We consider the system Applicative_first_order_05__30.

  Alphabet:

    !faccolon : [a * a] --> a 
    C : [] --> 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:

    !faccolon(!faccolon(!faccolon(!faccolon(C, x), y), z), u) => !faccolon(!faccolon(x, z), !faccolon(!faccolon(!faccolon(x, y), z), u)) 
    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] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   
    1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Z)   
    2] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(!faccolon(X, Y), Z), U)   
    3] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Y), Z)   
    4] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   
    5] map#(F, cons(X, Y)) =#> F(X)   
    6] map#(F, cons(X, Y)) =#> map#(F, Y)   
    7] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    8] filter#(F, cons(X, Y)) =#> F(X)   
    9] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    10] filter2#(false, F, X, Y) =#> filter#(F, Y)   

  Rules R_0:

    !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
    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 : 0, 1, 2, 3, 4 
    * 1 : 0, 1, 2, 3, 4 
    * 2 : 0, 1, 2, 3, 4 
    * 3 : 0, 1, 2, 3, 4 
    * 4 : 0, 1, 2, 3, 4 
    * 5 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 
    * 6 : 5, 6 
    * 7 : 9, 10 
    * 8 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 
    * 9 : 7, 8 
    * 10 : 7, 8 

This graph has the following strongly connected components:

  P_1:

    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   
    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Z)   
    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(!faccolon(X, Y), Z), U)   
    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Y), Z)   
    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   

  P_2:

    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) 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).

The formative rules of (P_2, 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_2, R_0, minimal, formative) by (P_2, R_1, minimal, formative).

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

We consider the dependency pair problem (P_2, 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_2, R_1, minimal, formative) by (P_3, R_1, minimal, formative), where P_3 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) and (P_3, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, 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_4:

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

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_3, R_1, m, f) by (P_4, R_1, m, f).

Thus, the original system is terminating if each of (P_1, 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 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_4, 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 (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 ::=

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

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:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(!faccolon(X, Y), Z), U) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Y), Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Y) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[!faccolon#(x_1, x_2)]] = !faccolon#(x_1) 

We choose Lex = {!faccolon} and Mul = {!faccolon#, C}, and the following precedence: !faccolon > !faccolon# > C

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(!faccolon(!faccolon(X, Y), Z)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Y)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z))  because !faccolon# in Mul and [2], by (Fun) 
  2] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [3] and [8], by (Fun) 
  3] !faccolon(!faccolon(C, X), Y) >= X  because [4], by (Star) 
  4] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  5] !faccolon(C, X) >= X  because [6], by (Star) 
  6] !faccolon*(C, X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] Z >= Z  by (Meta) 

  9] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because !faccolon# in Mul and [10], by (Fun) 
  10] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= X  because [11], by (Star) 
  11] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [3], by (Select) 

  12] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(!faccolon(!faccolon(X, Y), Z))  because [13], by definition 
  13] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(!faccolon(X, Y), Z))  because [14], by (Select) 
  14] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(!faccolon(!faccolon(X, Y), Z))  because [15], by (Star) 
  15] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(!faccolon(!faccolon(X, Y), Z))  because !faccolon > !faccolon# and [16], by (Copy) 
  16] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [17], [24] and [29], by (Stat) 
  17] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [18], by definition 
  18] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [19], [21] and [22], by (Stat) 
  19] !faccolon(C, X) > X  because [20], by definition 
  20] !faccolon*(C, X) >= X  because [7], by (Select) 
  21] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  22] !faccolon*(!faccolon(C, X), Y) >= Y  because [23], by (Select) 
  23] Y >= Y  by (Meta) 
  24] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [25], [26] and [27], by (Stat) 
  25] !faccolon(!faccolon(C, X), Y) > X  because [21], by definition 
  26] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [3], by (Select) 
  27] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Y  because [28], by (Select) 
  28] !faccolon(!faccolon(C, X), Y) >= Y  because [22], by (Star) 
  29] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [8], by (Select) 

  30] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Y))  because !faccolon# in Mul and [31], by (Fun) 
  31] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [24], by (Star) 

