We consider the system map.

  Alphabet:

    cons : [a * list] --> list 
    map : [list * a -> a] --> list 
    nil : [] --> list 

  Rules:

    map(nil, f) => nil 
    map(cons(x, y), f) => cons(f x, map(y, f)) 

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]):

  map(nil, F) >? nil 
  map(cons(X, Y), F) >? cons(F X, map(Y, F)) 

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

Argument functions:

  [[nil]] = _|_ 

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

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

  map(_|_, F) > _|_ 
  map(cons(X, Y), F) >= cons(@_{o -> o}(F, X), map(Y, F)) 

With these choices, we have:

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

  3] map(cons(X, Y), F) >= cons(@_{o -> o}(F, X), map(Y, F))  because [4], by (Star) 
  4] map*(cons(X, Y), F) >= cons(@_{o -> o}(F, X), map(Y, F))  because map > cons, [5] and [12], by (Copy) 
  5] map*(cons(X, Y), F) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [6] and [8], by (Copy) 
  6] map*(cons(X, Y), F) >= F  because [7], by (Select) 
  7] F >= F  by (Meta) 
  8] map*(cons(X, Y), F) >= X  because [9], by (Select) 
  9] cons(X, Y) >= X  because [10], by (Star) 
  10] cons*(X, Y) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] map*(cons(X, Y), F) >= map(Y, F)  because map in Mul, [13] and [16], by (Stat) 
  13] cons(X, Y) > Y  because [14], by definition 
  14] cons*(X, Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 
  16] F >= F  by (Meta) 

We can thus remove the following rules:

  map(nil, F) => nil 

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

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

  map(cons(X, Y), F) >? cons(F X, map(Y, F)) 

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

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

With these choices, we have:

  1] map(cons(X, Y), F) > cons(@_{o -> o}(F, X), map(Y, F))  because [2], by definition 
  2] map*(cons(X, Y), F) >= cons(@_{o -> o}(F, X), map(Y, F))  because map > cons, [3] and [10], by (Copy) 
  3] map*(cons(X, Y), F) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [4] and [6], by (Copy) 
  4] map*(cons(X, Y), F) >= F  because [5], by (Select) 
  5] F >= F  by (Meta) 
  6] map*(cons(X, Y), F) >= X  because [7], by (Select) 
  7] cons(X, Y) >= X  because [8], by (Star) 
  8] cons*(X, Y) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] map*(cons(X, Y), F) >= map(Y, F)  because map in Mul, [11] and [14], by (Stat) 
  11] cons(X, Y) > Y  because [12], by definition 
  12] cons*(X, Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 
  14] F >= F  by (Meta) 

We can thus remove the following rules:

  map(cons(X, Y), F) => cons(F X, map(Y, F)) 

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.