We consider the system Applicative_05__Ex9Maps.

  Alphabet:

    cons : [d * e] --> e 
    g : [] --> b 
    map!fac62201 : [d -> d * e] --> e 
    map!fac62202 : [d -> a -> d * a * e] --> e 
    map!fac62203 : [b -> d -> c -> d * b * c * e] --> e 

  Rules:

    map!fac62201(f, cons(x, y)) => cons(f x, map!fac62201(f, y)) 
    map!fac62202(f, x, cons(y, z)) => cons(f y x, map!fac62202(f, x, z)) 
    map!fac62203(f, g, x, cons(y, z)) => cons(f g y x, map!fac62203(f, g, x, z)) 

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!fac62201#(F, cons(X, Y)) =#> F(X)   
    1] map!fac62201#(F, cons(X, Y)) =#> map!fac62201#(F, Y)   
    2] map!fac62202#(F, X, cons(Y, Z)) =#> F(Y, X)   
    3] map!fac62202#(F, X, cons(Y, Z)) =#> map!fac62202#(F, X, Z)   
    4] map!fac62203#(F, g, X, cons(Y, Z)) =#> F(g, Y, X)   
    5] map!fac62203#(F, g, X, cons(Y, Z)) =#> map!fac62203#(F, g, X, Z)   

  Rules R_0:

    map!fac62201(F, cons(X, Y)) => cons(F X, map!fac62201(F, Y)) 
    map!fac62202(F, X, cons(Y, Z)) => cons(F Y X, map!fac62202(F, X, Z)) 
    map!fac62203(F, g, X, cons(Y, Z)) => cons(F g Y X, map!fac62203(F, g, X, Z)) 

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 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!fac62201#(F, cons(X, Y)) >? F(X) 
  map!fac62201#(F, cons(X, Y)) >? map!fac62201#(F, Y) 
  map!fac62202#(F, X, cons(Y, Z)) >? F(Y, X) 
  map!fac62202#(F, X, cons(Y, Z)) >? map!fac62202#(F, X, Z) 
  map!fac62203#(F, g, X, cons(Y, Z)) >? F(g, Y, X) 
  map!fac62203#(F, g, X, cons(Y, Z)) >? map!fac62203#(F, g, X, Z) 
  map!fac62201(F, cons(X, Y)) >= cons(F X, map!fac62201(F, Y)) 
  map!fac62202(F, X, cons(Y, Z)) >= cons(F Y X, map!fac62202(F, X, Z)) 
  map!fac62203(F, g, X, cons(Y, Z)) >= cons(F g Y X, map!fac62203(F, g, X, Z)) 

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 = {} and Mul = {@_{o -> o -> o -> o}, @_{o -> o -> o}, @_{o -> o}, cons, g, map!fac62201, map!fac62201#, map!fac62202, map!fac62202#, map!fac62203, map!fac62203#}, and the following precedence: map!fac62201 > map!fac62201# > map!fac62202 > map!fac62202# > map!fac62203# > map!fac62203 > g > cons > @_{o -> o -> o -> o} > @_{o -> o -> o} > @_{o -> o}

With these choices, we have:

  1] map!fac62201#(F, cons(X, Y)) > @_{o -> o}(F, X)  because [2], by definition 
  2] map!fac62201#*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map!fac62201# > @_{o -> o}, [3] and [5], by (Copy) 
  3] map!fac62201#*(F, cons(X, Y)) >= F  because [4], by (Select) 
  4] F >= F  by (Meta) 
  5] map!fac62201#*(F, cons(X, Y)) >= X  because [6], by (Select) 
  6] cons(X, Y) >= X  because [7], by (Star) 
  7] cons*(X, Y) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

  9] map!fac62201#(F, cons(X, Y)) >= map!fac62201#(F, Y)  because map!fac62201# in Mul, [10] and [11], by (Fun) 
  10] F >= F  by (Meta) 
  11] cons(X, Y) >= Y  because [12], by (Star) 
  12] cons*(X, Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 

  14] map!fac62202#(F, X, cons(Y, Z)) > @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because [15], by definition 
  15] map!fac62202#*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202# > @_{o -> o}, [16] and [23], by (Copy) 
  16] map!fac62202#*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202# > @_{o -> o -> o}, [17] and [19], by (Copy) 
  17] map!fac62202#*(F, X, cons(Y, Z)) >= F  because [18], by (Select) 
  18] F >= F  by (Meta) 
  19] map!fac62202#*(F, X, cons(Y, Z)) >= Y  because [20], by (Select) 
  20] cons(Y, Z) >= Y  because [21], by (Star) 
  21] cons*(Y, Z) >= Y  because [22], by (Select) 
  22] Y >= Y  by (Meta) 
  23] map!fac62202#*(F, X, cons(Y, Z)) >= X  because [24], by (Select) 
  24] X >= X  by (Meta) 

  25] map!fac62202#(F, X, cons(Y, Z)) > map!fac62202#(F, X, Z)  because [26], by definition 
  26] map!fac62202#*(F, X, cons(Y, Z)) >= map!fac62202#(F, X, Z)  because map!fac62202# in Mul, [27], [28] and [29], by (Stat) 
  27] F >= F  by (Meta) 
  28] X >= X  by (Meta) 
  29] cons(Y, Z) > Z  because [30], by definition 
  30] cons*(Y, Z) >= Z  because [31], by (Select) 
  31] Z >= Z  by (Meta) 

