We consider the system Applicative_05__Ex6Recursor.

  Alphabet:

    0 : [] --> a 
    rec : [a -> b -> b * b * a] --> b 
    s : [a] --> a 

  Rules:

    rec(f, x, 0) => x 
    rec(f, x, s(y)) => f s(y) rec(f, x, 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] rec#(F, X, s(Y)) =#> rec#(F, X, Y)   

  Rules R_0:

    rec(F, X, 0) => X 
    rec(F, X, s(Y)) => F s(Y) rec(F, 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 will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  rec#(F, X, s(Y)) >? rec#(F, X, Y) 
  rec(F, X, 0) >= X 
  rec(F, X, s(Y)) >= F s(Y) rec(F, X, Y) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

We choose Lex = {} and Mul = {0, @_{o -> o -> o}, @_{o -> o}, rec, rec#, s}, and the following precedence: 0 > rec# > rec = s > @_{o -> o -> o} > @_{o -> o}

With these choices, we have:

  1] rec#(F, X, s(Y)) > rec#(F, X, Y)  because [2], by definition 
  2] rec#*(F, X, s(Y)) >= rec#(F, X, Y)  because rec# in Mul, [3], [4] and [5], by (Stat) 
  3] F >= F  by (Meta) 
  4] X >= X  by (Meta) 
  5] s(Y) > Y  because [6], by definition 
  6] s*(Y) >= Y  because [7], by (Select) 
  7] Y >= Y  by (Meta) 

  8] rec(F, X, 0) >= X  because [9], by (Star) 
  9] rec*(F, X, 0) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 

  11] rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because [12], by (Star) 
  12] rec*(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because rec > @_{o -> o}, [13] and [16], by (Copy) 
  13] rec*(F, X, s(Y)) >= @_{o -> o -> o}(F, s(Y))  because rec > @_{o -> o -> o}, [14] and [15], by (Copy) 
  14] rec*(F, X, s(Y)) >= F  because [3], by (Select) 
  15] rec*(F, X, s(Y)) >= s(Y)  because rec = s, rec in Mul and [5], by (Stat) 
  16] rec*(F, X, s(Y)) >= rec(F, X, Y)  because rec in Mul, [3], [4] and [5], by (Stat) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_0, 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.