We consider the system 03minus.

  Alphabet:

    minus : [N * N] --> N 
    s : [N] --> N 
    z : [] --> N 

  Rules:

    minus(z, x) => z 
    minus(x, z) => x 
    minus(s(x), s(y)) => minus(x, y) 
    minus(x, x) => z 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

We thus obtain the following dependency pair problem (P_0, R_0, minimal, all):

  Dependency Pairs P_0:

    0] minus#(s(X), s(Y)) =#> minus#(X, Y)   

  Rules R_0:

    minus(z, X) => z 
    minus(X, z) => X 
    minus(s(X), s(Y)) => minus(X, Y) 
    minus(X, X) => z 

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

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

We apply the subterm criterion with the following projection function:

  nu(minus#) = 1 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_0, R_0, minimal, f) by ({}, R_0, minimal, 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 +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.