/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 19 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 31 ms]
(4) RelTRS
(5) RIsEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   top(ok(new(x))) -> top(check(x))
   top(ok(old(x))) -> top(check(x))

The relative TRS consists of the following S rules:

   bot -> new(bot)
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> ok(old(x))
   new(ok(x)) -> ok(new(x))
   old(ok(x)) -> ok(old(x))


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(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(bot) = 0
   POL(check(x_1)) = x_1
   POL(new(x_1)) = x_1
   POL(ok(x_1)) = x_1
   POL(old(x_1)) = 1 + x_1
   POL(top(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   top(ok(old(x))) -> top(check(x))
Rules from S:
none




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(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   top(ok(new(x))) -> top(check(x))

The relative TRS consists of the following S rules:

   bot -> new(bot)
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> ok(old(x))
   new(ok(x)) -> ok(new(x))
   old(ok(x)) -> ok(old(x))


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(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(top(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(ok(x_1)) =  	[[2], [2]] 	 +  	[[2, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(new(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
>>>

   <<<
 POL(check(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
>>>

   <<<
 POL(bot) =  	[[0], [0]]
>>>

   <<<
 POL(old(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   top(ok(new(x))) -> top(check(x))
Rules from S:
none




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(4)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   bot -> new(bot)
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> ok(old(x))
   new(ok(x)) -> ok(new(x))
   old(ok(x)) -> ok(old(x))


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(5) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(6)
YES