/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.

  app : [o * o] --> o 
  cons : [o] --> o 
  from : [o] --> o 
  nil : [] --> o 
  prefix : [o] --> o 
  zWadr : [o * o] --> o 

  app(nil, X) => X 
  app(cons(X), Y) => cons(X) 
  from(X) => cons(X) 
  zWadr(nil, X) => nil 
  zWadr(X, nil) => nil 
  zWadr(cons(X), cons(Y)) => cons(app(Y, cons(X))) 
  prefix(X) => cons(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(cons(X), Y) >? cons(X) 
  from(X) >? cons(X) 
  zWadr(nil, X) >? nil 
  zWadr(X, nil) >? nil 
  zWadr(cons(X), cons(Y)) >? cons(app(Y, cons(X))) 
  prefix(X) >? cons(nil) 

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

The following interpretation satisfies the requirements:

  app = \y0y1.2 + y0 + y1 
  cons = \y0.2 + 2y0 
  from = \y0.3 + 3y0 
  nil = 0 
  prefix = \y0.3 + 2y0 
  zWadr = \y0y1.3 + 3y0 + 3y1 

Using this interpretation, the requirements translate to:

  [[app(nil, _x0)]] = 2 + x0 > x0 = [[_x0]] 
  [[app(cons(_x0), _x1)]] = 4 + x1 + 2x0 > 2 + 2x0 = [[cons(_x0)]] 
  [[from(_x0)]] = 3 + 3x0 > 2 + 2x0 = [[cons(_x0)]] 
  [[zWadr(nil, _x0)]] = 3 + 3x0 > 0 = [[nil]] 
  [[zWadr(_x0, nil)]] = 3 + 3x0 > 0 = [[nil]] 
  [[zWadr(cons(_x0), cons(_x1))]] = 15 + 6x0 + 6x1 > 10 + 2x1 + 4x0 = [[cons(app(_x1, cons(_x0)))]] 
  [[prefix(_x0)]] = 3 + 2x0 > 2 = [[cons(nil)]] 

We can thus remove the following rules:

  app(nil, X) => X 
  app(cons(X), Y) => cons(X) 
  from(X) => cons(X) 
  zWadr(nil, X) => nil 
  zWadr(X, nil) => nil 
  zWadr(cons(X), cons(Y)) => cons(app(Y, cons(X))) 
  prefix(X) => cons(nil) 

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 +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.