/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 65 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   not(x) -> if(x, false, true)
   and(x, y) -> if(x, y, false)
   or(x, y) -> if(x, true, y)
   implies(x, y) -> if(x, y, true)
   =(x, x) -> true
   =(x, y) -> if(x, y, not(y))
   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, x, if(x, false, true)) -> true
   =(x, y) -> if(x, y, if(y, false, true))

Q is empty.

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

   POL(=(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(and(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(false) = 0
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(implies(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(not(x_1)) = x_1
   POL(or(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   and(x, y) -> if(x, y, false)
   or(x, y) -> if(x, true, y)
   implies(x, y) -> if(x, y, true)
   =(x, x) -> true
   =(x, y) -> if(x, y, not(y))
   =(x, y) -> if(x, y, if(y, false, true))




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

   not(x) -> if(x, false, true)
   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, x, if(x, false, true)) -> true

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:true > false > not_1 > if_3

and weight map:

   false=1
   true=1
   not_1=2
   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:

   not(x) -> if(x, false, true)
   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, x, if(x, false, true)) -> true




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES