We consider the system foldl.

  Alphabet:

    0 : [] --> nat 
    cons : [nat * list] --> list 
    foldl : [nat -> nat -> nat * nat * list] --> nat 
    nil : [] --> list 
    plus : [nat * nat] --> nat 
    plusc : [] --> nat -> nat -> nat 
    sum : [list] --> nat 

  Rules:

    foldl(f, x, nil) => x 
    foldl(f, x, cons(y, z)) => foldl(f, f x y, z) 
    plusc => /\x./\y.plus(x, y) 
    sum(x) => foldl(plusc, 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.

To start, the system is beta-saturated by adding the following rules:

  plusc X => /\x.plus(X, x) 
  plusc X Y => plus(X, Y) 

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] foldl#(F, X, cons(Y, Z)) =#> foldl#(F, F X Y, Z)   
    1] foldl#(F, X, cons(Y, Z)) =#> F(X, Y)   
    2] sum#(X) =#> foldl#(plusc, 0, X)   
    3] sum#(X) =#> plusc#   

  Rules R_0:

    foldl(F, X, nil) => X 
    foldl(F, X, cons(Y, Z)) => foldl(F, F X Y, Z) 
    plusc => /\x./\y.plus(x, y) 
    sum(X) => foldl(plusc, 0, X) 
    plusc X => /\x.plus(X, x) 
    plusc X Y => plus(X, 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 
    * 1 : 0, 1, 2, 3 
    * 2 : 0, 1 
    * 3 :  

This graph has the following strongly connected components:

  P_1:

    foldl#(F, X, cons(Y, Z)) =#> foldl#(F, F X Y, Z)   
    foldl#(F, X, cons(Y, Z)) =#> F(X, Y)   
    sum#(X) =#> foldl#(plusc, 0, 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).

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).

This combination (P_1, 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_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative).

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, 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#(F, X, cons(Y, Z)) >? foldl#(F, F X Y, Z) 
  foldl#(F, X, cons(Y, Z)) >? F(X, Y) 
  sum#(X) >? foldl#(plusc, 0, X) 

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

  pi( sum#(X) ) = #argfun-sum##(foldl#(plusc, 0, X)) 

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:

  [[0]] = _|_ 
  [[foldl#(x_1, x_2, x_3)]] = foldl#(x_3, x_1, x_2) 
  [[plusc]] = _|_ 
  [[sum#(x_1)]] = x_1 

We choose Lex = {foldl#} and Mul = {#argfun-sum##, @_{o -> o -> o}, @_{o -> o}, cons}, and the following precedence: #argfun-sum## > cons > foldl# > @_{o -> o -> o} > @_{o -> o}

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

  foldl#(F, X, cons(Y, Z)) >= foldl#(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z) 
  foldl#(F, X, cons(Y, Z)) > @_{o -> o}(@_{o -> o -> o}(F, X), Y) 
  #argfun-sum##(foldl#(_|_, _|_, X)) >= foldl#(_|_, _|_, X) 

With these choices, we have:

  1] foldl#(F, X, cons(Y, Z)) >= foldl#(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [2], by (Star) 
  2] foldl#*(F, X, cons(Y, Z)) >= foldl#(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [3], [6], [8] and [16], by (Stat) 
  3] cons(Y, Z) > Z  because [4], by definition 
  4] cons*(Y, Z) >= Z  because [5], by (Select) 
  5] Z >= Z  by (Meta) 
  6] foldl#*(F, X, cons(Y, Z)) >= F  because [7], by (Select) 
  7] F >= F  by (Meta) 
  8] foldl#*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because foldl# > @_{o -> o}, [9] and [12], by (Copy) 
  9] foldl#*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, X)  because foldl# > @_{o -> o -> o}, [6] and [10], by (Copy) 
  10] foldl#*(F, X, cons(Y, Z)) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] foldl#*(F, X, cons(Y, Z)) >= Y  because [13], by (Select) 
  13] cons(Y, Z) >= Y  because [14], by (Star) 
  14] cons*(Y, Z) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 
  16] foldl#*(F, X, cons(Y, Z)) >= Z  because [17], by (Select) 
  17] cons(Y, Z) >= Z  because [4], by (Star) 

  18] foldl#(F, X, cons(Y, Z)) > @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because [8], by definition 

  19] #argfun-sum##(foldl#(_|_, _|_, X)) >= foldl#(_|_, _|_, X)  because [20], by (Star) 
  20] #argfun-sum##*(foldl#(_|_, _|_, X)) >= foldl#(_|_, _|_, X)  because #argfun-sum## > foldl#, [21], [22] and [23], by (Copy) 
  21] #argfun-sum##*(foldl#(_|_, _|_, X)) >= _|_  by (Bot) 
  22] #argfun-sum##*(foldl#(_|_, _|_, X)) >= _|_  by (Bot) 
  23] #argfun-sum##*(foldl#(_|_, _|_, X)) >= X  because [24], by (Select) 
  24] foldl#(_|_, _|_, X) >= X  because [25], by (Star) 
  25] foldl#*(_|_, _|_, X) >= X  because [26], by (Select) 
  26] X >= X  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_1, minimal, formative) by (P_2, R_1, minimal, formative), where P_2 consists of:

  foldl#(F, X, cons(Y, Z)) =#> foldl#(F, F X Y, Z)   
  sum#(X) =#> foldl#(plusc, 0, X)   

Thus, the original system is terminating if (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_1, 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 

This graph has the following strongly connected components:

  P_3:

    foldl#(F, X, cons(Y, Z)) =#> foldl#(F, F X Y, Z)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_2, R_1, m, f) by (P_3, R_1, m, f).

Thus, the original system is terminating if (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(foldl#) = 3 

Thus, we can orient the dependency pairs as follows:

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

By [FuhKop19, Thm. 61], 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.

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


+++ Citations +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.