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 observe that the rules contain a first-order subset:

  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 

Moreover, the system is finitely branching.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS.

According to nattprover, this system is indeed Ce-terminating:

 || Input TRS:
 ||     1: dom(s(PeRCenTX),s(PeRCenTY),s(PeRCenTZ)) -> s(dom(PeRCenTX,PeRCenTY,PeRCenTZ))
 ||     2: dom(o(),s(PeRCenTX),s(PeRCenTY)) -> s(dom(o(),PeRCenTX,PeRCenTY))
 ||     3: dom(PeRCenTX,PeRCenTY,o()) -> PeRCenTX
 ||     4: dom(o(),o(),PeRCenTX) -> o()
 ||     5: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTX
 ||     6: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTY
 || Number of strict rules: 6
 || Direct POLO(bPol) ... removes: 4 1 3 5 6 2
 ||       TIlDePAIR	w: 2 * x1 + 2 * x2 + 1
 ||       s 	w: 2 * x1 + 4
 ||       o 	w: 3
 ||       dom	w: 2 * x1 + 2 * x2 + 2 * x3 + 3
 || Number of strict rules: 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)   

  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 
    * 1 :  

This graph has the following strongly connected components:

  P_1:

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

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).

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 +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.