We consider the system 470.

  Alphabet:

    L : [o] --> o 
    P : [o * o -> o * o -> o] --> o 
    R : [o] --> o 

  Rules:

    P(L(X), /\x.Y(x), /\y.Z(y)) => Y(X) 
    P(R(X), /\x.Y(x), /\y.Z(y)) => Z(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]):

  P(L(X), /\x.Y(x), /\y.Z(y)) >? Y(X) 
  P(R(X), /\x.Y(x), /\y.Z(y)) >? Z(X) 

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

We choose Lex = {} and Mul = {L, P, R}, and the following precedence: R > L > P

With these choices, we have:

  1] P(L(X), /\x.Y(x), /\y.Z(y)) >= Y(X)  because [2], by (Star) 
  2] P*(L(X), /\x.Y(x), /\y.Z(y)) >= Y(X)  because [3], by (Select) 
  3] Y(P*(L(X), /\x.Y(x), /\y.Z(y))) >= Y(X)  because [4], by (Meta) 
  4] P*(L(X), /\x.Y(x), /\y.Z(y)) >= X  because [5], by (Select) 
  5] L(X) >= X  because [6], by (Star) 
  6] L*(X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 

  8] P(R(X), /\x.Y(x), /\y.Z(y)) > Z(X)  because [9], by definition 
  9] P*(R(X), /\x.Y(x), /\y.Z(y)) >= Z(X)  because [10], by (Select) 
  10] Z(P*(R(X), /\x.Y(x), /\y.Z(y))) >= Z(X)  because [11], by (Meta) 
  11] P*(R(X), /\x.Y(x), /\y.Z(y)) >= X  because [12], by (Select) 
  12] R(X) >= X  because [13], by (Star) 
  13] R*(X) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 

We can thus remove the following rules:

  P(R(X), /\x.Y(x), /\y.Z(y)) => Z(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]):

  P(L(X), /\x.Y(x), /\y.Z(y)) >? Y(X) 

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

We choose Lex = {} and Mul = {L, P}, and the following precedence: L > P

With these choices, we have:

  1] P(L(X), /\x.Y(x), /\y.Z(y)) > Y(X)  because [2], by definition 
  2] P*(L(X), /\x.Y(x), /\y.Z(y)) >= Y(X)  because [3], by (Select) 
  3] Y(P*(L(X), /\x.Y(x), /\y.Z(y))) >= Y(X)  because [4], by (Meta) 
  4] P*(L(X), /\x.Y(x), /\y.Z(y)) >= X  because [5], by (Select) 
  5] L(X) >= X  because [6], by (Star) 
  6] L*(X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 

We can thus remove the following rules:

  P(L(X), /\x.Y(x), /\y.Z(y)) => Y(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.