We consider the system Applicative_05__TakeDropWhile.

  Alphabet:

    cons : [a * c] --> c 
    dropWhile : [a -> b * c] --> c 
    if : [b * c * c] --> c 
    nil : [] --> c 
    takeWhile : [a -> b * c] --> c 
    true : [] --> b 

  Rules:

    if(true, x, y) => x 
    if(true, x, y) => y 
    takeWhile(f, nil) => nil 
    takeWhile(f, cons(x, y)) => if(f x, cons(x, takeWhile(f, y)), nil) 
    dropWhile(f, nil) => nil 
    dropWhile(f, cons(x, y)) => if(f x, dropWhile(f, y), cons(x, y)) 

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:

  if(true, X, Y) => X 
  if(true, X, Y) => Y 

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:
 || 
 ||    if(true, %X, %Y) -> %X
 ||    if(true, %X, %Y) -> %Y
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(if(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3
 ||    POL(true) = 2
 ||    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:
 || 
 ||    if(true, %X, %Y) -> %X
 ||    if(true, %X, %Y) -> %Y
 ||    ~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:

    0] takeWhile#(F, cons(X, Y)) =#> if#(F X, cons(X, takeWhile(F, Y)), nil)   
    1] takeWhile#(F, cons(X, Y)) =#> takeWhile#(F, Y)   
    2] dropWhile#(F, cons(X, Y)) =#> if#(F X, dropWhile(F, Y), cons(X, Y))   
    3] dropWhile#(F, cons(X, Y)) =#> dropWhile#(F, Y)   

  Rules R_0:

    if(true, X, Y) => X 
    if(true, X, Y) => Y 
    takeWhile(F, nil) => nil 
    takeWhile(F, cons(X, Y)) => if(F X, cons(X, takeWhile(F, Y)), nil) 
    dropWhile(F, nil) => nil 
    dropWhile(F, cons(X, Y)) => if(F X, dropWhile(F, Y), cons(X, Y)) 

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 : 0, 1 
    * 2 :  
    * 3 : 2, 3 

This graph has the following strongly connected components:

  P_1:

    takeWhile#(F, cons(X, Y)) =#> takeWhile#(F, Y)   

  P_2:

    dropWhile#(F, cons(X, Y)) =#> dropWhile#(F, Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_2, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_2, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(dropWhile#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(dropWhile#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(dropWhile#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by ({}, R_0, computable, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_1, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_1, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(takeWhile#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(takeWhile#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(takeWhile#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, 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.
[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.