We consider the system AotoYamada_05__002.

  Alphabet:

    cons : [b * c] --> c 
    false : [] --> a 
    filter : [b -> a * c] --> c 
    filtersub : [a * b -> a * c] --> c 
    nil : [] --> c 
    true : [] --> a 

  Rules:

    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filtersub(f x, f, cons(x, y)) 
    filtersub(true, f, cons(x, y)) => cons(x, filter(f, y)) 
    filtersub(false, f, cons(x, y)) => filter(f, 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 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] filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y))   
    1] filter#(F, cons(X, Y)) =#> F(X)   
    2] filtersub#(true, F, cons(X, Y)) =#> filter#(F, Y)   
    3] filtersub#(false, F, cons(X, Y)) =#> filter#(F, Y)   

  Rules R_0:

    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filtersub(F X, F, cons(X, Y)) 
    filtersub(true, F, cons(X, Y)) => cons(X, filter(F, Y)) 
    filtersub(false, F, cons(X, Y)) => filter(F, Y) 

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).

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

  filter(F, cons(X, Y)) => filtersub(F X, F, cons(X, Y)) 
  filtersub(true, F, cons(X, Y)) => cons(X, filter(F, Y)) 
  filtersub(false, F, cons(X, Y)) => filter(F, Y) 

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

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

We consider the dependency pair problem (P_0, 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:

  filter#(F, cons(X, Y)) >? filtersub#(F X, F, cons(X, Y)) 
  filter#(F, cons(X, Y)) >? F(X) 
  filtersub#(true, F, cons(X, Y)) >? filter#(F, Y) 
  filtersub#(false, F, cons(X, Y)) >? filter#(F, Y) 
  filter(F, cons(X, Y)) >= filtersub(F X, F, cons(X, Y)) 
  filtersub(true, F, cons(X, Y)) >= cons(X, filter(F, Y)) 
  filtersub(false, F, cons(X, Y)) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.y0 + 2y1 
  false = 3 
  filter = \G0y1.y1 
  filter# = \G0y1.1 + G0(y1) 
  filtersub = \y0G1y2.y2 
  filtersub# = \y0G1y2.1 + G1(y2) 
  true = 3 

Using this interpretation, the requirements translate to:

  [[filter#(_F0, cons(_x1, _x2))]] = 1 + F0(x1 + 2x2) >= 1 + F0(x1 + 2x2) = [[filtersub#(_F0 _x1, _F0, cons(_x1, _x2))]] 
  [[filter#(_F0, cons(_x1, _x2))]] = 1 + F0(x1 + 2x2) > F0(x1) = [[_F0(_x1)]] 
  [[filtersub#(true, _F0, cons(_x1, _x2))]] = 1 + F0(x1 + 2x2) >= 1 + F0(x2) = [[filter#(_F0, _x2)]] 
  [[filtersub#(false, _F0, cons(_x1, _x2))]] = 1 + F0(x1 + 2x2) >= 1 + F0(x2) = [[filter#(_F0, _x2)]] 
  [[filter(_F0, cons(_x1, _x2))]] = x1 + 2x2 >= x1 + 2x2 = [[filtersub(_F0 _x1, _F0, cons(_x1, _x2))]] 
  [[filtersub(true, _F0, cons(_x1, _x2))]] = x1 + 2x2 >= x1 + 2x2 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filtersub(false, _F0, cons(_x1, _x2))]] = x1 + 2x2 >= x2 = [[filter(_F0, _x2)]] 

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

  filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y))   
  filtersub#(true, F, cons(X, Y)) =#> filter#(F, Y)   
  filtersub#(false, F, cons(X, Y)) =#> filter#(F, Y)   

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

We consider the dependency pair problem (P_1, R_1, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(filter#) = 2 
  nu(filtersub#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(filter#(F, cons(X, Y))) = cons(X, Y) = cons(X, Y) = nu(filtersub#(F X, F, cons(X, Y))) 
  nu(filtersub#(true, F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) 
  nu(filtersub#(false, F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_1, minimal, f) by (P_2, R_1, minimal, f), where P_2 contains:

  filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y))   

Thus, the original system is terminating if (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_1, 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 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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.