We consider the system Applicative_first_order_05__13.

  Alphabet:

    !facplus : [a * a] --> a 
    !factimes : [a * a] --> a 
    cons : [c * d] --> d 
    false : [] --> b 
    filter : [c -> b * d] --> d 
    filter2 : [b * c -> b * c * d] --> d 
    map : [c -> c * d] --> d 
    nil : [] --> d 
    true : [] --> b 

  Rules:

    !factimes(x, !facplus(y, z)) => !facplus(!factimes(x, y), !factimes(x, z)) 
    !factimes(!facplus(x, y), z) => !facplus(!factimes(z, x), !factimes(z, y)) 
    !factimes(!factimes(x, y), z) => !factimes(x, !factimes(y, z)) 
    !facplus(!facplus(x, y), z) => !facplus(x, !facplus(y, z)) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
    filter2(true, f, x, y) => cons(x, filter(f, y)) 
    filter2(false, f, x, y) => filter(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] !factimes#(X, !facplus(Y, Z)) =#> !facplus#(!factimes(X, Y), !factimes(X, Z))   
    1] !factimes#(X, !facplus(Y, Z)) =#> !factimes#(X, Y)   
    2] !factimes#(X, !facplus(Y, Z)) =#> !factimes#(X, Z)   
    3] !factimes#(!facplus(X, Y), Z) =#> !facplus#(!factimes(Z, X), !factimes(Z, Y))   
    4] !factimes#(!facplus(X, Y), Z) =#> !factimes#(Z, X)   
    5] !factimes#(!facplus(X, Y), Z) =#> !factimes#(Z, Y)   
    6] !factimes#(!factimes(X, Y), Z) =#> !factimes#(X, !factimes(Y, Z))   
    7] !factimes#(!factimes(X, Y), Z) =#> !factimes#(Y, Z)   
    8] !facplus#(!facplus(X, Y), Z) =#> !facplus#(X, !facplus(Y, Z))   
    9] !facplus#(!facplus(X, Y), Z) =#> !facplus#(Y, Z)   
    10] map#(F, cons(X, Y)) =#> F(X)   
    11] map#(F, cons(X, Y)) =#> map#(F, Y)   
    12] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    13] filter#(F, cons(X, Y)) =#> F(X)   
    14] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    15] filter2#(false, F, X, Y) =#> filter#(F, Y)   

  Rules R_0:

    !factimes(X, !facplus(Y, Z)) => !facplus(!factimes(X, Y), !factimes(X, Z)) 
    !factimes(!facplus(X, Y), Z) => !facplus(!factimes(Z, X), !factimes(Z, Y)) 
    !factimes(!factimes(X, Y), Z) => !factimes(X, !factimes(Y, Z)) 
    !facplus(!facplus(X, Y), Z) => !facplus(X, !facplus(Y, Z)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(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 : 8, 9 
    * 1 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 2 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 3 : 8, 9 
    * 4 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 5 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 6 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 7 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 8 : 8, 9 
    * 9 : 8, 9 
    * 10 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 
    * 11 : 10, 11 
    * 12 : 14, 15 
    * 13 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 
    * 14 : 12, 13 
    * 15 : 12, 13 

This graph has the following strongly connected components:

  P_1:

    !factimes#(X, !facplus(Y, Z)) =#> !factimes#(X, Y)   
    !factimes#(X, !facplus(Y, Z)) =#> !factimes#(X, Z)   
    !factimes#(!facplus(X, Y), Z) =#> !factimes#(Z, X)   
    !factimes#(!facplus(X, Y), Z) =#> !factimes#(Z, Y)   
    !factimes#(!factimes(X, Y), Z) =#> !factimes#(X, !factimes(Y, Z))   
    !factimes#(!factimes(X, Y), Z) =#> !factimes#(Y, Z)   

  P_2:

    !facplus#(!facplus(X, Y), Z) =#> !facplus#(X, !facplus(Y, Z))   
    !facplus#(!facplus(X, Y), Z) =#> !facplus#(Y, Z)   

  P_3:

    map#(F, cons(X, Y)) =#> F(X)   
    map#(F, cons(X, Y)) =#> map#(F, Y)   
    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter#(F, cons(X, Y)) =#> F(X)   
    filter2#(true, F, X, Y) =#> filter#(F, Y)   
    filter2#(false, F, X, Y) =#> filter#(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), (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 ::=

