We consider the system fuhkop12rta2.

  Alphabet:

    0 : [] --> nat 
    build : [nat] --> list 
    collapse : [list] --> nat 
    cons : [nat -> nat * list] --> list 
    diff : [nat * nat] --> nat 
    gcd : [nat * nat] --> nat 
    min : [nat * nat] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 

  Rules:

    min(x, 0) => 0 
    min(0, x) => 0 
    min(s(x), s(y)) => s(min(x, y)) 
    diff(x, 0) => x 
    diff(0, x) => x 
    diff(s(x), s(y)) => diff(x, y) 
    gcd(s(x), 0) => s(x) 
    gcd(0, s(x)) => s(x) 
    gcd(s(x), s(y)) => gcd(diff(x, y), s(min(x, y))) 
    collapse(nil) => 0 
    collapse(cons(f, x)) => f collapse(x) 
    build(0) => nil 
    build(s(x)) => cons(/\y.gcd(y, x), build(x)) 

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 [FuhKop19]).

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

  Dependency Pairs P_0:

    0] min#(s(X), s(Y)) =#> min#(X, Y)   
    1] diff#(s(X), s(Y)) =#> diff#(X, Y)   
    2] gcd#(s(X), s(Y)) =#> gcd#(diff(X, Y), s(min(X, Y)))   
    3] gcd#(s(X), s(Y)) =#> diff#(X, Y)   
    4] gcd#(s(X), s(Y)) =#> min#(X, Y)   
    5] collapse#(cons(F, X)) =#> collapse#(X)   
    6] build#(s(X)) =#> gcd#(Y, X)   
    7] build#(s(X)) =#> build#(X)   

  Rules R_0:

    min(X, 0) => 0 
    min(0, X) => 0 
    min(s(X), s(Y)) => s(min(X, Y)) 
    diff(X, 0) => X 
    diff(0, X) => X 
    diff(s(X), s(Y)) => diff(X, Y) 
    gcd(s(X), 0) => s(X) 
    gcd(0, s(X)) => s(X) 
    gcd(s(X), s(Y)) => gcd(diff(X, Y), s(min(X, Y))) 
    collapse(nil) => 0 
    collapse(cons(F, X)) => F collapse(X) 
    build(0) => nil 
    build(s(X)) => cons(/\x.gcd(x, X), build(X)) 

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

We consider the dependency pair problem (P_0, R_0, computable, 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, 4 
    * 3 : 1 
    * 4 : 0 
    * 5 : 5 
    * 6 : 2, 3, 4 
    * 7 : 6, 7 

This graph has the following strongly connected components:

  P_1:

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

  P_2:

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

  P_3:

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

  P_4:

    collapse#(cons(F, X)) =#> collapse#(X)   

  P_5:

    build#(s(X)) =#> build#(X)   

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), (P_4, R_0, m, f) and (P_5, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, computable, formative), (P_2, R_0, computable, formative), (P_3, R_0, computable, formative), (P_4, R_0, computable, formative) and (P_5, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_5, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(build#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(build#(s(X))) = s(X) |> X = nu(build#(X)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_5, R_0, computable, f) by ({}, R_0, computable, 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, computable, formative), (P_2, R_0, computable, formative), (P_3, R_0, computable, formative) and (P_4, R_0, computable, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(collapse#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(collapse#(cons(F, X))) = cons(F, X) |> X = nu(collapse#(X)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_4, R_0, computable, f) by ({}, R_0, computable, 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, computable, formative), (P_2, R_0, computable, formative) and (P_3, R_0, computable, formative) is finite.

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

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_3, R_0) are:

  min(X, 0) => 0 
  min(0, X) => 0 
  min(s(X), s(Y)) => s(min(X, Y)) 
  diff(X, 0) => X 
  diff(0, X) => X 
  diff(s(X), s(Y)) => diff(X, Y) 

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

  gcd#(s(X), s(Y)) >? gcd#(diff(X, Y), s(min(X, Y))) 
  min(X, 0) >= 0 
  min(0, X) >= 0 
  min(s(X), s(Y)) >= s(min(X, Y)) 
  diff(X, 0) >= X 
  diff(0, X) >= X 
  diff(s(X), s(Y)) >= diff(X, Y) 

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

The following interpretation satisfies the requirements:

  0 = 0 
  diff = \y0y1.y0 + y1 
  gcd# = \y0y1.y1 + 2y0 
  min = \y0y1.y0 
  s = \y0.1 + 2y0 

Using this interpretation, the requirements translate to:

  [[gcd#(s(_x0), s(_x1))]] = 3 + 2x1 + 4x0 > 1 + 2x1 + 4x0 = [[gcd#(diff(_x0, _x1), s(min(_x0, _x1)))]] 
  [[min(_x0, 0)]] = x0 >= 0 = [[0]] 
  [[min(0, _x0)]] = 0 >= 0 = [[0]] 
  [[min(s(_x0), s(_x1))]] = 1 + 2x0 >= 1 + 2x0 = [[s(min(_x0, _x1))]] 
  [[diff(_x0, 0)]] = x0 >= x0 = [[_x0]] 
  [[diff(0, _x0)]] = x0 >= x0 = [[_x0]] 
  [[diff(s(_x0), s(_x1))]] = 2 + 2x0 + 2x1 >= x0 + x1 = [[diff(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_3, 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 each of (P_1, R_0, computable, formative) and (P_2, R_0, computable, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(diff#) = 1 

Thus, we can orient the dependency pairs as follows:

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

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by ({}, R_0, computable, 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, computable, formative) is finite.

We consider the dependency pair problem (P_1, R_0, computable, 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 [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_0, computable, f) by ({}, R_0, computable, 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 +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[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.