We consider the system sort.

  Alphabet:

    0 : [] --> nat 
    ascending!6220sort : [list] --> list 
    cons : [nat * list] --> list 
    descending!6220sort : [list] --> list 
    insert : [nat * list * nat -> nat -> nat * nat -> nat -> nat] --> list 
    max : [nat * nat] --> nat 
    min : [nat * nat] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 
    sort : [list * nat -> nat -> nat * nat -> nat -> nat] --> list 

  Rules:

    max(0, x) => x 
    max(x, 0) => x 
    max(s(x), s(y)) => s(max(x, y)) 
    min(0, x) => 0 
    min(x, 0) => 0 
    min(s(x), s(y)) => s(min(x, y)) 
    insert(x, nil, f, g) => cons(x, nil) 
    insert(x, cons(y, z), f, g) => cons(f x y, insert(g x y, z, f, g)) 
    sort(nil, f, g) => nil 
    sort(cons(x, y), f, g) => insert(x, sort(y, f, g), f, g) 
    ascending!6220sort(x) => sort(x, /\y./\z.min(y, z), /\u./\v.max(u, v)) 
    descending!6220sort(x) => sort(x, /\y./\z.max(y, z), /\u./\v.min(u, v)) 

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] max#(s(X), s(Y)) =#> max#(X, Y)   
    1] min#(s(X), s(Y)) =#> min#(X, Y)   
    2] insert#(X, cons(Y, Z), F, G) =#> insert#(G X Y, Z, F, G)   
    3] sort#(cons(X, Y), F, G) =#> insert#(X, sort(Y, F, G), F, G)   
    4] sort#(cons(X, Y), F, G) =#> sort#(Y, F, G)   
    5] ascending!6220sort#(X) =#> sort#(X, /\x./\y.min(x, y), /\z./\u.max(z, u))   
    6] ascending!6220sort#(X) =#> min#(Y, Z)   
    7] ascending!6220sort#(X) =#> max#(Y, Z)   
    8] descending!6220sort#(X) =#> sort#(X, /\x./\y.max(x, y), /\z./\u.min(z, u))   
    9] descending!6220sort#(X) =#> max#(Y, Z)   
    10] descending!6220sort#(X) =#> min#(Y, Z)   

  Rules R_0:

    max(0, X) => X 
    max(X, 0) => X 
    max(s(X), s(Y)) => s(max(X, Y)) 
    min(0, X) => 0 
    min(X, 0) => 0 
    min(s(X), s(Y)) => s(min(X, Y)) 
    insert(X, nil, F, G) => cons(X, nil) 
    insert(X, cons(Y, Z), F, G) => cons(F X Y, insert(G X Y, Z, F, G)) 
    sort(nil, F, G) => nil 
    sort(cons(X, Y), F, G) => insert(X, sort(Y, F, G), F, G) 
    ascending!6220sort(X) => sort(X, /\x./\y.min(x, y), /\z./\u.max(z, u)) 
    descending!6220sort(X) => sort(X, /\x./\y.max(x, y), /\z./\u.min(z, u)) 

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 
    * 1 : 1 
    * 2 : 2 
    * 3 : 2 
    * 4 : 3, 4 
    * 5 : 3, 4 
    * 6 : 1 
    * 7 : 0 
    * 8 : 3, 4 
    * 9 : 0 
    * 10 : 1 

This graph has the following strongly connected components:

  P_1:

    max#(s(X), s(Y)) =#> max#(X, Y)   

  P_2:

    min#(s(X), s(Y)) =#> min#(X, Y)   

  P_3:

    insert#(X, cons(Y, Z), F, G) =#> insert#(G X Y, Z, F, G)   

  P_4:

    sort#(cons(X, Y), F, G) =#> sort#(Y, F, G)   

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), (P_3, R_0, m, f) and (P_4, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, static, formative), (P_2, R_0, static, formative), (P_3, R_0, static, formative) and (P_4, R_0, static, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(sort#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(sort#(cons(X, Y), F, G)) = cons(X, Y) |> Y = nu(sort#(Y, F, G)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_4, 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 each of (P_1, R_0, static, formative), (P_2, R_0, static, formative) and (P_3, R_0, static, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(insert#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(insert#(X, cons(Y, Z), F, G)) = cons(Y, Z) |> Z = nu(insert#(G X Y, Z, F, G)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_3, 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 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(min#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(min#(s(X), s(Y))) = s(X) |> X = nu(min#(X, 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(max#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(max#(s(X), s(Y))) = s(X) |> X = nu(max#(X, Y)) 

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.