  32] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because !faccolon# in Mul and [10], by (Fun) 

  33] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [34], by (Star) 
  34] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [35], [37] and [38], by (Stat) 
  35] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [36], by definition 
  36] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [25], [26] and [29], by (Stat) 
  37] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [2], by (Select) 
  38] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [39], [40] and [45], by (Stat) 
  39] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [16], by definition 
  40] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [41], by (Select) 
  41] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [42] and [8], by (Fun) 
  42] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [43] and [44], by (Fun) 
  43] !faccolon(C, X) >= X  because [20], by (Star) 
  44] Y >= Y  by (Meta) 
  45] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [46], by (Select) 
  46] U >= U  by (Meta) 

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_5, R_2, minimal, formative), where P_5 consists of:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Z)   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Y), Z)   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   

Thus, the original system is terminating if (P_5, R_2, minimal, formative) is finite.

We consider the dependency pair problem (P_5, 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:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Y), Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Y) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

We choose Lex = {!faccolon, !faccolon#} and Mul = {C}, and the following precedence: !faccolon = !faccolon# > C

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) > !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [2], by definition 
  2] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [3], [14] and [20], by (Stat) 
  3] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [4], by definition 
  4] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [5], [10] and [12], by (Stat) 
  5] !faccolon(!faccolon(C, X), Y) > X  because [6], by definition 
  6] !faccolon*(!faccolon(C, X), Y) >= X  because [7], by (Select) 
  7] !faccolon(C, X) >= X  because [8], by (Star) 
  8] !faccolon*(C, X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [11], by (Select) 
  11] !faccolon(!faccolon(C, X), Y) >= X  because [6], by (Star) 
  12] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [13], by (Select) 
  13] Z >= Z  by (Meta) 
  14] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because !faccolon# = !faccolon, [15], [16] and [18], by (Stat) 
  15] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > X  because [10], by definition 
  16] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= X  because [17], by (Select) 
  17] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= X  because [10], by (Star) 
  18] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= Z  because [19], by (Select) 
  19] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [12], by (Star) 
  20] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because !faccolon# = !faccolon, [21], [33] and [40], by (Stat) 
  21] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [22], by definition 
  22] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [23], [29] and [12], by (Stat) 
  23] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [24], by definition 
  24] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [25], [6] and [27], by (Stat) 
  25] !faccolon(C, X) > X  because [26], by definition 
  26] !faccolon*(C, X) >= X  because [9], by (Select) 
  27] !faccolon*(!faccolon(C, X), Y) >= Y  because [28], by (Select) 
  28] Y >= Y  by (Meta) 
  29] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [30], by (Select) 
  30] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [31] and [32], by (Fun) 
  31] !faccolon(C, X) >= X  because [26], by (Star) 
  32] Y >= Y  by (Meta) 
  33] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because !faccolon# = !faccolon, [34], [35] and [18], by (Stat) 
  34] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Y)  because [29], by definition 
  35] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Y)  because !faccolon# = !faccolon, [15], [16] and [36], by (Stat) 
  36] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= Y  because [37], by (Select) 
  37] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= Y  because [38], by (Star) 
  38] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Y  because [39], by (Select) 
  39] !faccolon(!faccolon(C, X), Y) >= Y  because [27], by (Star) 
  40] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [41], by (Select) 
  41] U >= U  by (Meta) 

  42] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Z)  because [43], by (Star) 
  43] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Z)  because [15], [16] and [18], by (Stat) 

  44] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(!faccolon(X, Y), Z)  because [45], by (Star) 
  45] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(!faccolon(X, Y), Z)  because [34], [35] and [18], by (Stat) 

  46] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Y)  because [47], by (Star) 
  47] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Y)  because [15], [16] and [36], by (Stat) 

