We consider the system 459.

  Alphabet:

    and : [form * form] --> form 
    exists : [term -> form] --> form 
    forall : [term -> form] --> form 
    neg : [form] --> form 
    or : [form * form] --> form 

  Rules:

    neg(neg(X)) => X 
    neg(and(X, Y)) => or(neg(X), neg(Y)) 
    neg(or(X, Y)) => and(neg(X), neg(Y)) 
    neg(forall(/\x.X(x))) => exists(/\y.neg(X(y))) 
    neg(exists(/\x.X(x))) => forall(/\y.neg(X(y))) 

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] neg#(and(X, Y)) =#> neg#(X)   
    1] neg#(and(X, Y)) =#> neg#(Y)   
    2] neg#(or(X, Y)) =#> neg#(X)   
    3] neg#(or(X, Y)) =#> neg#(Y)   
    4] neg#(forall(/\x.X(x))) =#> neg#(X(Y))   
    5] neg#(exists(/\x.X(x))) =#> neg#(X(Y))   

  Rules R_0:

    neg(neg(X)) => X 
    neg(and(X, Y)) => or(neg(X), neg(Y)) 
    neg(or(X, Y)) => and(neg(X), neg(Y)) 
    neg(forall(/\x.X(x))) => exists(/\y.neg(X(y))) 
    neg(exists(/\x.X(x))) => forall(/\y.neg(X(y))) 

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 accessible subterm criterion with the following projection function:

  nu(neg#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(neg#(and(X, Y))) = and(X, Y) [>] X = nu(neg#(X)) 
  nu(neg#(and(X, Y))) = and(X, Y) [>] Y = nu(neg#(Y)) 
  nu(neg#(or(X, Y))) = or(X, Y) [>] X = nu(neg#(X)) 
  nu(neg#(or(X, Y))) = or(X, Y) [>] Y = nu(neg#(Y)) 
  nu(neg#(forall(/\x.X(x)))) = forall(/\y.X(y)) [>] X(Y) = nu(neg#(X(Y))) 
  nu(neg#(exists(/\x.X(x)))) = exists(/\y.X(y)) [>] X(Y) = nu(neg#(X(Y))) 

By [FuhKop19, Thm. 63], we may replace a dependency pair problem (P_0, 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.