We consider the system average.

  Alphabet:

    0 : [] --> nat 
    apply : [nat * nat] --> nat 
    avg : [nat * nat] --> nat 
    check : [nat] --> nat 
    fun : [nat -> nat] --> nat 
    s : [nat] --> nat 

  Rules:

    avg(s(x), y) => avg(x, s(y)) 
    avg(x, s(s(s(y)))) => s(avg(s(x), y)) 
    avg(0, 0) => 0 
    avg(0, s(0)) => 0 
    avg(0, s(s(0))) => s(0) 
    apply(fun(f), x) => f check(x) 
    check(s(x)) => s(check(x)) 
    check(0) => 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 dynamic dependency pairs.

After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] avg#(s(X), Y) =#> avg#(X, s(Y))   
    1] avg#(X, s(s(s(Y)))) =#> avg#(s(X), Y)   
    2] apply#(fun(F), X) =#> F(check(X))   
    3] apply#(fun(F), X) =#> check#(X)   
    4] check#(s(X)) =#> check#(X)   

  Rules R_0:

    avg(s(X), Y) => avg(X, s(Y)) 
    avg(X, s(s(s(Y)))) => s(avg(s(X), Y)) 
    avg(0, 0) => 0 
    avg(0, s(0)) => 0 
    avg(0, s(s(0))) => s(0) 
    apply(fun(F), X) => F check(X) 
    check(s(X)) => s(check(X)) 
    check(0) => 0 

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

We consider the dependency pair problem (P_0, R_0, minimal, 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 : 0, 1 
    * 2 : 0, 1, 2, 3, 4 
    * 3 : 4 
    * 4 : 4 

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    apply#(fun(F), X) =#> F(check(X))   

  P_3:

    check#(s(X)) =#> check#(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) and (P_3, R_0, m, f).

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

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

We apply the subterm criterion with the following projection function:

  nu(check#) = 1 

Thus, we can orient the dependency pairs as follows:

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

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

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

The formative rules of (P_2, R_0) are R_1 ::=

  apply(fun(F), X) => F check(X) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_2, R_0, minimal, formative) by (P_2, R_1, minimal, formative).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

  apply#(fun(F), X) >? F(check(X)) 
  apply(fun(F), X) >= F check(X) 

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

The following interpretation satisfies the requirements:

  apply = \y0y1.3 + 3y0 
  apply# = \y0y1.3 + y0 
  check = \y0.0 
  fun = \G0.3 + G0(0) 

Using this interpretation, the requirements translate to:

  [[apply#(fun(_F0), _x1)]] = 6 + F0(0) > F0(0) = [[_F0(check(_x1))]] 
  [[apply(fun(_F0), _x1)]] = 12 + 3F0(0) >= F0(0) = [[_F0 check(_x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_2, R_1) by ({}, R_1).  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, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, 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:

  avg#(s(X), Y) >? avg#(X, s(Y)) 
  avg#(X, s(s(s(Y)))) >? avg#(s(X), Y) 

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

The following interpretation satisfies the requirements:

  avg# = \y0y1.y1 + 2y0 
  s = \y0.2 + y0 

Using this interpretation, the requirements translate to:

  [[avg#(s(_x0), _x1)]] = 4 + x1 + 2x0 > 2 + x1 + 2x0 = [[avg#(_x0, s(_x1))]] 
  [[avg#(_x0, s(s(s(_x1))))]] = 6 + x1 + 2x0 > 4 + x1 + 2x0 = [[avg#(s(_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 +++

[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.