2.62/1.46	YES
2.78/1.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.78/1.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.78/1.47	
2.78/1.47	
2.78/1.47	Termination of the given CSR could be proven:
2.78/1.47	
2.78/1.47	(0) CSR
2.78/1.47	(1) CSRRRRProof [EQUIVALENT, 23 ms]
2.78/1.47	(2) CSR
2.78/1.47	(3) CSRRRRProof [EQUIVALENT, 0 ms]
2.78/1.47	(4) CSR
2.78/1.47	(5) RisEmptyProof [EQUIVALENT, 0 ms]
2.78/1.47	(6) YES
2.78/1.47	
2.78/1.47	
2.78/1.47	----------------------------------------
2.78/1.47	
2.78/1.47	(0)
2.78/1.47	Obligation:
2.78/1.47	Context-sensitive rewrite system:
2.78/1.47	The TRS R consists of the following rules:
2.78/1.47	
2.78/1.47	   zeros -> cons(0, zeros)
2.78/1.47	   tail(cons(X, XS)) -> XS
2.78/1.47	
2.78/1.47	The replacement map contains the following entries:
2.78/1.47	
2.78/1.47	zeros: empty set
2.78/1.47	cons: {1}
2.78/1.47	0: empty set
2.78/1.47	tail: {1}
2.78/1.47	
2.78/1.47	----------------------------------------
2.78/1.47	
2.78/1.47	(1) CSRRRRProof (EQUIVALENT)
2.78/1.47	The following CSR is given: Context-sensitive rewrite system:
2.78/1.47	The TRS R consists of the following rules:
2.78/1.47	
2.78/1.47	   zeros -> cons(0, zeros)
2.78/1.47	   tail(cons(X, XS)) -> XS
2.78/1.47	
2.78/1.47	The replacement map contains the following entries:
2.78/1.47	
2.78/1.47	zeros: empty set
2.78/1.47	cons: {1}
2.78/1.47	0: empty set
2.78/1.47	tail: {1}
2.78/1.47	Used ordering:
2.78/1.47	Polynomial interpretation [POLO]:
2.78/1.47	
2.78/1.47	   POL(0) = 0
2.78/1.47	   POL(cons(x_1, x_2)) = x_1 + x_2
2.78/1.47	   POL(tail(x_1)) = 1 + x_1
2.78/1.47	   POL(zeros) = 0
2.78/1.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.78/1.47	
2.78/1.47	   tail(cons(X, XS)) -> XS
2.78/1.47	
2.78/1.47	
2.78/1.47	
2.78/1.47	
2.78/1.47	----------------------------------------
2.78/1.47	
2.78/1.47	(2)
2.78/1.47	Obligation:
2.78/1.47	Context-sensitive rewrite system:
2.78/1.47	The TRS R consists of the following rules:
2.78/1.47	
2.78/1.47	   zeros -> cons(0, zeros)
2.78/1.47	
2.78/1.47	The replacement map contains the following entries:
2.78/1.47	
2.78/1.47	zeros: empty set
2.78/1.47	cons: {1}
2.78/1.47	0: empty set
2.78/1.47	
2.78/1.47	----------------------------------------
2.78/1.47	
2.78/1.47	(3) CSRRRRProof (EQUIVALENT)
2.78/1.47	The following CSR is given: Context-sensitive rewrite system:
2.78/1.47	The TRS R consists of the following rules:
2.78/1.47	
2.78/1.47	   zeros -> cons(0, zeros)
2.78/1.47	
2.78/1.47	The replacement map contains the following entries:
2.78/1.47	
2.78/1.47	zeros: empty set
2.78/1.47	cons: {1}
2.78/1.47	0: empty set
2.78/1.47	Used ordering:
2.78/1.47	Polynomial interpretation [POLO]:
2.78/1.47	
2.78/1.47	   POL(0) = 0
2.78/1.47	   POL(cons(x_1, x_2)) = x_1
2.78/1.47	   POL(zeros) = 1
2.78/1.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.78/1.47	
2.78/1.47	   zeros -> cons(0, zeros)
2.78/1.47	
2.78/1.47	
2.78/1.47	
2.78/1.47	
2.78/1.47	----------------------------------------
2.78/1.47	
2.78/1.47	(4)
2.78/1.47	Obligation:
2.78/1.47	Context-sensitive rewrite system:
2.78/1.47	R is empty.
2.78/1.47	
2.78/1.47	----------------------------------------
2.78/1.47	
2.78/1.47	(5) RisEmptyProof (EQUIVALENT)
2.78/1.47	The CSR R is empty. Hence, termination is trivially proven.
2.78/1.47	----------------------------------------
2.78/1.47	
2.78/1.47	(6)
2.78/1.47	YES
2.81/1.49	EOF