We consider the system kop12thesis_ex7.23.

  Alphabet:

    0 : [] --> o 
    either : [o * o] --> o 
    f : [o -> o * o * o] --> o 
    g : [o * o] --> o 
    s : [o] --> o 

  Rules:

    f(h, x, 0) => 0 
    f(h, x, s(y)) => g(y, either(y, h x)) 
    g(x, y) => f(/\z.s(0), y, x) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [FuhKop19]).

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

  Dependency Pairs P_0:

    0] f#(F, X, s(Y)) =#> g#(Y, either(Y, F X))   
    1] g#(X, Y) =#> f#(/\x.s(0), Y, X)   

  Rules R_0:

    f(F, X, 0) => 0 
    f(F, X, s(Y)) => g(Y, either(Y, F X)) 
    g(X, Y) => f(/\x.s(0), Y, X) 

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

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

We apply the subterm criterion with the following projection function:

  nu(f#) = 3 
  nu(g#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(f#(F, X, s(Y))) = s(Y) |> Y = nu(g#(Y, either(Y, F X))) 
  nu(g#(X, Y)) = X = X = nu(f#(/\x.s(0), Y, X)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_0, R_0, computable, f) by (P_1, R_0, computable, f), where P_1 contains:

  g#(X, Y) =#> f#(/\x.s(0), Y, X)   

Thus, the original system is terminating if (P_1, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_1, R_0, computable, formative).

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

    * 0 :  

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 +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[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.