We consider the system AotoYamada_05__026.

  Alphabet:

    comp : [c -> c * c -> c] --> c -> c 
    cons : [a * b] --> b 
    map : [a -> a * b] --> b 
    nil : [] --> b 
    twice : [c -> c] --> c -> c 

  Rules:

    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    comp(f, g) x => f (g x) 
    twice(f) => comp(f, f) 

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] map#(F, cons(X, Y)) =#> F(X)   
    1] map#(F, cons(X, Y)) =#> map#(F, Y)   
    2] comp(F, G) X =#> F(G X)   
    3] comp(F, G) X =#> G(X)   
    4] twice(F) X =#> comp(F, F) X   
    5] twice#(F) =#> comp#(F, F)   

  Rules R_0:

    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    comp(F, G) X => F (G X) 
    twice(F) => comp(F, F) 

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, 2, 3, 4, 5 
    * 1 : 0, 1 
    * 2 : 0, 1, 2, 3, 4, 5 
    * 3 : 0, 1, 2, 3, 4, 5 
    * 4 : 2, 3 
    * 5 :  

This graph has the following strongly connected components:

  P_1:

    map#(F, cons(X, Y)) =#> F(X)   
    map#(F, cons(X, Y)) =#> map#(F, Y)   
    comp(F, G) X =#> F(G X)   
    comp(F, G) X =#> G(X)   
    twice(F) X =#> comp(F, F) 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).

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).

The formative rules of (P_1, R_0) are R_1 ::=

  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative).

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 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:

  map#(F, cons(X, Y)) >? F(X) 
  map#(F, cons(X, Y)) >? map#(F, Y) 
  comp(F, G, X) >? F(G X) 
  comp(F, G, X) >? G(X) 
  twice(F, X) >? comp(F, F, X) 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( comp(F, G, X) ) = #argfun-comp#(F (G X), G X) 
  pi( twice(F, X) ) = #argfun-twice#(#argfun-comp#(F (F X), F X)) 

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

The following interpretation satisfies the requirements:

  #argfun-comp# = \y0y1.3 + max(y0, y1) 
  #argfun-twice# = \y0.3 + y0 
  comp = \G0G1y2.0 
  cons = \y0y1.3 + y0 + y1 
  map = \G0y1.y1 + 3y1G0(y1) 
  map# = \G0y1.3 + 2G0(y1) + 3y1G0(y1) 
  twice = \G0y1.0 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 3 + 3x1F0(3 + x1 + x2) + 3x2F0(3 + x1 + x2) + 11F0(3 + x1 + x2) > F0(x1) = [[_F0(_x1)]] 
  [[map#(_F0, cons(_x1, _x2))]] = 3 + 3x1F0(3 + x1 + x2) + 3x2F0(3 + x1 + x2) + 11F0(3 + x1 + x2) >= 3 + 2F0(x2) + 3x2F0(x2) = [[map#(_F0, _x2)]] 
  [[#argfun-comp#(_F0 (_F1 _x2), _F1 _x2)]] = 3 + max(x2, F0(max(x2, F1(x2))), F1(x2)) > F0(max(x2, F1(x2))) = [[_F0(_F1 _x2)]] 
  [[#argfun-comp#(_F0 (_F1 _x2), _F1 _x2)]] = 3 + max(x2, F0(max(x2, F1(x2))), F1(x2)) > F1(x2) = [[_F1(_x2)]] 
  [[#argfun-twice#(#argfun-comp#(_F0 (_F0 _x1), _F0 _x1))]] = 6 + max(x1, F0(x1), F0(max(x1, F0(x1)))) > 3 + max(x1, F0(x1), F0(max(x1, F0(x1)))) = [[#argfun-comp#(_F0 (_F0 _x1), _F0 _x1)]] 
  [[map(_F0, cons(_x1, _x2))]] = 3 + x1 + x2 + 3x1F0(3 + x1 + x2) + 3x2F0(3 + x1 + x2) + 9F0(3 + x1 + x2) >= 3 + x2 + 3x2F0(x2) + max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_1, minimal, formative) by (P_2, R_1, minimal, formative), where P_2 consists of:

  map#(F, cons(X, Y)) =#> map#(F, 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 apply the subterm criterion with the following projection function:

  nu(map#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_1, minimal, f) by ({}, R_1, minimal, 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.