We consider the system 427.

  Alphabet:

    all : [T -> F] --> F 
    and : [F * F] --> F 
    ex : [T -> F] --> F 
    not : [F] --> F 
    or : [F * F] --> F 

  Rules:

    and(X, all(/\x.Y(x))) => all(/\y.and(X, Y(y))) 
    and(all(/\x.X(x)), Y) => all(/\y.and(X(y), Y)) 
    or(X, all(/\x.Y(x))) => all(/\y.or(X, Y(y))) 
    or(all(/\x.X(x)), Y) => all(/\y.or(X(y), Y)) 
    and(X, ex(/\x.Y(x))) => ex(/\y.and(X, Y(y))) 
    and(ex(/\x.X(x)), Y) => ex(/\y.and(X(y), Y)) 
    or(X, ex(/\x.Y(x))) => ex(/\y.or(X, Y(y))) 
    or(ex(/\x.X(x)), Y) => ex(/\y.or(X(y), Y)) 
    not(all(/\x.X(x))) => ex(/\y.not(X(y))) 
    not(ex(/\x.X(x))) => all(/\y.not(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] and#(X, all(/\x.Y(x))) =#> and#(X, Y(Z))   
    1] and#(all(/\x.X(x)), Y) =#> and#(X(Z), Y)   
    2] or#(X, all(/\x.Y(x))) =#> or#(X, Y(Z))   
    3] or#(all(/\x.X(x)), Y) =#> or#(X(Z), Y)   
    4] and#(X, ex(/\x.Y(x))) =#> and#(X, Y(Z))   
    5] and#(ex(/\x.X(x)), Y) =#> and#(X(Z), Y)   
    6] or#(X, ex(/\x.Y(x))) =#> or#(X, Y(Z))   
    7] or#(ex(/\x.X(x)), Y) =#> or#(X(Z), Y)   
    8] not#(all(/\x.X(x))) =#> not#(X(Y))   
    9] not#(ex(/\x.X(x))) =#> not#(X(Y))   

  Rules R_0:

    and(X, all(/\x.Y(x))) => all(/\y.and(X, Y(y))) 
    and(all(/\x.X(x)), Y) => all(/\y.and(X(y), Y)) 
    or(X, all(/\x.Y(x))) => all(/\y.or(X, Y(y))) 
    or(all(/\x.X(x)), Y) => all(/\y.or(X(y), Y)) 
    and(X, ex(/\x.Y(x))) => ex(/\y.and(X, Y(y))) 
    and(ex(/\x.X(x)), Y) => ex(/\y.and(X(y), Y)) 
    or(X, ex(/\x.Y(x))) => ex(/\y.or(X, Y(y))) 
    or(ex(/\x.X(x)), Y) => ex(/\y.or(X(y), Y)) 
    not(all(/\x.X(x))) => ex(/\y.not(X(y))) 
    not(ex(/\x.X(x))) => all(/\y.not(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 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, 4, 5 
    * 1 : 0, 1, 4, 5 
    * 2 : 2, 3, 6, 7 
    * 3 : 2, 3, 6, 7 
    * 4 : 0, 1, 4, 5 
    * 5 : 0, 1, 4, 5 
    * 6 : 2, 3, 6, 7 
    * 7 : 2, 3, 6, 7 
    * 8 : 8, 9 
    * 9 : 8, 9 

This graph has the following strongly connected components:

  P_1:

    and#(X, all(/\x.Y(x))) =#> and#(X, Y(Z))   
    and#(all(/\x.X(x)), Y) =#> and#(X(Z), Y)   
    and#(X, ex(/\x.Y(x))) =#> and#(X, Y(Z))   
    and#(ex(/\x.X(x)), Y) =#> and#(X(Z), Y)   

  P_2:

    or#(X, all(/\x.Y(x))) =#> or#(X, Y(Z))   
    or#(all(/\x.X(x)), Y) =#> or#(X(Z), Y)   
    or#(X, ex(/\x.Y(x))) =#> or#(X, Y(Z))   
    or#(ex(/\x.X(x)), Y) =#> or#(X(Z), Y)   

  P_3:

    not#(all(/\x.X(x))) =#> not#(X(Y))   
    not#(ex(/\x.X(x))) =#> not#(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), (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, computable, formative), (P_2, R_0, computable, formative) and (P_3, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_3, R_0, computable, formative).

We apply the accessible subterm criterion with the following projection function:

  nu(not#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(not#(all(/\x.X(x)))) = all(/\y.X(y)) [>] X(Y) = nu(not#(X(Y))) 
  nu(not#(ex(/\x.X(x)))) = ex(/\y.X(y)) [>] X(Y) = nu(not#(X(Y))) 

By [FuhKop19, Thm. 63], we may replace a dependency pair problem (P_3, 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 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 accessible subterm criterion with the following projection function:

  nu(or#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(or#(X, all(/\x.Y(x)))) = all(/\y.Y(y)) [>] Y(Z) = nu(or#(X, Y(Z))) 
  nu(or#(all(/\x.X(x)), Y)) = Y = Y = nu(or#(X(Z), Y)) 
  nu(or#(X, ex(/\x.Y(x)))) = ex(/\y.Y(y)) [>] Y(Z) = nu(or#(X, Y(Z))) 
  nu(or#(ex(/\x.X(x)), Y)) = Y = Y = nu(or#(X(Z), Y)) 

By [FuhKop19, Thm. 7.6], we may replace a dependency pair problem (P_2, R_0, computable, f) by (P_4, R_0, computable, f), where P_4 contains:

  or#(all(/\x.X(x)), Y) =#> or#(X(Z), Y)   
  or#(ex(/\x.X(x)), Y) =#> or#(X(Z), Y)   

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_4, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_4, R_0, computable, formative).

We apply the accessible subterm criterion with the following projection function:

  nu(or#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(or#(all(/\x.X(x)), Y)) = all(/\y.X(y)) [>] X(Z) = nu(or#(X(Z), Y)) 
  nu(or#(ex(/\x.X(x)), Y)) = ex(/\y.X(y)) [>] X(Z) = nu(or#(X(Z), Y)) 

By [FuhKop19, Thm. 63], we may replace a dependency pair problem (P_4, 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 accessible subterm criterion with the following projection function:

  nu(and#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(and#(X, all(/\x.Y(x)))) = all(/\y.Y(y)) [>] Y(Z) = nu(and#(X, Y(Z))) 
  nu(and#(all(/\x.X(x)), Y)) = Y = Y = nu(and#(X(Z), Y)) 
  nu(and#(X, ex(/\x.Y(x)))) = ex(/\y.Y(y)) [>] Y(Z) = nu(and#(X, Y(Z))) 
  nu(and#(ex(/\x.X(x)), Y)) = Y = Y = nu(and#(X(Z), Y)) 

By [FuhKop19, Thm. 7.6], we may replace a dependency pair problem (P_1, R_0, computable, f) by (P_5, R_0, computable, f), where P_5 contains:

  and#(all(/\x.X(x)), Y) =#> and#(X(Z), Y)   
  and#(ex(/\x.X(x)), Y) =#> and#(X(Z), Y)   

Thus, the original system is terminating if (P_5, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_5, R_0, computable, formative).

We apply the accessible subterm criterion with the following projection function:

  nu(and#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(and#(all(/\x.X(x)), Y)) = all(/\y.X(y)) [>] X(Z) = nu(and#(X(Z), Y)) 
  nu(and#(ex(/\x.X(x)), Y)) = ex(/\y.X(y)) [>] X(Z) = nu(and#(X(Z), Y)) 

By [FuhKop19, Thm. 63], we may replace a dependency pair problem (P_5, 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.