We consider the system plode.

  Alphabet:

    cons : [nat * list] --> list 
    explode : [list * nat -> nat * nat] --> nat 
    implode : [list * nat -> nat * nat] --> nat 
    nil : [] --> list 
    op : [nat -> nat * nat -> nat] --> nat -> nat 

  Rules:

    op(f, g) x => f (g x) 
    implode(nil, f, x) => x 
    implode(cons(x, y), f, z) => implode(y, f, f z) 
    explode(nil, f, x) => x 
    explode(cons(x, y), f, z) => explode(y, op(f, f), f 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] op(F, G) X =#> F(G X)   
    1] op(F, G) X =#> G(X)   
    2] implode#(cons(X, Y), F, Z) =#> implode#(Y, F, F Z)   
    3] implode#(cons(X, Y), F, Z) =#> F(Z)   
    4] explode#(cons(X, Y), F, Z) =#> explode#(Y, op(F, F), F Z)   
    5] explode#(cons(X, Y), F, Z) =#> op#(F, F)   
    6] explode#(cons(X, Y), F, Z) =#> F(Z)   

  Rules R_0:

    op(F, G) X => F (G X) 
    implode(nil, F, X) => X 
    implode(cons(X, Y), F, Z) => implode(Y, F, F Z) 
    explode(nil, F, X) => X 
    explode(cons(X, Y), F, Z) => explode(Y, op(F, F), F 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 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, 5, 6 
    * 1 : 0, 1, 2, 3, 4, 5, 6 
    * 2 : 2, 3 
    * 3 : 0, 1, 2, 3, 4, 5, 6 
    * 4 : 4, 5, 6 
    * 5 :  
    * 6 : 0, 1, 2, 3, 4, 5, 6 

This graph has the following strongly connected components:

  P_1:

    op(F, G) X =#> F(G X)   
    op(F, G) X =#> G(X)   
    implode#(cons(X, Y), F, Z) =#> implode#(Y, F, F Z)   
    implode#(cons(X, Y), F, Z) =#> F(Z)   
    explode#(cons(X, Y), F, Z) =#> explode#(Y, op(F, F), F Z)   
    explode#(cons(X, Y), F, Z) =#> F(Z)   

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).

This combination (P_1, R_0) has no formative rules!  We will name the empty set of rules:R_1.

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:

  op(F, G) X >? F(G X) 
  op(F, G) X >? G(X) 
  implode#(cons(X, Y), F, Z) >? implode#(Y, F, F Z) 
  implode#(cons(X, Y), F, Z) >? F(Z) 
  explode#(cons(X, Y), F, Z) >? explode#(Y, op(F, F), F Z) 
  explode#(cons(X, Y), F, Z) >? F(Z) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( op(F, G) ) = #argfun-op#(/\x.F (G x), /\y.G 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].

Argument functions:

  [[explode#(x_1, x_2, x_3)]] = explode#(x_1, x_3, x_2) 

We choose Lex = {explode#, implode#} and Mul = {#argfun-op#, @_{o -> o}, cons, op}, and the following precedence: explode# > #argfun-op# > cons > implode# > @_{o -> o} > op

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

  @_{o -> o}(#argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) 
  @_{o -> o}(#argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)), X) > @_{o -> o}(G, X) 
  implode#(cons(X, Y), F, Z) >= implode#(Y, F, @_{o -> o}(F, Z)) 
  implode#(cons(X, Y), F, Z) >= @_{o -> o}(F, Z) 
  explode#(cons(X, Y), F, Z) > explode#(Y, #argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(F, x)), /\y.@_{o -> o}(F, y)), @_{o -> o}(F, Z)) 
  explode#(cons(X, Y), F, Z) > @_{o -> o}(F, Z) 

With these choices, we have:

  1] @_{o -> o}(#argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [2], by (Star) 
  2] @_{o -> o}*(#argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [3], by (Select) 
  3] #argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)) @_{o -> o}*(#argfun-op#(/\z.@_{o -> o}(F, @_{o -> o}(G, z)), /\u.@_{o -> o}(G, u)), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [4] 
  4] #argfun-op#*(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y), @_{o -> o}*(#argfun-op#(/\z.@_{o -> o}(F, @_{o -> o}(G, z)), /\u.@_{o -> o}(G, u)), X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [5], by (Select) 
  5] @_{o -> o}(F, @_{o -> o}(G, #argfun-op#*(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y), @_{o -> o}*(#argfun-op#(/\z.@_{o -> o}(F, @_{o -> o}(G, z)), /\u.@_{o -> o}(G, u)), X)))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because @_{o -> o} in Mul, [6] and [7], by (Fun) 
  6] F >= F  by (Meta) 
  7] @_{o -> o}(G, #argfun-op#*(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y), @_{o -> o}*(#argfun-op#(/\z.@_{o -> o}(F, @_{o -> o}(G, z)), /\u.@_{o -> o}(G, u)), X))) >= @_{o -> o}(G, X)  because [8], by (Star) 
  8] @_{o -> o}*(G, #argfun-op#*(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y), @_{o -> o}*(#argfun-op#(/\z.@_{o -> o}(F, @_{o -> o}(G, z)), /\u.@_{o -> o}(G, u)), X))) >= @_{o -> o}(G, X)  because [9], by (Select) 
  9] #argfun-op#*(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y), @_{o -> o}*(#argfun-op#(/\z.@_{o -> o}(F, @_{o -> o}(G, z)), /\u.@_{o -> o}(G, u)), X)) >= @_{o -> o}(G, X)  because [10], by (Select) 
  10] @_{o -> o}*(#argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)), X) >= @_{o -> o}(G, X)  because @_{o -> o} in Mul, [11] and [15], by (Stat) 
  11] #argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)) > G  because [12], by definition 
  12] #argfun-op#*(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)) >= G  because [13], by (Select) 
  13] /\x.@_{o -> o}(G, x) >= G  because [14], by (Eta)[Kop13:2] 
  14] G >= G  by (Meta) 
  15] X >= X  by (Meta) 

  16] @_{o -> o}(#argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)), X) > @_{o -> o}(G, X)  because [17], by definition 
  17] @_{o -> o}*(#argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(G, x)), /\y.@_{o -> o}(G, y)), X) >= @_{o -> o}(G, X)  because @_{o -> o} in Mul, [11] and [15], by (Stat) 

  18] implode#(cons(X, Y), F, Z) >= implode#(Y, F, @_{o -> o}(F, Z))  because [19], by (Star) 
  19] implode#*(cons(X, Y), F, Z) >= implode#(Y, F, @_{o -> o}(F, Z))  because [20], [23], [25] and [27], by (Stat) 
  20] cons(X, Y) > Y  because [21], by definition 
  21] cons*(X, Y) >= Y  because [22], by (Select) 
  22] Y >= Y  by (Meta) 
  23] implode#*(cons(X, Y), F, Z) >= Y  because [24], by (Select) 
  24] cons(X, Y) >= Y  because [21], by (Star) 
  25] implode#*(cons(X, Y), F, Z) >= F  because [26], by (Select) 
  26] F >= F  by (Meta) 
  27] implode#*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode# > @_{o -> o}, [25] and [28], by (Copy) 
  28] implode#*(cons(X, Y), F, Z) >= Z  because [29], by (Select) 
  29] Z >= Z  by (Meta) 

  30] implode#(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because [27], by (Star) 

  31] explode#(cons(X, Y), F, Z) > explode#(Y, #argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(F, x)), /\y.@_{o -> o}(F, y)), @_{o -> o}(F, Z))  because [32], by definition 
  32] explode#*(cons(X, Y), F, Z) >= explode#(Y, #argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(F, x)), /\y.@_{o -> o}(F, y)), @_{o -> o}(F, Z))  because [33], [36], [38] and [51], by (Stat) 
  33] cons(X, Y) > Y  because [34], by definition 
  34] cons*(X, Y) >= Y  because [35], by (Select) 
  35] Y >= Y  by (Meta) 
  36] explode#*(cons(X, Y), F, Z) >= Y  because [37], by (Select) 
  37] cons(X, Y) >= Y  because [34], by (Star) 
  38] explode#*(cons(X, Y), F, Z) >= #argfun-op#(/\x.@_{o -> o}(F, @_{o -> o}(F, x)), /\y.@_{o -> o}(F, y))  because explode# > #argfun-op#, [39] and [46], by (Copy) 
  39] explode#*(cons(X, Y), F, Z) >= /\y.@_{o -> o}(F, @_{o -> o}(F, y))  because [40], by (F-Abs) 
  40] explode#*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, @_{o -> o}(F, x))  because explode# > @_{o -> o}, [41] and [43], by (Copy) 
  41] explode#*(cons(X, Y), F, Z, x) >= F  because [42], by (Select) 
  42] F >= F  by (Meta) 
  43] explode#*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, x)  because explode# > @_{o -> o}, [41] and [44], by (Copy) 
  44] explode#*(cons(X, Y), F, Z, x) >= x  because [45], by (Select) 
  45] x >= x  by (Var) 
  46] explode#*(cons(X, Y), F, Z) >= /\z.@_{o -> o}(F, z)  because [47], by (F-Abs) 
  47] explode#*(cons(X, Y), F, Z, y) >= @_{o -> o}(F, y)  because explode# > @_{o -> o}, [48] and [49], by (Copy) 
  48] explode#*(cons(X, Y), F, Z, y) >= F  because [42], by (Select) 
  49] explode#*(cons(X, Y), F, Z, y) >= y  because [50], by (Select) 
  50] y >= y  by (Var) 
  51] explode#*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because explode# > @_{o -> o}, [52] and [53], by (Copy) 
  52] explode#*(cons(X, Y), F, Z) >= F  because [42], by (Select) 
  53] explode#*(cons(X, Y), F, Z) >= Z  because [54], by (Select) 
  54] Z >= Z  by (Meta) 