  32] map!fac62203#(F, g, X, cons(Y, Z)) > @_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y), X)  because [33], by definition 
  33] map!fac62203#*(F, g, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y), X)  because map!fac62203# > @_{o -> o}, [34] and [43], by (Copy) 
  34] map!fac62203#*(F, g, X, cons(Y, Z)) >= @_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y)  because map!fac62203# > @_{o -> o -> o}, [35] and [39], by (Copy) 
  35] map!fac62203#*(F, g, X, cons(Y, Z)) >= @_{o -> o -> o -> o}(F, g)  because map!fac62203# > @_{o -> o -> o -> o}, [36] and [38], by (Copy) 
  36] map!fac62203#*(F, g, X, cons(Y, Z)) >= F  because [37], by (Select) 
  37] F >= F  by (Meta) 
  38] map!fac62203#*(F, g, X, cons(Y, Z)) >= g  because map!fac62203# > g, by (Copy) 
  39] map!fac62203#*(F, g, X, cons(Y, Z)) >= Y  because [40], by (Select) 
  40] cons(Y, Z) >= Y  because [41], by (Star) 
  41] cons*(Y, Z) >= Y  because [42], by (Select) 
  42] Y >= Y  by (Meta) 
  43] map!fac62203#*(F, g, X, cons(Y, Z)) >= X  because [44], by (Select) 
  44] X >= X  by (Meta) 

  45] map!fac62203#(F, g, X, cons(Y, Z)) >= map!fac62203#(F, g, X, Z)  because [46], by (Star) 
  46] map!fac62203#*(F, g, X, cons(Y, Z)) >= map!fac62203#(F, g, X, Z)  because map!fac62203# in Mul, [47], [48], [49] and [50], by (Stat) 
  47] F >= F  by (Meta) 
  48] g >= g  by (Fun) 
  49] X >= X  by (Meta) 
  50] cons(Y, Z) > Z  because [51], by definition 
  51] cons*(Y, Z) >= Z  because [52], by (Select) 
  52] Z >= Z  by (Meta) 

  53] map!fac62201(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because [54], by (Star) 
  54] map!fac62201*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because map!fac62201 > cons, [55] and [58], by (Copy) 
  55] map!fac62201*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map!fac62201 > @_{o -> o}, [56] and [57], by (Copy) 
  56] map!fac62201*(F, cons(X, Y)) >= F  because [10], by (Select) 
  57] map!fac62201*(F, cons(X, Y)) >= X  because [6], by (Select) 
  58] map!fac62201*(F, cons(X, Y)) >= map!fac62201(F, Y)  because map!fac62201 in Mul, [10] and [59], by (Stat) 
  59] cons(X, Y) > Y  because [60], by definition 
  60] cons*(X, Y) >= Y  because [13], by (Select) 

  61] map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because [62], by (Star) 
  62] map!fac62202*(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because map!fac62202 > cons, [63] and [68], by (Copy) 
  63] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202 > @_{o -> o}, [64] and [67], by (Copy) 
  64] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202 > @_{o -> o -> o}, [65] and [66], by (Copy) 
  65] map!fac62202*(F, X, cons(Y, Z)) >= F  because [27], by (Select) 
  66] map!fac62202*(F, X, cons(Y, Z)) >= Y  because [20], by (Select) 
  67] map!fac62202*(F, X, cons(Y, Z)) >= X  because [28], by (Select) 
  68] map!fac62202*(F, X, cons(Y, Z)) >= map!fac62202(F, X, Z)  because map!fac62202 in Mul, [27], [28] and [29], by (Stat) 

  69] map!fac62203(F, g, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y), X), map!fac62203(F, g, X, Z))  because [70], by (Star) 
  70] map!fac62203*(F, g, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y), X), map!fac62203(F, g, X, Z))  because map!fac62203 > cons, [71] and [78], by (Copy) 
  71] map!fac62203*(F, g, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y), X)  because map!fac62203 > @_{o -> o}, [72] and [77], by (Copy) 
  72] map!fac62203*(F, g, X, cons(Y, Z)) >= @_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y)  because map!fac62203 > @_{o -> o -> o}, [73] and [76], by (Copy) 
  73] map!fac62203*(F, g, X, cons(Y, Z)) >= @_{o -> o -> o -> o}(F, g)  because map!fac62203 > @_{o -> o -> o -> o}, [74] and [75], by (Copy) 
  74] map!fac62203*(F, g, X, cons(Y, Z)) >= F  because [47], by (Select) 
  75] map!fac62203*(F, g, X, cons(Y, Z)) >= g  because map!fac62203 > g, by (Copy) 
  76] map!fac62203*(F, g, X, cons(Y, Z)) >= Y  because [40], by (Select) 
  77] map!fac62203*(F, g, X, cons(Y, Z)) >= X  because [49], by (Select) 
  78] map!fac62203*(F, g, X, cons(Y, Z)) >= map!fac62203(F, g, X, Z)  because map!fac62203 in Mul, [47], [48], [49] and [50], by (Stat) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_1, R_0, minimal, formative), where P_1 consists of:

  map!fac62201#(F, cons(X, Y)) =#> map!fac62201#(F, Y)   
  map!fac62203#(F, g, X, cons(Y, Z)) =#> map!fac62203#(F, g, X, Z)   

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 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_2:

    map!fac62201#(F, cons(X, Y)) =#> map!fac62201#(F, Y)   

  P_3:

    map!fac62203#(F, g, X, cons(Y, Z)) =#> map!fac62203#(F, g, X, Z)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_1, R_0, m, f) by (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (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).

We apply the subterm criterion with the following projection function:

  nu(map!fac62203#) = 4 

Thus, we can orient the dependency pairs as follows:

  nu(map!fac62203#(F, g, X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(map!fac62203#(F, g, X, Z)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_3, 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_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(map!fac62201#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(map!fac62201#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map!fac62201#(F, Y)) 

By [FuhKop19, Thm. 61], 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.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.