We consider the system kop12thesis_ex2.11.

  Alphabet:

    cons : [nat * list] --> list 
    emap : [nat -> nat * list] --> list 
    nil : [] --> list 
    twice : [nat -> nat] --> nat -> nat 

  Rules:

    emap(f, nil) => nil 
    emap(f, cons(x, y)) => cons(f x, emap(/\z.twice(f) z, y)) 
    twice(f) x => f (f 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] emap#(F, cons(X, Y)) =#> emap#(/\x.twice(F, x), Y)   
    1] emap#(F, cons(X, Y)) =#> twice#(F, Z)   

  Rules R_0:

    emap(F, nil) => nil 
    emap(F, cons(X, Y)) => cons(F X, emap(/\x.twice(F, x), Y)) 
    twice(F, X) => F (F 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 place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0, 1 
    * 1 :  

This graph has the following strongly connected components:

  P_1:

    emap#(F, cons(X, Y)) =#> emap#(/\x.twice(F, x), Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).

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 apply the subterm criterion with the following projection function:

  nu(emap#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(emap#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(emap#(/\x.twice(F, x), Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_0, computable, f) by ({}, R_0, computable, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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.