We consider the system 447.

  Alphabet:

    a : [] --> A 
    b : [] --> A 
    f : [A] --> A 
    g : [A] --> A 
    h : [A] --> A 

  Rules:

    f(X) => g(X) 
    g(X) => h(X) 
    a => b 

We observe that the rules contain a first-order subset:

  f(X) => g(X) 
  g(X) => h(X) 
  a => b 

Moreover, the system is orthogonal.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is terminating when seen as a many-sorted first-order TRS.

According to the external first-order termination prover, this system is indeed terminating:

 || proof of resources/system.trs
 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
 || 
 || 
 || Termination w.r.t. Q of the given QTRS could be proven:
 || 
 || (0) QTRS
 || (1) QTRS Reverse [EQUIVALENT]
 || (2) QTRS
 || (3) RFCMatchBoundsTRSProof [EQUIVALENT]
 || (4) YES
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    f(%X) -> g(%X)
 ||    g(%X) -> h(%X)
 ||    a -> b
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRS Reverse (EQUIVALENT)
 || We applied the QTRS Reverse Processor [REVERSE].
 || ----------------------------------------
 || 
 || (2)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    f(x) -> g(x)
 ||    g(x) -> h(x)
 ||    a'(x) -> b'(x)
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (3) RFCMatchBoundsTRSProof (EQUIVALENT)
 || Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
 || The following rules were used to construct the certificate:
 || 
 ||    f(x) -> g(x)
 ||    g(x) -> h(x)
 ||    a'(x) -> b'(x)
 || 
 || The certificate found is represented by the following graph.
 || The certificate consists of the following enumerated nodes:
 || 2, 4
 || 
 || Node 2 is start node and node 4 is final node.
 || 
 || Those nodes are connected through the following edges:
 || 
 || * 2 to 4 labelled g_1(0), h_1(0), b'_1(0), h_1(1)* 4 to 4 labelled #_1(0)
 || 
 || 
 || ----------------------------------------
 || 
 || (4)
 || YES
 || 
We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]).

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:


  Rules R_0:

    f(X) => g(X) 
    g(X) => h(X) 
    a => b 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:


This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.