We consider the system 07ordinal.

  Alphabet:

    lim : [] --> (N -> O) -> O 
    plus : [] --> O -> O -> O 
    s : [] --> O -> O 
    z : [] --> O 

  Rules:

    plus z x => x 
    plus (s x) y => s (plus x y) 
    plus (lim (/\x.f x)) y => lim (/\u.plus (f u) y) 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    lim : [N -> O] --> O 
    plus : [O * O] --> O 
    s : [O] --> O 
    z : [] --> O 
    ~AP1 : [N -> O * N] --> O 

  Rules:

    plus(z, X) => X 
    plus(s(X), Y) => s(plus(X, Y)) 
    plus(lim(/\x.~AP1(F, x)), X) => lim(/\y.plus(~AP1(F, y), X)) 
    ~AP1(F, X) => F X 

Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:

  Alphabet:

    lim : [N -> O] --> O 
    plus : [O * O] --> O 
    s : [O] --> O 
    z : [] --> O 

  Rules:

    plus(z, X) => X 
    plus(s(X), Y) => s(plus(X, Y)) 
    plus(lim(/\x.X(x)), Y) => lim(/\y.plus(X(y), Y)) 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] plus#(s(X), Y) =#> plus#(X, Y)   
    1] plus#(lim(/\x.X(x)), Y) =#> plus#(X(y), Y)   
    2] plus#(lim(/\x.X(x)), Y) =#> X(y)   

  Rules R_0:

    plus(z, X) => X 
    plus(s(X), Y) => s(plus(X, Y)) 
    plus(lim(/\x.X(x)), Y) => lim(/\y.plus(X(y), 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 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:

  plus#(s(X), Y) >? plus#(X, Y) 
  plus#(lim(/\x.X(x)), Y) >? plus#(X(~c0), Y) 
  plus#(lim(/\x.X(x)), Y) >? X(~c1) 
  plus(z, X) >= X 
  plus(s(X), Y) >= s(plus(X, Y)) 
  plus(lim(/\x.X(x)), Y) >= lim(/\y.plus-(X(y), Y)) 
  plus-(X, Y) >= plus(X, Y) 
  plus-(X, Y) >= plus#(X, Y) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  lim = \G0.G0(0) 
  plus = \y0y1.y1 + 2y0 
  plus- = \y0y1.y1 + 2y0 
  plus# = \y0y1.2y0 
  s = \y0.1 + y0 
  z = 0 
  ~c0 = 0 
  ~c1 = 0 

Using this interpretation, the requirements translate to:

  [[plus#(s(_x0), _x1)]] = 2 + 2x0 > 2x0 = [[plus#(_x0, _x1)]] 
  [[plus#(lim(/\x._x0(x)), _x1)]] = 2F0(0) >= 2F0(0) = [[plus#(_x0(~c0), _x1)]] 
  [[plus#(lim(/\x._x0(x)), _x1)]] = 2F0(0) >= F0(0) = [[_x0(~c1)]] 
  [[plus(z, _x0)]] = x0 >= x0 = [[_x0]] 
  [[plus(s(_x0), _x1)]] = 2 + x1 + 2x0 >= 1 + x1 + 2x0 = [[s(plus(_x0, _x1))]] 
  [[plus(lim(/\x._x0(x)), _x1)]] = x1 + 2F0(0) >= x1 + 2F0(0) = [[lim(/\x.plus-(_x0(x), _x1))]] 
  [[plus-(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[plus(_x0, _x1)]] 
  [[plus-(_x0, _x1)]] = x1 + 2x0 >= 2x0 = [[plus#(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_1, R_0, minimal, formative), where P_1 consists of:

  plus#(lim(/\x.X(x)), Y) =#> plus#(X(y), Y)   
  plus#(lim(/\x.X(x)), Y) =#> X(y)   

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).

The formative rules of (P_1, R_0) are R_1 ::=

  plus(z, X) => X 
  plus(lim(/\x.X(x)), Y) => lim(/\y.plus(X(y), Y)) 

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:

  plus#(lim(/\x.X(x)), Y) >? plus#(X(~c0), Y) 
  plus#(lim(/\x.X(x)), Y) >? X(~c1) 
  plus(z, X) >= X 
  plus(lim(/\x.X(x)), Y) >= lim(/\y.plus-(X(y), Y)) 
  plus-(X, Y) >= plus(X, Y) 
  plus-(X, Y) >= plus#(X, Y) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  lim = \G0.2 + G0(0) 
  plus = \y0y1.y0 + y1 
  plus- = \y0y1.y0 + y1 
  plus# = \y0y1.y0 
  z = 0 
  ~c0 = 0 
  ~c1 = 0 

Using this interpretation, the requirements translate to:

  [[plus#(lim(/\x._x0(x)), _x1)]] = 2 + F0(0) > F0(0) = [[plus#(_x0(~c0), _x1)]] 
  [[plus#(lim(/\x._x0(x)), _x1)]] = 2 + F0(0) > F0(0) = [[_x0(~c1)]] 
  [[plus(z, _x0)]] = x0 >= x0 = [[_x0]] 
  [[plus(lim(/\x._x0(x)), _x1)]] = 2 + x1 + F0(0) >= 2 + x1 + F0(0) = [[lim(/\x.plus-(_x0(x), _x1))]] 
  [[plus-(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] 
  [[plus-(_x0, _x1)]] = x0 + x1 >= x0 = [[plus#(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_1) by ({}, R_1).  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 +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.