We consider the system prenex_modif1.

  Alphabet:

    and : [form * form] --> form 
    exists : [term -> form] --> form 
    forall : [term -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 

  Rules:

    and(x, forall(/\y.f y)) => forall(/\z.and(x, f z)) 
    and(forall(/\x.f x), y) => forall(/\z.and(f z, y)) 
    and(x, exists(/\y.f y)) => exists(/\z.and(x, f z)) 
    and(exists(/\x.f x), y) => exists(/\z.and(f z, y)) 
    or(x, forall(/\y.f y)) => forall(/\z.or(x, f z)) 
    or(forall(/\x.f x), y) => forall(/\z.or(f z, y)) 
    or(x, exists(/\y.f y)) => exists(/\z.or(x, f z)) 
    or(exists(/\x.f x), y) => exists(/\z.or(f z, y)) 
    not(forall(/\x.f x)) => exists(/\y.not(f y)) 
    not(exists(/\x.f x)) => forall(/\y.not(f y)) 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    and : [form * form] --> form 
    exists : [term -> form] --> form 
    forall : [term -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 
    ~AP1 : [term -> form * term] --> form 

  Rules:

    and(X, forall(/\x.~AP1(F, x))) => forall(/\y.and(X, ~AP1(F, y))) 
    and(forall(/\x.~AP1(F, x)), X) => forall(/\y.and(~AP1(F, y), X)) 
    and(X, exists(/\x.~AP1(F, x))) => exists(/\y.and(X, ~AP1(F, y))) 
    and(exists(/\x.~AP1(F, x)), X) => exists(/\y.and(~AP1(F, y), X)) 
    or(X, forall(/\x.~AP1(F, x))) => forall(/\y.or(X, ~AP1(F, y))) 
    or(forall(/\x.~AP1(F, x)), X) => forall(/\y.or(~AP1(F, y), X)) 
    or(X, exists(/\x.~AP1(F, x))) => exists(/\y.or(X, ~AP1(F, y))) 
    or(exists(/\x.~AP1(F, x)), X) => exists(/\y.or(~AP1(F, y), X)) 
    not(forall(/\x.~AP1(F, x))) => exists(/\y.not(~AP1(F, y))) 
    not(exists(/\x.~AP1(F, x))) => forall(/\y.not(~AP1(F, y))) 
    ~AP1(F, X) => F X 

Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:

  Alphabet:

    and : [form * form] --> form 
    exists : [term -> form] --> form 
    forall : [term -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 

  Rules:

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

  Rules R_0:

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

This graph has the following strongly connected components:

  P_1:

    and#(X, forall(/\x.Y(x))) =#> and#(X, Y(Z))   
    and#(forall(/\x.X(x)), Y) =#> and#(X(Z), Y)   
    and#(X, exists(/\x.Y(x))) =#> and#(X, Y(Z))   
    and#(exists(/\x.X(x)), Y) =#> and#(X(Z), Y)   

  P_2:

    or#(X, forall(/\x.Y(x))) =#> or#(X, Y(Z))   
    or#(forall(/\x.X(x)), Y) =#> or#(X(Z), Y)   
    or#(X, exists(/\x.Y(x))) =#> or#(X, Y(Z))   
    or#(exists(/\x.X(x)), Y) =#> or#(X(Z), Y)   

  P_3:

    not#(forall(/\x.X(x))) =#> not#(X(Y))   
    not#(exists(/\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#(forall(/\x.X(x)))) = forall(/\y.X(y)) [>] X(Y) = nu(not#(X(Y))) 
  nu(not#(exists(/\x.X(x)))) = exists(/\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, forall(/\x.Y(x)))) = forall(/\y.Y(y)) [>] Y(Z) = nu(or#(X, Y(Z))) 
  nu(or#(forall(/\x.X(x)), Y)) = Y = Y = nu(or#(X(Z), Y)) 
  nu(or#(X, exists(/\x.Y(x)))) = exists(/\y.Y(y)) [>] Y(Z) = nu(or#(X, Y(Z))) 
  nu(or#(exists(/\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#(forall(/\x.X(x)), Y) =#> or#(X(Z), Y)   
  or#(exists(/\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#(forall(/\x.X(x)), Y)) = forall(/\y.X(y)) [>] X(Z) = nu(or#(X(Z), Y)) 
  nu(or#(exists(/\x.X(x)), Y)) = exists(/\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, forall(/\x.Y(x)))) = forall(/\y.Y(y)) [>] Y(Z) = nu(and#(X, Y(Z))) 
  nu(and#(forall(/\x.X(x)), Y)) = Y = Y = nu(and#(X(Z), Y)) 
  nu(and#(X, exists(/\x.Y(x)))) = exists(/\y.Y(y)) [>] Y(Z) = nu(and#(X, Y(Z))) 
  nu(and#(exists(/\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#(forall(/\x.X(x)), Y) =#> and#(X(Z), Y)   
  and#(exists(/\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#(forall(/\x.X(x)), Y)) = forall(/\y.X(y)) [>] X(Z) = nu(and#(X(Z), Y)) 
  nu(and#(exists(/\x.X(x)), Y)) = exists(/\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.
[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[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.