We consider the system Applicative_first_order_05__31.

  Alphabet:

    !faccolon : [] --> b -> b -> b 
    !facplus : [] --> b -> b -> b 
    a : [] --> b 
    cons : [] --> d -> e -> e 
    false : [] --> c 
    filter : [] --> (d -> c) -> e -> e 
    filter2 : [] --> c -> (d -> c) -> d -> e -> e 
    g : [] --> b -> a -> b 
    map : [] --> (d -> d) -> e -> e 
    nil : [] --> e 
    true : [] --> c 

  Rules:

    !faccolon (!faccolon x y) z => !faccolon x (!faccolon y z) 
    !faccolon (!facplus x y) z => !facplus (!faccolon x z) (!faccolon y z) 
    !faccolon x (!facplus y (f z)) => !faccolon (g x z) (!facplus y a) 
    map f nil => nil 
    map f (cons x y) => cons (f x) (map f y) 
    filter f nil => nil 
    filter f (cons x y) => filter2 (f x) f x y 
    filter2 true f x y => cons x (filter f y) 
    filter2 false f x y => filter f y 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    !faccolon : [b * b] --> b 
    !facplus : [b * b] --> b 
    a : [] --> b 
    cons : [d * e] --> e 
    false : [] --> c 
    filter : [d -> c * e] --> e 
    filter2 : [c * d -> c * d * e] --> e 
    g : [b] --> a -> b 
    map : [d -> d * e] --> e 
    nil : [] --> e 
    true : [] --> c 
    ~AP1 : [a -> b * a] --> b 

  Rules:

    !faccolon(!faccolon(X, Y), Z) => !faccolon(X, !faccolon(Y, Z)) 
    !faccolon(!facplus(X, Y), Z) => !facplus(!faccolon(X, Z), !faccolon(Y, Z)) 
    !faccolon(X, !facplus(Y, ~AP1(F, Z))) => !faccolon(~AP1(g(X), Z), !facplus(Y, a)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, Y) 
    ~AP1(F, X) => F X 

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]).

In order to do so, we start by eta-expanding the system, which gives:

  !faccolon(!faccolon(X, Y), Z) => !faccolon(X, !faccolon(Y, Z)) 
  !faccolon(!facplus(X, Y), Z) => !facplus(!faccolon(X, Z), !faccolon(Y, Z)) 
  !faccolon(X, !facplus(Y, ~AP1(F, Z))) => !faccolon(~AP1(/\x.g(X, x), Z), !facplus(Y, a)) 
  map(F, nil) => nil 
  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
  filter(F, nil) => nil 
  filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) => filter(F, Y) 
  ~AP1(F, X) => F X 

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

  Dependency Pairs P_0:

    0] !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(X, !faccolon(Y, Z))   
    1] !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(Y, Z)   
    2] !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(X, Z)   
    3] !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(Y, Z)   
    4] !faccolon#(X, !facplus(Y, ~AP1(F, Z))) =#> !faccolon#(~AP1(/\x.g(X, x), Z), !facplus(Y, a))   
    5] !faccolon#(X, !facplus(Y, ~AP1(F, Z))) =#> ~AP1#(/\x.g(X, x), Z)   
    6] map#(F, cons(X, Y)) =#> map#(F, Y)   
    7] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    8] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    9] filter2#(false, F, X, Y) =#> filter#(F, Y)   

  Rules R_0:

    !faccolon(!faccolon(X, Y), Z) => !faccolon(X, !faccolon(Y, Z)) 
    !faccolon(!facplus(X, Y), Z) => !facplus(!faccolon(X, Z), !faccolon(Y, Z)) 
    !faccolon(X, !facplus(Y, ~AP1(F, Z))) => !faccolon(~AP1(/\x.g(X, x), Z), !facplus(Y, a)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, Y) 
    ~AP1(F, X) => F 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, 2, 3, 4, 5 
    * 1 : 0, 1, 2, 3, 4, 5 
    * 2 : 0, 1, 2, 3, 4, 5 
    * 3 : 0, 1, 2, 3, 4, 5 
    * 4 : 0, 1, 2, 3 
    * 5 :  
    * 6 : 6 
    * 7 : 8, 9 
    * 8 : 7 
    * 9 : 7 

This graph has the following strongly connected components:

  P_1:

    !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(X, !faccolon(Y, Z))   
    !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(Y, Z)   
    !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(X, Z)   
    !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(Y, Z)   
    !faccolon#(X, !facplus(Y, ~AP1(F, Z))) =#> !faccolon#(~AP1(/\x.g(X, x), Z), !facplus(Y, a))   

  P_2:

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

  P_3:

    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter2#(true, F, X, Y) =#> filter#(F, Y)   
    filter2#(false, F, X, Y) =#> filter#(F, Y)   

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, 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 apply the subterm criterion with the following projection function:

  nu(filter2#) = 4 
  nu(filter#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_3, R_0, computable, f) by (P_4, R_0, computable, f), where P_4 contains:

  filter2#(true, F, X, Y) =#> filter#(F, Y)   
  filter2#(false, F, X, Y) =#> filter#(F, Y)   

Thus, the original system is terminating if each of (P_1, R_0, computable, formative), (P_2, 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 place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  
    * 1 :  

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

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(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_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).

