We consider the system 05height.

  Alphabet:

    cons : [t * f] --> f 
    heightf : [f] --> N 
    heightt : [t] --> N 
    leaf : [] --> t 
    max : [N * N] --> N 
    nil : [] --> f 
    node : [f] --> t 
    s : [N] --> N 
    z : [] --> N 

  Rules:

    heightf(nil) => z 
    heightf(cons(x, y)) => max(heightt(x), heightf(y)) 
    heightt(leaf) => z 
    heightt(node(x)) => s(heightf(x)) 

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]):

  heightf(nil) >? z 
  heightf(cons(X, Y)) >? max(heightt(X), heightf(Y)) 
  heightt(leaf) >? z 
  heightt(node(X)) >? s(heightf(X)) 

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

Argument functions:

  [[z]] = _|_ 

We choose Lex = {} and Mul = {cons, heightf, heightt, leaf, max, nil, node, s}, and the following precedence: cons > leaf > max > nil > heightf = node > heightt = s

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

  heightf(nil) > _|_ 
  heightf(cons(X, Y)) > max(heightt(X), heightf(Y)) 
  heightt(leaf) >= _|_ 
  heightt(node(X)) >= s(heightf(X)) 

With these choices, we have:

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

  3] heightf(cons(X, Y)) > max(heightt(X), heightf(Y))  because [4], by definition 
  4] heightf*(cons(X, Y)) >= max(heightt(X), heightf(Y))  because [5], by (Select) 
  5] cons(X, Y) >= max(heightt(X), heightf(Y))  because [6], by (Star) 
  6] cons*(X, Y) >= max(heightt(X), heightf(Y))  because cons > max, [7] and [10], by (Copy) 
  7] cons*(X, Y) >= heightt(X)  because cons > heightt and [8], by (Copy) 
  8] cons*(X, Y) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] cons*(X, Y) >= heightf(Y)  because cons > heightf and [11], by (Copy) 
  11] cons*(X, Y) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 

  13] heightt(leaf) >= _|_  by (Bot) 

  14] heightt(node(X)) >= s(heightf(X))  because heightt = s, heightt in Mul and [15], by (Fun) 
  15] node(X) >= heightf(X)  because node = heightf, node in Mul and [16], by (Fun) 
  16] X >= X  by (Meta) 

We can thus remove the following rules:

  heightf(nil) => z 
  heightf(cons(X, Y)) => max(heightt(X), heightf(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]):

  heightt(leaf) >? z 
  heightt(node(X)) >? s(heightf(X)) 

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

Argument functions:

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

We choose Lex = {} and Mul = {heightf, heightt, leaf, node}, and the following precedence: heightf = node > leaf > heightt

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

  heightt(leaf) >= _|_ 
  heightt(node(X)) > heightf(X) 

With these choices, we have:

  1] heightt(leaf) >= _|_  by (Bot) 

  2] heightt(node(X)) > heightf(X)  because [3], by definition 
  3] heightt*(node(X)) >= heightf(X)  because [4], by (Select) 
  4] node(X) >= heightf(X)  because node = heightf, node in Mul and [5], by (Fun) 
  5] X >= X  by (Meta) 

We can thus remove the following rules:

  heightt(node(X)) => s(heightf(X)) 

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

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

  heightt(leaf) >? z 

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

Argument functions:

  [[z]] = _|_ 

We choose Lex = {} and Mul = {heightt, leaf}, and the following precedence: heightt > leaf

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

  heightt(leaf) > _|_ 

With these choices, we have:

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

We can thus remove the following rules:

  heightt(leaf) => 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.