/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.

  !colon : [o * o] --> o 
  e : [] --> o 
  i : [o] --> o 

  !colon(X, X) => e 
  !colon(X, e) => X 
  i(!colon(X, Y)) => !colon(Y, X) 
  !colon(!colon(X, Y), Z) => !colon(X, !colon(Z, i(Y))) 
  !colon(e, X) => i(X) 
  i(i(X)) => X 
  i(e) => e 
  !colon(X, !colon(Y, i(X))) => i(Y) 
  !colon(X, !colon(Y, !colon(i(X), Z))) => !colon(i(Z), Y) 
  !colon(i(X), !colon(Y, X)) => i(Y) 
  !colon(i(X), !colon(Y, !colon(X, Z))) => !colon(i(Z), Y) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  !colon(X, X) >? e 
  !colon(X, e) >? X 
  i(!colon(X, Y)) >? !colon(Y, X) 
  !colon(!colon(X, Y), Z) >? !colon(X, !colon(Z, i(Y))) 
  !colon(e, X) >? i(X) 
  i(i(X)) >? X 
  i(e) >? e 
  !colon(X, !colon(Y, i(X))) >? i(Y) 
  !colon(X, !colon(Y, !colon(i(X), Z))) >? !colon(i(Z), Y) 
  !colon(i(X), !colon(Y, X)) >? i(Y) 
  !colon(i(X), !colon(Y, !colon(X, Z))) >? !colon(i(Z), Y) 

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

The following interpretation satisfies the requirements:

  !colon = \y0y1.1 + y0 + y1 
  e = 0 
  i = \y0.y0 

Using this interpretation, the requirements translate to:

  [[!colon(_x0, _x0)]] = 1 + 2x0 > 0 = [[e]] 
  [[!colon(_x0, e)]] = 1 + x0 > x0 = [[_x0]] 
  [[i(!colon(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[!colon(_x1, _x0)]] 
  [[!colon(!colon(_x0, _x1), _x2)]] = 2 + x0 + x1 + x2 >= 2 + x0 + x1 + x2 = [[!colon(_x0, !colon(_x2, i(_x1)))]] 
  [[!colon(e, _x0)]] = 1 + x0 > x0 = [[i(_x0)]] 
  [[i(i(_x0))]] = x0 >= x0 = [[_x0]] 
  [[i(e)]] = 0 >= 0 = [[e]] 
  [[!colon(_x0, !colon(_x1, i(_x0)))]] = 2 + x1 + 2x0 > x1 = [[i(_x1)]] 
  [[!colon(_x0, !colon(_x1, !colon(i(_x0), _x2)))]] = 3 + x1 + x2 + 2x0 > 1 + x1 + x2 = [[!colon(i(_x2), _x1)]] 
  [[!colon(i(_x0), !colon(_x1, _x0))]] = 2 + x1 + 2x0 > x1 = [[i(_x1)]] 
  [[!colon(i(_x0), !colon(_x1, !colon(_x0, _x2)))]] = 3 + x1 + x2 + 2x0 > 1 + x1 + x2 = [[!colon(i(_x2), _x1)]] 

We can thus remove the following rules:

  !colon(X, X) => e 
  !colon(X, e) => X 
  !colon(e, X) => i(X) 
  !colon(X, !colon(Y, i(X))) => i(Y) 
  !colon(X, !colon(Y, !colon(i(X), Z))) => !colon(i(Z), Y) 
  !colon(i(X), !colon(Y, X)) => i(Y) 
  !colon(i(X), !colon(Y, !colon(X, Z))) => !colon(i(Z), Y) 

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] i#(!colon(X, Y)) =#> !colon#(Y, X)   
    1] !colon#(!colon(X, Y), Z) =#> !colon#(X, !colon(Z, i(Y)))   
    2] !colon#(!colon(X, Y), Z) =#> !colon#(Z, i(Y))   
    3] !colon#(!colon(X, Y), Z) =#> i#(Y)   

  Rules R_0:

    i(!colon(X, Y)) => !colon(Y, X) 
    !colon(!colon(X, Y), Z) => !colon(X, !colon(Z, i(Y))) 
    i(i(X)) => X 
    i(e) => e 

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).

The formative rules of (P_0, R_0) are R_1 ::=

  i(!colon(X, Y)) => !colon(Y, X) 
  !colon(!colon(X, Y), Z) => !colon(X, !colon(Z, i(Y))) 
  i(i(X)) => X 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative).

Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_1, 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:

  i#(!colon(X, Y)) >? !colon#(Y, X) 
  !colon#(!colon(X, Y), Z) >? !colon#(X, !colon(Z, i(Y))) 
  !colon#(!colon(X, Y), Z) >? !colon#(Z, i(Y)) 
  !colon#(!colon(X, Y), Z) >? i#(Y) 
  i(!colon(X, Y)) >= !colon(Y, X) 
  !colon(!colon(X, Y), Z) >= !colon(X, !colon(Z, i(Y))) 
  i(i(X)) >= X 

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

The following interpretation satisfies the requirements:

  !colon = \y0y1.2 + y0 + y1 
  !colon# = \y0y1.2y0 + 2y1 
  i = \y0.y0 
  i# = \y0.2y0 

Using this interpretation, the requirements translate to:

  [[i#(!colon(_x0, _x1))]] = 4 + 2x0 + 2x1 > 2x0 + 2x1 = [[!colon#(_x1, _x0)]] 
  [[!colon#(!colon(_x0, _x1), _x2)]] = 4 + 2x0 + 2x1 + 2x2 >= 4 + 2x0 + 2x1 + 2x2 = [[!colon#(_x0, !colon(_x2, i(_x1)))]] 
  [[!colon#(!colon(_x0, _x1), _x2)]] = 4 + 2x0 + 2x1 + 2x2 > 2x1 + 2x2 = [[!colon#(_x2, i(_x1))]] 
  [[!colon#(!colon(_x0, _x1), _x2)]] = 4 + 2x0 + 2x1 + 2x2 > 2x1 = [[i#(_x1)]] 
  [[i(!colon(_x0, _x1))]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[!colon(_x1, _x0)]] 
  [[!colon(!colon(_x0, _x1), _x2)]] = 4 + x0 + x1 + x2 >= 4 + x0 + x1 + x2 = [[!colon(_x0, !colon(_x2, i(_x1)))]] 
  [[i(i(_x0))]] = x0 >= x0 = [[_x0]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_1, minimal, formative) by (P_1, R_1, minimal, formative), where P_1 consists of:

  !colon#(!colon(X, Y), Z) =#> !colon#(X, !colon(Z, i(Y)))   

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_1, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(!colon#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(!colon#(!colon(X, Y), Z)) = !colon(X, Y) |> X = nu(!colon#(X, !colon(Z, i(Y)))) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_1, minimal, f) by ({}, R_1, 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.