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

  !faccolon(!faccolon(X, Y), Z) => !faccolon(X, !faccolon(Y, Z)) 
  !faccolon(!facplus(X, Y), Z) => !facplus(!faccolon(X, Z), !faccolon(Y, Z)) 
  !faccolon(X, !facplus(Y, ~AP1(F, Z))) => !faccolon(~AP1(/\x.g(X, x), Z), !facplus(Y, a)) 
  ~AP1(F, X) => F X 

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

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

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

  !faccolon#(!faccolon(X, Y), Z) >? !faccolon#(X, !faccolon(Y, Z)) 
  !faccolon#(!faccolon(X, Y), Z) >? !faccolon#(Y, Z) 
  !faccolon#(!facplus(X, Y), Z) >? !faccolon#(X, Z) 
  !faccolon#(!facplus(X, Y), Z) >? !faccolon#(Y, Z) 
  !faccolon#(X, !facplus(Y, ~AP1(F, Z))) >? !faccolon#(~AP1(/\x.g(X, x), Z), !facplus(Y, a)) 
  !faccolon(!faccolon(X, Y), Z) >= !faccolon(X, !faccolon(Y, Z)) 
  !faccolon(!facplus(X, Y), Z) >= !facplus(!faccolon(X, Z), !faccolon(Y, Z)) 
  !faccolon(X, !facplus(Y, ~AP1(F, Z))) >= !faccolon(~AP1(/\x.g(X, x), Z), !facplus(Y, a)) 
  ~AP1(F, X) >= F X 

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

The following interpretation satisfies the requirements:

  !faccolon = \y0y1.0 
  !faccolon# = \y0y1.2y1 
  !facplus = \y0y1.y1 
  a = 0 
  g = \y0y1.0 
  ~AP1 = \G0y1.2 + G0(y1) 

Using this interpretation, the requirements translate to:

  [[!faccolon#(!faccolon(_x0, _x1), _x2)]] = 2x2 >= 0 = [[!faccolon#(_x0, !faccolon(_x1, _x2))]] 
  [[!faccolon#(!faccolon(_x0, _x1), _x2)]] = 2x2 >= 2x2 = [[!faccolon#(_x1, _x2)]] 
  [[!faccolon#(!facplus(_x0, _x1), _x2)]] = 2x2 >= 2x2 = [[!faccolon#(_x0, _x2)]] 
  [[!faccolon#(!facplus(_x0, _x1), _x2)]] = 2x2 >= 2x2 = [[!faccolon#(_x1, _x2)]] 
  [[!faccolon#(_x0, !facplus(_x1, ~AP1(_F2, _x3)))]] = 4 + 2F2(x3) > 0 = [[!faccolon#(~AP1(/\x.g(_x0, x), _x3), !facplus(_x1, a))]] 
  [[!faccolon(!faccolon(_x0, _x1), _x2)]] = 0 >= 0 = [[!faccolon(_x0, !faccolon(_x1, _x2))]] 
  [[!faccolon(!facplus(_x0, _x1), _x2)]] = 0 >= 0 = [[!facplus(!faccolon(_x0, _x2), !faccolon(_x1, _x2))]] 
  [[!faccolon(_x0, !facplus(_x1, ~AP1(_F2, _x3)))]] = 0 >= 0 = [[!faccolon(~AP1(/\x.g(_x0, x), _x3), !facplus(_x1, a))]] 
  [[~AP1(_F0, _x1)]] = 2 + F0(x1) >= F0(x1) = [[_F0 _x1]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_1, computable, formative) by (P_5, R_1, computable, formative), where P_5 consists of:

  !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(X, !faccolon(Y, Z))   
  !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(Y, Z)   
  !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(X, Z)   
  !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(Y, Z)   

Thus, the original system is terminating if (P_5, R_1, computable, formative) is finite.

We consider the dependency pair problem (P_5, R_1, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(!faccolon#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(!faccolon#(!faccolon(X, Y), Z)) = !faccolon(X, Y) |> X = nu(!faccolon#(X, !faccolon(Y, Z))) 
  nu(!faccolon#(!faccolon(X, Y), Z)) = !faccolon(X, Y) |> Y = nu(!faccolon#(Y, Z)) 
  nu(!faccolon#(!facplus(X, Y), Z)) = !facplus(X, Y) |> X = nu(!faccolon#(X, Z)) 
  nu(!faccolon#(!facplus(X, Y), Z)) = !facplus(X, Y) |> Y = nu(!faccolon#(Y, Z)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_5, R_1, computable, f) by ({}, R_1, 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.
[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[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.