  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
  filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) => filter(F, 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 [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) 
  filter#(F, cons(X, Y)) >? filter2#(F X, F, X, Y) 
  filter#(F, cons(X, Y)) >? F(X) 
  filter2#(true, F, X, Y) >? filter#(F, Y) 
  filter2#(false, F, X, Y) >? filter#(F, Y) 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.2 + y1 + 2y0 
  false = 3 
  filter = \G0y1.y1 + 2y1G0(y1) 
  filter2 = \y0G1y2y3.2 + y3 + 2y2 + 2y3G1(y3) 
  filter2# = \y0G1y2y3.y3 + 2y3G1(y3) + 2G1(y3) 
  filter# = \G0y1.y1 + 2y1G0(y1) + 2G0(y1) 
  map = \G0y1.2y1 + y1G0(y1) 
  map# = \G0y1.3 + 2G0(y1) + y1G0(y1) 
  true = 3 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 3 + 2x1F0(2 + x2 + 2x1) + 4F0(2 + x2 + 2x1) + x2F0(2 + x2 + 2x1) > F0(x1) = [[_F0(_x1)]] 
  [[map#(_F0, cons(_x1, _x2))]] = 3 + 2x1F0(2 + x2 + 2x1) + 4F0(2 + x2 + 2x1) + x2F0(2 + x2 + 2x1) >= 3 + 2F0(x2) + x2F0(x2) = [[map#(_F0, _x2)]] 
  [[filter#(_F0, cons(_x1, _x2))]] = 2 + x2 + 2x1 + 2x2F0(2 + x2 + 2x1) + 4x1F0(2 + x2 + 2x1) + 6F0(2 + x2 + 2x1) > x2 + 2x2F0(x2) + 2F0(x2) = [[filter2#(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter#(_F0, cons(_x1, _x2))]] = 2 + x2 + 2x1 + 2x2F0(2 + x2 + 2x1) + 4x1F0(2 + x2 + 2x1) + 6F0(2 + x2 + 2x1) > F0(x1) = [[_F0(_x1)]] 
  [[filter2#(true, _F0, _x1, _x2)]] = x2 + 2x2F0(x2) + 2F0(x2) >= x2 + 2x2F0(x2) + 2F0(x2) = [[filter#(_F0, _x2)]] 
  [[filter2#(false, _F0, _x1, _x2)]] = x2 + 2x2F0(x2) + 2F0(x2) >= x2 + 2x2F0(x2) + 2F0(x2) = [[filter#(_F0, _x2)]] 
  [[map(_F0, cons(_x1, _x2))]] = 4 + 2x2 + 4x1 + 2x1F0(2 + x2 + 2x1) + 2F0(2 + x2 + 2x1) + x2F0(2 + x2 + 2x1) >= 2 + 2x2 + x2F0(x2) + 2max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, cons(_x1, _x2))]] = 2 + x2 + 2x1 + 2x2F0(2 + x2 + 2x1) + 4x1F0(2 + x2 + 2x1) + 4F0(2 + x2 + 2x1) >= 2 + x2 + 2x1 + 2x2F0(x2) = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 2 + x2 + 2x1 + 2x2F0(x2) >= 2 + x2 + 2x1 + 2x2F0(x2) = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 2 + x2 + 2x1 + 2x2F0(x2) >= x2 + 2x2F0(x2) = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_1, minimal, formative) by (P_4, R_1, minimal, formative), where P_4 consists of:

  map#(F, cons(X, Y)) =#> map#(F, Y)   
  filter2#(true, F, X, Y) =#> filter#(F, Y)   
  filter2#(false, F, X, Y) =#> filter#(F, Y)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_4, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_4, R_1, 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 :  

This graph has the following strongly connected components:

  P_5:

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

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_4, R_1, m, f) by (P_5, R_1, 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_5, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_1, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(map#) = 2 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_5, R_1, minimal, f) by ({}, R_1, 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(!facplus#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(!facplus#(!facplus(X, Y), Z)) = !facplus(X, Y) |> X = nu(!facplus#(X, !facplus(Y, Z))) 
  nu(!facplus#(!facplus(X, Y), Z)) = !facplus(X, Y) |> Y = nu(!facplus#(Y, Z)) 

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 ::=

  !factimes(X, !facplus(Y, Z)) => !facplus(!factimes(X, Y), !factimes(X, Z)) 
  !factimes(!facplus(X, Y), Z) => !facplus(!factimes(Z, X), !factimes(Z, Y)) 
  !factimes(!factimes(X, Y), Z) => !factimes(X, !factimes(Y, Z)) 
  !facplus(!facplus(X, Y), Z) => !facplus(X, !facplus(Y, Z)) 

