We consider the system Applicative_05__Ex3Lists.

  Alphabet:

    append : [b * b] --> b 
    cons : [a * b] --> b 
    map : [a -> a * b] --> b 
    nil : [] --> b 

  Rules:

    append(nil, x) => x 
    append(cons(x, y), z) => cons(x, append(y, z)) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    append(append(x, y), z) => append(x, append(y, z)) 
    map(f, append(x, y)) => append(map(f, x), map(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 [Kop13]).

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

  Dependency Pairs P_0:

    0] append#(cons(X, Y), Z) =#> append#(Y, Z)   
    1] map#(F, cons(X, Y)) =#> map#(F, Y)   
    2] append#(append(X, Y), Z) =#> append#(X, append(Y, Z))   
    3] append#(append(X, Y), Z) =#> append#(Y, Z)   
    4] map#(F, append(X, Y)) =#> append#(map(F, X), map(F, Y))   
    5] map#(F, append(X, Y)) =#> map#(F, X)   
    6] map#(F, append(X, Y)) =#> map#(F, Y)   

  Rules R_0:

    append(nil, X) => X 
    append(cons(X, Y), Z) => cons(X, append(Y, Z)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    append(append(X, Y), Z) => append(X, append(Y, Z)) 
    map(F, append(X, Y)) => append(map(F, X), map(F, Y)) 

Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.

We consider the dependency pair problem (P_0, R_0, static, 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, 2, 3 
    * 1 : 1, 4, 5, 6 
    * 2 : 0, 2, 3 
    * 3 : 0, 2, 3 
    * 4 : 0, 2, 3 
    * 5 : 1, 4, 5, 6 
    * 6 : 1, 4, 5, 6 

This graph has the following strongly connected components:

  P_1:

    append#(cons(X, Y), Z) =#> append#(Y, Z)   
    append#(append(X, Y), Z) =#> append#(X, append(Y, Z))   
    append#(append(X, Y), Z) =#> append#(Y, Z)   

  P_2:

    map#(F, cons(X, Y)) =#> map#(F, Y)   
    map#(F, append(X, Y)) =#> map#(F, X)   
    map#(F, append(X, Y)) =#> map#(F, 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) and (P_2, R_0, m, f).

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

We consider the dependency pair problem (P_2, R_0, static, formative).

We apply the subterm criterion with the following projection function:

  nu(map#) = 2 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_2, R_0, static, f) by ({}, R_0, static, 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, static, formative) is finite.

We consider the dependency pair problem (P_1, R_0, static, formative).

We apply the subterm criterion with the following projection function:

  nu(append#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(append#(cons(X, Y), Z)) = cons(X, Y) |> Y = nu(append#(Y, Z)) 
  nu(append#(append(X, Y), Z)) = append(X, Y) |> X = nu(append#(X, append(Y, Z))) 
  nu(append#(append(X, Y), Z)) = append(X, Y) |> Y = nu(append#(Y, Z)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by ({}, R_0, static, 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 +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[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.