We consider the system Applicative_05__mapDivMinus.

  Alphabet:

    0 : [] --> c 
    cons : [a * b] --> b 
    div : [c * c] --> c 
    map : [a -> a * b] --> b 
    minus : [c * c] --> c 
    nil : [] --> b 
    s : [c] --> c 

  Rules:

    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    minus(x, 0) => x 
    minus(s(x), s(y)) => minus(x, y) 
    div(0, s(x)) => 0 
    div(s(x), s(y)) => s(div(minus(x, y), s(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] map#(F, cons(X, Y)) =#> F(X)   
    1] map#(F, cons(X, Y)) =#> map#(F, Y)   
    2] minus#(s(X), s(Y)) =#> minus#(X, Y)   
    3] div#(s(X), s(Y)) =#> div#(minus(X, Y), s(Y))   
    4] div#(s(X), s(Y)) =#> minus#(X, Y)   

  Rules R_0:

    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    minus(X, 0) => X 
    minus(s(X), s(Y)) => minus(X, Y) 
    div(0, s(X)) => 0 
    div(s(X), s(Y)) => s(div(minus(X, Y), s(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 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, 4 
    * 1 : 0, 1 
    * 2 : 2 
    * 3 : 3, 4 
    * 4 : 2 

This graph has the following strongly connected components:

  P_1:

    map#(F, cons(X, Y)) =#> F(X)   
    map#(F, cons(X, Y)) =#> map#(F, Y)   

  P_2:

    minus#(s(X), s(Y)) =#> minus#(X, Y)   

  P_3:

    div#(s(X), s(Y)) =#> div#(minus(X, Y), s(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).

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

  minus(X, 0) => X 
  minus(s(X), s(Y)) => minus(X, Y) 
  div(0, s(X)) => 0 
  div(s(X), s(Y)) => s(div(minus(X, Y), s(Y))) 

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 with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_3, R_1) are:

  minus(X, 0) => X 
  minus(s(X), s(Y)) => minus(X, Y) 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  div#(s(X), s(Y)) >? div#(minus(X, Y), s(Y)) 
  minus(X, 0) >= X 
  minus(s(X), s(Y)) >= minus(X, Y) 

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

The following interpretation satisfies the requirements:

  0 = 0 
  div# = \y0y1.3y0 
  minus = \y0y1.y0 
  s = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[div#(s(_x0), s(_x1))]] = 9 + 9x0 > 3x0 = [[div#(minus(_x0, _x1), s(_x1))]] 
  [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] 
  [[minus(s(_x0), s(_x1))]] = 3 + 3x0 >= x0 = [[minus(_x0, _x1)]] 

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(minus#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(minus#(s(X), s(Y))) = s(X) |> X = nu(minus#(X, Y)) 

By [Kop12, Thm. 7.35], 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_2 ::=

  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_2, minimal, formative).

Thus, the original system is terminating if (P_1, R_2, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_2, 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:

  map#(F, cons(X, Y)) >? F(X) 
  map#(F, cons(X, Y)) >? map#(F, Y) 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.1 + y0 + 2y1 
  map = \G0y1.y1 + 2y1G0(y1) 
  map# = \G0y1.3 + 2y1 + G0(y1) 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 5 + 2x1 + 4x2 + F0(1 + x1 + 2x2) > F0(x1) = [[_F0(_x1)]] 
  [[map#(_F0, cons(_x1, _x2))]] = 5 + 2x1 + 4x2 + F0(1 + x1 + 2x2) > 3 + 2x2 + F0(x2) = [[map#(_F0, _x2)]] 
  [[map(_F0, cons(_x1, _x2))]] = 1 + x1 + 2x2 + 2x1F0(1 + x1 + 2x2) + 2F0(1 + x1 + 2x2) + 4x2F0(1 + x1 + 2x2) >= 1 + 2x2 + 4x2F0(x2) + max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_2) by ({}, R_2).  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.