We consider the system monad.

  Alphabet:

    bind : [] --> Ta -> (a -> Ta) -> Ta 
    return : [] --> a -> Ta 

  Rules:

    bind (return x) (/\y.f y) => f x 
    bind x (/\y.return y) => x 
    bind (bind x (/\y.f y)) (/\z.g z) => bind x (/\u.bind (f u) (/\v.g v)) 

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:

    bind : [Ta * a -> Ta] --> Ta 
    return : [a] --> Ta 
    ~AP1 : [a -> Ta * a] --> Ta 

  Rules:

    bind(return(X), /\x.~AP1(F, x)) => ~AP1(F, X) 
    bind(X, /\x.return(x)) => X 
    bind(bind(X, /\x.~AP1(F, x)), /\y.~AP1(G, y)) => bind(X, /\z.bind(~AP1(F, z), /\u.~AP1(G, u))) 
    bind(return(X), /\x.return(x)) => return(X) 
    bind(bind(X, /\x.return(x)), /\y.~AP1(F, y)) => bind(X, /\z.bind(return(z), /\u.~AP1(F, u))) 
    bind(bind(X, /\x.~AP1(F, x)), /\y.return(y)) => bind(X, /\z.bind(~AP1(F, z), /\u.return(u))) 
    bind(bind(X, /\x.return(x)), /\y.return(y)) => bind(X, /\z.bind(return(z), /\u.return(u))) 
    ~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:

    bind : [Ta * a -> Ta] --> Ta 
    return : [a] --> Ta 

  Rules:

    bind(return(X), /\x.Y(x)) => Y(X) 
    bind(X, /\x.return(x)) => X 
    bind(bind(X, /\x.Y(x)), /\y.Z(y)) => bind(X, /\z.bind(Y(z), /\u.Z(u))) 

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]).

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

  Dependency Pairs P_0:

    0] bind#(bind(X, /\x.Y(x)), /\y.Z(y)) =#> bind#(X, /\z.bind(Y(z), /\u.Z(u)))   
    1] bind#(bind(X, /\x.Y(x)), /\y.Z(y)) =#> bind#(Y(U), /\z.Z(z))   

  Rules R_0:

    bind(return(X), /\x.Y(x)) => Y(X) 
    bind(X, /\x.return(x)) => X 
    bind(bind(X, /\x.Y(x)), /\y.Z(y)) => bind(X, /\z.bind(Y(z), /\u.Z(u))) 

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 apply the accessible subterm criterion with the following projection function:

  nu(bind#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(bind#(bind(X, /\x.Y(x)), /\y.Z(y))) = bind(X, /\z.Y(z)) [>] X = nu(bind#(X, /\x.bind(Y(x), /\u.Z(u)))) 
  nu(bind#(bind(X, /\x.Y(x)), /\y.Z(y))) = bind(X, /\z.Y(z)) [>] Y(U) = nu(bind#(Y(U), /\x.Z(x))) 

By [FuhKop19, Thm. 63], we may replace a dependency pair problem (P_0, R_0, computable, f) by ({}, R_0, computable, 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.
[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.