7.79/2.90	YES
7.91/2.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.91/2.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.91/2.91	
7.91/2.91	
7.91/2.91	Termination of the given RelTRS could be proven:
7.91/2.91	
7.91/2.91	(0) RelTRS
7.91/2.91	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
7.91/2.91	(2) RelTRS
7.91/2.91	(3) RelTRSRRRProof [EQUIVALENT, 26 ms]
7.91/2.91	(4) RelTRS
7.91/2.91	(5) RelTRSRRRProof [EQUIVALENT, 96 ms]
7.91/2.91	(6) RelTRS
7.91/2.91	(7) RelTRSRRRProof [EQUIVALENT, 0 ms]
7.91/2.91	(8) RelTRS
7.91/2.91	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
7.91/2.91	(10) YES
7.91/2.91	
7.91/2.91	
7.91/2.91	----------------------------------------
7.91/2.91	
7.91/2.91	(0)
7.91/2.91	Obligation:
7.91/2.91	Relative term rewrite system:
7.91/2.91	The relative TRS consists of the following R rules:
7.91/2.91	
7.91/2.91	   o(l(x1)) -> r(x1)
7.91/2.91	   n(l(o(x1))) -> r(o(x1))
7.91/2.91	   L(l(o(x1))) -> L(r(o(x1)))
7.91/2.91	   r(o(x1)) -> l(x1)
7.91/2.91	   o(r(n(x1))) -> o(l(x1))
7.91/2.91	   o(r(R(x1))) -> o(l(R(x1)))
7.91/2.91	
7.91/2.91	The relative TRS consists of the following S rules:
7.91/2.91	
7.91/2.91	   L(x1) -> L(n(x1))
7.91/2.91	   R(x1) -> n(R(x1))
7.91/2.91	
7.91/2.91	
7.91/2.91	----------------------------------------
7.91/2.91	
7.91/2.91	(1) RelTRS Reverse (EQUIVALENT)
7.91/2.91	We have reversed the following relative TRS [REVERSE]:
7.91/2.91	The set of rules R is 
7.91/2.91	   o(l(x1)) -> r(x1)
7.91/2.91	   n(l(o(x1))) -> r(o(x1))
7.91/2.91	   L(l(o(x1))) -> L(r(o(x1)))
7.91/2.91	   r(o(x1)) -> l(x1)
7.91/2.91	   o(r(n(x1))) -> o(l(x1))
7.91/2.91	   o(r(R(x1))) -> o(l(R(x1)))
7.91/2.91	
7.91/2.91	The set of rules S is 
7.91/2.91	   L(x1) -> L(n(x1))
7.91/2.91	   R(x1) -> n(R(x1))
7.91/2.91	
7.91/2.91	We have obtained the following relative TRS:
7.91/2.91	The set of rules R is 
7.91/2.91	   l(o(x1)) -> r(x1)
7.91/2.91	   o(l(n(x1))) -> o(r(x1))
7.91/2.91	   o(l(L(x1))) -> o(r(L(x1)))
7.91/2.91	   o(r(x1)) -> l(x1)
7.91/2.91	   n(r(o(x1))) -> l(o(x1))
7.91/2.91	   R(r(o(x1))) -> R(l(o(x1)))
7.91/2.91	
7.91/2.91	The set of rules S is 
7.91/2.91	   L(x1) -> n(L(x1))
7.91/2.91	   R(x1) -> R(n(x1))
7.91/2.91	
7.91/2.91	
7.91/2.91	----------------------------------------
7.91/2.91	
7.91/2.91	(2)
7.91/2.91	Obligation:
7.91/2.91	Relative term rewrite system:
7.91/2.91	The relative TRS consists of the following R rules:
7.91/2.91	
7.91/2.91	   l(o(x1)) -> r(x1)
7.91/2.91	   o(l(n(x1))) -> o(r(x1))
7.91/2.91	   o(l(L(x1))) -> o(r(L(x1)))
7.91/2.91	   o(r(x1)) -> l(x1)
7.91/2.91	   n(r(o(x1))) -> l(o(x1))
7.91/2.91	   R(r(o(x1))) -> R(l(o(x1)))
7.91/2.91	
7.91/2.91	The relative TRS consists of the following S rules:
7.91/2.91	
7.91/2.91	   L(x1) -> n(L(x1))
7.91/2.91	   R(x1) -> R(n(x1))
7.91/2.91	
7.91/2.91	
7.91/2.91	----------------------------------------
7.91/2.91	
7.91/2.91	(3) RelTRSRRRProof (EQUIVALENT)
7.91/2.91	We used the following monotonic ordering for rule removal:
7.91/2.91	Polynomial interpretation [POLO]:
7.91/2.91	
7.91/2.91	   POL(L(x_1)) = x_1
7.91/2.91	   POL(R(x_1)) = x_1
7.91/2.91	   POL(l(x_1)) = x_1
7.91/2.91	   POL(n(x_1)) = x_1
7.91/2.91	   POL(o(x_1)) = 1 + x_1
7.91/2.91	   POL(r(x_1)) = x_1
7.91/2.91	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.91/2.91	Rules from R:
7.91/2.91	
7.91/2.91	   l(o(x1)) -> r(x1)
7.91/2.91	   o(r(x1)) -> l(x1)
7.91/2.91	Rules from S:
7.91/2.91	none
7.91/2.91	
7.91/2.91	
7.91/2.91	
7.91/2.91	
