We consider the system h57.

  Alphabet:

    0 : [] --> nat 
    cons : [] --> nat -> list -> list 
    foldr : [] --> (nat -> nat -> nat) -> nat -> list -> nat 
    length : [] --> list -> nat 
    nil : [] --> list 
    s : [] --> nat -> nat 

  Rules:

    foldr (/\x./\y.f x y) z nil => z 
    foldr (/\x./\y.f x y) z (cons u v) => f u (foldr (/\w./\x'.f w x') z v) 
    length x => foldr (/\y./\z.s z) 0 x 

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 : [] --> nat 
    cons : [nat * list] --> list 
    foldr : [nat -> nat -> nat * nat * list] --> nat 
    length : [list] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 
    ~AP1 : [nat -> nat -> nat * nat] --> nat -> nat 

  Rules:

    foldr(/\x./\y.~AP1(F, x) y, X, nil) => X 
    foldr(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => ~AP1(F, Y) foldr(/\z./\u.~AP1(F, z) u, X, Z) 
    length(X) => foldr(/\x./\y.s(y), 0, 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.  This gives:

  Alphabet:

    0 : [] --> nat 
    cons : [nat * list] --> list 
    foldr : [nat -> nat -> nat * nat * list] --> nat 
    length : [list] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 

  Rules:

    foldr(/\x./\y.X(x, y), Y, nil) => Y 
    foldr(/\x./\y.X(x, y), Y, cons(Z, U)) => X(Z, foldr(/\z./\u.X(z, u), Y, U)) 
    length(X) => foldr(/\x./\y.s(y), 0, 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] foldr#(/\x./\y.X(x, y), Y, cons(Z, U)) =#> X(Z, foldr(/\z./\u.X(z, u), Y, U))   
    1] foldr#(/\x./\y.X(x, y), Y, cons(Z, U)) =#> foldr#(/\z./\u.X(z, u), Y, U) {X : 2}  
    2] foldr#(/\x./\y.X(x, y), Y, cons(Z, U)) =#> X(z, u) {X : 2}  
    3] length#(X) =#> foldr#(/\x./\y.s(y), 0, X)   

  Rules R_0:

    foldr(/\x./\y.X(x, y), Y, nil) => Y 
    foldr(/\x./\y.X(x, y), Y, cons(Z, U)) => X(Z, foldr(/\z./\u.X(z, u), Y, U)) 
    length(X) => foldr(/\x./\y.s(y), 0, 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:

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

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

  pi( length#(X) ) = #argfun-length##(foldr#(/\x./\y.s-(y), 0, X)) 

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

The following interpretation satisfies the requirements:

  #argfun-length## = \y0.3 + y0 
  0 = 0 
  cons = \y0y1.3 + y0 + 3y1 
  foldr = \G0y1y2.0 
  foldr# = \G0y1y2.3 + 3y2 + 2y2y2G0(y2,y2) + 3y1y2G0(y1,y2) + 3y1y2G0(y2,y1) 
  length# = \y0.0 
  s = \y0.0 
  s- = \y0.3 + 3y0 
  ~c0 = 0 
  ~c1 = 0 

Using this interpretation, the requirements translate to:

  [[foldr#(/\x./\y._x0(x, y), _x1, cons(_x2, _x3))]] = 12 + 3x2 + 9x3 + 2x2x2F0(3 + x2 + 3x3,3 + x2 + 3x3) + 3x1x2F0(x1,3 + x2 + 3x3) + 3x1x2F0(3 + x2 + 3x3,x1) + 9x1x3F0(x1,3 + x2 + 3x3) + 9x1x3F0(3 + x2 + 3x3,x1) + 9x1F0(x1,3 + x2 + 3x3) + 9x1F0(3 + x2 + 3x3,x1) + 12x2x3F0(3 + x2 + 3x3,3 + x2 + 3x3) + 12x2F0(3 + x2 + 3x3,3 + x2 + 3x3) + 18x3x3F0(3 + x2 + 3x3,3 + x2 + 3x3) + 18F0(3 + x2 + 3x3,3 + x2 + 3x3) + 36x3F0(3 + x2 + 3x3,3 + x2 + 3x3) > F0(x2,0) = [[_x0(_x2, foldr(/\x./\y._x0(x, y), _x1, _x3))]] 
  [[foldr#(/\x./\y._x0(x, y), _x1, cons(_x2, _x3))]] = 12 + 3x2 + 9x3 + 2x2x2F0(3 + x2 + 3x3,3 + x2 + 3x3) + 3x1x2F0(x1,3 + x2 + 3x3) + 3x1x2F0(3 + x2 + 3x3,x1) + 9x1x3F0(x1,3 + x2 + 3x3) + 9x1x3F0(3 + x2 + 3x3,x1) + 9x1F0(x1,3 + x2 + 3x3) + 9x1F0(3 + x2 + 3x3,x1) + 12x2x3F0(3 + x2 + 3x3,3 + x2 + 3x3) + 12x2F0(3 + x2 + 3x3,3 + x2 + 3x3) + 18x3x3F0(3 + x2 + 3x3,3 + x2 + 3x3) + 18F0(3 + x2 + 3x3,3 + x2 + 3x3) + 36x3F0(3 + x2 + 3x3,3 + x2 + 3x3) > 3 + 3x3 + 2x3x3F0(x3,x3) + 3x1x3F0(x1,x3) + 3x1x3F0(x3,x1) = [[foldr#(/\x./\y._x0(x, y), _x1, _x3)]] 
  [[foldr#(/\x./\y._x0(x, y), _x1, cons(_x2, _x3))]] = 12 + 3x2 + 9x3 + 2x2x2F0(3 + x2 + 3x3,3 + x2 + 3x3) + 3x1x2F0(x1,3 + x2 + 3x3) + 3x1x2F0(3 + x2 + 3x3,x1) + 9x1x3F0(x1,3 + x2 + 3x3) + 9x1x3F0(3 + x2 + 3x3,x1) + 9x1F0(x1,3 + x2 + 3x3) + 9x1F0(3 + x2 + 3x3,x1) + 12x2x3F0(3 + x2 + 3x3,3 + x2 + 3x3) + 12x2F0(3 + x2 + 3x3,3 + x2 + 3x3) + 18x3x3F0(3 + x2 + 3x3,3 + x2 + 3x3) + 18F0(3 + x2 + 3x3,3 + x2 + 3x3) + 36x3F0(3 + x2 + 3x3,3 + x2 + 3x3) > F0(0,0) = [[_x0(~c0, ~c1)]] 
  [[#argfun-length##(foldr#(/\x./\y.s-(y), 0, _x0))]] = 6 + 3x0 + 6x0x0 + 6x0x0x0 > 3 + 3x0 + 6x0x0 + 6x0x0x0 = [[foldr#(/\x./\y.s-(y), 0, _x0)]] 
  [[s-(_x0)]] = 3 + 3x0 >= 0 = [[s(_x0)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_0, 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 +++

[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.