We consider the system Applicative_05__ReverseLastInit.

  Alphabet:

    compose : [a -> a * a -> a] --> a -> a 
    cons : [a * a] --> a 
    hd : [] --> a -> a 
    init : [] --> a -> a 
    last : [] --> a -> a 
    nil : [] --> a 
    reverse : [] --> a -> a 
    reverse2 : [a * a] --> a 
    tl : [] --> a -> a 

  Rules:

    compose(f, g) x => g (f x) 
    reverse x => reverse2(x, nil) 
    reverse2(nil, x) => x 
    reverse2(cons(x, y), z) => reverse2(y, cons(x, z)) 
    hd cons(x, y) => x 
    tl cons(x, y) => y 
    last => compose(hd, reverse) 
    init => compose(reverse, compose(tl, reverse)) 

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:

  reverse(X) => reverse2(X, nil) 
  reverse2(nil, X) => X 
  reverse2(cons(X, Y), Z) => reverse2(Y, cons(X, Z)) 
  hd(cons(X, Y)) => X 
  tl(cons(X, Y)) => Y 

Moreover, the system is orthogonal.  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 terminating when seen as a many-sorted first-order TRS.

According to the external first-order termination prover, this system is indeed 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) QTRSRRRProof [EQUIVALENT]
 || (4) QTRS
 || (5) RisEmptyProof [EQUIVALENT]
 || (6) YES
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    reverse(%X) -> reverse2(%X, nil)
 ||    reverse2(nil, %X) -> %X
 ||    reverse2(cons(%X, %Y), %Z) -> reverse2(%Y, cons(%X, %Z))
 ||    hd(cons(%X, %Y)) -> %X
 ||    tl(cons(%X, %Y)) -> %Y
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(cons(x_1, x_2)) = 1 + x_1 + x_2
 ||    POL(hd(x_1)) = 2 + x_1
 ||    POL(nil) = 2
 ||    POL(reverse(x_1)) = 2 + 2*x_1
 ||    POL(reverse2(x_1, x_2)) = 2*x_1 + x_2
 ||    POL(tl(x_1)) = 2 + x_1
 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
 || 
 ||    reverse2(nil, %X) -> %X
 ||    reverse2(cons(%X, %Y), %Z) -> reverse2(%Y, cons(%X, %Z))
 ||    hd(cons(%X, %Y)) -> %X
 ||    tl(cons(%X, %Y)) -> %Y
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (2)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    reverse(%X) -> reverse2(%X, nil)
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (3) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(nil) = 0
 ||    POL(reverse(x_1)) = 2 + 2*x_1
 ||    POL(reverse2(x_1, x_2)) = 1 + x_1 + 2*x_2
 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
 || 
 ||    reverse(%X) -> reverse2(%X, nil)
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (4)
 || Obligation:
 || Q restricted rewrite system:
 || R is empty.
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (5) RisEmptyProof (EQUIVALENT)
 || The TRS R is empty. Hence, termination is trivially proven.
 || ----------------------------------------
 || 
 || (6)
 || 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]).

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

  compose(F, G, X) => G (F X) 
  reverse(X) => reverse2(X, nil) 
  reverse2(nil, X) => X 
  reverse2(cons(X, Y), Z) => reverse2(Y, cons(X, Z)) 
  hd(cons(X, Y)) => X 
  tl(cons(X, Y)) => Y 
  last(X) => compose(/\x.hd(x), /\y.reverse(y), X) 
  init(X) => compose(/\x.reverse(x), /\y.compose(/\z.tl(z), /\u.reverse(u), y), X) 

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

  Dependency Pairs P_0:

    0] last#(X) =#> compose#(/\x.hd(x), /\y.reverse(y), X)   
    1] last#(X) =#> hd#(Y)   
    2] last#(X) =#> reverse#(Y)   
    3] init#(X) =#> compose#(/\x.reverse(x), /\y.compose(/\z.tl(z), /\u.reverse(u), y), X)   
    4] init#(X) =#> reverse#(Y)   
    5] init#(X) =#> compose#(/\x.tl(x), /\y.reverse(y), Y)   
    6] init#(X) =#> tl#(Y)   
    7] init#(X) =#> reverse#(Y)   

  Rules R_0:

    compose(F, G, X) => G (F X) 
    reverse(X) => reverse2(X, nil) 
    reverse2(nil, X) => X 
    reverse2(cons(X, Y), Z) => reverse2(Y, cons(X, Z)) 
    hd(cons(X, Y)) => X 
    tl(cons(X, Y)) => Y 
    last(X) => compose(/\x.hd(x), /\y.reverse(y), X) 
    init(X) => compose(/\x.reverse(x), /\y.compose(/\z.tl(z), /\u.reverse(u), y), 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 :  
    * 2 :  
    * 3 :  
    * 4 :  
    * 5 :  
    * 6 :  
    * 7 :  

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.