3.40/1.45	YES
3.49/1.46	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.49/1.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.49/1.46	
3.49/1.46	
3.49/1.46	Termination of the given CSR could be proven:
3.49/1.46	
3.49/1.46	(0) CSR
3.49/1.46	(1) CSRRRRProof [EQUIVALENT, 60 ms]
3.49/1.46	(2) CSR
3.49/1.46	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.49/1.46	(4) CSR
3.49/1.46	(5) CSRRRRProof [EQUIVALENT, 6 ms]
3.49/1.46	(6) CSR
3.49/1.46	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.49/1.46	(8) CSR
3.49/1.46	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.49/1.46	(10) YES
3.49/1.46	
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(0)
3.49/1.46	Obligation:
3.49/1.46	Context-sensitive rewrite system:
3.49/1.46	The TRS R consists of the following rules:
3.49/1.46	
3.49/1.46	   nats -> cons(0, incr(nats))
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	   odds -> incr(pairs)
3.49/1.46	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.49/1.46	   head(cons(X, XS)) -> X
3.49/1.46	   tail(cons(X, XS)) -> XS
3.49/1.46	
3.49/1.46	The replacement map contains the following entries:
3.49/1.46	
3.49/1.46	nats: empty set
3.49/1.46	cons: {1}
3.49/1.46	0: empty set
3.49/1.46	incr: {1}
3.49/1.46	pairs: empty set
3.49/1.46	odds: empty set
3.49/1.46	s: {1}
3.49/1.46	head: {1}
3.49/1.46	tail: {1}
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(1) CSRRRRProof (EQUIVALENT)
3.49/1.46	The following CSR is given: Context-sensitive rewrite system:
3.49/1.46	The TRS R consists of the following rules:
3.49/1.46	
3.49/1.46	   nats -> cons(0, incr(nats))
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	   odds -> incr(pairs)
3.49/1.46	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.49/1.46	   head(cons(X, XS)) -> X
3.49/1.46	   tail(cons(X, XS)) -> XS
3.49/1.46	
3.49/1.46	The replacement map contains the following entries:
3.49/1.46	
3.49/1.46	nats: empty set
3.49/1.46	cons: {1}
3.49/1.46	0: empty set
3.49/1.46	incr: {1}
3.49/1.46	pairs: empty set
3.49/1.46	odds: empty set
3.49/1.46	s: {1}
3.49/1.46	head: {1}
3.49/1.46	tail: {1}
3.49/1.46	Used ordering:
3.49/1.46	Polynomial interpretation [POLO]:
3.49/1.46	
3.49/1.46	   POL(0) = 0
3.49/1.46	   POL(cons(x_1, x_2)) = 2*x_1 + x_2
3.49/1.46	   POL(head(x_1)) = 1 + 2*x_1
3.49/1.46	   POL(incr(x_1)) = x_1
3.49/1.46	   POL(nats) = 0
3.49/1.46	   POL(odds) = 0
3.49/1.46	   POL(pairs) = 0
3.49/1.46	   POL(s(x_1)) = x_1
3.49/1.46	   POL(tail(x_1)) = 2*x_1
3.49/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.49/1.46	
3.49/1.46	   head(cons(X, XS)) -> X
3.49/1.46	
3.49/1.46	
3.49/1.46	
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(2)
3.49/1.46	Obligation:
3.49/1.46	Context-sensitive rewrite system:
3.49/1.46	The TRS R consists of the following rules:
3.49/1.46	
3.49/1.46	   nats -> cons(0, incr(nats))
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	   odds -> incr(pairs)
3.49/1.46	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.49/1.46	   tail(cons(X, XS)) -> XS
3.49/1.46	
3.49/1.46	The replacement map contains the following entries:
3.49/1.46	
3.49/1.46	nats: empty set
3.49/1.46	cons: {1}
3.49/1.46	0: empty set
3.49/1.46	incr: {1}
3.49/1.46	pairs: empty set
3.49/1.46	odds: empty set
3.49/1.46	s: {1}
3.49/1.46	tail: {1}
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(3) CSRRRRProof (EQUIVALENT)
3.49/1.46	The following CSR is given: Context-sensitive rewrite system:
3.49/1.46	The TRS R consists of the following rules:
3.49/1.46	
3.49/1.46	   nats -> cons(0, incr(nats))
