We consider the system uncurry.

  Alphabet:

    f : [] --> a -> b -> c 
    f1 : [a] --> b -> c 
    f2 : [a * b] --> c 

  Rules:

    f x => f1(x) 
    f1(x) y => f2(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]):

  f(X) >? f1(X) 
  f1(X) Y >? f2(X, Y) 

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

We choose Lex = {} and Mul = {@_{o -> o}, f, f1, f2}, and the following precedence: f > f1 > @_{o -> o} > f2

With these choices, we have:

  1] f(X) >= f1(X)  because [2], by (Star) 
  2] f*(X) >= f1(X)  because f > f1 and [3], by (Copy) 
  3] f*(X) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 

  5] @_{o -> o}(f1(X), Y) > f2(X, Y)  because [6], by definition 
  6] @_{o -> o}*(f1(X), Y) >= f2(X, Y)  because [7], by (Select) 
  7] f1(X) @_{o -> o}*(f1(X), Y) >= f2(X, Y)  because [8] 
  8] f1*(X, @_{o -> o}*(f1(X), Y)) >= f2(X, Y)  because [9], by (Select) 
  9] @_{o -> o}*(f1(X), Y) >= f2(X, Y)  because @_{o -> o} > f2, [10] and [14], by (Copy) 
  10] @_{o -> o}*(f1(X), Y) >= X  because [11], by (Select) 
  11] f1(X) @_{o -> o}*(f1(X), Y) >= X  because [12] 
  12] f1*(X, @_{o -> o}*(f1(X), Y)) >= X  because [13], by (Select) 
  13] X >= X  by (Meta) 
  14] @_{o -> o}*(f1(X), Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 

We can thus remove the following rules:

  f1(X) Y => f2(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]):

  f(X) >? f1(X) 

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

We choose Lex = {} and Mul = {f, f1}, and the following precedence: f > f1

With these choices, we have:

  1] f(X) > f1(X)  because [2], by definition 
  2] f*(X) >= f1(X)  because f > f1 and [3], by (Copy) 
  3] f*(X) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 

We can thus remove the following rules:

  f(X) => f1(X) 

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.