/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 
  !times : [o * o] --> o 

  !times(X, !plus(Y, Z)) => !plus(!times(X, Y), !times(X, Z)) 

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:

  !plus : [r * r] --> r 
  !times : [d * r] --> r 

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

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

  !times(X, !plus(Y, Z)) >? !plus(!times(X, Y), !times(X, Z)) 

about to try horpo

We use a recursive path ordering as defined in [Kop12, Chapter 5].

We choose Lex = {} and Mul = {!plus, !times}, and the following precedence: !times > !plus

With these choices, we have:

  1] !times(X, !plus(Y, Z)) > !plus(!times(X, Y), !times(X, Z))  because [2], by definition 
  2] !times*(X, !plus(Y, Z)) >= !plus(!times(X, Y), !times(X, Z))  because !times > !plus, [3] and [8], by (Copy) 
  3] !times*(X, !plus(Y, Z)) >= !times(X, Y)  because !times in Mul, [4] and [5], by (Stat) 
  4] X >= X  by (Meta) 
  5] !plus(Y, Z) > Y  because [6], by definition 
  6] !plus*(Y, Z) >= Y  because [7], by (Select) 
  7] Y >= Y  by (Meta) 
  8] !times*(X, !plus(Y, Z)) >= !times(X, Z)  because !times in Mul, [4] and [9], by (Stat) 
  9] !plus(Y, Z) > Z  because [10], by definition 
  10] !plus*(Y, Z) >= Z  because [11], by (Select) 
  11] Z >= Z  by (Meta) 

We can thus remove the following rules:

  !times(X, !plus(Y, Z)) => !plus(!times(X, Y), !times(X, Z)) 

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.