We consider the system h17.

  Alphabet:

    cons : [] --> a -> alist -> alist 
    map : [] --> (a -> a) -> alist -> alist 
    nil : [] --> alist 
    o : [] --> (a -> a) -> (a -> a) -> a -> a 

  Rules:

    map (/\x.f x) nil => nil 
    map (/\x.f x) (cons y z) => cons (f y) (map (/\u.f u) z) 
    map (/\x.f x) (map (/\y.g y) z) => map (/\u.o (/\v.f v) (/\w.g w) u) z 
    o (/\x.f x) (/\y.g y) z => f (g z) 

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:

    cons : [a * alist] --> alist 
    map : [a -> a * alist] --> alist 
    nil : [] --> alist 
    o : [a -> a * a -> a * a] --> a 
    ~AP1 : [a -> a * a] --> a 

  Rules:

    map(/\x.~AP1(F, x), nil) => nil 
    map(/\x.~AP1(F, x), cons(X, Y)) => cons(~AP1(F, X), map(/\y.~AP1(F, y), Y)) 
    map(/\x.~AP1(F, x), map(/\y.~AP1(G, y), X)) => map(/\z.o(/\u.~AP1(F, u), /\v.~AP1(G, v), z), X) 
    o(/\x.~AP1(F, x), /\y.~AP1(G, y), X) => ~AP1(F, ~AP1(G, X)) 
    map(/\x.o(F, G, x), nil) => nil 
    map(/\x.o(F, G, x), cons(X, Y)) => cons(o(F, G, X), map(/\y.o(F, G, y), Y)) 
    map(/\x.o(F, G, x), map(/\y.~AP1(H, y), X)) => map(/\z.o(/\u.o(F, G, u), /\v.~AP1(H, v), z), X) 
    map(/\x.~AP1(F, x), map(/\y.o(G, H, y), X)) => map(/\z.o(/\u.~AP1(F, u), /\v.o(G, H, v), z), X) 
    o(/\x.o(F, G, x), /\y.~AP1(H, y), X) => o(F, G, ~AP1(H, X)) 
    o(/\x.~AP1(F, x), /\y.o(G, H, y), X) => ~AP1(F, o(G, H, X)) 
    map(/\x.o(F, G, x), map(/\y.o(H, I, y), X)) => map(/\z.o(/\u.o(F, G, u), /\v.o(H, I, v), z), X) 
    o(/\x.o(F, G, x), /\y.o(H, I, y), X) => o(F, G, o(H, I, X)) 
    ~AP1(F, X) => F X 

Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  Additionally, we can remove some (now-)redundant rules.  This gives:

  Alphabet:

    cons : [a * alist] --> alist 
    map : [a -> a * alist] --> alist 
    nil : [] --> alist 
    o : [a -> a * a -> a * a] --> a 

  Rules:

    map(/\x.X(x), nil) => nil 
    map(/\x.X(x), cons(Y, Z)) => cons(X(Y), map(/\y.X(y), Z)) 
    map(/\x.X(x), map(/\y.Y(y), Z)) => map(/\z.o(/\u.X(u), /\v.Y(v), z), Z) 
    o(/\x.X(x), /\y.Y(y), Z) => X(Y(Z)) 

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] map#(/\x.X(x), cons(Y, Z)) =#> map#(/\y.X(y), Z)   
    1] map#(/\x.X(x), map(/\y.Y(y), Z)) =#> map#(/\z.o(/\u.X(u), /\v.Y(v), z), Z)   
    2] map#(/\x.X(x), map(/\y.Y(y), Z)) =#> o#(/\z.X(z), /\u.Y(u), U)   

  Rules R_0:

    map(/\x.X(x), nil) => nil 
    map(/\x.X(x), cons(Y, Z)) => cons(X(Y), map(/\y.X(y), Z)) 
    map(/\x.X(x), map(/\y.Y(y), Z)) => map(/\z.o(/\u.X(u), /\v.Y(v), z), Z) 
    o(/\x.X(x), /\y.Y(y), Z) => X(Y(Z)) 

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 
    * 1 : 0, 1, 2 
    * 2 :  

This graph has the following strongly connected components:

  P_1:

    map#(/\x.X(x), cons(Y, Z)) =#> map#(/\y.X(y), Z)   
    map#(/\x.X(x), map(/\y.Y(y), Z)) =#> map#(/\z.o(/\u.X(u), /\v.Y(v), z), Z)   

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, 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(map#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(map#(/\x.X(x), cons(Y, Z))) = cons(Y, Z) |> Z = nu(map#(/\y.X(y), Z)) 
  nu(map#(/\x.X(x), map(/\y.Y(y), Z))) = map(/\z.Y(z), Z) |> Z = nu(map#(/\y.o(/\u.X(u), /\v.Y(v), y), Z)) 

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.
[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.