7.91/2.91	----------------------------------------
7.91/2.91	
7.91/2.91	(4)
7.91/2.91	Obligation:
7.91/2.91	Relative term rewrite system:
7.91/2.91	The relative TRS consists of the following R rules:
7.91/2.91	
7.91/2.91	   o(l(n(x1))) -> o(r(x1))
7.91/2.91	   o(l(L(x1))) -> o(r(L(x1)))
7.91/2.91	   n(r(o(x1))) -> l(o(x1))
7.91/2.91	   R(r(o(x1))) -> R(l(o(x1)))
7.91/2.91	
7.91/2.91	The relative TRS consists of the following S rules:
7.91/2.91	
7.91/2.91	   L(x1) -> n(L(x1))
7.91/2.91	   R(x1) -> R(n(x1))
7.91/2.91	
7.91/2.91	
7.91/2.91	----------------------------------------
7.91/2.91	
7.91/2.91	(5) RelTRSRRRProof (EQUIVALENT)
7.91/2.91	We used the following monotonic ordering for rule removal:
7.91/2.91	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.91/2.91	
7.91/2.91	   <<<
7.91/2.91	 POL(o(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [0, 0]] 	* 	x_1
7.91/2.91	>>>
7.91/2.91	
7.91/2.91	   <<<
7.91/2.91	 POL(l(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
7.91/2.91	>>>
7.91/2.91	
7.91/2.91	   <<<
7.91/2.91	 POL(n(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
7.91/2.91	>>>
7.91/2.91	
7.91/2.91	   <<<
7.91/2.91	 POL(r(x_1)) =  	[[1], [2]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
7.91/2.91	>>>
7.91/2.91	
7.91/2.91	   <<<
7.91/2.91	 POL(L(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
7.91/2.91	>>>
7.91/2.91	
7.91/2.91	   <<<
7.91/2.91	 POL(R(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 2]] 	* 	x_1
7.91/2.91	>>>
7.91/2.91	
7.91/2.91	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.91/2.91	Rules from R:
7.91/2.91	
7.91/2.91	   n(r(o(x1))) -> l(o(x1))
7.91/2.91	   R(r(o(x1))) -> R(l(o(x1)))
7.91/2.91	Rules from S:
7.91/2.91	none
7.91/2.91	
7.91/2.91	
7.91/2.91	
7.91/2.91	
7.91/2.91	----------------------------------------
7.91/2.91	
7.91/2.91	(6)
7.91/2.91	Obligation:
7.91/2.91	Relative term rewrite system:
7.91/2.91	The relative TRS consists of the following R rules:
7.91/2.91	
7.91/2.91	   o(l(n(x1))) -> o(r(x1))
7.91/2.91	   o(l(L(x1))) -> o(r(L(x1)))
7.91/2.91	
7.91/2.91	The relative TRS consists of the following S rules:
7.91/2.91	
7.91/2.91	   L(x1) -> n(L(x1))
7.91/2.91	   R(x1) -> R(n(x1))
7.91/2.91	
7.91/2.91	
7.91/2.91	----------------------------------------
7.91/2.91	
7.91/2.91	(7) RelTRSRRRProof (EQUIVALENT)
7.91/2.91	We used the following monotonic ordering for rule removal:
7.91/2.91	Polynomial interpretation [POLO]:
7.91/2.91	
7.91/2.91	   POL(L(x_1)) = x_1
7.91/2.91	   POL(R(x_1)) = x_1
7.91/2.91	   POL(l(x_1)) = 1 + x_1
7.91/2.91	   POL(n(x_1)) = x_1
7.91/2.92	   POL(o(x_1)) = x_1
7.91/2.92	   POL(r(x_1)) = x_1
7.91/2.92	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.91/2.92	Rules from R:
7.91/2.92	
7.91/2.92	   o(l(n(x1))) -> o(r(x1))
7.91/2.92	   o(l(L(x1))) -> o(r(L(x1)))
7.91/2.92	Rules from S:
7.91/2.92	none
7.91/2.92	
7.91/2.92	
7.91/2.92	
7.91/2.92	
7.91/2.92	----------------------------------------
7.91/2.92	
7.91/2.92	(8)
7.91/2.92	Obligation:
7.91/2.92	Relative term rewrite system:
7.91/2.92	R is empty.
7.91/2.92	The relative TRS consists of the following S rules:
7.91/2.92	
7.91/2.92	   L(x1) -> n(L(x1))
7.91/2.92	   R(x1) -> R(n(x1))
7.91/2.92	
7.91/2.92	
7.91/2.92	----------------------------------------
7.91/2.92	
7.91/2.92	(9) RIsEmptyProof (EQUIVALENT)
7.91/2.92	The TRS R is empty. Hence, termination is trivially proven.
7.91/2.92	----------------------------------------
7.91/2.92	
7.91/2.92	(10)
7.91/2.92	YES
8.02/2.97	EOF