We consider the system lambda_prod.

  Alphabet:

    app : [] --> arrab -> a -> b 
    lam : [] --> (a -> b) -> arrab 
    pair : [] --> a -> b -> prodab 
    pia : [] --> prodab -> a 
    pib : [] --> prodab -> b 

  Rules:

    app (lam (/\x.f x)) y => f y 
    lam (/\x.app y x) => y 
    pia (pair x y) => x 
    pib (pair x y) => y 
    pair (pia x) (pib x) => 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:

    app : [arrab * a] --> b 
    lam : [a -> b] --> arrab 
    pair : [a * b] --> prodab 
    pia : [prodab] --> a 
    pib : [prodab] --> b 
    ~AP1 : [a -> b * a] --> b 

  Rules:

    app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) 
    lam(/\x.app(X, x)) => X 
    pia(pair(X, Y)) => X 
    pib(pair(X, Y)) => Y 
    pair(pia(X), pib(X)) => X 
    app(lam(/\x.app(X, x)), Y) => app(X, Y) 
    ~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:

    app : [arrab * a] --> b 
    lam : [a -> b] --> arrab 
    pair : [a * b] --> prodab 
    pia : [prodab] --> a 
    pib : [prodab] --> b 

  Rules:

    app(lam(/\x.X(x)), Y) => X(Y) 
    lam(/\x.app(X, x)) => X 
    pia(pair(X, Y)) => X 
    pib(pair(X, Y)) => Y 
    pair(pia(X), pib(X)) => X 

We observe that the rules contain a first-order subset:

  pia(pair(X, Y)) => X 
  pib(pair(X, Y)) => Y 
  pair(pia(X), pib(X)) => X 

Moreover, the system is finitely branching.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS.

According to the external first-order termination prover, this system is indeed Ce-terminating:

 || proof of resources/system.trs
 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
 || 
 || 
 || Termination w.r.t. Q of the given QTRS could be proven:
 || 
 || (0) QTRS
 || (1) QTRSRRRProof [EQUIVALENT]
 || (2) QTRS
 || (3) RisEmptyProof [EQUIVALENT]
 || (4) YES
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    pia(pair(%X, %Y)) -> %X
 ||    pib(pair(%X, %Y)) -> %Y
 ||    pair(pia(%X), pib(%X)) -> %X
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(pair(x_1, x_2)) = 1 + x_1 + x_2
 ||    POL(pia(x_1)) = 1 + 2*x_1
 ||    POL(pib(x_1)) = 1 + x_1
 ||    POL(~PAIR(x_1, x_2)) = 2 + x_1 + x_2
 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
 || 
 ||    pia(pair(%X, %Y)) -> %X
 ||    pib(pair(%X, %Y)) -> %Y
 ||    pair(pia(%X), pib(%X)) -> %X
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (2)
 || Obligation:
 || Q restricted rewrite system:
 || R is empty.
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (3) RisEmptyProof (EQUIVALENT)
 || The TRS R is empty. Hence, termination is trivially proven.
 || ----------------------------------------
 || 
 || (4)
 || YES
 || 
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, all):

  Dependency Pairs P_0:


  Rules R_0:

    app(lam(/\x.X(x)), Y) => X(Y) 
    lam(/\x.app(X, x)) => X 
    pia(pair(X, Y)) => X 
    pib(pair(X, Y)) => Y 
    pair(pia(X), pib(X)) => X 

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

We consider the dependency pair problem (P_0, R_0, computable, all).

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


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.