We consider the system Applicative_05__TreeMap.

  Alphabet:

    cons : [c * b] --> b 
    map : [c -> c * b] --> b 
    nil : [] --> b 
    node : [a * b] --> c 
    treemap : [a -> a] --> c -> c 

  Rules:

    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    treemap(f) node(x, y) => node(f x, map(treemap(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] map#(F, cons(X, Y)) =#> F(X)   
    1] map#(F, cons(X, Y)) =#> map#(F, Y)   
    2] treemap(F) node(X, Y) =#> F(X)   
    3] treemap(F) node(X, Y) =#> map#(treemap(F), Y)   
    4] treemap(F) node(X, Y) =#> treemap#(F)   

  Rules R_0:

    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    treemap(F) node(X, Y) => node(F X, map(treemap(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 : 0, 1, 2, 3, 4 
    * 3 : 0, 1 
    * 4 :  

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)   
    treemap(F) node(X, Y) =#> F(X)   
    treemap(F) node(X, Y) =#> map#(treemap(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).

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_1 ::=

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

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative).

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, 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) 
  treemap(F) node(X, Y) >? F(X) 
  treemap(F) node(X, Y) >? map#(treemap(F), Y) 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  treemap(F) node(X, Y) >= node(F X, map(treemap(F), Y)) 

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}, cons, map, map#, node, treemap}, and the following precedence: @_{o -> o} = map = map# > cons > node = treemap

With these choices, we have:

  1] map#(F, cons(X, Y)) > @_{o -> o}(F, X)  because [2], by definition 
  2] map#*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map# = @_{o -> o}, map# in Mul, [3] and [4], by (Stat) 
  3] F >= F  by (Meta) 
  4] cons(X, Y) > X  because [5], by definition 
  5] cons*(X, Y) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 

  7] map#(F, cons(X, Y)) >= map#(F, Y)  because [8], by (Star) 
  8] map#*(F, cons(X, Y)) >= map#(F, Y)  because map# in Mul, [3] and [9], by (Stat) 
  9] cons(X, Y) > Y  because [10], by definition 
  10] cons*(X, Y) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 

  12] @_{o -> o}(treemap(F), node(X, Y)) > @_{o -> o}(F, X)  because [13], by definition 
  13] @_{o -> o}*(treemap(F), node(X, Y)) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [14] and [17], by (Stat) 
  14] treemap(F) > F  because [15], by definition 
  15] treemap*(F) >= F  because [16], by (Select) 
  16] F >= F  by (Meta) 
  17] node(X, Y) > X  because [18], by definition 
  18] node*(X, Y) >= X  because [19], by (Select) 
  19] X >= X  by (Meta) 

  20] @_{o -> o}(treemap(F), node(X, Y)) >= map#(treemap(F), Y)  because @_{o -> o} = map#, @_{o -> o} in Mul, [21] and [23], by (Fun) 
  21] treemap(F) >= treemap(F)  because treemap in Mul and [22], by (Fun) 
  22] F >= F  by (Meta) 
  23] node(X, Y) >= Y  because [24], by (Star) 
  24] node*(X, Y) >= Y  because [25], by (Select) 
  25] Y >= Y  by (Meta) 

  26] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [27], by (Star) 
  27] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [28] and [29], by (Copy) 
  28] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map = @_{o -> o}, map in Mul, [3] and [4], by (Stat) 
  29] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [3] and [9], by (Stat) 

  30] @_{o -> o}(treemap(F), node(X, Y)) >= node(@_{o -> o}(F, X), map(treemap(F), Y))  because [31], by (Star) 
  31] @_{o -> o}*(treemap(F), node(X, Y)) >= node(@_{o -> o}(F, X), map(treemap(F), Y))  because @_{o -> o} > node, [32] and [36], by (Copy) 
  32] @_{o -> o}*(treemap(F), node(X, Y)) >= @_{o -> o}(F, X)  because [33], by (Select) 
  33] treemap(F) @_{o -> o}*(treemap(F), node(X, Y)) >= @_{o -> o}(F, X)  because [34] 
  34] treemap*(F, @_{o -> o}*(treemap(F), node(X, Y))) >= @_{o -> o}(F, X)  because [35], by (Select) 
  35] @_{o -> o}*(treemap(F), node(X, Y)) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [14] and [17], by (Stat) 
  36] @_{o -> o}*(treemap(F), node(X, Y)) >= map(treemap(F), Y)  because @_{o -> o} = map, @_{o -> o} in Mul, [21] and [37], by (Stat) 
  37] node(X, Y) > Y  because [38], by definition 
  38] node*(X, Y) >= Y  because [25], by (Select) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_1, minimal, formative) by (P_2, R_1, minimal, formative), where P_2 consists of:

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

Thus, the original system is terminating if (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, 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 : 0 

This graph has the following strongly connected components:

  P_3:

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

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_2, R_1, m, f) by (P_3, R_1, m, f).

Thus, the original system is terminating if (P_3, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, 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 [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_3, R_1, minimal, f) by ({}, R_1, 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.