We consider the system AotoYamada_05__020.

  Alphabet:

    0 : [] --> a 
    comp : [b -> b * b -> b] --> b -> b 
    plus : [a * a] --> a 
    s : [a] --> a 
    times : [a * a] --> a 
    twice : [b -> b] --> b -> b 

  Rules:

    plus(0, x) => x 
    plus(s(x), y) => s(plus(x, y)) 
    times(0, x) => 0 
    times(s(x), y) => plus(times(x, y), y) 
    comp(f, g) x => f (g x) 
    twice(f) => comp(f, 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]):

  plus(0, X) >? X 
  plus(s(X), Y) >? s(plus(X, Y)) 
  times(0, X) >? 0 
  times(s(X), Y) >? plus(times(X, Y), Y) 
  comp(F, G) X >? F (G X) 
  twice(F) >? comp(F, F) 

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

We choose Lex = {plus, times} and Mul = {0, @_{o -> o}, comp, s, twice}, and the following precedence: 0 > twice > comp > times > plus > @_{o -> o} > s

With these choices, we have:

  1] plus(0, X) > X  because [2], by definition 
  2] plus*(0, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] plus(s(X), Y) >= s(plus(X, Y))  because [5], by (Star) 
  5] plus*(s(X), Y) >= s(plus(X, Y))  because plus > s and [6], by (Copy) 
  6] plus*(s(X), Y) >= plus(X, Y)  because [7], [10] and [12], by (Stat) 
  7] s(X) > X  because [8], by definition 
  8] s*(X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] plus*(s(X), Y) >= X  because [11], by (Select) 
  11] s(X) >= X  because [8], by (Star) 
  12] plus*(s(X), Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 

  14] times(0, X) > 0  because [15], by definition 
  15] times*(0, X) >= 0  because [16], by (Select) 
  16] 0 >= 0  by (Fun) 

  17] times(s(X), Y) > plus(times(X, Y), Y)  because [18], by definition 
  18] times*(s(X), Y) >= plus(times(X, Y), Y)  because times > plus, [19] and [25], by (Copy) 
  19] times*(s(X), Y) >= times(X, Y)  because [20], [23] and [25], by (Stat) 
  20] s(X) > X  because [21], by definition 
  21] s*(X) >= X  because [22], by (Select) 
  22] X >= X  by (Meta) 
  23] times*(s(X), Y) >= X  because [24], by (Select) 
  24] s(X) >= X  because [21], by (Star) 
  25] times*(s(X), Y) >= Y  because [26], by (Select) 
  26] Y >= Y  by (Meta) 

  27] @_{o -> o}(comp(F, G), X) > @_{o -> o}(F, @_{o -> o}(G, X))  because [28], by definition 
  28] @_{o -> o}*(comp(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [29], by (Select) 
  29] comp(F, G) @_{o -> o}*(comp(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [30] 
  30] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because comp > @_{o -> o}, [31] and [33], by (Copy) 
  31] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= F  because [32], by (Select) 
  32] F >= F  by (Meta) 
  33] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= @_{o -> o}(G, X)  because comp > @_{o -> o}, [34] and [36], by (Copy) 
  34] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= G  because [35], by (Select) 
  35] G >= G  by (Meta) 
  36] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= X  because [37], by (Select) 
  37] @_{o -> o}*(comp(F, G), X) >= X  because [38], by (Select) 
  38] X >= X  by (Meta) 

  39] twice(F) >= comp(F, F)  because [40], by (Star) 
  40] twice*(F) >= comp(F, F)  because twice > comp, [41] and [41], by (Copy) 
  41] twice*(F) >= F  because [42], by (Select) 
  42] F >= F  by (Meta) 

We can thus remove the following rules:

  plus(0, X) => X 
  times(0, X) => 0 
  times(s(X), Y) => plus(times(X, Y), Y) 
  comp(F, G) X => F (G 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]):

  plus(s(X), Y) >? s(plus(X, Y)) 
  twice(F) >? comp(F, F) 

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

We choose Lex = {} and Mul = {comp, plus, s, twice}, and the following precedence: twice > comp = plus > s

With these choices, we have:

  1] plus(s(X), Y) >= s(plus(X, Y))  because [2], by (Star) 
  2] plus*(s(X), Y) >= s(plus(X, Y))  because plus > s and [3], by (Copy) 
  3] plus*(s(X), Y) >= plus(X, Y)  because plus in Mul, [4] and [7], by (Stat) 
  4] s(X) > X  because [5], by definition 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] Y >= Y  by (Meta) 

  8] twice(F) > comp(F, F)  because [9], by definition 
  9] twice*(F) >= comp(F, F)  because twice > comp, [10] and [10], by (Copy) 
  10] twice*(F) >= F  because [11], by (Select) 
  11] F >= F  by (Meta) 

We can thus remove the following rules:

  twice(F) => comp(F, F) 

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

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

  plus(s(X), Y) >? s(plus(X, Y)) 

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

We choose Lex = {} and Mul = {plus, s}, and the following precedence: plus > s

With these choices, we have:

  1] plus(s(X), Y) > s(plus(X, Y))  because [2], by definition 
  2] plus*(s(X), Y) >= s(plus(X, Y))  because plus > s and [3], by (Copy) 
  3] plus*(s(X), Y) >= plus(X, Y)  because plus in Mul, [4] and [7], by (Stat) 
  4] s(X) > X  because [5], by definition 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] Y >= Y  by (Meta) 

We can thus remove the following rules:

  plus(s(X), Y) => s(plus(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.