We consider the system h02.

  Alphabet:

    0 : [] --> c 
    1 : [] --> c 
    add : [] --> c -> a -> c 
    cons : [] --> a -> b -> b 
    fold : [] --> (c -> a -> c) -> b -> c -> c 
    mul : [] --> c -> a -> c 
    nil : [] --> b 
    prod : [] --> b -> c 
    sum : [] --> b -> c 

  Rules:

    fold (/\x./\y.f x y) nil z => z 
    fold (/\x./\y.f x y) (cons z u) v => fold (/\w./\x'.f w x') u (f v z) 
    sum x => fold (/\y./\z.add y z) x 0 
    fold (/\x./\y.mul x y) z 1 => prod 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:

    0 : [] --> c 
    1 : [] --> c 
    add : [] --> c -> a -> c 
    cons : [a * b] --> b 
    fold : [c -> a -> c * b * c] --> c 
    mul : [c * a] --> c 
    nil : [] --> b 
    prod : [b] --> c 
    sum : [b] --> c 
    ~AP1 : [c -> a -> c * c] --> a -> c 

  Rules:

    fold(/\x./\y.~AP1(F, x) y, nil, X) => X 
    fold(/\x./\y.~AP1(F, x) y, cons(X, Y), Z) => fold(/\z./\u.~AP1(F, z) u, Y, ~AP1(F, Z) X) 
    sum(X) => fold(/\x./\y.~AP1(add, x) y, X, 0) 
    fold(/\x./\y.mul(x, y), X, 1) => prod(X) 
    fold(/\x./\y.mul(x, y), nil, X) => X 
    fold(/\x./\y.mul(x, y), cons(X, Y), Z) => fold(/\z./\u.mul(z, u), Y, mul(Z, 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:

    0 : [] --> c 
    1 : [] --> c 
    add : [c * a] --> c 
    cons : [a * b] --> b 
    fold : [c -> a -> c * b * c] --> c 
    mul : [c * a] --> c 
    nil : [] --> b 
    prod : [b] --> c 
    sum : [b] --> c 

  Rules:

    fold(/\x./\y.X(x, y), nil, Y) => Y 
    fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) 
    sum(X) => fold(/\x./\y.add(x, y), X, 0) 
    fold(/\x./\y.mul(x, y), X, 1) => prod(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]).

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

  Dependency Pairs P_0:

    0] fold#(/\x./\y.X(x, y), cons(Y, Z), U) =#> fold#(/\z./\u.X(z, u), Z, X(U, Y))   
    1] sum#(X) =#> fold#(/\x./\y.add(x, y), X, 0)   

  Rules R_0:

    fold(/\x./\y.X(x, y), nil, Y) => Y 
    fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) 
    sum(X) => fold(/\x./\y.add(x, y), X, 0) 
    fold(/\x./\y.mul(x, y), X, 1) => prod(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 : 0 

This graph has the following strongly connected components:

  P_1:

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

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(fold#) = 2 

Thus, we can orient the dependency pairs as follows:

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

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.