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

  !factimes#(X, !facplus(Y, Z)) >? !factimes#(X, Y) 
  !factimes#(X, !facplus(Y, Z)) >? !factimes#(X, Z) 
  !factimes#(!facplus(X, Y), Z) >? !factimes#(Z, X) 
  !factimes#(!facplus(X, Y), Z) >? !factimes#(Z, Y) 
  !factimes#(!factimes(X, Y), Z) >? !factimes#(X, !factimes(Y, Z)) 
  !factimes#(!factimes(X, Y), Z) >? !factimes#(Y, Z) 
  !factimes(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z)) 
  !factimes(!facplus(X, Y), Z) >= !facplus(!factimes(Z, X), !factimes(Z, Y)) 
  !factimes(!factimes(X, Y), Z) >= !factimes(X, !factimes(Y, Z)) 
  !facplus(!facplus(X, Y), Z) >= !facplus(X, !facplus(Y, Z)) 

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

The following interpretation satisfies the requirements:

  !facplus = \y0y1.2 + y1 + 2y0 
  !factimes = \y0y1.y0 + y1 + 2y0y1 
  !factimes# = \y0y1.y0 + y1 + 2y0y1 

Using this interpretation, the requirements translate to:

  [[!factimes#(_x0, !facplus(_x1, _x2))]] = 2 + x2 + 2x0x2 + 2x1 + 4x0x1 + 5x0 > x0 + x1 + 2x0x1 = [[!factimes#(_x0, _x1)]] 
  [[!factimes#(_x0, !facplus(_x1, _x2))]] = 2 + x2 + 2x0x2 + 2x1 + 4x0x1 + 5x0 > x0 + x2 + 2x0x2 = [[!factimes#(_x0, _x2)]] 
  [[!factimes#(!facplus(_x0, _x1), _x2)]] = 2 + x1 + 2x0 + 2x1x2 + 4x0x2 + 5x2 > x0 + x2 + 2x0x2 = [[!factimes#(_x2, _x0)]] 
  [[!factimes#(!facplus(_x0, _x1), _x2)]] = 2 + x1 + 2x0 + 2x1x2 + 4x0x2 + 5x2 > x1 + x2 + 2x1x2 = [[!factimes#(_x2, _x1)]] 
  [[!factimes#(!factimes(_x0, _x1), _x2)]] = x0 + x1 + x2 + 2x0x1 + 2x0x2 + 2x1x2 + 4x0x1x2 >= x0 + x1 + x2 + 2x0x1 + 2x0x2 + 2x1x2 + 4x0x1x2 = [[!factimes#(_x0, !factimes(_x1, _x2))]] 
  [[!factimes#(!factimes(_x0, _x1), _x2)]] = x0 + x1 + x2 + 2x0x1 + 2x0x2 + 2x1x2 + 4x0x1x2 >= x1 + x2 + 2x1x2 = [[!factimes#(_x1, _x2)]] 
  [[!factimes(_x0, !facplus(_x1, _x2))]] = 2 + x2 + 2x0x2 + 2x1 + 4x0x1 + 5x0 >= 2 + x2 + 2x0x2 + 2x1 + 3x0 + 4x0x1 = [[!facplus(!factimes(_x0, _x1), !factimes(_x0, _x2))]] 
  [[!factimes(!facplus(_x0, _x1), _x2)]] = 2 + x1 + 2x0 + 2x1x2 + 4x0x2 + 5x2 >= 2 + x1 + 2x0 + 2x1x2 + 3x2 + 4x0x2 = [[!facplus(!factimes(_x2, _x0), !factimes(_x2, _x1))]] 
  [[!factimes(!factimes(_x0, _x1), _x2)]] = x0 + x1 + x2 + 2x0x1 + 2x0x2 + 2x1x2 + 4x0x1x2 >= x0 + x1 + x2 + 2x0x1 + 2x0x2 + 2x1x2 + 4x0x1x2 = [[!factimes(_x0, !factimes(_x1, _x2))]] 
  [[!facplus(!facplus(_x0, _x1), _x2)]] = 6 + x2 + 2x1 + 4x0 >= 4 + x2 + 2x0 + 2x1 = [[!facplus(_x0, !facplus(_x1, _x2))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_2, minimal, formative) by (P_6, R_2, minimal, formative), where P_6 consists of:

  !factimes#(!factimes(X, Y), Z) =#> !factimes#(X, !factimes(Y, Z))   
  !factimes#(!factimes(X, Y), Z) =#> !factimes#(Y, Z)   

Thus, the original system is terminating if (P_6, R_2, minimal, formative) is finite.

We consider the dependency pair problem (P_6, R_2, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(!factimes#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(!factimes#(!factimes(X, Y), Z)) = !factimes(X, Y) |> X = nu(!factimes#(X, !factimes(Y, Z))) 
  nu(!factimes#(!factimes(X, Y), Z)) = !factimes(X, Y) |> Y = nu(!factimes#(Y, Z)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_6, R_2, minimal, f) by ({}, R_2, 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.