We consider the system 468.

  Alphabet:

    0 : [] --> nat 
    add : [nat * nat] --> nat 
    rec : [nat -> nat -> nat * nat * nat] --> nat 
    s : [nat] --> nat 

  Rules:

    rec(/\x./\y.X(x, y), Y, 0) => Y 
    rec(/\x./\y.X(x, y), Y, s(Z)) => X(Z, rec(/\z./\u.X(z, u), Y, Z)) 
    add(X, Y) => rec(/\x./\y.s(y), X, Y) 
    add(X, 0) => X 
    add(X, s(Y)) => s(add(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]):

  rec(/\x./\y.X(x, y), Y, 0) >? Y 
  rec(/\x./\y.X(x, y), Y, s(Z)) >? X(Z, rec(/\z./\u.X(z, u), Y, Z)) 
  add(X, Y) >? rec(/\x./\y.s(y), X, Y) 
  add(X, 0) >? X 
  add(X, s(Y)) >? s(add(X, Y)) 

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

We choose Lex = {} and Mul = {0, add, rec, s}, and the following precedence: 0 > add > rec > s

With these choices, we have:

  1] rec(/\x./\y.X(x, y), Y, 0) >= Y  because [2], by (Star) 
  2] rec*(/\x./\y.X(x, y), Y, 0) >= Y  because [3], by (Select) 
  3] Y >= Y  by (Meta) 

  4] rec(/\x./\y.X(x, y), Y, s(Z)) >= X(Z, rec(/\x./\y.X(x, y), Y, Z))  because [5], by (Star) 
  5] rec*(/\x./\y.X(x, y), Y, s(Z)) >= X(Z, rec(/\x./\y.X(x, y), Y, Z))  because [6], by (Select) 
  6] X(rec*(/\x./\y.X(x, y), Y, s(Z)), rec*(/\z./\u.X(z, u), Y, s(Z))) >= X(Z, rec(/\x./\y.X(x, y), Y, Z))  because [7] and [11], by (Meta) 
  7] rec*(/\x./\y.X(x, y), Y, s(Z)) >= Z  because [8], by (Select) 
  8] s(Z) >= Z  because [9], by (Star) 
  9] s*(Z) >= Z  because [10], by (Select) 
  10] Z >= Z  by (Meta) 
  11] rec*(/\x./\y.X(x, y), Y, s(Z)) >= rec(/\x./\y.X(x, y), Y, Z)  because rec in Mul, [12], [17] and [18], by (Stat) 
  12] /\x./\z.X(x, z) >= /\x./\z.X(x, z)  because [13], by (Abs) 
  13] /\z.X(y, z) >= /\z.X(y, z)  because [14], by (Abs) 
  14] X(y, x) >= X(y, x)  because [15] and [16], by (Meta) 
  15] y >= y  by (Var) 
  16] x >= x  by (Var) 
  17] Y >= Y  by (Meta) 
  18] s(Z) > Z  because [19], by definition 
  19] s*(Z) >= Z  because [10], by (Select) 

  20] add(X, Y) >= rec(/\x./\y.s(y), X, Y)  because [21], by (Star) 
  21] add*(X, Y) >= rec(/\x./\y.s(y), X, Y)  because add > rec, [22], [27] and [29], by (Copy) 
  22] add*(X, Y) >= /\y./\z.s(z)  because [23], by (F-Abs) 
  23] add*(X, Y, x) >= /\z.s(z)  because [24], by (F-Abs) 
  24] add*(X, Y, x, y) >= s(y)  because add > s and [25], by (Copy) 
  25] add*(X, Y, x, y) >= y  because [26], by (Select) 
  26] y >= y  by (Var) 
  27] add*(X, Y) >= X  because [28], by (Select) 
  28] X >= X  by (Meta) 
  29] add*(X, Y) >= Y  because [30], by (Select) 
  30] Y >= Y  by (Meta) 

  31] add(X, 0) > X  because [32], by definition 
  32] add*(X, 0) >= X  because [33], by (Select) 
  33] X >= X  by (Meta) 

