We consider the system 06plusmult.

  Alphabet:

    mult : [N * N] --> N 
    plus : [N * N] --> N 
    s : [N] --> N 
    z : [] --> N 

  Rules:

    plus(z, x) => x 
    plus(s(x), y) => plus(x, s(y)) 
    plus(plus(x, y), u) => plus(x, plus(y, u)) 
    mult(z, x) => z 
    mult(s(x), y) => plus(mult(x, y), y) 
    mult(plus(x, y), u) => plus(mult(x, u), mult(y, u)) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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

  plus(z, X) => X 
  plus(s(X), Y) => plus(X, s(Y)) 
  plus(plus(X, Y), Z) => plus(X, plus(Y, Z)) 
  mult(z, X) => z 
  mult(s(X), Y) => plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) => plus(mult(X, Z), mult(Y, Z)) 

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:
 || 
 ||    plus(z, %X) -> %X
 ||    plus(s(%X), %Y) -> plus(%X, s(%Y))
 ||    plus(plus(%X, %Y), %Z) -> plus(%X, plus(%Y, %Z))
 ||    mult(z, %X) -> z
 ||    mult(s(%X), %Y) -> plus(mult(%X, %Y), %Y)
 ||    mult(plus(%X, %Y), %Z) -> plus(mult(%X, %Z), mult(%Y, %Z))
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Quasi precedence:
 || mult_2 > plus_2 > s_1
 || mult_2 > z > s_1
 || ~PAIR_2 > s_1
 || 
 || 
 || Status:
 || plus_2: [1,2]
 || z: multiset status
 || s_1: multiset status
 || mult_2: [1,2]
 || ~PAIR_2: multiset status
 || 
 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
 || 
 ||    plus(z, %X) -> %X
 ||    plus(s(%X), %Y) -> plus(%X, s(%Y))
 ||    plus(plus(%X, %Y), %Z) -> plus(%X, plus(%Y, %Z))
 ||    mult(z, %X) -> z
 ||    mult(s(%X), %Y) -> plus(mult(%X, %Y), %Y)
 ||    mult(plus(%X, %Y), %Z) -> plus(mult(%X, %Z), mult(%Y, %Z))
 ||    ~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, formative):

  Dependency Pairs P_0:


  Rules R_0:

    plus(z, X) => X 
    plus(s(X), Y) => plus(X, s(Y)) 
    plus(plus(X, Y), Z) => plus(X, plus(Y, Z)) 
    mult(z, X) => z 
    mult(s(X), Y) => plus(mult(X, Y), Y) 
    mult(plus(X, Y), Z) => plus(mult(X, Z), mult(Y, Z)) 

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:


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.
[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.