We consider the system plode.

  Alphabet:

    cons : [nat * list] --> list 
    explode : [list * nat -> nat * nat] --> nat 
    implode : [list * nat -> nat * nat] --> nat 
    nil : [] --> list 
    op : [nat -> nat * nat -> nat] --> nat -> nat 

  Rules:

    op(f, g) x => f (g x) 
    implode(nil, f, x) => x 
    implode(cons(x, y), f, z) => implode(y, f, f z) 
    explode(nil, f, x) => x 
    explode(cons(x, y), f, z) => explode(y, op(f, f), f z) 

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]):

  op(F, G) X >? F (G X) 
  implode(nil, F, X) >? X 
  implode(cons(X, Y), F, Z) >? implode(Y, F, F Z) 
  explode(nil, F, X) >? X 
  explode(cons(X, Y), F, Z) >? explode(Y, op(F, F), F Z) 

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

Argument functions:

  [[explode(x_1, x_2, x_3)]] = explode(x_1, x_3, x_2) 
  [[implode(x_1, x_2, x_3)]] = implode(x_1, x_3, x_2) 

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

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

  @_{o -> o}(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) 
  implode(nil, F, X) >= X 
  implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z)) 
  explode(nil, F, X) > X 
  explode(cons(X, Y), F, Z) > explode(Y, op(F, F), @_{o -> o}(F, Z)) 

With these choices, we have:

  1] @_{o -> o}(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [2], by (Star) 
  2] @_{o -> o}*(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [3], by (Select) 
  3] op(F, G) @_{o -> o}*(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [4] 
  4] op*(F, G, @_{o -> o}*(op(F, G), X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because op > @_{o -> o}, [5] and [7], by (Copy) 
  5] op*(F, G, @_{o -> o}*(op(F, G), X)) >= F  because [6], by (Select) 
  6] F >= F  by (Meta) 
  7] op*(F, G, @_{o -> o}*(op(F, G), X)) >= @_{o -> o}(G, X)  because op > @_{o -> o}, [8] and [10], by (Copy) 
  8] op*(F, G, @_{o -> o}*(op(F, G), X)) >= G  because [9], by (Select) 
  9] G >= G  by (Meta) 
  10] op*(F, G, @_{o -> o}*(op(F, G), X)) >= X  because [11], by (Select) 
  11] @_{o -> o}*(op(F, G), X) >= X  because [12], by (Select) 
  12] X >= X  by (Meta) 

  13] implode(nil, F, X) >= X  because [14], by (Star) 
  14] implode*(nil, F, X) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 

  16] implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [17], by (Star) 
  17] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [18], [21], [23], [25] and [28], by (Stat) 
  18] cons(X, Y) > Y  because [19], by definition 
  19] cons*(X, Y) >= Y  because [20], by (Select) 
  20] Y >= Y  by (Meta) 
  21] implode*(cons(X, Y), F, Z) >= Y  because [22], by (Select) 
  22] cons(X, Y) >= Y  because [19], by (Star) 
  23] implode*(cons(X, Y), F, Z) >= F  because [24], by (Select) 
  24] F >= F  by (Meta) 
  25] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [23] and [26], by (Copy) 
  26] implode*(cons(X, Y), F, Z) >= Z  because [27], by (Select) 
  27] Z >= Z  by (Meta) 
  28] F >= F  by (Meta) 

  29] explode(nil, F, X) > X  because [30], by definition 
  30] explode*(nil, F, X) >= X  because [31], by (Select) 
  31] X >= X  by (Meta) 

  32] explode(cons(X, Y), F, Z) > explode(Y, op(F, F), @_{o -> o}(F, Z))  because [33], by definition 
  33] explode*(cons(X, Y), F, Z) >= explode(Y, op(F, F), @_{o -> o}(F, Z))  because [34], [37], [39] and [42], by (Stat) 
  34] cons(X, Y) > Y  because [35], by definition 
  35] cons*(X, Y) >= Y  because [36], by (Select) 
  36] Y >= Y  by (Meta) 
  37] explode*(cons(X, Y), F, Z) >= Y  because [38], by (Select) 
  38] cons(X, Y) >= Y  because [35], by (Star) 
  39] explode*(cons(X, Y), F, Z) >= op(F, F)  because explode > op, [40] and [40], by (Copy) 
  40] explode*(cons(X, Y), F, Z) >= F  because [41], by (Select) 
  41] F >= F  by (Meta) 
  42] explode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because explode > @_{o -> o}, [40] and [43], by (Copy) 
  43] explode*(cons(X, Y), F, Z) >= Z  because [44], by (Select) 
  44] Z >= Z  by (Meta) 

We can thus remove the following rules:

  explode(nil, F, X) => X 
  explode(cons(X, Y), F, Z) => explode(Y, op(F, F), F Z) 

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

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

  op(F, G, X) >? F (G X) 
  implode(nil, F, X) >? X 
  implode(cons(X, Y), F, Z) >? implode(Y, F, F Z) 

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

Argument functions:

  [[implode(x_1, x_2, x_3)]] = implode(x_2, x_1, x_3) 

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

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

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

With these choices, we have:

  1] op(F, G, X) > @_{o -> o}(F, @_{o -> o}(G, X))  because [2], by definition 
  2] op*(F, G, X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because op > @_{o -> o}, [3] and [5], by (Copy) 
  3] op*(F, G, X) >= F  because [4], by (Select) 
  4] F >= F  by (Meta) 
  5] op*(F, G, X) >= @_{o -> o}(G, X)  because op > @_{o -> o}, [6] and [8], by (Copy) 
  6] op*(F, G, X) >= G  because [7], by (Select) 
  7] G >= G  by (Meta) 
  8] op*(F, G, X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 

  10] implode(nil, F, X) > X  because [11], by definition 
  11] implode*(nil, F, X) >= X  because [12], by (Select) 
  12] X >= X  by (Meta) 

  13] implode(cons(X, Y), F, Z) > implode(Y, F, @_{o -> o}(F, Z))  because [14], by definition 
  14] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [15], [18], [19], [21], [22] and [18], by (Stat) 
  15] cons(X, Y) > Y  because [16], by definition 
  16] cons*(X, Y) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 
  18] F >= F  by (Meta) 
  19] implode*(cons(X, Y), F, Z) >= Y  because [20], by (Select) 
  20] cons(X, Y) >= Y  because [16], by (Star) 
  21] implode*(cons(X, Y), F, Z) >= F  because [18], by (Select) 
  22] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [21] and [23], by (Copy) 
  23] implode*(cons(X, Y), F, Z) >= Z  because [24], by (Select) 
  24] Z >= Z  by (Meta) 

We can thus remove the following rules:

  op(F, G, X) => F (G X) 
  implode(nil, F, X) => X 
  implode(cons(X, Y), F, Z) => implode(Y, F, F 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.