/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 60 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 10 ms]
(4) QTRS
(5) AAECC Innermost [EQUIVALENT, 0 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) TRUE


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(0) -> cons(0, n__f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X

Q is empty.

----------------------------------------

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(activate(x_1)) = 2 + 2*x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(f(x_1)) = 1 + 2*x_1
   POL(n__f(x_1)) = x_1
   POL(p(x_1)) = x_1
   POL(s(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   p(s(0)) -> 0
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X




----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(0) -> cons(0, n__f(s(0)))
   f(s(0)) -> f(p(s(0)))

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = 2*x_1 + x_2
   POL(f(x_1)) = 1 + 2*x_1
   POL(n__f(x_1)) = 2*x_1
   POL(p(x_1)) = x_1
   POL(s(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f(0) -> cons(0, n__f(s(0)))




----------------------------------------

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(s(0)) -> f(p(s(0)))

Q is empty.

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(5) AAECC Innermost (EQUIVALENT)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none

The TRS R 2 is 
   f(s(0)) -> f(p(s(0)))

The signature Sigma is {f_1}
----------------------------------------

(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(s(0)) -> f(p(s(0)))

The set Q consists of the following terms:

   f(s(0))


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(7) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(0)) -> F(p(s(0)))

The TRS R consists of the following rules:

   f(s(0)) -> f(p(s(0)))

The set Q consists of the following terms:

   f(s(0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
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(10)
TRUE