  48] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [49], by (Star) 
  49] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [3], [50] and [53], by (Stat) 
  50] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [15], [51] and [52], by (Stat) 
  51] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= X  because [17], by (Select) 
  52] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= Z  because [19], by (Select) 
  53] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [21], [54] and [57], by (Stat) 
  54] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [34], [55] and [52], by (Stat) 
  55] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Y)  because [15], [51] and [56], by (Stat) 
  56] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= Y  because [37], by (Select) 
  57] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [41], by (Select) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_2, minimal, formative) by (P_6, R_2, minimal, formative), where P_6 consists of:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Z)   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Y), Z)   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   

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 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:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Y), Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Y) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

We choose Lex = {!faccolon} and Mul = {!faccolon#, C}, and the following precedence: !faccolon > C > !faccolon#

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Z)  because [2], by (Star) 
  2] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Z)  because [3], by (Select) 
  3] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X, Z)  because [4], by (Star) 
  4] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X, Z)  because !faccolon > !faccolon#, [5] and [11], by (Copy) 
  5] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [6], by (Select) 
  6] !faccolon(!faccolon(C, X), Y) >= X  because [7], by (Star) 
  7] !faccolon*(!faccolon(C, X), Y) >= X  because [8], by (Select) 
  8] !faccolon(C, X) >= X  because [9], by (Star) 
  9] !faccolon*(C, X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [12], by (Select) 
  12] Z >= Z  by (Meta) 

  13] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) > !faccolon#(!faccolon(X, Y), Z)  because [14], by definition 
  14] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(!faccolon(X, Y), Z)  because [15], by (Select) 
  15] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(!faccolon(X, Y), Z)  because [16], by (Star) 
  16] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(!faccolon(X, Y), Z)  because !faccolon > !faccolon#, [17] and [11], by (Copy) 
  17] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [18], [5] and [20], by (Stat) 
  18] !faccolon(!faccolon(C, X), Y) > X  because [19], by definition 
  19] !faccolon*(!faccolon(C, X), Y) >= X  because [8], by (Select) 
  20] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Y  because [21], by (Select) 
  21] !faccolon(!faccolon(C, X), Y) >= Y  because [22], by (Star) 
  22] !faccolon*(!faccolon(C, X), Y) >= Y  because [23], by (Select) 
  23] Y >= Y  by (Meta) 

  24] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Y)  because [25], by (Star) 
  25] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Y)  because [26], by (Select) 
  26] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X, Y)  because [27], by (Star) 
  27] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X, Y)  because !faccolon > !faccolon#, [5] and [20], by (Copy) 

  28] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [29], by (Star) 
  29] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [30], [32] and [36], by (Stat) 
  30] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [31], by definition 
  31] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [18], [5] and [11], by (Stat) 
  32] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [33], by (Select) 
  33] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [34] and [35], by (Fun) 
  34] !faccolon(!faccolon(C, X), Y) >= X  because [19], by (Star) 
  35] Z >= Z  by (Meta) 
  36] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [37], [44] and [49], by (Stat) 
  37] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [38], by definition 
  38] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [39], [17] and [11], by (Stat) 
  39] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [40], by definition 
  40] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [41], [19] and [43], by (Stat) 
  41] !faccolon(C, X) > X  because [42], by definition 
  42] !faccolon*(C, X) >= X  because [10], by (Select) 
  43] !faccolon*(!faccolon(C, X), Y) >= Y  because [23], by (Select) 
  44] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [45], by (Select) 
  45] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [46] and [35], by (Fun) 
  46] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [47] and [48], by (Fun) 
  47] !faccolon(C, X) >= X  because [42], by (Star) 
  48] Y >= Y  by (Meta) 
  49] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [50], by (Select) 
  50] U >= U  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_6, R_2, minimal, formative) by (P_7, R_2, minimal, formative), where P_7 consists of:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Z)   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   

Thus, the original system is terminating if (P_7, R_2, minimal, formative) is finite.

We consider the dependency pair problem (P_7, R_2, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(!faccolon#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(!faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U)) = !faccolon(!faccolon(!faccolon(C, X), Y), Z) |> X = nu(!faccolon#(X, Z)) 
  nu(!faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U)) = !faccolon(!faccolon(!faccolon(C, X), Y), Z) |> X = nu(!faccolon#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_7, 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.