We consider the system length.

  Alphabet:

    0 : [] --> nat 
    cons : [nat * list] --> list 
    foldr : [nat -> nat -> nat * nat * list] --> nat 
    length : [list] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 

  Rules:

    foldr(f, x, nil) => x 
    foldr(f, x, cons(y, z)) => f y foldr(f, x, z) 
    length(x) => foldr(/\y./\z.s(z), 0, x) 

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] foldr#(F, X, cons(Y, Z)) =#> F(Y, foldr(F, X, Z))   
    1] foldr#(F, X, cons(Y, Z)) =#> foldr#(F, X, Z)   
    2] length#(X) =#> foldr#(/\x./\y.s(y), 0, X)   

  Rules R_0:

    foldr(F, X, nil) => X 
    foldr(F, X, cons(Y, Z)) => F Y foldr(F, X, Z) 
    length(X) => foldr(/\x./\y.s(y), 0, X) 

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).

This combination (P_0, 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_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative).

Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_0, 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:

  foldr#(F, X, cons(Y, Z)) >? F(Y, foldr(F, X, Z)) 
  foldr#(F, X, cons(Y, Z)) >? foldr#(F, X, Z) 
  length#(X) >? foldr#(/\x./\y.s-(y), 0, X) 
  s-(X) >= s(X) 

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

  pi( length#(X) ) = #argfun-length##(foldr#(/\x./\y.s-(y), 0, X)) 

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

The following interpretation satisfies the requirements:

  #argfun-length## = \y0.3 + y0 
  0 = 0 
  cons = \y0y1.3 + y1 + 2y0 
  foldr = \G0y1y2.0 
  foldr# = \G0y1y2.3 + y2 + 2G0(y2,y1) 
  length# = \y0.0 
  s = \y0.0 
  s- = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[foldr#(_F0, _x1, cons(_x2, _x3))]] = 6 + x3 + 2x2 + 2F0(3 + x3 + 2x2,x1) > F0(x2,0) = [[_F0(_x2, foldr(_F0, _x1, _x3))]] 
  [[foldr#(_F0, _x1, cons(_x2, _x3))]] = 6 + x3 + 2x2 + 2F0(3 + x3 + 2x2,x1) > 3 + x3 + 2F0(x3,x1) = [[foldr#(_F0, _x1, _x3)]] 
  [[#argfun-length##(foldr#(/\x./\y.s-(y), 0, _x0))]] = 12 + x0 > 9 + x0 = [[foldr#(/\x./\y.s-(y), 0, _x0)]] 
  [[s-(_x0)]] = 3 + 3x0 >= 0 = [[s(_x0)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_0, R_1) by ({}, R_1).  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.