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 the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] plus#(s(X), Y) =#> plus#(X, Y)   
    1] times#(s(X), Y) =#> plus#(times(X, Y), Y)   
    2] times#(s(X), Y) =#> times#(X, Y)   
    3] comp(F, G) X =#> F(G X)   
    4] comp(F, G) X =#> G(X)   
    5] twice(F) X =#> comp(F, F) X   
    6] twice#(F) =#> comp#(F, F)   

  Rules R_0:

    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) 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0 
    * 1 : 0 
    * 2 : 1, 2 
    * 3 : 0, 1, 2, 3, 4, 5, 6 
    * 4 : 0, 1, 2, 3, 4, 5, 6 
    * 5 : 3, 4 
    * 6 :  

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    times#(s(X), Y) =#> times#(X, Y)   

  P_3:

    comp(F, G) X =#> F(G X)   
    comp(F, G) X =#> G(X)   
    twice(F) X =#> comp(F, F) X   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f), (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_3, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_0, minimal, formative).

This combination (P_3, R_0) has no formative rules!  We will name the empty set of rules:R_1.

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_3, R_0, minimal, formative) by (P_3, R_1, minimal, formative).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_3, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

  comp(F, G, X) >? F(G X) 
  comp(F, G, X) >? G(X) 
  twice(F, X) >? comp(F, F, X) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( comp(F, G, X) ) = #argfun-comp#(F (G X), G X) 
  pi( twice(F, X) ) = #argfun-twice#(#argfun-comp#(F (F X), F X)) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  #argfun-comp# = \y0y1.3 + max(y0, y1) 
  #argfun-twice# = \y0.3 + y0 
  comp = \G0G1y2.0 
  twice = \G0y1.0 

Using this interpretation, the requirements translate to:

  [[#argfun-comp#(_F0 (_F1 _x2), _F1 _x2)]] = 3 + max(x2, F0(max(x2, F1(x2))), F1(x2)) > F0(max(x2, F1(x2))) = [[_F0(_F1 _x2)]] 
  [[#argfun-comp#(_F0 (_F1 _x2), _F1 _x2)]] = 3 + max(x2, F0(max(x2, F1(x2))), F1(x2)) > F1(x2) = [[_F1(_x2)]] 
  [[#argfun-twice#(#argfun-comp#(_F0 (_F0 _x1), _F0 _x1))]] = 6 + max(x1, F0(x1), F0(max(x1, F0(x1)))) > 3 + max(x1, F0(x1), F0(max(x1, F0(x1)))) = [[#argfun-comp#(_F0 (_F0 _x1), _F0 _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_3, R_1) by ({}, R_1).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(times#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(times#(s(X), Y)) = s(X) |> X = nu(times#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(plus#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(plus#(s(X), Y)) = s(X) |> X = nu(plus#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.