We consider the system AotoYamada_05__022.

  Alphabet:

    cons : [b * c] --> c 
    leaf : [a] --> b 
    mapt : [a -> a * b] --> b 
    maptlist : [a -> a * c] --> c 
    nil : [] --> c 
    node : [c] --> b 

  Rules:

    mapt(f, leaf(x)) => leaf(f x) 
    mapt(f, node(x)) => node(maptlist(f, x)) 
    maptlist(f, nil) => nil 
    maptlist(f, cons(x, y)) => cons(mapt(f, x), maptlist(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 [FuhKop19]).

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

  Dependency Pairs P_0:

    0] mapt#(F, node(X)) =#> maptlist#(F, X)   
    1] maptlist#(F, cons(X, Y)) =#> mapt#(F, X)   
    2] maptlist#(F, cons(X, Y)) =#> maptlist#(F, Y)   

  Rules R_0:

    mapt(F, leaf(X)) => leaf(F X) 
    mapt(F, node(X)) => node(maptlist(F, X)) 
    maptlist(F, nil) => nil 
    maptlist(F, cons(X, Y)) => cons(mapt(F, X), maptlist(F, 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 subterm criterion with the following projection function:

  nu(mapt#) = 2 
  nu(maptlist#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(mapt#(F, node(X))) = node(X) |> X = nu(maptlist#(F, X)) 
  nu(maptlist#(F, cons(X, Y))) = cons(X, Y) |> X = nu(mapt#(F, X)) 
  nu(maptlist#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(maptlist#(F, Y)) 

By [FuhKop19, Thm. 61], 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.