/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.

  add : [o * o] --> o 
  app : [o * o] --> o 
  nil : [] --> o 
  reverse : [o] --> o 
  shuffle : [o] --> o 

  app(nil, X) => X 
  app(add(X, Y), Z) => add(X, app(Y, Z)) 
  reverse(nil) => nil 
  reverse(add(X, Y)) => app(reverse(Y), add(X, nil)) 
  shuffle(nil) => nil 
  shuffle(add(X, Y)) => add(X, shuffle(reverse(Y))) 

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:

  add : [q * fb] --> fb 
  app : [fb * fb] --> fb 
  nil : [] --> fb 
  reverse : [fb] --> fb 
  shuffle : [fb] --> fb 

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

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

  app(nil, X) >? X 
  app(add(X, Y), Z) >? add(X, app(Y, Z)) 
  reverse(nil) >? nil 
  reverse(add(X, Y)) >? app(reverse(Y), add(X, nil)) 
  shuffle(nil) >? nil 
  shuffle(add(X, Y)) >? add(X, shuffle(reverse(Y))) 

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

The following interpretation satisfies the requirements:

  add = \y0y1.y0 + y1 
  app = \y0y1.y0 + y1 
  nil = 0 
  reverse = \y0.y0 
  shuffle = \y0.2 + 2y0 

Using this interpretation, the requirements translate to:

  [[app(nil, _x0)]] = x0 >= x0 = [[_x0]] 
  [[app(add(_x0, _x1), _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[add(_x0, app(_x1, _x2))]] 
  [[reverse(nil)]] = 0 >= 0 = [[nil]] 
  [[reverse(add(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[app(reverse(_x1), add(_x0, nil))]] 
  [[shuffle(nil)]] = 2 > 0 = [[nil]] 
  [[shuffle(add(_x0, _x1))]] = 2 + 2x0 + 2x1 >= 2 + x0 + 2x1 = [[add(_x0, shuffle(reverse(_x1)))]] 

We can thus remove the following rules:

  shuffle(nil) => nil 

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

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

  app(nil, X) >? X 
  app(add(X, Y), Z) >? add(X, app(Y, Z)) 
  reverse(nil) >? nil 
  reverse(add(X, Y)) >? app(reverse(Y), add(X, nil)) 
  shuffle(add(X, Y)) >? add(X, shuffle(reverse(Y))) 

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

The following interpretation satisfies the requirements:

  add = \y0y1.1 + y1 + 2y0 
  app = \y0y1.y0 + y1 
  nil = 0 
  reverse = \y0.y0 
  shuffle = \y0.2 + 2y0 

Using this interpretation, the requirements translate to:

  [[app(nil, _x0)]] = x0 >= x0 = [[_x0]] 
  [[app(add(_x0, _x1), _x2)]] = 1 + x1 + x2 + 2x0 >= 1 + x1 + x2 + 2x0 = [[add(_x0, app(_x1, _x2))]] 
  [[reverse(nil)]] = 0 >= 0 = [[nil]] 
  [[reverse(add(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[app(reverse(_x1), add(_x0, nil))]] 
  [[shuffle(add(_x0, _x1))]] = 4 + 2x1 + 4x0 > 3 + 2x0 + 2x1 = [[add(_x0, shuffle(reverse(_x1)))]] 

We can thus remove the following rules:

  shuffle(add(X, Y)) => add(X, shuffle(reverse(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]):

  app(nil, X) >? X 
  app(add(X, Y), Z) >? add(X, app(Y, Z)) 
  reverse(nil) >? nil 
  reverse(add(X, Y)) >? app(reverse(Y), add(X, nil)) 

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

The following interpretation satisfies the requirements:

  add = \y0y1.y0 + y1 
  app = \y0y1.y0 + 2y1 
  nil = 0 
  reverse = \y0.1 + 2y0 

Using this interpretation, the requirements translate to:

  [[app(nil, _x0)]] = 2x0 >= x0 = [[_x0]] 
  [[app(add(_x0, _x1), _x2)]] = x0 + x1 + 2x2 >= x0 + x1 + 2x2 = [[add(_x0, app(_x1, _x2))]] 
  [[reverse(nil)]] = 1 > 0 = [[nil]] 
  [[reverse(add(_x0, _x1))]] = 1 + 2x0 + 2x1 >= 1 + 2x0 + 2x1 = [[app(reverse(_x1), add(_x0, nil))]] 

We can thus remove the following rules:

  reverse(nil) => nil 

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

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

  app(nil, X) >? X 
  app(add(X, Y), Z) >? add(X, app(Y, Z)) 
  reverse(add(X, Y)) >? app(reverse(Y), add(X, nil)) 

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

The following interpretation satisfies the requirements:

  add = \y0y1.1 + y0 + y1 
  app = \y0y1.y0 + y1 
  nil = 0 
  reverse = \y0.2y0 

Using this interpretation, the requirements translate to:

  [[app(nil, _x0)]] = x0 >= x0 = [[_x0]] 
  [[app(add(_x0, _x1), _x2)]] = 1 + x0 + x1 + x2 >= 1 + x0 + x1 + x2 = [[add(_x0, app(_x1, _x2))]] 
  [[reverse(add(_x0, _x1))]] = 2 + 2x0 + 2x1 > 1 + x0 + 2x1 = [[app(reverse(_x1), add(_x0, nil))]] 

We can thus remove the following rules:

  reverse(add(X, Y)) => app(reverse(Y), add(X, nil)) 

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

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

  app(nil, X) >? X 
  app(add(X, Y), Z) >? add(X, app(Y, Z)) 

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

The following interpretation satisfies the requirements:

  add = \y0y1.3 + y0 + y1 
  app = \y0y1.3 + y1 + 3y0 
  nil = 3 

Using this interpretation, the requirements translate to:

  [[app(nil, _x0)]] = 12 + x0 > x0 = [[_x0]] 
  [[app(add(_x0, _x1), _x2)]] = 12 + x2 + 3x0 + 3x1 > 6 + x0 + x2 + 3x1 = [[add(_x0, app(_x1, _x2))]] 

We can thus remove the following rules:

  app(nil, X) => X 
  app(add(X, Y), Z) => add(X, app(Y, 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.