We consider the system Applicative_05__Ex9Maps.

  Alphabet:

    cons : [d * e] --> e 
    g : [] --> b 
    map!fac62201 : [d -> d * e] --> e 
    map!fac62202 : [d -> a -> d * a * e] --> e 
    map!fac62203 : [b -> d -> c -> d * b * c * e] --> e 

  Rules:

    map!fac62201(f, cons(x, y)) => cons(f x, map!fac62201(f, y)) 
    map!fac62202(f, x, cons(y, z)) => cons(f y x, map!fac62202(f, x, z)) 
    map!fac62203(f, g, x, cons(y, z)) => cons(f g y x, map!fac62203(f, g, x, z)) 

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]).

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

  Dependency Pairs P_0:

    0] map!fac62201#(F, cons(X, Y)) =#> map!fac62201#(F, Y)   
    1] map!fac62202#(F, X, cons(Y, Z)) =#> map!fac62202#(F, X, Z)   
    2] map!fac62203#(F, g, X, cons(Y, Z)) =#> map!fac62203#(F, g, X, Z)   

  Rules R_0:

    map!fac62201(F, cons(X, Y)) => cons(F X, map!fac62201(F, Y)) 
    map!fac62202(F, X, cons(Y, Z)) => cons(F Y X, map!fac62202(F, X, Z)) 
    map!fac62203(F, g, X, cons(Y, Z)) => cons(F g Y X, map!fac62203(F, g, X, Z)) 

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 : 1 
    * 2 : 2 

This graph has the following strongly connected components:

  P_1:

    map!fac62201#(F, cons(X, Y)) =#> map!fac62201#(F, Y)   

  P_2:

    map!fac62202#(F, X, cons(Y, Z)) =#> map!fac62202#(F, X, Z)   

  P_3:

    map!fac62203#(F, g, X, cons(Y, Z)) =#> map!fac62203#(F, g, X, 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), (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 subterm criterion with the following projection function:

  nu(map!fac62203#) = 4 

Thus, we can orient the dependency pairs as follows:

  nu(map!fac62203#(F, g, X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(map!fac62203#(F, g, X, Z)) 

By [FuhKop19, Thm. 61], 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 subterm criterion with the following projection function:

  nu(map!fac62202#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(map!fac62202#(F, X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(map!fac62202#(F, X, Z)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, 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 subterm criterion with the following projection function:

  nu(map!fac62201#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(map!fac62201#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map!fac62201#(F, Y)) 

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.