We consider the system AotoYamada_05__023.

  Alphabet:

    0 : [] --> a 
    id : [] --> a -> a 
    plus : [a] --> a -> a 
    s : [a] --> a 

  Rules:

    id x => x 
    plus(0) => id 
    plus(s(x)) y => s(plus(x) y) 

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]):

  id X >? X 
  plus(0) >? id 
  plus(s(X)) Y >? s(plus(X) Y) 

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

Argument functions:

  [[id]] = _|_ 
  [[s(x_1)]] = x_1 

We choose Lex = {} and Mul = {0, @_{o -> o}, plus}, and the following precedence: 0 > @_{o -> o} > plus

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

  @_{o -> o}(_|_, X) > X 
  plus(0) > _|_ 
  @_{o -> o}(plus(X), Y) >= @_{o -> o}(plus(X), Y) 

With these choices, we have:

  1] @_{o -> o}(_|_, X) > X  because [2], by definition 
  2] @_{o -> o}*(_|_, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] plus(0) > _|_  because [5], by definition 
  5] plus*(0) >= _|_  by (Bot) 

  6] @_{o -> o}(plus(X), Y) >= @_{o -> o}(plus(X), Y)  because @_{o -> o} in Mul, [7] and [9], by (Fun) 
  7] plus(X) >= plus(X)  because plus in Mul and [8], by (Fun) 
  8] X >= X  by (Meta) 
  9] Y >= Y  by (Meta) 

We can thus remove the following rules:

  id X => X 
  plus(0) => id 

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

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

  plus(s(X), Y) >? s(plus(X, Y)) 

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

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

With these choices, we have:

  1] plus(s(X), Y) > s(plus(X, Y))  because [2], by definition 
  2] plus*(s(X), Y) >= s(plus(X, Y))  because plus > s and [3], by (Copy) 
  3] plus*(s(X), Y) >= plus(X, Y)  because plus in Mul, [4] and [7], by (Stat) 
  4] s(X) > X  because [5], by definition 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] Y >= Y  by (Meta) 

We can thus remove the following rules:

  plus(s(X), Y) => s(plus(X, Y)) 

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.