We consider the system h02.

  Alphabet:

    0 : [] --> c 
    1 : [] --> c 
    add : [] --> c -> a -> c 
    cons : [] --> a -> b -> b 
    fold : [] --> (c -> a -> c) -> b -> c -> c 
    mul : [] --> c -> a -> c 
    nil : [] --> b 
    prod : [] --> b -> c 
    sum : [] --> b -> c 

  Rules:

    fold (/\x./\y.f x y) nil z => z 
    fold (/\x./\y.f x y) (cons z u) v => fold (/\w./\x'.f w x') u (f v z) 
    sum x => fold (/\y./\z.add y z) x 0 
    fold (/\x./\y.mul x y) z 1 => prod z 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    0 : [] --> c 
    1 : [] --> c 
    add : [] --> c -> a -> c 
    cons : [a * b] --> b 
    fold : [c -> a -> c * b * c] --> c 
    mul : [c * a] --> c 
    nil : [] --> b 
    prod : [b] --> c 
    sum : [b] --> c 
    ~AP1 : [c -> a -> c * c] --> a -> c 

  Rules:

    fold(/\x./\y.~AP1(F, x) y, nil, X) => X 
    fold(/\x./\y.~AP1(F, x) y, cons(X, Y), Z) => fold(/\z./\u.~AP1(F, z) u, Y, ~AP1(F, Z) X) 
    sum(X) => fold(/\x./\y.~AP1(add, x) y, X, 0) 
    fold(/\x./\y.mul(x, y), X, 1) => prod(X) 
    fold(/\x./\y.mul(x, y), nil, X) => X 
    fold(/\x./\y.mul(x, y), cons(X, Y), Z) => fold(/\z./\u.mul(z, u), Y, mul(Z, X)) 
    ~AP1(F, X) => F X 

Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  Additionally, we can remove some (now-)redundant rules.  This gives:

  Alphabet:

    0 : [] --> c 
    1 : [] --> c 
    add : [c * a] --> c 
    cons : [a * b] --> b 
    fold : [c -> a -> c * b * c] --> c 
    mul : [c * a] --> c 
    nil : [] --> b 
    prod : [b] --> c 
    sum : [b] --> c 

  Rules:

    fold(/\x./\y.X(x, y), nil, Y) => Y 
    fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) 
    sum(X) => fold(/\x./\y.add(x, y), X, 0) 
    fold(/\x./\y.mul(x, y), X, 1) => prod(X) 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] fold#(/\x./\y.X(x, y), cons(Y, Z), U) =#> fold#(/\z./\u.X(z, u), Z, X(U, Y))   
    1] fold#(/\x./\y.X(x, y), cons(Y, Z), U) =#> X(z, u)   
    2] fold#(/\x./\y.X(x, y), cons(Y, Z), U) =#> X(U, Y)   
    3] sum#(X) =#> fold#(/\x./\y.add(x, y), X, 0)   

  Rules R_0:

    fold(/\x./\y.X(x, y), nil, Y) => Y 
    fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) 
    sum(X) => fold(/\x./\y.add(x, y), X, 0) 
    fold(/\x./\y.mul(x, y), X, 1) => prod(X) 

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).

This combination (P_0, 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_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative).

Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_0, 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:

  fold#(/\x./\y.X(x, y), cons(Y, Z), U) >? fold#(/\z./\u.X(z, u), Z, X(U, Y)) 
  fold#(/\x./\y.X(x, y), cons(Y, Z), U) >? X(~c0, ~c1) 
  fold#(/\x./\y.X(x, y), cons(Y, Z), U) >? X(U, Y) 
  sum#(X) >? fold#(/\x./\y.add-(x, y), X, 0) 
  add-(X, Y) >= add(X, Y) 

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

  pi( sum#(X) ) = #argfun-sum##(fold#(/\x./\y.add-(x, y), X, 0)) 

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:

  [[0]] = _|_ 
  [[add(x_1, x_2)]] = _|_ 
  [[fold#(x_1, x_2, x_3)]] = fold#(x_2, x_3, x_1) 
  [[sum#(x_1)]] = x_1 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 

We choose Lex = {fold#} and Mul = {#argfun-sum##, add-, cons}, and the following precedence: cons > #argfun-sum## > fold# > add-

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

  fold#(/\x./\y.X(x, y), cons(Y, Z), U) > fold#(/\x./\y.X(x, y), Z, X(U, Y)) 
  fold#(/\x./\y.X(x, y), cons(Y, Z), U) > X(_|_, _|_) 
  fold#(/\x./\y.X(x, y), cons(Y, Z), U) > X(U, Y) 
  #argfun-sum##(fold#(/\x./\y.add-(x, y), X, _|_)) >= fold#(/\x./\y.add-(x, y), X, _|_) 
  add-(X, Y) >= _|_ 

With these choices, we have:

