We consider the system plode.

  Alphabet:

    cons : [nat * list] --> list 
    explode : [list * nat -> nat * nat] --> nat 
    implode : [list * nat -> nat * nat] --> nat 
    nil : [] --> list 
    op : [nat -> nat * nat -> nat] --> nat -> nat 

  Rules:

    op(f, g) x => f (g x) 
    implode(nil, f, x) => x 
    implode(cons(x, y), f, z) => implode(y, f, f z) 
    explode(nil, f, x) => x 
    explode(cons(x, y), f, z) => explode(y, op(f, f), f z) 

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]).

In order to do so, we start by eta-expanding the system, which gives:

  op(F, G, X) => F (G X) 
  implode(nil, F, X) => X 
  implode(cons(X, Y), F, Z) => implode(Y, F, F Z) 
  explode(nil, F, X) => X 
  explode(cons(X, Y), F, Z) => explode(Y, /\x.op(F, F, x), F Z) 

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

  Dependency Pairs P_0:

    0] implode#(cons(X, Y), F, Z) =#> implode#(Y, F, F Z)   
    1] explode#(cons(X, Y), F, Z) =#> explode#(Y, /\x.op(F, F, x), F Z)   
    2] explode#(cons(X, Y), F, Z) =#> op#(F, F, U)   

  Rules R_0:

    op(F, G, X) => F (G X) 
    implode(nil, F, X) => X 
    implode(cons(X, Y), F, Z) => implode(Y, F, F Z) 
    explode(nil, F, X) => X 
    explode(cons(X, Y), F, Z) => explode(Y, /\x.op(F, F, x), F Z) 

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, 2 
    * 2 :  

This graph has the following strongly connected components:

  P_1:

    implode#(cons(X, Y), F, Z) =#> implode#(Y, F, F Z)   

  P_2:

    explode#(cons(X, Y), F, Z) =#> explode#(Y, /\x.op(F, F, x), F Z)   

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) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_2, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_2, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(explode#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(explode#(cons(X, Y), F, Z)) = cons(X, Y) |> Y = nu(explode#(Y, /\x.op(F, F, x), F Z)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by ({}, R_0, computable, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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(implode#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(implode#(cons(X, Y), F, Z)) = cons(X, Y) |> Y = nu(implode#(Y, F, F Z)) 

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.