/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  plus : [o * o] --> o 
  s : [o] --> o 

  plus(s(X), plus(Y, Z)) => plus(X, plus(s(s(Y)), Z)) 
  plus(s(X), plus(Y, plus(Z, U))) => plus(X, plus(Z, plus(Y, 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 in [Kop12, Ch. 7.8]).

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

  Dependency Pairs P_0:

    0] plus#(s(X), plus(Y, Z)) =#> plus#(X, plus(s(s(Y)), Z))   
    1] plus#(s(X), plus(Y, Z)) =#> plus#(s(s(Y)), Z)   
    2] plus#(s(X), plus(Y, plus(Z, U))) =#> plus#(X, plus(Z, plus(Y, U)))   
    3] plus#(s(X), plus(Y, plus(Z, U))) =#> plus#(Z, plus(Y, U))   
    4] plus#(s(X), plus(Y, plus(Z, U))) =#> plus#(Y, U)   

  Rules R_0:

    plus(s(X), plus(Y, Z)) => plus(X, plus(s(s(Y)), Z)) 
    plus(s(X), plus(Y, plus(Z, U))) => plus(X, plus(Z, plus(Y, U))) 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  plus#(s(X), plus(Y, Z)) >? plus#(X, plus(s(s(Y)), Z)) 
  plus#(s(X), plus(Y, Z)) >? plus#(s(s(Y)), Z) 
  plus#(s(X), plus(Y, plus(Z, U))) >? plus#(X, plus(Z, plus(Y, U))) 
  plus#(s(X), plus(Y, plus(Z, U))) >? plus#(Z, plus(Y, U)) 
  plus#(s(X), plus(Y, plus(Z, U))) >? plus#(Y, U) 
  plus(s(X), plus(Y, Z)) >= plus(X, plus(s(s(Y)), Z)) 
  plus(s(X), plus(Y, plus(Z, U))) >= plus(X, plus(Z, plus(Y, U))) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  plus = \y0y1.3 + y1 
  plus# = \y0y1.2y1 
  s = \y0.0 

Using this interpretation, the requirements translate to:

  [[plus#(s(_x0), plus(_x1, _x2))]] = 6 + 2x2 >= 6 + 2x2 = [[plus#(_x0, plus(s(s(_x1)), _x2))]] 
  [[plus#(s(_x0), plus(_x1, _x2))]] = 6 + 2x2 > 2x2 = [[plus#(s(s(_x1)), _x2)]] 
  [[plus#(s(_x0), plus(_x1, plus(_x2, _x3)))]] = 12 + 2x3 >= 12 + 2x3 = [[plus#(_x0, plus(_x2, plus(_x1, _x3)))]] 
  [[plus#(s(_x0), plus(_x1, plus(_x2, _x3)))]] = 12 + 2x3 > 6 + 2x3 = [[plus#(_x2, plus(_x1, _x3))]] 
  [[plus#(s(_x0), plus(_x1, plus(_x2, _x3)))]] = 12 + 2x3 > 2x3 = [[plus#(_x1, _x3)]] 
  [[plus(s(_x0), plus(_x1, _x2))]] = 6 + x2 >= 6 + x2 = [[plus(_x0, plus(s(s(_x1)), _x2))]] 
  [[plus(s(_x0), plus(_x1, plus(_x2, _x3)))]] = 9 + x3 >= 9 + x3 = [[plus(_x0, plus(_x2, plus(_x1, _x3)))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_1, R_0, minimal, formative), where P_1 consists of:

  plus#(s(X), plus(Y, Z)) =#> plus#(X, plus(s(s(Y)), Z))   
  plus#(s(X), plus(Y, plus(Z, U))) =#> plus#(X, plus(Z, plus(Y, U)))   

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

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

We apply the subterm criterion with the following projection function:

  nu(plus#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(plus#(s(X), plus(Y, Z))) = s(X) |> X = nu(plus#(X, plus(s(s(Y)), Z))) 
  nu(plus#(s(X), plus(Y, plus(Z, U)))) = s(X) |> X = nu(plus#(X, plus(Z, plus(Y, U)))) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, 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 +++

[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.