We consider the system AotoYamada_05__005.

  Alphabet:

    0 : [] --> a 
    add : [] --> a -> a -> a 
    curry : [a -> a -> a] --> a -> a -> a 
    plus : [] --> a -> a -> a 
    s : [a] --> a 

  Rules:

    plus 0 x => x 
    plus s(x) y => s(plus x y) 
    curry(f) x y => f x y 
    add => curry(plus) 

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] plus s(X) Y =#> plus X   
    2] plus s(X) Y =#> plus#   
    3] curry(F) X Y =#> F(X, Y)   
    4] add X =#> curry(plus) X   
    5] add X Y =#> curry(plus) X Y   
    6] add# =#> curry#(plus)   
    7] add# =#> plus#   

  Rules R_0:

    plus 0 X => X 
    plus s(X) Y => s(plus X Y) 
    curry(F) X Y => F X Y 
    add => curry(plus) 

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, 2 
    * 1 :  
    * 2 :  
    * 3 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 4 :  
    * 5 : 3 
    * 6 :  
    * 7 :  

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    curry(F) X Y =#> F(X, Y)   
    add X Y =#> curry(plus) X Y   

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) and (P_2, R_0, m, f).

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).

This combination (P_2, 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_2, R_0, minimal, formative) by (P_2, R_1, minimal, formative).

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

We consider the dependency pair problem (P_2, 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:

  curry(F, X, Y) >? F(X, Y) 
  add(X, Y) >? curry(plus, X, Y) 

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

  pi( curry(F, X, Y) ) = #argfun-curry#(F X Y) 
  pi( add(X, Y) ) = #argfun-add#(#argfun-curry#(plus X Y)) 

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

The following interpretation satisfies the requirements:

  #argfun-add# = \y0.3 + y0 
  #argfun-curry# = \y0.3 + y0 
  add = \y0y1.0 
  curry = \G0y1y2.0 
  plus = \y0y1.0 

Using this interpretation, the requirements translate to:

  [[#argfun-curry#(_F0 _x1 _x2)]] = 3 + max(x1, x2, F0(x1,x2)) > F0(x1,x2) = [[_F0(_x1, _x2)]] 
  [[#argfun-add#(#argfun-curry#(plus _x0 _x1))]] = 6 + max(x0, x1) > 3 + max(x0, x1) = [[#argfun-curry#(plus _x0 _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_2, 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 (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.