We consider the system 726.

  Alphabet:

    0 : [] --> nat 
    dom : [nat * nat * nat] --> nat 
    eval : [nat * nat] --> nat 
    fun : [nat -> nat * nat * nat] --> nat 
    s : [nat] --> nat 

  Rules:

    dom(s(X), s(Y), s(Z)) => s(dom(X, Y, Z)) 
    dom(0, s(X), s(Y)) => s(dom(0, X, Y)) 
    dom(X, Y, 0) => X 
    dom(0, 0, X) => 0 
    eval(fun(/\x.X(x), Y, Z), U) => X(dom(Y, Z, U)) 

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] dom#(s(X), s(Y), s(Z)) =#> dom#(X, Y, Z)   
    1] dom#(0, s(X), s(Y)) =#> dom#(0, X, Y)   
    2] eval#(fun(/\x.X(x), Y, Z), U) =#> X(dom(Y, Z, U))   
    3] eval#(fun(/\x.X(x), Y, Z), U) =#> dom#(Y, Z, U) {X : 1}  

  Rules R_0:

    dom(s(X), s(Y), s(Z)) => s(dom(X, Y, Z)) 
    dom(0, s(X), s(Y)) => s(dom(0, X, Y)) 
    dom(X, Y, 0) => X 
    dom(0, 0, X) => 0 
    eval(fun(/\x.X(x), Y, Z), U) => X(dom(Y, Z, U)) 

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

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    dom#(0, s(X), s(Y)) =#> dom#(0, X, Y)   

  P_3:

    eval#(fun(/\x.X(x), Y, Z), U) =#> X(dom(Y, Z, U))   

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).

The formative rules of (P_3, R_0) are R_1 ::=

  dom(X, Y, 0) => X 
  dom(0, 0, X) => 0 
  eval(fun(/\x.X(x), Y, Z), U) => X(dom(Y, Z, U)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_3, R_0, minimal, formative) by (P_3, R_1, minimal, formative).

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_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, 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(/\x.X(x), Y, Z), U) >? X(dom(Y, Z, U)) 
  dom(X, Y, 0) >= X 
  dom(0, 0, X) >= 0 
  eval(fun(/\x.X(x), Y, Z), U) >= X(dom(Y, Z, U)) 

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

The following interpretation satisfies the requirements:

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

Using this interpretation, the requirements translate to:

  [[eval#(fun(/\x._x0(x), _x1, _x2), _x3)]] = 6 + F0(x1) > F0(x1) = [[_x0(dom(_x1, _x2, _x3))]] 
  [[dom(_x0, _x1, 0)]] = x0 >= x0 = [[_x0]] 
  [[dom(0, 0, _x0)]] = 0 >= 0 = [[0]] 
  [[eval(fun(/\x._x0(x), _x1, _x2), _x3)]] = 3 + F0(x1) >= F0(x1) = [[_x0(dom(_x1, _x2, _x3))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_3, R_1) by ({}, R_1).  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#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(dom#(0, s(X), s(Y))) = s(X) |> X = nu(dom#(0, X, Y)) 

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).

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_1, R_0, minimal, f) by ({}, R_0, 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.