We consider the system 468.

  Alphabet:

    0 : [] --> nat 
    add : [nat * nat] --> nat 
    rec : [nat -> nat -> nat * nat * nat] --> nat 
    s : [nat] --> nat 

  Rules:

    rec(/\x./\y.X(x, y), Y, 0) => Y 
    rec(/\x./\y.X(x, y), Y, s(Z)) => X(Z, rec(/\z./\u.X(z, u), Y, Z)) 
    add(X, Y) => rec(/\x./\y.s(y), X, Y) 
    add(X, 0) => X 
    add(X, s(Y)) => s(add(X, 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, formative):

  Dependency Pairs P_0:

    0] rec#(/\x./\y.X(x, y), Y, s(Z)) =#> rec#(/\z./\u.X(z, u), Y, Z) {X : 2}  
    1] add#(X, Y) =#> rec#(/\x./\y.s(y), X, Y)   
    2] add#(X, s(Y)) =#> add#(X, Y)   

  Rules R_0:

    rec(/\x./\y.X(x, y), Y, 0) => Y 
    rec(/\x./\y.X(x, y), Y, s(Z)) => X(Z, rec(/\z./\u.X(z, u), Y, Z)) 
    add(X, Y) => rec(/\x./\y.s(y), X, Y) 
    add(X, 0) => X 
    add(X, s(Y)) => s(add(X, Y)) 

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 
    * 2 : 1, 2 

This graph has the following strongly connected components:

  P_1:

    rec#(/\x./\y.X(x, y), Y, s(Z)) =#> rec#(/\z./\u.X(z, u), Y, Z) {X : 2}  

  P_2:

    add#(X, s(Y)) =#> add#(X, 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) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_2, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_2, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(add#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(add#(X, s(Y))) = s(Y) |> Y = nu(add#(X, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by ({}, R_0, computable, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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(rec#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(rec#(/\x./\y.X(x, y), Y, s(Z))) = s(Z) |> Z = nu(rec#(/\z./\u.X(z, u), 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.
[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.