We consider the system h51.

  Alphabet:

    cons : [] --> a -> alist -> alist 
    foldl : [] --> (a -> a -> a) -> a -> alist -> a 
    nil : [] --> alist 

  Rules:

    foldl (/\x./\y.f x y) z nil => z 
    foldl (/\x./\y.f x y) z (cons u v) => foldl (/\w./\x'.f w x') (f z u) v 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    cons : [a * alist] --> alist 
    foldl : [a -> a -> a * a * alist] --> a 
    nil : [] --> alist 
    ~AP1 : [a -> a -> a * a] --> a -> a 

  Rules:

    foldl(/\x./\y.~AP1(F, x) y, X, nil) => X 
    foldl(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => foldl(/\z./\u.~AP1(F, z) u, ~AP1(F, X) Y, Z) 
    ~AP1(F, X) => F X 

Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:

  Alphabet:

    cons : [a * alist] --> alist 
    foldl : [a -> a -> a * a * alist] --> a 
    nil : [] --> alist 

  Rules:

    foldl(/\x./\y.X(x, y), Y, nil) => Y 
    foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) 

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

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) =#> foldl#(/\z./\u.X(z, u), X(Y, Z), U)   
    1] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) =#> X(z, u)   
    2] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) =#> X(Y, Z)   

  Rules R_0:

    foldl(/\x./\y.X(x, y), Y, nil) => Y 
    foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) 

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:

  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) >? foldl#(/\z./\u.X(z, u), X(Y, Z), U) 
  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) >? X(~c0, ~c1) 
  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) >? X(Y, Z) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

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

Argument functions:

  [[foldl#(x_1, x_2, x_3)]] = foldl#(x_1, x_3, x_2) 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 

We choose Lex = {foldl#} and Mul = {cons}, and the following precedence: foldl# > cons

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > foldl#(/\x./\y.X(x, y), X(Y, Z), U) 
  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > X(_|_, _|_) 
  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > X(Y, Z) 

With these choices, we have:

  1] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > foldl#(/\x./\y.X(x, y), X(Y, Z), U)  because [2], by definition 
  2] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl#(/\x./\y.X(x, y), X(Y, Z), U)  because [3], [8], [11], [12] and [20], by (Stat) 
  3] /\x./\z.X(x, z) >= /\x./\z.X(x, z)  because [4], by (Abs) 
  4] /\z.X(y, z) >= /\z.X(y, z)  because [5], by (Abs) 
  5] X(y, x) >= X(y, x)  because [6] and [7], by (Meta) 
  6] y >= y  by (Var) 
  7] x >= x  by (Var) 
  8] cons(Z, U) > U  because [9], by definition 
  9] cons*(Z, U) >= U  because [10], by (Select) 
  10] U >= U  by (Meta) 
  11] foldl#*(/\z./\u.X(z, u), Y, cons(Z, U)) >= /\z./\u.X(z, u)  because [3], by (Select) 
  12] foldl#*(/\z./\u.X(z, u), Y, cons(Z, U)) >= X(Y, Z)  because [13], by (Select) 
  13] X(foldl#*(/\z./\u.X(z, u), Y, cons(Z, U)), foldl#*(/\v./\w.X(v, w), Y, cons(Z, U))) >= X(Y, Z)  because [14] and [16], by (Meta) 
  14] foldl#*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 
  16] foldl#*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Z  because [17], by (Select) 
  17] cons(Z, U) >= Z  because [18], by (Star) 
  18] cons*(Z, U) >= Z  because [19], by (Select) 
  19] Z >= Z  by (Meta) 
  20] foldl#*(/\z./\u.X(z, u), Y, cons(Z, U)) >= U  because [21], by (Select) 
  21] cons(Z, U) >= U  because [9], by (Star) 

  22] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > X(_|_, _|_)  because [23], by definition 
  23] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(_|_, _|_)  because [24], by (Select) 
  24] X(foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)), foldl#*(/\z./\u.X(z, u), Y, cons(Z, U))) >= X(_|_, _|_)  because [25] and [26], by (Meta) 
  25] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= _|_  by (Bot) 
  26] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= _|_  by (Bot) 

  27] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > X(Y, Z)  because [12], by definition 

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 +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.