We consider the system AotoYamada_05__028.

  Alphabet:

    cons : [a * c] --> c 
    consif : [b * a * c] --> c 
    false : [] --> b 
    filter : [a -> b * c] --> c 
    nil : [] --> c 
    true : [] --> b 

  Rules:

    consif(true, x, y) => cons(x, y) 
    consif(false, x, y) => y 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => consif(f x, x, filter(f, 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]):

  consif(true, X, Y) >? cons(X, Y) 
  consif(false, X, Y) >? Y 
  filter(F, nil) >? nil 
  filter(F, cons(X, Y)) >? consif(F X, X, filter(F, Y)) 

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

Argument functions:

  [[nil]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, cons, consif, false, filter, true}, and the following precedence: false > filter > consif > cons > true > @_{o -> o}

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

  consif(true, X, Y) >= cons(X, Y) 
  consif(false, X, Y) > Y 
  filter(F, _|_) >= _|_ 
  filter(F, cons(X, Y)) >= consif(@_{o -> o}(F, X), X, filter(F, Y)) 

With these choices, we have:

  1] consif(true, X, Y) >= cons(X, Y)  because [2], by (Star) 
  2] consif*(true, X, Y) >= cons(X, Y)  because consif > cons, [3] and [5], by (Copy) 
  3] consif*(true, X, Y) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 
  5] consif*(true, X, Y) >= Y  because [6], by (Select) 
  6] Y >= Y  by (Meta) 

  7] consif(false, X, Y) > Y  because [8], by definition 
  8] consif*(false, X, Y) >= Y  because [9], by (Select) 
  9] Y >= Y  by (Meta) 

  10] filter(F, _|_) >= _|_  by (Bot) 

  11] filter(F, cons(X, Y)) >= consif(@_{o -> o}(F, X), X, filter(F, Y))  because [12], by (Star) 
  12] filter*(F, cons(X, Y)) >= consif(@_{o -> o}(F, X), X, filter(F, Y))  because filter > consif, [13], [16] and [20], by (Copy) 
  13] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [14] and [16], by (Copy) 
  14] filter*(F, cons(X, Y)) >= F  because [15], by (Select) 
  15] F >= F  by (Meta) 
  16] filter*(F, cons(X, Y)) >= X  because [17], by (Select) 
  17] cons(X, Y) >= X  because [18], by (Star) 
  18] cons*(X, Y) >= X  because [19], by (Select) 
  19] X >= X  by (Meta) 
  20] filter*(F, cons(X, Y)) >= filter(F, Y)  because filter in Mul, [21] and [22], by (Stat) 
  21] F >= F  by (Meta) 
  22] cons(X, Y) > Y  because [23], by definition 
  23] cons*(X, Y) >= Y  because [24], by (Select) 
  24] Y >= Y  by (Meta) 

We can thus remove the following rules:

  consif(false, X, Y) => 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]):

  consif(true, X, Y) >? cons(X, Y) 
  filter(F, nil) >? nil 
  filter(F, cons(X, Y)) >? consif(F X, X, filter(F, Y)) 

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

Argument functions:

  [[nil]] = _|_ 

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

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

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

With these choices, we have:

  1] consif(true, X, Y) >= cons(X, Y)  because [2], by (Star) 
  2] consif*(true, X, Y) >= cons(X, Y)  because consif > cons, [3] and [5], by (Copy) 
  3] consif*(true, X, Y) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 
  5] consif*(true, X, Y) >= Y  because [6], by (Select) 
  6] Y >= Y  by (Meta) 

  7] filter(F, _|_) > _|_  because [8], by definition 
  8] filter*(F, _|_) >= _|_  by (Bot) 

  9] filter(F, cons(X, Y)) >= consif(@_{o -> o}(F, X), X, filter(F, Y))  because [10], by (Star) 
  10] filter*(F, cons(X, Y)) >= consif(@_{o -> o}(F, X), X, filter(F, Y))  because filter > consif, [11], [16] and [18], by (Copy) 
  11] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter = @_{o -> o}, filter in Mul, [12] and [13], by (Stat) 
  12] F >= F  by (Meta) 
  13] cons(X, Y) > X  because [14], by definition 
  14] cons*(X, Y) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 
  16] filter*(F, cons(X, Y)) >= X  because [17], by (Select) 
  17] cons(X, Y) >= X  because [14], by (Star) 
  18] filter*(F, cons(X, Y)) >= filter(F, Y)  because filter in Mul, [12] and [19], by (Stat) 
  19] cons(X, Y) > Y  because [20], by definition 
  20] cons*(X, Y) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 

We can thus remove the following rules:

  filter(F, nil) => 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]):

  consif(true, X, Y) >? cons(X, Y) 
  filter(F, cons(X, Y)) >? consif(F X, X, filter(F, Y)) 

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

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

With these choices, we have:

  1] consif(true, X, Y) >= cons(X, Y)  because [2], by (Star) 
  2] consif*(true, X, Y) >= cons(X, Y)  because consif > cons, [3] and [5], by (Copy) 
  3] consif*(true, X, Y) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 
  5] consif*(true, X, Y) >= Y  because [6], by (Select) 
  6] Y >= Y  by (Meta) 

  7] filter(F, cons(X, Y)) > consif(@_{o -> o}(F, X), X, filter(F, Y))  because [8], by definition 
  8] filter*(F, cons(X, Y)) >= consif(@_{o -> o}(F, X), X, filter(F, Y))  because filter > consif, [9], [12] and [16], by (Copy) 
  9] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [10] and [12], by (Copy) 
  10] filter*(F, cons(X, Y)) >= F  because [11], by (Select) 
  11] F >= F  by (Meta) 
  12] filter*(F, cons(X, Y)) >= X  because [13], by (Select) 
  13] cons(X, Y) >= X  because [14], by (Star) 
  14] cons*(X, Y) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 
  16] filter*(F, cons(X, Y)) >= filter(F, Y)  because filter in Mul, [17] and [18], by (Stat) 
  17] F >= F  by (Meta) 
  18] cons(X, Y) > Y  because [19], by definition 
  19] cons*(X, Y) >= Y  because [20], by (Select) 
  20] Y >= Y  by (Meta) 

We can thus remove the following rules:

  filter(F, cons(X, Y)) => consif(F X, X, filter(F, 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]):

  consif(true, X, Y) >? cons(X, Y) 

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

We choose Lex = {} and Mul = {cons, consif, true}, and the following precedence: consif > cons > true

With these choices, we have:

  1] consif(true, X, Y) > cons(X, Y)  because [2], by definition 
  2] consif*(true, X, Y) >= cons(X, Y)  because consif > cons, [3] and [5], by (Copy) 
  3] consif*(true, X, Y) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 
  5] consif*(true, X, Y) >= Y  because [6], by (Select) 
  6] Y >= Y  by (Meta) 

We can thus remove the following rules:

  consif(true, X, Y) => cons(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.