We consider the system extrec.

  Alphabet:

    !plus : [nat * nat] --> nat 
    !times : [nat * nat] --> nat 
    0 : [] --> nat 
    rec : [nat * nat * nat -> nat -> nat] --> nat 
    s : [nat] --> nat 

  Rules:

    !plus(x, 0) => x 
    !plus(x, s(y)) => s(!plus(x, y)) 
    rec(0, x, f) => x 
    rec(s(x), y, f) => f x rec(x, y, f) 
    !times(x, y) => rec(y, 0, /\z./\u.!plus(x, u)) 

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] !plus#(X, s(Y)) =#> !plus#(X, Y)   
    1] rec#(s(X), Y, F) =#> rec#(X, Y, F)   
    2] !times#(X, Y) =#> rec#(Y, 0, /\x./\y.!plus(X, y))   
    3] !times#(X, Y) =#> !plus#(X, Z)   

  Rules R_0:

    !plus(X, 0) => X 
    !plus(X, s(Y)) => s(!plus(X, Y)) 
    rec(0, X, F) => X 
    rec(s(X), Y, F) => F X rec(X, Y, F) 
    !times(X, Y) => rec(Y, 0, /\x./\y.!plus(X, 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 
    * 1 : 1 
    * 2 : 1 
    * 3 : 0 

This graph has the following strongly connected components:

  P_1:

    !plus#(X, s(Y)) =#> !plus#(X, Y)   

  P_2:

    rec#(s(X), Y, F) =#> rec#(X, Y, F)   

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 will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_2, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  rec#(s(X), Y, F) >? rec#(X, Y, F) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  rec# = \y0y1G2.y0 
  s = \y0.3 + 2y0 

Using this interpretation, the requirements translate to:

  [[rec#(s(_x0), _x1, _F2)]] = 3 + 2x0 > x0 = [[rec#(_x0, _x1, _F2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_2, R_0) by ({}, R_0).  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 will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_1, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  !plus#(X, s(Y)) >? !plus#(X, Y) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  !plus# = \y0y1.y1 
  s = \y0.1 + 2y0 

Using this interpretation, the requirements translate to:

  [[!plus#(_x0, s(_x1))]] = 1 + 2x1 > x1 = [[!plus#(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_0) by ({}, R_0).  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.