/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.

  concat : [o * o] --> o 
  cons : [o * o] --> o 
  false : [] --> o 
  leaf : [] --> o 
  lessleaves : [o * o] --> o 
  true : [] --> o 

  concat(leaf, X) => X 
  concat(cons(X, Y), Z) => cons(X, concat(Y, Z)) 
  lessleaves(X, leaf) => false 
  lessleaves(leaf, cons(X, Y)) => true 
  lessleaves(cons(X, Y), cons(Z, U)) => lessleaves(concat(X, Y), concat(Z, U)) 

As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95].  Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows:

  concat : [fb * fb] --> fb 
  cons : [fb * fb] --> fb 
  false : [] --> gb 
  leaf : [] --> fb 
  lessleaves : [fb * fb] --> gb 
  true : [] --> gb 

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

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

  concat(leaf, X) >? X 
  concat(cons(X, Y), Z) >? cons(X, concat(Y, Z)) 
  lessleaves(X, leaf) >? false 
  lessleaves(leaf, cons(X, Y)) >? true 
  lessleaves(cons(X, Y), cons(Z, U)) >? lessleaves(concat(X, Y), concat(Z, U)) 

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

The following interpretation satisfies the requirements:

  concat = \y0y1.y0 + y1 
  cons = \y0y1.y1 + 3y0 
  false = 0 
  leaf = 3 
  lessleaves = \y0y1.3 + y1 + 3y0 
  true = 0 

Using this interpretation, the requirements translate to:

  [[concat(leaf, _x0)]] = 3 + x0 > x0 = [[_x0]] 
  [[concat(cons(_x0, _x1), _x2)]] = x1 + x2 + 3x0 >= x1 + x2 + 3x0 = [[cons(_x0, concat(_x1, _x2))]] 
  [[lessleaves(_x0, leaf)]] = 6 + 3x0 > 0 = [[false]] 
  [[lessleaves(leaf, cons(_x0, _x1))]] = 12 + x1 + 3x0 > 0 = [[true]] 
  [[lessleaves(cons(_x0, _x1), cons(_x2, _x3))]] = 3 + x3 + 3x1 + 3x2 + 9x0 >= 3 + x2 + x3 + 3x0 + 3x1 = [[lessleaves(concat(_x0, _x1), concat(_x2, _x3))]] 

We can thus remove the following rules:

  concat(leaf, X) => X 
  lessleaves(X, leaf) => false 
  lessleaves(leaf, cons(X, Y)) => true 

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

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

  concat(cons(X, Y), Z) >? cons(X, concat(Y, Z)) 
  lessleaves(cons(X, Y), cons(Z, U)) >? lessleaves(concat(X, Y), concat(Z, U)) 

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

The following interpretation satisfies the requirements:

  concat = \y0y1.2 + y1 + 2y0 
  cons = \y0y1.3 + y1 + 2y0 
  lessleaves = \y0y1.3y0 + 3y1 

Using this interpretation, the requirements translate to:

  [[concat(cons(_x0, _x1), _x2)]] = 8 + x2 + 2x1 + 4x0 > 5 + x2 + 2x0 + 2x1 = [[cons(_x0, concat(_x1, _x2))]] 
  [[lessleaves(cons(_x0, _x1), cons(_x2, _x3))]] = 18 + 3x1 + 3x3 + 6x0 + 6x2 > 12 + 3x1 + 3x3 + 6x0 + 6x2 = [[lessleaves(concat(_x0, _x1), concat(_x2, _x3))]] 

We can thus remove the following rules:

  concat(cons(X, Y), Z) => cons(X, concat(Y, Z)) 
  lessleaves(cons(X, Y), cons(Z, U)) => lessleaves(concat(X, Y), concat(Z, U)) 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[FuhGieParSchSwi11]  C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski.  Proving Termination by Dependency Pairs and Inductive Theorem Proving.  In volume 47(2) of Journal of Automated Reasoning.  133--160, 2011.
[Gra95]  B. Gramlich.  Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems.  In volume 24(1-2) of Fundamentae Informaticae.  3--23, 1995.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.