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 rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  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) 

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

Argument functions:

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

We choose Lex = {foldl} and Mul = {@_{o -> o -> o}, @_{o -> o}, cons, nil, plus, plusc, sum}, and the following precedence: plusc = sum > foldl > plus > @_{o -> o} > nil > cons > @_{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, nil) >= X 
  foldl(F, X, cons(Y, Z)) >= foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z) 
  plusc > /\x./\y.plus(x, y) 
  sum(X) > foldl(plusc, _|_, X) 

With these choices, we have:

  1] foldl(F, X, nil) >= X  because [2], by (Star) 
  2] foldl*(F, X, nil) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] foldl(F, X, cons(Y, Z)) >= foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [5], by (Star) 
  5] foldl*(F, X, cons(Y, Z)) >= foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [6], [7], [10], [11] and [19], by (Stat) 
  6] F >= F  by (Meta) 
  7] cons(Y, Z) > Z  because [8], by definition 
  8] cons*(Y, Z) >= Z  because [9], by (Select) 
  9] Z >= Z  by (Meta) 
  10] foldl*(F, X, cons(Y, Z)) >= F  because [6], by (Select) 
  11] foldl*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because foldl > @_{o -> o}, [12] and [15], by (Copy) 
  12] foldl*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, X)  because foldl > @_{o -> o -> o}, [10] and [13], by (Copy) 
  13] foldl*(F, X, cons(Y, Z)) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 
  15] foldl*(F, X, cons(Y, Z)) >= Y  because [16], by (Select) 
  16] cons(Y, Z) >= Y  because [17], by (Star) 
  17] cons*(Y, Z) >= Y  because [18], by (Select) 
  18] Y >= Y  by (Meta) 
  19] foldl*(F, X, cons(Y, Z)) >= Z  because [20], by (Select) 
  20] cons(Y, Z) >= Z  because [8], by (Star) 

  21] plusc > /\x./\y.plus(x, y)  because [22], by definition 
  22] plusc* >= /\y./\z.plus(y, z)  because [23], by (F-Abs) 
  23] plusc*(x) >= /\z.plus(x, z)  because [24], by (F-Abs) 
  24] plusc*(x, y) >= plus(x, y)  because plusc > plus, [25] and [27], by (Copy) 
  25] plusc*(x, y) >= x  because [26], by (Select) 
  26] x >= x  by (Var) 
  27] plusc*(x, y) >= y  because [28], by (Select) 
  28] y >= y  by (Var) 

  29] sum(X) > foldl(plusc, _|_, X)  because [30], by definition 
  30] sum*(X) >= foldl(plusc, _|_, X)  because sum > foldl, [31], [32] and [33], by (Copy) 
  31] sum*(X) >= plusc  because sum = plusc and sum in Mul, by (Stat) 
  32] sum*(X) >= _|_  by (Bot) 
  33] sum*(X) >= X  because [34], by (Select) 
  34] X >= X  by (Meta) 

We can thus remove the following rules:

  plusc => /\x./\y.plus(x, y) 
  sum(X) => foldl(plusc, 0, X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  foldl(F, X, nil) >? X 
  foldl(F, X, cons(Y, Z)) >? foldl(F, F X Y, Z) 

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

Argument functions:

  [[@_{o -> o}(x_1, x_2)]] = @_{o -> o}(x_2, x_1) 
  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 

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

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

  foldl(F, X, nil) > X 
  foldl(F, X, cons(Y, Z)) > foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z) 

With these choices, we have:

  1] foldl(F, X, nil) > X  because [2], by definition 
  2] foldl*(F, X, nil) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] foldl(F, X, cons(Y, Z)) > foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [5], by definition 
  5] foldl*(F, X, cons(Y, Z)) >= foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [6], [9], [11] and [20], by (Stat) 
  6] cons(Y, Z) > Z  because [7], by definition 
  7] cons*(Y, Z) >= Z  because [8], by (Select) 
  8] Z >= Z  by (Meta) 
  9] foldl*(F, X, cons(Y, Z)) >= F  because [10], by (Select) 
  10] F >= F  by (Meta) 
  11] foldl*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because foldl = @_{o -> o}, [12], [15] and [18], by (Stat) 
  12] cons(Y, Z) > Y  because [13], by definition 
  13] cons*(Y, Z) >= Y  because [14], by (Select) 
  14] Y >= Y  by (Meta) 
  15] foldl*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, X)  because foldl > @_{o -> o -> o}, [9] and [16], by (Copy) 
  16] foldl*(F, X, cons(Y, Z)) >= X  because [17], by (Select) 
  17] X >= X  by (Meta) 
  18] foldl*(F, X, cons(Y, Z)) >= Y  because [19], by (Select) 
  19] cons(Y, Z) >= Y  because [13], by (Star) 
  20] foldl*(F, X, cons(Y, Z)) >= Z  because [21], by (Select) 
  21] cons(Y, Z) >= Z  because [7], by (Star) 

We can thus remove the following rules:

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

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.