We consider the system foldl.

  Alphabet:

    0 : [] --> nat 
    cons : [nat * list] --> list 
    foldl : [nat -> nat -> nat * nat * list] --> nat 
    nil : [] --> list 
    plus : [nat * nat] --> nat 
    plusc : [] --> nat -> nat -> nat 
    sum : [list] --> nat 

  Rules:

    foldl(f, x, nil) => x 
    foldl(f, x, cons(y, z)) => foldl(f, f x y, z) 
    plusc => /\x./\y.plus(x, y) 
    sum(x) => foldl(plusc, 0, 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]).

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

  foldl(F, X, nil) => X 
  foldl(F, X, cons(Y, Z)) => foldl(F, F X Y, Z) 
  plusc(X, Y) => (/\x./\y.plus x y) X Y 
  sum(X) => foldl(/\x./\y.plusc(x, y), 0, X) 

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

  Dependency Pairs P_0:

    0] foldl#(F, X, cons(Y, Z)) =#> foldl#(F, F X Y, Z)   
    1] sum#(X) =#> foldl#(/\x./\y.plusc(x, y), 0, X)   
    2] sum#(X) =#> plusc#(Y, Z)   

  Rules R_0:

    foldl(F, X, nil) => X 
    foldl(F, X, cons(Y, Z)) => foldl(F, F X Y, Z) 
    plusc(X, Y) => (/\x./\y.plus x y) X Y 
    sum(X) => foldl(/\x./\y.plusc(x, y), 0, 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 : 0 
    * 2 :  

This graph has the following strongly connected components:

  P_1:

    foldl#(F, X, cons(Y, Z)) =#> foldl#(F, F X Y, 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).

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(foldl#) = 3 

Thus, we can orient the dependency pairs as follows:

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