We consider the system Applicative_05__Ex6Recursor.

  Alphabet:

    0 : [] --> a 
    rec : [a -> b -> b * b * a] --> b 
    s : [a] --> a 

  Rules:

    rec(f, x, 0) => x 
    rec(f, x, s(y)) => f s(y) rec(f, x, y) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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

After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] rec#(F, X, s(Y)) =#> F(s(Y), rec(F, X, Y))   
    1] rec#(F, X, s(Y)) =#> rec#(F, X, Y)   

  Rules R_0:

    rec(F, X, 0) => X 
    rec(F, X, s(Y)) => F s(Y) rec(F, X, 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).

This combination (P_0, R_0) has no formative rules!  We will name the empty set of rules:R_1.

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative).

Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_0, 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:

  rec#(F, X, s(Y)) >? F(s(Y), rec(F, X, Y)) 
  rec#(F, X, s(Y)) >? rec#(F, X, Y) 

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

The following interpretation satisfies the requirements:

  rec = \G0y1y2.0 
  rec# = \G0y1y2.3 + G0(0,0) 
  s = \y0.0 

Using this interpretation, the requirements translate to:

  [[rec#(_F0, _x1, s(_x2))]] = 3 + F0(0,0) > F0(0,0) = [[_F0(s(_x2), rec(_F0, _x1, _x2))]] 
  [[rec#(_F0, _x1, s(_x2))]] = 3 + F0(0,0) >= 3 + F0(0,0) = [[rec#(_F0, _x1, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_1, minimal, formative) by (P_1, R_1, minimal, formative), where P_1 consists of:

  rec#(F, X, s(Y)) =#> rec#(F, X, Y)   

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 apply the subterm criterion with the following projection function:

  nu(rec#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(rec#(F, X, s(Y))) = s(Y) |> Y = nu(rec#(F, X, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_1, minimal, f) by ({}, R_1, minimal, f).  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 +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.