We consider the system h55.

  Alphabet:

    cons : [] --> a -> b -> b 
    foldr : [] --> (a -> b -> b) -> b -> b -> b 
    nil : [] --> b 

  Rules:

    foldr (/\x./\y.f x y) z nil => z 
    foldr (/\x./\y.f x y) z (cons u v) => f u (foldr (/\w./\x'.f w x') z v) 

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 * b] --> b 
    foldr : [a -> b -> b * b * b] --> b 
    nil : [] --> b 
    ~AP1 : [a -> b -> b * a] --> b -> b 

  Rules:

    foldr(/\x./\y.~AP1(F, x) y, X, nil) => X 
    foldr(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => ~AP1(F, Y) foldr(/\z./\u.~AP1(F, z) u, X, Z) 
    foldr(/\x./\y.cons(x, y), X, nil) => X 
    foldr(/\x./\y.cons(x, y), X, cons(Y, Z)) => cons(Y, foldr(/\z./\u.cons(z, u), X, Z)) 
    ~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 * b] --> b 
    foldr : [a -> b -> b * b * b] --> b 
    nil : [] --> b 

  Rules:

    foldr(/\x./\y.X(x, y), Y, nil) => Y 
    foldr(/\x./\y.X(x, y), Y, cons(Z, U)) => X(Z, foldr(/\z./\u.X(z, u), Y, U)) 

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] foldr#(/\x./\y.X(x, y), Y, cons(Z, U)) =#> foldr#(/\z./\u.X(z, u), Y, U) {X : 2}  

  Rules R_0:

    foldr(/\x./\y.X(x, y), Y, nil) => Y 
    foldr(/\x./\y.X(x, y), Y, cons(Z, U)) => X(Z, foldr(/\z./\u.X(z, u), Y, U)) 

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 apply the subterm criterion with the following projection function:

  nu(foldr#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(foldr#(/\x./\y.X(x, y), Y, cons(Z, U))) = cons(Z, U) |> U = nu(foldr#(/\z./\u.X(z, u), Y, U)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_0, 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.