  34] add(X, s(Y)) > s(add(X, Y))  because [35], by definition 
  35] add*(X, s(Y)) >= s(add(X, Y))  because add > s and [36], by (Copy) 
  36] add*(X, s(Y)) >= add(X, Y)  because add in Mul, [37] and [38], by (Stat) 
  37] X >= X  by (Meta) 
  38] s(Y) > Y  because [39], by definition 
  39] s*(Y) >= Y  because [40], by (Select) 
  40] Y >= Y  by (Meta) 

We can thus remove the following rules:

  add(X, 0) => X 
  add(X, s(Y)) => s(add(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]):

  rec(/\x./\y.X(x, y), Y, 0) >? Y 
  rec(/\x./\y.X(x, y), Y, s(Z)) >? X(Z, rec(/\z./\u.X(z, u), Y, Z)) 
  add(X, Y) >? rec(/\x./\y.s(y), X, Y) 

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

We choose Lex = {} and Mul = {0, add, rec, s}, and the following precedence: add > rec > s > 0

With these choices, we have:

  1] rec(/\x./\y.X(x, y), Y, 0) > Y  because [2], by definition 
  2] rec*(/\x./\y.X(x, y), Y, 0) >= Y  because [3], by (Select) 
  3] Y >= Y  by (Meta) 

  4] rec(/\x./\y.X(x, y), Y, s(Z)) > X(Z, rec(/\x./\y.X(x, y), Y, Z))  because [5], by definition 
  5] rec*(/\x./\y.X(x, y), Y, s(Z)) >= X(Z, rec(/\x./\y.X(x, y), Y, Z))  because [6], by (Select) 
  6] X(rec*(/\x./\y.X(x, y), Y, s(Z)), rec*(/\z./\u.X(z, u), Y, s(Z))) >= X(Z, rec(/\x./\y.X(x, y), Y, Z))  because [7] and [11], by (Meta) 
  7] rec*(/\x./\y.X(x, y), Y, s(Z)) >= Z  because [8], by (Select) 
  8] s(Z) >= Z  because [9], by (Star) 
  9] s*(Z) >= Z  because [10], by (Select) 
  10] Z >= Z  by (Meta) 
  11] rec*(/\x./\y.X(x, y), Y, s(Z)) >= rec(/\x./\y.X(x, y), Y, Z)  because rec in Mul, [12], [17] and [18], by (Stat) 
  12] /\x./\z.X(x, z) >= /\x./\z.X(x, z)  because [13], by (Abs) 
  13] /\z.X(y, z) >= /\z.X(y, z)  because [14], by (Abs) 
  14] X(y, x) >= X(y, x)  because [15] and [16], by (Meta) 
  15] y >= y  by (Var) 
  16] x >= x  by (Var) 
  17] Y >= Y  by (Meta) 
  18] s(Z) > Z  because [19], by definition 
  19] s*(Z) >= Z  because [10], by (Select) 

  20] add(X, Y) > rec(/\x./\y.s(y), X, Y)  because [21], by definition 
  21] add*(X, Y) >= rec(/\x./\y.s(y), X, Y)  because add > rec, [22], [27] and [29], by (Copy) 
  22] add*(X, Y) >= /\y./\z.s(z)  because [23], by (F-Abs) 
  23] add*(X, Y, x) >= /\z.s(z)  because [24], by (F-Abs) 
  24] add*(X, Y, x, y) >= s(y)  because add > s and [25], by (Copy) 
  25] add*(X, Y, x, y) >= y  because [26], by (Select) 
  26] y >= y  by (Var) 
  27] add*(X, Y) >= X  because [28], by (Select) 
  28] X >= X  by (Meta) 
  29] add*(X, Y) >= Y  because [30], by (Select) 
  30] Y >= Y  by (Meta) 

We can thus remove the following rules:

  rec(/\x./\y.X(x, y), Y, 0) => Y 
  rec(/\x./\y.X(x, y), Y, s(Z)) => X(Z, rec(/\z./\u.X(z, u), Y, Z)) 
  add(X, Y) => rec(/\x./\y.s(y), X, Y) 

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.