We consider the system Applicative_05__Ex3Lists.

  Alphabet:

    append : [b * b] --> b 
    cons : [a * b] --> b 
    map : [a -> a * b] --> b 
    nil : [] --> b 

  Rules:

    append(nil, x) => x 
    append(cons(x, y), z) => cons(x, append(y, z)) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    append(append(x, y), z) => append(x, append(y, z)) 
    map(f, append(x, y)) => append(map(f, x), map(f, 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] append#(cons(X, Y), Z) =#> append#(Y, Z)   
    1] map#(F, cons(X, Y)) =#> F(X)   
    2] map#(F, cons(X, Y)) =#> map#(F, Y)   
    3] append#(append(X, Y), Z) =#> append#(X, append(Y, Z))   
    4] append#(append(X, Y), Z) =#> append#(Y, Z)   
    5] map#(F, append(X, Y)) =#> append#(map(F, X), map(F, Y))   
    6] map#(F, append(X, Y)) =#> map#(F, X)   
    7] map#(F, append(X, Y)) =#> map#(F, Y)   

  Rules R_0:

    append(nil, X) => X 
    append(cons(X, Y), Z) => cons(X, append(Y, Z)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    append(append(X, Y), Z) => append(X, append(Y, Z)) 
    map(F, append(X, Y)) => append(map(F, X), map(F, 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, 3, 4 
    * 1 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 2 : 1, 2, 5, 6, 7 
    * 3 : 0, 3, 4 
    * 4 : 0, 3, 4 
    * 5 : 0, 3, 4 
    * 6 : 1, 2, 5, 6, 7 
    * 7 : 1, 2, 5, 6, 7 

This graph has the following strongly connected components:

  P_1:

    append#(cons(X, Y), Z) =#> append#(Y, Z)   
    append#(append(X, Y), Z) =#> append#(X, append(Y, Z))   
    append#(append(X, Y), Z) =#> append#(Y, Z)   

  P_2:

    map#(F, cons(X, Y)) =#> F(X)   
    map#(F, cons(X, Y)) =#> map#(F, Y)   
    map#(F, append(X, Y)) =#> map#(F, X)   
    map#(F, append(X, Y)) =#> map#(F, 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) and (P_2, R_0, m, f).

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

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

The following interpretation satisfies the requirements:

  append = \y0y1.2 + y0 + y1 
  cons = \y0y1.2 + y0 + y1 
  map = \G0y1.y1 + G0(y1) + 2y1G0(y1) 
  map# = \G0y1.3 + y1 + 2G0(y1) 
  nil = 0 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 5 + x1 + x2 + 2F0(2 + x1 + x2) > F0(x1) = [[_F0(_x1)]] 
  [[map#(_F0, cons(_x1, _x2))]] = 5 + x1 + x2 + 2F0(2 + x1 + x2) > 3 + x2 + 2F0(x2) = [[map#(_F0, _x2)]] 
  [[map#(_F0, append(_x1, _x2))]] = 5 + x1 + x2 + 2F0(2 + x1 + x2) > 3 + x1 + 2F0(x1) = [[map#(_F0, _x1)]] 
  [[map#(_F0, append(_x1, _x2))]] = 5 + x1 + x2 + 2F0(2 + x1 + x2) > 3 + x2 + 2F0(x2) = [[map#(_F0, _x2)]] 
  [[append(nil, _x0)]] = 2 + x0 >= x0 = [[_x0]] 
  [[append(cons(_x0, _x1), _x2)]] = 4 + x0 + x1 + x2 >= 4 + x0 + x1 + x2 = [[cons(_x0, append(_x1, _x2))]] 
  [[map(_F0, nil)]] = F0(0) >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 2 + x1 + x2 + 2x1F0(2 + x1 + x2) + 2x2F0(2 + x1 + x2) + 5F0(2 + x1 + x2) >= 2 + x2 + F0(x2) + 2x2F0(x2) + max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[append(append(_x0, _x1), _x2)]] = 4 + x0 + x1 + x2 >= 4 + x0 + x1 + x2 = [[append(_x0, append(_x1, _x2))]] 
  [[map(_F0, append(_x1, _x2))]] = 2 + x1 + x2 + 2x1F0(2 + x1 + x2) + 2x2F0(2 + x1 + x2) + 5F0(2 + x1 + x2) >= 2 + x1 + x2 + F0(x1) + F0(x2) + 2x1F0(x1) + 2x2F0(x2) = [[append(map(_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_2, R_0) by ({}, R_0).  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(append#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(append#(cons(X, Y), Z)) = cons(X, Y) |> Y = nu(append#(Y, Z)) 
  nu(append#(append(X, Y), Z)) = append(X, Y) |> X = nu(append#(X, append(Y, Z))) 
  nu(append#(append(X, Y), Z)) = append(X, Y) |> Y = nu(append#(Y, Z)) 

By [Kop12, Thm. 7.35], 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 +++

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