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. This implies Q-termination of R.
The following rules were used to construct the certificate:

f(f(X)) → n__f(g(n__f(c(X))))
c(X) → activate(d(X))
h(X) → n__d(c(X))
f(X) → n__f(X)
d(X) → n__d(X)
n__f(activate(X)) → f(X)
n__d(activate(X)) → d(X)
activate(X) → X

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

2, 3, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19

Node 2 is start node and node 3 is final node.

Those nodes are connected through the following edges:

  • 2 to 9 labelled n__f_1(0)
  • 2 to 12 labelled activate_1(0)
  • 2 to 11 labelled n__d_1(0)
  • 2 to 3 labelled n__f_1(0), n__d_1(0), f_1(0), d_1(0), g_1(0), c_1(0), activate_1(0), h_1(0), n__f_1(1), n__d_1(1), d_1(1), f_1(1), g_1(1), c_1(1), activate_1(1), h_1(1), n__d_1(2), n__f_1(2), c_1(2), d_1(2), n__d_1(3), d_1(3), n__d_1(4), d_1(4), n__d_1(5)
  • 2 to 13 labelled activate_1(1), d_1(1), n__d_1(2)
  • 2 to 14 labelled n__f_1(1)
  • 2 to 17 labelled activate_1(2), f_1(1), d_1(1), activate_1(1), f_1(2), d_1(2), d_1(3), f_1(3), n__f_1(2), n__d_1(2), n__f_1(3), n__d_1(3), n__d_1(4), n__f_1(4), n__d_1(1), d_1(4), n__d_1(5)
  • 2 to 19 labelled activate_1(3), f_1(1), d_1(1), activate_1(1), f_1(2), d_1(2), activate_1(2), d_1(3), f_1(3), d_1(4), n__f_1(2), n__d_1(2), n__f_1(3), n__d_1(3), n__d_1(4), n__f_1(4), n__d_1(5), n__d_1(1)
  • 3 to 3 labelled #_1(0), c_1(1), c_1(2), d_1(3), n__d_1(4), n__d_1(3), d_1(4), n__d_1(5), d_1(1), d_1(2), n__d_1(2)
  • 3 to 17 labelled activate_1(2), d_1(3), n__d_1(4), n__d_1(3), d_1(4), n__d_1(5)
  • 3 to 19 labelled activate_1(3), d_1(4), d_1(3), n__d_1(5), n__d_1(4), n__d_1(3)
  • 9 to 10 labelled g_1(0)
  • 10 to 11 labelled n__f_1(0)
  • 10 to 13 labelled f_1(1), n__f_1(2)
  • 11 to 3 labelled c_1(0), d_1(2), n__d_1(3), n__d_1(2), d_1(1)
  • 11 to 13 labelled activate_1(1)
  • 11 to 17 labelled d_1(3), d_1(2), n__d_1(4), n__d_1(3), n__d_1(2)
  • 11 to 19 labelled d_1(4), d_1(3), n__d_1(5), n__d_1(4), d_1(2), n__d_1(3), n__d_1(2)
  • 12 to 3 labelled d_1(0), n__d_1(1), d_1(1), n__d_1(2)
  • 12 to 17 labelled d_1(2), n__d_1(3), d_1(3), n__d_1(4)
  • 12 to 19 labelled d_1(2), d_1(3), n__d_1(3), n__d_1(4)
  • 13 to 3 labelled d_1(1), n__d_1(2)
  • 13 to 17 labelled d_1(3), n__d_1(4)
  • 13 to 19 labelled d_1(3), n__d_1(4)
  • 14 to 15 labelled g_1(1)
  • 15 to 16 labelled n__f_1(1)
  • 15 to 17 labelled f_1(2), n__f_1(3)
  • 16 to 3 labelled c_1(1), d_1(3), n__d_1(4), n__d_1(3), d_1(1), d_1(2), n__d_1(2)
  • 16 to 17 labelled activate_1(2), d_1(3), n__d_1(4), n__d_1(3)
  • 16 to 19 labelled d_1(4), d_1(3), n__d_1(5), n__d_1(4), n__d_1(3)
  • 17 to 3 labelled d_1(2), n__d_1(3), d_1(1), n__d_1(2)
  • 17 to 17 labelled d_1(3), n__d_1(4)
  • 17 to 19 labelled d_1(4), n__d_1(5), d_1(3), n__d_1(4)
  • 19 to 3 labelled d_1(3), n__d_1(4), d_1(1), n__d_1(2)
  • 19 to 17 labelled d_1(3), n__d_1(4)
  • 19 to 19 labelled d_1(4), n__d_1(5), d_1(3), n__d_1(4)

(4) YES