  55] explode#(cons(X, Y), F, Z) > @_{o -> o}(F, Z)  because [51], by definition 

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:

  op(F, G) X =#> F(G X)   
  implode#(cons(X, Y), F, Z) =#> implode#(Y, F, F Z)   
  implode#(cons(X, Y), F, Z) =#> F(Z)   

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 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:

  op(F, G, X) >? F(G X) 
  implode#(cons(X, Y), F, Z) >? implode#(Y, F, F Z) 
  implode#(cons(X, Y), F, Z) >? F(Z) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( op(F, G, X) ) = #argfun-op#(F (G X)) 

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:

  [[implode#(x_1, x_2, x_3)]] = implode#(x_1, x_3, x_2) 

We choose Lex = {implode#} and Mul = {#argfun-op#, @_{o -> o}, cons, op}, and the following precedence: #argfun-op# > cons > implode# > @_{o -> o} > op

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

  #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X)) 
  implode#(cons(X, Y), F, Z) >= implode#(Y, F, @_{o -> o}(F, Z)) 
  implode#(cons(X, Y), F, Z) > @_{o -> o}(F, Z) 

With these choices, we have:

  1] #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [2], by (Star) 
  2] #argfun-op#*(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [3], by (Select) 
  3] @_{o -> o}(F, @_{o -> o}(G, X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because @_{o -> o} in Mul, [4] and [5], by (Fun) 
  4] F >= F  by (Meta) 
  5] @_{o -> o}(G, X) >= @_{o -> o}(G, X)  because @_{o -> o} in Mul, [6] and [7], by (Fun) 
  6] G >= G  by (Meta) 
  7] X >= X  by (Meta) 

  8] implode#(cons(X, Y), F, Z) >= implode#(Y, F, @_{o -> o}(F, Z))  because [9], by (Star) 
  9] implode#*(cons(X, Y), F, Z) >= implode#(Y, F, @_{o -> o}(F, Z))  because [10], [13], [15] and [17], by (Stat) 
  10] cons(X, Y) > Y  because [11], by definition 
  11] cons*(X, Y) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 
  13] implode#*(cons(X, Y), F, Z) >= Y  because [14], by (Select) 
  14] cons(X, Y) >= Y  because [11], by (Star) 
  15] implode#*(cons(X, Y), F, Z) >= F  because [16], by (Select) 
  16] F >= F  by (Meta) 
  17] implode#*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode# > @_{o -> o}, [15] and [18], by (Copy) 
  18] implode#*(cons(X, Y), F, Z) >= Z  because [19], by (Select) 
  19] Z >= Z  by (Meta) 

  20] implode#(cons(X, Y), F, Z) > @_{o -> o}(F, Z)  because [17], by definition 

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:

  op(F, G) X =#> F(G X)   
  implode#(cons(X, Y), F, Z) =#> implode#(Y, F, F Z)   

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 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 : 1 

This graph has the following strongly connected components:

  P_4:

    op(F, G) X =#> F(G X)   

  P_5:

    implode#(cons(X, Y), F, Z) =#> implode#(Y, F, F Z)   

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) and (P_5, R_1, m, f).

Thus, the original system is terminating if each of (P_4, R_1, 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(implode#) = 1 

Thus, we can orient the dependency pairs as follows:

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

By [FuhKop19, Thm. 61], 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 (P_4, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_4, 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:

  op(F, G, X) >? F(G X) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( op(F, G, X) ) = #argfun-op#(F (G X)) 

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

The following interpretation satisfies the requirements:

  #argfun-op# = \y0.1 + y0 
  op = \G0G1y2.0 

Using this interpretation, the requirements translate to:

  [[#argfun-op#(_F0 (_F1 _x2))]] = 1 + max(x2, F0(max(x2, F1(x2))), F1(x2)) > F0(max(x2, F1(x2))) = [[_F0(_F1 _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_4, R_1) by ({}, R_1).  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.
[Kop13:2]  C. Kop.  StarHorpo with an Eta-Rule.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/etahorpo.pdf, 2013.