YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRS Reverse [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 22 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 83 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 3 ms]
(8) RelTRS
(9) RIsEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   o(l(x1)) -> r(x1)
   n(l(o(x1))) -> r(o(x1))
   L(l(o(x1))) -> L(r(o(x1)))
   r(o(x1)) -> l(x1)
   o(r(n(x1))) -> o(l(x1))
   o(r(R(x1))) -> o(l(R(x1)))

The relative TRS consists of the following S rules:

   L(x1) -> L(n(x1))
   R(x1) -> n(R(x1))


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(1) RelTRS Reverse (EQUIVALENT)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is 
   o(l(x1)) -> r(x1)
   n(l(o(x1))) -> r(o(x1))
   L(l(o(x1))) -> L(r(o(x1)))
   r(o(x1)) -> l(x1)
   o(r(n(x1))) -> o(l(x1))
   o(r(R(x1))) -> o(l(R(x1)))

The set of rules S is 
   L(x1) -> L(n(x1))
   R(x1) -> n(R(x1))

We have obtained the following relative TRS:
The set of rules R is 
   l(o(x1)) -> r(x1)
   o(l(n(x1))) -> o(r(x1))
   o(l(L(x1))) -> o(r(L(x1)))
   o(r(x1)) -> l(x1)
   n(r(o(x1))) -> l(o(x1))
   R(r(o(x1))) -> R(l(o(x1)))

The set of rules S is 
   L(x1) -> n(L(x1))
   R(x1) -> R(n(x1))


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

   l(o(x1)) -> r(x1)
   o(l(n(x1))) -> o(r(x1))
   o(l(L(x1))) -> o(r(L(x1)))
   o(r(x1)) -> l(x1)
   n(r(o(x1))) -> l(o(x1))
   R(r(o(x1))) -> R(l(o(x1)))

The relative TRS consists of the following S rules:

   L(x1) -> n(L(x1))
   R(x1) -> R(n(x1))


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

   POL(L(x_1)) = x_1
   POL(R(x_1)) = x_1
   POL(l(x_1)) = x_1
   POL(n(x_1)) = x_1
   POL(o(x_1)) = 1 + x_1
   POL(r(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   l(o(x1)) -> r(x1)
   o(r(x1)) -> l(x1)
Rules from S:
none




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

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

   o(l(n(x1))) -> o(r(x1))
   o(l(L(x1))) -> o(r(L(x1)))
   n(r(o(x1))) -> l(o(x1))
   R(r(o(x1))) -> R(l(o(x1)))

The relative TRS consists of the following S rules:

   L(x1) -> n(L(x1))
   R(x1) -> R(n(x1))


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

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

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

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

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

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

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

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

   n(r(o(x1))) -> l(o(x1))
   R(r(o(x1))) -> R(l(o(x1)))
Rules from S:
none




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

(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   o(l(n(x1))) -> o(r(x1))
   o(l(L(x1))) -> o(r(L(x1)))

The relative TRS consists of the following S rules:

   L(x1) -> n(L(x1))
   R(x1) -> R(n(x1))


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

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

   POL(L(x_1)) = x_1
   POL(R(x_1)) = x_1
   POL(l(x_1)) = 1 + x_1
   POL(n(x_1)) = x_1
   POL(o(x_1)) = x_1
   POL(r(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   o(l(n(x1))) -> o(r(x1))
   o(l(L(x1))) -> o(r(L(x1)))
Rules from S:
none




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

(8)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   L(x1) -> n(L(x1))
   R(x1) -> R(n(x1))


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