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 rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

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

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[z]] = _|_ 

We choose Lex = {} and Mul = {minus, s}, and the following precedence: s > minus

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  minus(_|_, X) >= _|_ 
  minus(X, _|_) > X 
  minus(s(X), s(Y)) > minus(X, Y) 
  minus(X, X) >= _|_ 

With these choices, we have:

  1] minus(_|_, X) >= _|_  by (Bot) 

  2] minus(X, _|_) > X  because [3], by definition 
  3] minus*(X, _|_) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 

  5] minus(s(X), s(Y)) > minus(X, Y)  because [6], by definition 
  6] minus*(s(X), s(Y)) >= minus(X, Y)  because minus in Mul, [7] and [10], by (Stat) 
  7] s(X) > X  because [8], by definition 
  8] s*(X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] s(Y) >= Y  because [11], by (Star) 
  11] s*(Y) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 

  13] minus(X, X) >= _|_  by (Bot) 

We can thus remove the following rules:

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

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  minus(z, X) >? z 
  minus(X, X) >? z 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[z]] = _|_ 

We choose Lex = {} and Mul = {minus}, and the following precedence: minus

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  minus(_|_, X) > _|_ 
  minus(X, X) >= _|_ 

With these choices, we have:

  1] minus(_|_, X) > _|_  because [2], by definition 
  2] minus*(_|_, X) >= _|_  by (Bot) 

  3] minus(X, X) >= _|_  by (Bot) 

We can thus remove the following rules:

  minus(z, X) => z 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  minus(X, X) >? z 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[z]] = _|_ 

We choose Lex = {} and Mul = {minus}, and the following precedence: minus

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  minus(X, X) > _|_ 

With these choices, we have:

  1] minus(X, X) > _|_  because [2], by definition 
  2] minus*(X, X) >= _|_  by (Bot) 

We can thus remove the following rules:

  minus(X, X) => z 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

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