YES Termination w.r.t. Q proof of Der95_03.ari

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

0 QTRS
↳1 Strip Symbols Proof (⇒, 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)) → g(f(x))

Q is empty.

(1) Strip Symbols Proof (SOUND transformation)

We were given the following TRS:

f(f(x)) → g(f(x))

By stripping symbols from the only rule of the system, we obtained the following TRS [ENDRULLIS]:

f(x) → g(x)

(2) Obligation:

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

f(x) → g(x)

Q is empty.

(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 0. This implies Q-termination of R.
The following rules were used to construct the certificate:

f(x) → g(x)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 4

Node 1 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 1 to 4 labelled g_1(0)
  • 4 to 4 labelled #_1(0)

(4) YES