6.41/2.41	YES
6.41/2.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.41/2.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.41/2.46	
6.41/2.46	
6.41/2.46	Termination of the given RelTRS could be proven:
6.41/2.46	
6.41/2.46	(0) RelTRS
6.41/2.46	(1) RelTRSRRRProof [EQUIVALENT, 46 ms]
6.41/2.46	(2) RelTRS
6.41/2.46	(3) RelTRSRRRProof [EQUIVALENT, 34 ms]
6.41/2.46	(4) RelTRS
6.41/2.46	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
6.41/2.46	(6) YES
6.41/2.46	
6.41/2.46	
6.41/2.46	----------------------------------------
6.41/2.46	
6.41/2.46	(0)
6.41/2.46	Obligation:
6.41/2.46	Relative term rewrite system:
6.41/2.46	The relative TRS consists of the following R rules:
6.41/2.46	
6.41/2.46	   n(s(x1)) -> s(x1)
6.41/2.46	   o(s(x1)) -> s(x1)
6.41/2.46	
6.41/2.46	The relative TRS consists of the following S rules:
6.41/2.46	
6.41/2.46	   t(x1) -> t(c(n(x1)))
6.41/2.46	   c(n(x1)) -> n(c(x1))
6.41/2.46	   c(o(x1)) -> o(c(x1))
6.41/2.46	   c(o(x1)) -> o(x1)
6.41/2.46	
6.41/2.46	
6.41/2.46	----------------------------------------
6.41/2.46	
6.41/2.46	(1) RelTRSRRRProof (EQUIVALENT)
6.41/2.46	We used the following monotonic ordering for rule removal:
6.41/2.46	Polynomial interpretation [POLO]:
6.41/2.46	
6.41/2.46	   POL(c(x_1)) = x_1
6.41/2.46	   POL(n(x_1)) = x_1
6.41/2.46	   POL(o(x_1)) = 1 + x_1
6.41/2.46	   POL(s(x_1)) = x_1
6.41/2.46	   POL(t(x_1)) = x_1
6.41/2.46	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.41/2.46	Rules from R:
6.41/2.46	
6.41/2.46	   o(s(x1)) -> s(x1)
6.41/2.46	Rules from S:
6.41/2.46	none
6.41/2.46	
6.41/2.46	
6.41/2.46	
6.41/2.46	
6.41/2.46	----------------------------------------
6.41/2.46	
6.41/2.46	(2)
6.41/2.46	Obligation:
6.41/2.46	Relative term rewrite system:
6.41/2.46	The relative TRS consists of the following R rules:
6.41/2.46	
6.41/2.46	   n(s(x1)) -> s(x1)
6.41/2.46	
6.41/2.46	The relative TRS consists of the following S rules:
6.41/2.46	
6.41/2.46	   t(x1) -> t(c(n(x1)))
6.41/2.46	   c(n(x1)) -> n(c(x1))
6.41/2.46	   c(o(x1)) -> o(c(x1))
6.41/2.46	   c(o(x1)) -> o(x1)
6.41/2.46	
6.41/2.46	
6.41/2.46	----------------------------------------
6.41/2.46	
6.41/2.46	(3) RelTRSRRRProof (EQUIVALENT)
6.41/2.46	We used the following monotonic ordering for rule removal:
6.41/2.46	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.41/2.46	
6.41/2.46	   <<<
6.41/2.46	 POL(n(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 2]] 	* 	x_1
6.41/2.46	>>>
6.41/2.46	
6.41/2.46	   <<<
6.41/2.46	 POL(s(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
6.41/2.46	>>>
6.41/2.46	
6.41/2.46	   <<<
6.41/2.46	 POL(t(x_1)) =  	[[2], [2]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
6.41/2.46	>>>
6.41/2.46	
6.41/2.46	   <<<
6.41/2.46	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
6.41/2.46	>>>
6.41/2.46	
6.41/2.46	   <<<
6.41/2.46	 POL(o(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
6.41/2.46	>>>
6.41/2.46	
6.41/2.46	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.41/2.46	Rules from R:
6.41/2.46	
6.41/2.46	   n(s(x1)) -> s(x1)
6.41/2.46	Rules from S:
6.41/2.46	none
6.41/2.46	
6.41/2.46	
6.41/2.46	
6.41/2.46	
6.41/2.46	----------------------------------------
6.41/2.46	
6.41/2.46	(4)
6.41/2.46	Obligation:
6.41/2.46	Relative term rewrite system:
6.41/2.46	R is empty.
6.41/2.46	The relative TRS consists of the following S rules:
6.41/2.46	
6.41/2.46	   t(x1) -> t(c(n(x1)))
6.41/2.46	   c(n(x1)) -> n(c(x1))
6.41/2.46	   c(o(x1)) -> o(c(x1))
6.41/2.46	   c(o(x1)) -> o(x1)
6.41/2.46	
6.41/2.46	
6.41/2.46	----------------------------------------
6.41/2.46	
6.41/2.46	(5) RIsEmptyProof (EQUIVALENT)
6.41/2.46	The TRS R is empty. Hence, termination is trivially proven.
6.41/2.46	----------------------------------------
6.41/2.46	
6.41/2.46	(6)
6.41/2.46	YES
6.68/2.56	EOF