YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 61 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 9 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   f(X) -> if(X, c, n__f(n__true))
   if(true, X, Y) -> X
   if(false, X, Y) -> activate(Y)
   f(X) -> n__f(X)
   true -> n__true
   activate(n__f(X)) -> f(activate(X))
   activate(n__true) -> true
   activate(X) -> X

Q is empty.

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

   POL(activate(x_1)) = x_1
   POL(c) = 0
   POL(f(x_1)) = 2*x_1
   POL(false) = 1
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
   POL(n__f(x_1)) = 2*x_1
   POL(n__true) = 0
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   if(false, X, Y) -> activate(Y)




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

   f(X) -> if(X, c, n__f(n__true))
   if(true, X, Y) -> X
   f(X) -> n__f(X)
   true -> n__true
   activate(n__f(X)) -> f(activate(X))
   activate(n__true) -> true
   activate(X) -> X

Q is empty.

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

   POL(activate(x_1)) = 2 + 2*x_1
   POL(c) = 0
   POL(f(x_1)) = 2 + 2*x_1
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(n__f(x_1)) = 2 + 2*x_1
   POL(n__true) = 0
   POL(true) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   if(true, X, Y) -> X
   true -> n__true
   activate(n__true) -> true
   activate(X) -> X




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

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

   f(X) -> if(X, c, n__f(n__true))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(activate(X))

Q is empty.

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

   POL(activate(x_1)) = 2*x_1
   POL(c) = 0
   POL(f(x_1)) = 2 + 2*x_1
   POL(if(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
   POL(n__f(x_1)) = 2 + 2*x_1
   POL(n__true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   activate(n__f(X)) -> f(activate(X))




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

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

   f(X) -> if(X, c, n__f(n__true))
   f(X) -> n__f(X)

Q is empty.

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(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:n__true > n__f_1 > c > f_1 > if_3

and weight map:

   c=1
   n__true=1
   f_1=3
   n__f_1=1
   if_3=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f(X) -> if(X, c, n__f(n__true))
   f(X) -> n__f(X)




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

(8)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(9) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(10)
YES