We consider the system AotoYamada_05__004.

  Alphabet:

    0 : [] --> b 
    cons : [a * b] --> b 
    nil : [] --> b 
    plus : [b * b] --> b 
    s : [b] --> b 
    sumwith : [a -> b * b] --> b 

  Rules:

    plus(0, x) => x 
    plus(s(x), y) => s(plus(x, y)) 
    sumwith(f, nil) => nil 
    sumwith(f, cons(x, y)) => plus(f x, sumwith(f, y)) 

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] sumwith#(F, cons(X, Y)) =#> plus#(F X, sumwith(F, Y))   
    2] sumwith#(F, cons(X, Y)) =#> F(X)   
    3] sumwith#(F, cons(X, Y)) =#> sumwith#(F, Y)   

  Rules R_0:

    plus(0, X) => X 
    plus(s(X), Y) => s(plus(X, Y)) 
    sumwith(F, nil) => nil 
    sumwith(F, cons(X, Y)) => plus(F X, sumwith(F, Y)) 

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 : 0, 1, 2, 3 
    * 3 : 1, 2, 3 

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    sumwith#(F, cons(X, Y)) =#> F(X)   
    sumwith#(F, cons(X, Y)) =#> sumwith#(F, 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).

The formative rules of (P_2, R_0) are R_1 ::=

  plus(0, X) => X 
  sumwith(F, cons(X, Y)) => plus(F X, sumwith(F, Y)) 

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:

  sumwith#(F, cons(X, Y)) >? F(X) 
  sumwith#(F, cons(X, Y)) >? sumwith#(F, Y) 
  plus(0, X) >= X 
  sumwith(F, cons(X, Y)) >= plus(F X, sumwith(F, Y)) 

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

The following interpretation satisfies the requirements:

  0 = 0 
  cons = \y0y1.3 + y0 + y1 
  plus = \y0y1.y1 
  sumwith = \G0y1.0 
  sumwith# = \G0y1.3 + 2y1G0(y1) 

Using this interpretation, the requirements translate to:

  [[sumwith#(_F0, cons(_x1, _x2))]] = 3 + 2x1F0(3 + x1 + x2) + 2x2F0(3 + x1 + x2) + 6F0(3 + x1 + x2) > F0(x1) = [[_F0(_x1)]] 
  [[sumwith#(_F0, cons(_x1, _x2))]] = 3 + 2x1F0(3 + x1 + x2) + 2x2F0(3 + x1 + x2) + 6F0(3 + x1 + x2) >= 3 + 2x2F0(x2) = [[sumwith#(_F0, _x2)]] 
  [[plus(0, _x0)]] = x0 >= x0 = [[_x0]] 
  [[sumwith(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[plus(_F0 _x1, sumwith(_F0, _x2))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_1, minimal, formative) by (P_3, R_1, minimal, formative), where P_3 consists of:

  sumwith#(F, cons(X, Y)) =#> sumwith#(F, Y)   

Thus, the original system is terminating if each of (P_1, 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 apply the subterm criterion with the following projection function:

  nu(sumwith#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(sumwith#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(sumwith#(F, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_3, R_1, minimal, f) by ({}, R_1, 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.