We consider the system extrec.

  Alphabet:

    !plus : [nat * nat] --> nat 
    !times : [nat * nat] --> nat 
    0 : [] --> nat 
    rec : [nat * nat * nat -> nat -> nat] --> nat 
    s : [nat] --> nat 

  Rules:

    !plus(x, 0) => x 
    !plus(x, s(y)) => s(!plus(x, y)) 
    rec(0, x, f) => x 
    rec(s(x), y, f) => f x rec(x, y, f) 
    !times(x, y) => rec(y, 0, /\z./\u.!plus(x, u)) 

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]):

  !plus(X, 0) >? X 
  !plus(X, s(Y)) >? s(!plus(X, Y)) 
  rec(0, X, F) >? X 
  rec(s(X), Y, F) >? F X rec(X, Y, F) 
  !times(X, Y) >? rec(Y, 0, /\x./\y.!plus(X, y)) 

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

Argument functions:

  [[0]] = _|_ 

We choose Lex = {} and Mul = {!plus, !times, @_{o -> o -> o}, @_{o -> o}, rec, s}, and the following precedence: !times > !plus > @_{o -> o -> o} = rec > @_{o -> o} = s

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

  !plus(X, _|_) >= X 
  !plus(X, s(Y)) > s(!plus(X, Y)) 
  rec(_|_, X, F) > X 
  rec(s(X), Y, F) >= @_{o -> o}(@_{o -> o -> o}(F, X), rec(X, Y, F)) 
  !times(X, Y) > rec(Y, _|_, /\x./\y.!plus(X, y)) 

With these choices, we have:

  1] !plus(X, _|_) >= X  because [2], by (Star) 
  2] !plus*(X, _|_) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] !plus(X, s(Y)) > s(!plus(X, Y))  because [5], by definition 
  5] !plus*(X, s(Y)) >= s(!plus(X, Y))  because !plus > s and [6], by (Copy) 
  6] !plus*(X, s(Y)) >= !plus(X, Y)  because !plus in Mul, [7] and [8], by (Stat) 
  7] X >= X  by (Meta) 
  8] s(Y) > Y  because [9], by definition 
  9] s*(Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 

  11] rec(_|_, X, F) > X  because [12], by definition 
  12] rec*(_|_, X, F) >= X  because [13], by (Select) 
  13] X >= X  by (Meta) 

  14] rec(s(X), Y, F) >= @_{o -> o}(@_{o -> o -> o}(F, X), rec(X, Y, F))  because [15], by (Star) 
  15] rec*(s(X), Y, F) >= @_{o -> o}(@_{o -> o -> o}(F, X), rec(X, Y, F))  because rec > @_{o -> o}, [16] and [21], by (Copy) 
  16] rec*(s(X), Y, F) >= @_{o -> o -> o}(F, X)  because rec = @_{o -> o -> o}, rec in Mul, [17] and [20], by (Stat) 
  17] s(X) > X  because [18], by definition 
  18] s*(X) >= X  because [19], by (Select) 
  19] X >= X  by (Meta) 
  20] F >= F  by (Meta) 
  21] rec*(s(X), Y, F) >= rec(X, Y, F)  because rec in Mul, [17], [22] and [20], by (Stat) 
  22] Y >= Y  by (Meta) 

  23] !times(X, Y) > rec(Y, _|_, /\x./\y.!plus(X, y))  because [24], by definition 
  24] !times*(X, Y) >= rec(Y, _|_, /\x./\y.!plus(X, y))  because !times > rec, [25], [27] and [28], by (Copy) 
  25] !times*(X, Y) >= Y  because [26], by (Select) 
  26] Y >= Y  by (Meta) 
  27] !times*(X, Y) >= _|_  by (Bot) 
  28] !times*(X, Y) >= /\y./\z.!plus(X, z)  because [29], by (F-Abs) 
  29] !times*(X, Y, x) >= /\z.!plus(X, z)  because [30], by (F-Abs) 
  30] !times*(X, Y, x, y) >= !plus(X, y)  because !times > !plus, [31] and [33], by (Copy) 
  31] !times*(X, Y, x, y) >= X  because [32], by (Select) 
  32] X >= X  by (Meta) 
  33] !times*(X, Y, x, y) >= y  because [34], by (Select) 
  34] y >= y  by (Var) 

We can thus remove the following rules:

  !plus(X, s(Y)) => s(!plus(X, Y)) 
  rec(0, X, F) => X 
  !times(X, Y) => rec(Y, 0, /\x./\y.!plus(X, y)) 

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

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

  !plus(X, 0) >? X 
  rec(s(X), Y, F) >? F X rec(X, Y, F) 

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

We choose Lex = {} and Mul = {!plus, 0, @_{o -> o -> o}, @_{o -> o}, rec, s}, and the following precedence: rec > s > !plus > 0 > @_{o -> o -> o} > @_{o -> o}

With these choices, we have:

  1] !plus(X, 0) > X  because [2], by definition 
  2] !plus*(X, 0) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

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

We can thus remove the following rules:

  !plus(X, 0) => X 
  rec(s(X), Y, F) => F X rec(X, Y, F) 

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.