  1] fold#(/\x./\y.X(x, y), cons(Y, Z), U) > fold#(/\x./\y.X(x, y), Z, X(U, Y))  because [2], by definition 
  2] fold#*(/\x./\y.X(x, y), cons(Y, Z), U) >= fold#(/\x./\y.X(x, y), Z, X(U, Y))  because [3], [6], [12] and [14], by (Stat) 
  3] cons(Y, Z) > Z  because [4], by definition 
  4] cons*(Y, Z) >= Z  because [5], by (Select) 
  5] Z >= Z  by (Meta) 
  6] fold#*(/\x./\y.X(x, y), cons(Y, Z), U) >= /\x./\y.X(x, y)  because [7], by (Select) 
  7] /\x./\z.X(x, z) >= /\x./\z.X(x, z)  because [8], by (Abs) 
  8] /\z.X(y, z) >= /\z.X(y, z)  because [9], by (Abs) 
  9] X(y, x) >= X(y, x)  because [10] and [11], by (Meta) 
  10] y >= y  by (Var) 
  11] x >= x  by (Var) 
  12] fold#*(/\z./\u.X(z, u), cons(Y, Z), U) >= Z  because [13], by (Select) 
  13] cons(Y, Z) >= Z  because [4], by (Star) 
  14] fold#*(/\z./\u.X(z, u), cons(Y, Z), U) >= X(U, Y)  because [15], by (Select) 
  15] X(fold#*(/\z./\u.X(z, u), cons(Y, Z), U), fold#*(/\v./\w.X(v, w), cons(Y, Z), U)) >= X(U, Y)  because [16] and [18], by (Meta) 
  16] fold#*(/\z./\u.X(z, u), cons(Y, Z), U) >= U  because [17], by (Select) 
  17] U >= U  by (Meta) 
  18] fold#*(/\z./\u.X(z, u), cons(Y, Z), U) >= Y  because [19], by (Select) 
  19] cons(Y, Z) >= Y  because [20], by (Star) 
  20] cons*(Y, Z) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 

  22] fold#(/\x./\y.X(x, y), cons(Y, Z), U) > X(_|_, _|_)  because [23], by definition 
  23] fold#*(/\x./\y.X(x, y), cons(Y, Z), U) >= X(_|_, _|_)  because [24], by (Select) 
  24] X(fold#*(/\x./\y.X(x, y), cons(Y, Z), U), fold#*(/\z./\u.X(z, u), cons(Y, Z), U)) >= X(_|_, _|_)  because [25] and [26], by (Meta) 
  25] fold#*(/\x./\y.X(x, y), cons(Y, Z), U) >= _|_  by (Bot) 
  26] fold#*(/\x./\y.X(x, y), cons(Y, Z), U) >= _|_  by (Bot) 

  27] fold#(/\x./\y.X(x, y), cons(Y, Z), U) > X(U, Y)  because [14], by definition 

  28] #argfun-sum##(fold#(/\x./\y.add-(x, y), X, _|_)) >= fold#(/\x./\y.add-(x, y), X, _|_)  because [29], by (Star) 
  29] #argfun-sum##*(fold#(/\x./\y.add-(x, y), X, _|_)) >= fold#(/\x./\y.add-(x, y), X, _|_)  because #argfun-sum## > fold#, [30], [37] and [44], by (Copy) 
  30] #argfun-sum##*(fold#(/\x./\y.add-(x, y), X, _|_)) >= /\x./\y.add-(x, y)  because [31], by (F-Abs) 
  31] #argfun-sum##*(fold#(/\x./\y.add-(x, y), X, _|_), z) >= /\x.add-(z, x)  because [32], by (F-Abs) 
  32] #argfun-sum##*(fold#(/\x./\y.add-(x, y), X, _|_), z, u) >= add-(z, u)  because #argfun-sum## > add-, [33] and [35], by (Copy) 
  33] #argfun-sum##*(fold#(/\x./\y.add-(x, y), X, _|_), z, u) >= z  because [34], by (Select) 
  34] z >= z  by (Var) 
  35] #argfun-sum##*(fold#(/\x./\y.add-(x, y), X, _|_), z, u) >= u  because [36], by (Select) 
  36] u >= u  by (Var) 
  37] #argfun-sum##*(fold#(/\x./\y.add-(x, y), X, _|_)) >= X  because [38], by (Select) 
  38] fold#(/\x./\y.add-(x, y), X, _|_) >= X  because [39], by (Star) 
  39] fold#*(/\x./\y.add-(x, y), X, _|_) >= X  because [40], by (Select) 
  40] add-(fold#*(/\x./\y.add-(x, y), X, _|_), fold#*(/\v./\w.add-(v, w), X, _|_)) >= X  because [41], by (Star) 
  41] add-*(fold#*(/\x./\y.add-(x, y), X, _|_), fold#*(/\v./\w.add-(v, w), X, _|_)) >= X  because [42], by (Select) 
  42] fold#*(/\x./\y.add-(x, y), X, _|_) >= X  because [43], by (Select) 
  43] X >= X  by (Meta) 
  44] #argfun-sum##*(fold#(/\x./\y.add-(x, y), X, _|_)) >= _|_  by (Bot) 

  45] add-(X, Y) >= _|_  by (Bot) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_1, minimal, formative) by (P_1, R_1, minimal, formative), where P_1 consists of:

  sum#(X) =#> fold#(/\x./\y.add(x, y), X, 0)   

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 place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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


+++ Citations +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.