We consider the system eval.

  Alphabet:

    dom : [N * N * N] --> N 
    eval : [N * N] --> N 
    fun : [N -> N * N * N] --> N 
    o : [] --> N 
    s : [N] --> N 

  Rules:

    eval(fun(f, x, y), z) => f dom(x, y, z) 
    dom(s(x), s(y), s(z)) => s(dom(x, y, z)) 
    dom(o, s(x), s(y)) => s(dom(o, x, y)) 
    dom(x, y, o) => x 
    dom(o, o, x) => o 

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] eval#(fun(F, X, Y), Z) =#> F(dom(X, Y, Z))   
    1] eval#(fun(F, X, Y), Z) =#> dom#(X, Y, Z)   
    2] dom#(s(X), s(Y), s(Z)) =#> dom#(X, Y, Z)   
    3] dom#(o, s(X), s(Y)) =#> dom#(o, X, Y)   

  Rules R_0:

    eval(fun(F, X, Y), Z) => F dom(X, Y, Z) 
    dom(s(X), s(Y), s(Z)) => s(dom(X, Y, Z)) 
    dom(o, s(X), s(Y)) => s(dom(o, X, Y)) 
    dom(X, Y, o) => X 
    dom(o, o, X) => o 

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 place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0, 1, 2, 3 
    * 1 : 2, 3 
    * 2 : 2, 3 
    * 3 : 3 

This graph has the following strongly connected components:

  P_1:

    eval#(fun(F, X, Y), Z) =#> F(dom(X, Y, Z))   

  P_2:

    dom#(s(X), s(Y), s(Z)) =#> dom#(X, Y, Z)   

  P_3:

    dom#(o, s(X), s(Y)) =#> dom#(o, X, Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f), (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_3, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(dom#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(dom#(o, s(X), s(Y))) = s(X) |> X = nu(dom#(o, X, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_3, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(dom#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(dom#(s(X), s(Y), s(Z))) = s(X) |> X = nu(dom#(X, Y, Z)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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 ::=

  eval(fun(F, X, Y), Z) => F dom(X, Y, Z) 
  dom(X, Y, o) => X 
  dom(o, o, X) => o 

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:

  eval#(fun(F, X, Y), Z) >? F(dom(X, Y, Z)) 
  eval(fun(F, X, Y), Z) >= F dom(X, Y, Z) 
  dom(X, Y, o) >= X 
  dom(o, o, X) >= o 

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

The following interpretation satisfies the requirements:

  dom = \y0y1y2.y0 
  eval = \y0y1.3 + 3y0 
  eval# = \y0y1.3 + y0 
  fun = \G0y1y2.3 + y1 + G0(y1) 
  o = 0 

Using this interpretation, the requirements translate to:

  [[eval#(fun(_F0, _x1, _x2), _x3)]] = 6 + x1 + F0(x1) > F0(x1) = [[_F0(dom(_x1, _x2, _x3))]] 
  [[eval(fun(_F0, _x1, _x2), _x3)]] = 12 + 3x1 + 3F0(x1) >= max(x1, F0(x1)) = [[_F0 dom(_x1, _x2, _x3)]] 
  [[dom(_x0, _x1, o)]] = x0 >= x0 = [[_x0]] 
  [[dom(o, o, _x0)]] = 0 >= 0 = [[o]] 

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 +++

[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.