We consider the system h09.

  Alphabet:

    0 : [] --> N 
    false : [] --> B 
    g : [] --> N -> B 
    h : [] --> N -> (N -> B) -> N -> B 
    iszero : [] --> N -> N -> B 
    rec : [] --> (N -> (N -> B) -> N -> B) -> B -> N -> B 
    true : [] --> B 

  Rules:

    rec f (i 0) => i 
    g x => true 
    h x f y => false 
    iszero x y => rec h (g x) y 

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 : [] --> N 
    false : [] --> B 
    g : [] --> N -> B 
    h : [] --> N -> (N -> B) -> N -> B 
    iszero : [N] --> N -> B 
    rec : [N -> (N -> B) -> N -> B * B] --> N -> B 
    true : [] --> B 
    ~AP1 : [N -> B * N] --> B 

  Rules:

    rec(F, ~AP1(G, 0)) => G 
    g X => true 
    h X F Y => false 
    iszero(X) Y => ~AP1(rec(h, g X), Y) 
    rec(F, g 0) => g 
    rec(F, h X G 0) => h X G 
    rec(F, iszero(X) 0) => iszero(X) 
    ~AP1(F, X) => F X 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [FuhKop19]).

In order to do so, we start by eta-expanding the system, which gives:

  rec(F, ~AP1(G, 0), X) => G X 
  g(X) => true 
  h(X, F, Y) => false 
  iszero(X, Y) => ~AP1(/\x.rec(/\y./\f./\z.h(y, /\u.f u, z), g(X), x), Y) 
  rec(F, g(0), X) => g(X) 
  rec(F, h(X, G, 0), Y) => h(X, G, Y) 
  rec(F, iszero(X, 0), Y) => iszero(X, Y) 
  ~AP1(F, X) => F X 

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

  Dependency Pairs P_0:

    0] iszero#(X, Y) =#> ~AP1#(/\x.rec(/\y./\f./\z.h(y, /\u.f u, z), g(X), x), Y)   
    1] iszero#(X, Y) =#> rec#(/\x./\f./\y.h(x, /\z.f z, y), g(X), Z)   
    2] iszero#(X, Y) =#> h#(Z, /\x.F x, U)   
    3] iszero#(X, Y) =#> g#(X)   
    4] rec#(F, g(0), X) =#> g#(X)   
    5] rec#(F, h(X, G, 0), Y) =#> h#(X, G, Y)   
    6] rec#(F, iszero(X, 0), Y) =#> iszero#(X, Y)   

  Rules R_0:

    rec(F, ~AP1(G, 0), X) => G X 
    g(X) => true 
    h(X, F, Y) => false 
    iszero(X, Y) => ~AP1(/\x.rec(/\y./\f./\z.h(y, /\u.f u, z), g(X), x), Y) 
    rec(F, g(0), X) => g(X) 
    rec(F, h(X, G, 0), Y) => h(X, G, Y) 
    rec(F, iszero(X, 0), Y) => iszero(X, Y) 
    ~AP1(F, X) => F X 

Thus, the original system is terminating if (P_0, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_0, R_0, computable, 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 :  
    * 1 : 4 
    * 2 :  
    * 3 :  
    * 4 :  
    * 5 :  
    * 6 : 0, 1, 2, 3 

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 +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[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.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.