We consider the system AotoYamada_05__009.

  Alphabet:

    and : [c * c] --> c 
    cons : [a * b] --> b 
    false : [] --> c 
    forall : [a -> c * b] --> c 
    forsome : [a -> c * b] --> c 
    nil : [] --> b 
    or : [c * c] --> c 
    true : [] --> c 

  Rules:

    and(true, true) => true 
    and(true, false) => false 
    and(false, true) => false 
    and(false, false) => false 
    or(true, true) => true 
    or(true, false) => true 
    or(false, true) => true 
    or(false, false) => false 
    forall(f, nil) => true 
    forall(f, cons(x, y)) => and(f x, forall(f, y)) 
    forsome(f, nil) => false 
    forsome(f, cons(x, y)) => or(f x, forsome(f, y)) 

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 [Kop13]).

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

  Dependency Pairs P_0:

    0] forall#(F, cons(X, Y)) =#> and#(F X, forall(F, Y))   
    1] forall#(F, cons(X, Y)) =#> forall#(F, Y)   
    2] forsome#(F, cons(X, Y)) =#> or#(F X, forsome(F, Y))   
    3] forsome#(F, cons(X, Y)) =#> forsome#(F, Y)   

  Rules R_0:

    and(true, true) => true 
    and(true, false) => false 
    and(false, true) => false 
    and(false, false) => false 
    or(true, true) => true 
    or(true, false) => true 
    or(false, true) => true 
    or(false, false) => false 
    forall(F, nil) => true 
    forall(F, cons(X, Y)) => and(F X, forall(F, Y)) 
    forsome(F, nil) => false 
    forsome(F, cons(X, Y)) => or(F X, forsome(F, Y)) 

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

We consider the dependency pair problem (P_0, R_0, static, 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 :  
    * 1 : 0, 1 
    * 2 :  
    * 3 : 2, 3 

This graph has the following strongly connected components:

  P_1:

    forall#(F, cons(X, Y)) =#> forall#(F, Y)   

  P_2:

    forsome#(F, cons(X, Y)) =#> forsome#(F, 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) and (P_2, R_0, m, f).

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

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

We apply the subterm criterion with the following projection function:

  nu(forsome#) = 2 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_2, R_0, static, f) by ({}, R_0, static, 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, static, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(forall#) = 2 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by ({}, R_0, static, 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 +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[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.