We consider the system if.

  Alphabet:

    0 : [] --> nat 
    false : [] --> bool 
    h : [nat -> bool * nat -> nat * nat] --> nat 
    if : [bool * nat * nat] --> nat 
    s : [nat] --> nat 
    true : [] --> bool 

  Rules:

    if(true, x, y) => x 
    if(false, x, y) => y 
    h(f, g, 0) => 0 
    h(f, g, s(x)) => g h(f, g, if(f x, x, 0)) 

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] h#(F, G, s(X)) =#> h#(F, G, if(F X, X, 0))   
    1] h#(F, G, s(X)) =#> if#(F X, X, 0)   

  Rules R_0:

    if(true, X, Y) => X 
    if(false, X, Y) => Y 
    h(F, G, 0) => 0 
    h(F, G, s(X)) => G h(F, G, if(F X, X, 0)) 

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).

The formative rules of (P_0, R_0) are R_1 ::=

  if(true, X, Y) => X 
  if(false, X, Y) => Y 
  h(F, G, s(X)) => G h(F, G, if(F X, X, 0)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_0, R_0, static, formative) by (P_0, R_1, static, formative).

Thus, the original system is terminating if (P_0, R_1, static, formative) is finite.

We consider the dependency pair problem (P_0, R_1, 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:

  h#(F, G, s(X)) >? h#(F, G, if(F X, X, 0)) 
  h#(F, G, s(X)) >? if#(F X, X, 0) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  h(F, G, s(X)) >= G h(F, G, if(F X, X, 0)) 

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].

Argument functions:

  [[0]] = _|_ 
  [[if(x_1, x_2, x_3)]] = if(x_2, x_3) 

We choose Lex = {} and Mul = {@_{o -> o}, false, h, h#, if, if#, s, true}, and the following precedence: s > h > false > if > h# > if# > @_{o -> o} > true

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  h#(F, G, s(X)) > h#(F, G, if(X, _|_)) 
  h#(F, G, s(X)) >= if#(@_{o -> o}(F, X), X, _|_) 
  if(X, Y) >= X 
  if(X, Y) >= Y 
  h(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_))) 

With these choices, we have:

  1] h#(F, G, s(X)) > h#(F, G, if(X, _|_))  because [2], by definition 
  2] h#*(F, G, s(X)) >= h#(F, G, if(X, _|_))  because h# in Mul, [3], [4] and [5], by (Stat) 
  3] F >= F  by (Meta) 
  4] G >= G  by (Meta) 
  5] s(X) > if(X, _|_)  because [6], by definition 
  6] s*(X) >= if(X, _|_)  because s > if, [7] and [9], by (Copy) 
  7] s*(X) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 
  9] s*(X) >= _|_  by (Bot) 

  10] h#(F, G, s(X)) >= if#(@_{o -> o}(F, X), X, _|_)  because [11], by (Star) 
  11] h#*(F, G, s(X)) >= if#(@_{o -> o}(F, X), X, _|_)  because h# > if#, [12], [14] and [16], by (Copy) 
  12] h#*(F, G, s(X)) >= @_{o -> o}(F, X)  because h# > @_{o -> o}, [13] and [14], by (Copy) 
  13] h#*(F, G, s(X)) >= F  because [3], by (Select) 
  14] h#*(F, G, s(X)) >= X  because [15], by (Select) 
  15] s(X) >= X  because [7], by (Star) 
  16] h#*(F, G, s(X)) >= _|_  by (Bot) 

  17] if(X, Y) >= X  because [18], by (Star) 
  18] if*(X, Y) >= X  because [19], by (Select) 
  19] X >= X  by (Meta) 

  20] if(X, Y) >= Y  because [21], by (Star) 
  21] if*(X, Y) >= Y  because [22], by (Select) 
  22] Y >= Y  by (Meta) 

  23] h(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_)))  because [24], by (Star) 
  24] h*(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_)))  because h > @_{o -> o}, [25] and [26], by (Copy) 
  25] h*(F, G, s(X)) >= G  because [4], by (Select) 
  26] h*(F, G, s(X)) >= h(F, G, if(X, _|_))  because h in Mul, [3], [4] and [5], by (Stat) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_1, static, formative) by (P_1, R_1, static, formative), where P_1 consists of:

  h#(F, G, s(X)) =#> if#(F X, X, 0)   

Thus, the original system is terminating if (P_1, R_1, static, formative) is finite.

We consider the dependency pair problem (P_1, R_1, 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:

  h#(F, G, s(X)) >? if#(F X, X, 0) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  h(F, G, s(X)) >= G h(F, G, if(F X, X, 0)) 

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].

Argument functions:

  [[0]] = _|_ 
  [[if(x_1, x_2, x_3)]] = if(x_2, x_3) 
  [[if#(x_1, x_2, x_3)]] = x_3 

We choose Lex = {} and Mul = {@_{o -> o}, false, h, h#, if, s, true}, and the following precedence: false > h > @_{o -> o} > h# > s > if > true

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  h#(F, G, s(X)) > _|_ 
  if(X, Y) >= X 
  if(X, Y) >= Y 
  h(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_))) 

With these choices, we have:

  1] h#(F, G, s(X)) > _|_  because [2], by definition 
  2] h#*(F, G, s(X)) >= _|_  by (Bot) 

  3] if(X, Y) >= X  because [4], by (Star) 
  4] if*(X, Y) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] if(X, Y) >= Y  because [7], by (Star) 
  7] if*(X, Y) >= Y  because [8], by (Select) 
  8] Y >= Y  by (Meta) 

  9] h(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_)))  because [10], by (Star) 
  10] h*(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_)))  because h > @_{o -> o}, [11] and [13], by (Copy) 
  11] h*(F, G, s(X)) >= G  because [12], by (Select) 
  12] G >= G  by (Meta) 
  13] h*(F, G, s(X)) >= h(F, G, if(X, _|_))  because h in Mul, [14], [15] and [16], by (Stat) 
  14] F >= F  by (Meta) 
  15] G >= G  by (Meta) 
  16] s(X) > if(X, _|_)  because [17], by definition 
  17] s*(X) >= if(X, _|_)  because s > if, [18] and [20], by (Copy) 
  18] s*(X) >= X  because [19], by (Select) 
  19] X >= X  by (Meta) 
  20] s*(X) >= _|_  by (Bot) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_1) by ({}, R_1).  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.