We consider the system SystemT.

  Alphabet:

    app : [] --> arrab -> a -> b 
    lam : [] --> (a -> b) -> arrab 
    rec : [] --> Nat -> a -> (Nat -> a -> a) -> a 
    succ : [] --> Nat -> Nat 
    zero : [] --> Nat 

  Rules:

    app (lam (/\x.f x)) y => f y 
    lam (/\x.app y x) => y 
    rec zero x (/\y.f y) => x 
    rec (succ x) y (/\z.f z) => f x (rec x y (/\u.f u)) 

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 
    lam : [a -> b] --> arrab 
    rec : [Nat * a * Nat -> a -> a] --> a 
    succ : [Nat] --> Nat 
    zero : [] --> Nat 
    ~AP1 : [a -> b * a] --> b 
    ~AP2 : [Nat -> a -> a * Nat] --> a -> a 

  Rules:

    app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) 
    lam(/\x.app(X, x)) => X 
    rec(zero, X, /\x.~AP2(F, x)) => X 
    rec(succ(X), Y, /\x.~AP2(F, x)) => ~AP2(F, X) rec(X, Y, /\y.~AP2(F, y)) 
    app(lam(/\x.app(X, x)), Y) => app(X, Y) 
    ~AP1(F, X) => F X 
    ~AP2(F, X) => F X 

Symbols ~AP1, and ~AP2 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 
    lam : [a -> b] --> arrab 
    rec : [Nat * a * Nat -> a -> a] --> a 
    succ : [Nat] --> Nat 
    zero : [] --> Nat 

  Rules:

    app(lam(/\x.X(x)), Y) => X(Y) 
    lam(/\x.app(X, x)) => X 
    rec(zero, X, /\x.F(x)) => X 
    rec(succ(X), Y, /\x.F(x)) => F(X) rec(X, Y, /\y.F(y)) 

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:

    0] rec#(succ(X), Y, /\x.F(x)) =#> rec#(X, Y, /\y.F(y))   

  Rules R_0:

    app(lam(/\x.X(x)), Y) => X(Y) 
    lam(/\x.app(X, x)) => X 
    rec(zero, X, /\x.F(x)) => X 
    rec(succ(X), Y, /\x.F(x)) => F(X) rec(X, Y, /\y.F(y)) 

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 apply the subterm criterion with the following projection function:

  nu(rec#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(rec#(succ(X), Y, /\x.F(x))) = succ(X) |> X = nu(rec#(X, Y, /\y.F(y))) 

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.