We consider the system lambda_sum.

  Alphabet:

    app : [] --> arrAB -> A -> B 
    case : [] --> SAB -> (A -> C) -> (B -> C) -> C 
    inl : [] --> A -> SAB 
    inr : [] --> B -> SAB 
    lam : [] --> (A -> B) -> arrAB 

  Rules:

    app (lam (/\x.f x)) y => f y 
    lam (/\x.app y x) => y 
    case (inl x) (/\y.f y) (/\z.g z) => f x 
    case (inr x) (/\y.f y) (/\z.g z) => g x 

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:

    app : [arrAB * A] --> B 
    case : [SAB * A -> C * B -> C] --> C 
    inl : [A] --> SAB 
    inr : [B] --> SAB 
    lam : [A -> B] --> arrAB 
    ~AP1 : [A -> B * A] --> B 
    ~AP2 : [A -> C * A] --> C 
    ~AP3 : [B -> C * B] --> C 

  Rules:

    app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) 
    lam(/\x.app(X, x)) => X 
    case(inl(X), /\x.~AP2(F, x), /\y.~AP3(G, y)) => ~AP2(F, X) 
    case(inr(X), /\x.~AP2(F, x), /\y.~AP3(G, y)) => ~AP3(G, X) 
    app(lam(/\x.app(X, x)), Y) => app(X, Y) 
    ~AP1(F, X) => F X 
    ~AP2(F, X) => F X 
    ~AP3(F, X) => F X 

Symbols ~AP1, ~AP2, and ~AP3 are encodings for application that are only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing them.

  Additionally, we can remove some (now-)redundant rules.  This gives:

  Alphabet:

    app : [arrAB * A] --> B 
    case : [SAB * A -> C * B -> C] --> C 
    inl : [A] --> SAB 
    inr : [B] --> SAB 
    lam : [A -> B] --> arrAB 

  Rules:

    app(lam(/\x.X(x)), Y) => X(Y) 
    lam(/\x.app(X, x)) => X 
    case(inl(X), /\x.Y(x), /\y.Z(y)) => Y(X) 
    case(inr(X), /\x.Y(x), /\y.Z(y)) => Z(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, all):

  Dependency Pairs P_0:


  Rules R_0:

    app(lam(/\x.X(x)), Y) => X(Y) 
    lam(/\x.app(X, x)) => X 
    case(inl(X), /\x.Y(x), /\y.Z(y)) => Y(X) 
    case(inr(X), /\x.Y(x), /\y.Z(y)) => Z(X) 

Thus, the original system is terminating if (P_0, R_0, computable, all) is finite.

We consider the dependency pair problem (P_0, R_0, computable, all).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:


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

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.