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	   odds -> incr(pairs)
3.49/1.46	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.49/1.46	   tail(cons(X, XS)) -> XS
3.49/1.46	
3.49/1.46	The replacement map contains the following entries:
3.49/1.46	
3.49/1.46	nats: empty set
3.49/1.46	cons: {1}
3.49/1.46	0: empty set
3.49/1.46	incr: {1}
3.49/1.46	pairs: empty set
3.49/1.46	odds: empty set
3.49/1.46	s: {1}
3.49/1.46	tail: {1}
3.49/1.46	Used ordering:
3.49/1.46	Polynomial interpretation [POLO]:
3.49/1.46	
3.49/1.46	   POL(0) = 0
3.49/1.46	   POL(cons(x_1, x_2)) = x_1 + x_2
3.49/1.46	   POL(incr(x_1)) = x_1
3.49/1.46	   POL(nats) = 0
3.49/1.46	   POL(odds) = 0
3.49/1.46	   POL(pairs) = 0
3.49/1.46	   POL(s(x_1)) = x_1
3.49/1.46	   POL(tail(x_1)) = 1 + x_1
3.49/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.49/1.46	
3.49/1.46	   tail(cons(X, XS)) -> XS
3.49/1.46	
3.49/1.46	
3.49/1.46	
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(4)
3.49/1.46	Obligation:
3.49/1.46	Context-sensitive rewrite system:
3.49/1.46	The TRS R consists of the following rules:
3.49/1.46	
3.49/1.46	   nats -> cons(0, incr(nats))
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	   odds -> incr(pairs)
3.49/1.46	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.49/1.46	
3.49/1.46	The replacement map contains the following entries:
3.49/1.46	
3.49/1.46	nats: empty set
3.49/1.46	cons: {1}
3.49/1.46	0: empty set
3.49/1.46	incr: {1}
3.49/1.46	pairs: empty set
3.49/1.46	odds: empty set
3.49/1.46	s: {1}
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(5) CSRRRRProof (EQUIVALENT)
3.49/1.46	The following CSR is given: Context-sensitive rewrite system:
3.49/1.46	The TRS R consists of the following rules:
3.49/1.46	
3.49/1.46	   nats -> cons(0, incr(nats))
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	   odds -> incr(pairs)
3.49/1.46	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.49/1.46	
3.49/1.46	The replacement map contains the following entries:
3.49/1.46	
3.49/1.46	nats: empty set
3.49/1.46	cons: {1}
3.49/1.46	0: empty set
3.49/1.46	incr: {1}
3.49/1.46	pairs: empty set
3.49/1.46	odds: empty set
3.49/1.46	s: {1}
3.49/1.46	Used ordering:
3.49/1.46	Polynomial interpretation [POLO]:
3.49/1.46	
3.49/1.46	   POL(0) = 0
3.49/1.46	   POL(cons(x_1, x_2)) = 2*x_1
3.49/1.46	   POL(incr(x_1)) = 1 + x_1
3.49/1.46	   POL(nats) = 1
3.49/1.46	   POL(odds) = 2
3.49/1.46	   POL(pairs) = 0
3.49/1.46	   POL(s(x_1)) = x_1
3.49/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.49/1.46	
3.49/1.46	   nats -> cons(0, incr(nats))
3.49/1.46	   odds -> incr(pairs)
3.49/1.46	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.49/1.46	
3.49/1.46	
3.49/1.46	
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(6)
3.49/1.46	Obligation:
3.49/1.46	Context-sensitive rewrite system:
3.49/1.46	The TRS R consists of the following rules:
3.49/1.46	
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	
3.49/1.46	The replacement map contains the following entries:
3.49/1.46	
3.49/1.46	cons: {1}
3.49/1.46	0: empty set
3.49/1.46	incr: {1}
3.49/1.46	pairs: empty set
3.49/1.46	odds: empty set
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(7) CSRRRRProof (EQUIVALENT)
3.49/1.46	The following CSR is given: Context-sensitive rewrite system:
3.49/1.46	The TRS R consists of the following rules:
3.49/1.46	
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	
3.49/1.46	The replacement map contains the following entries:
3.49/1.46	
3.49/1.46	cons: {1}
3.49/1.46	0: empty set
3.49/1.46	incr: {1}
3.49/1.46	pairs: empty set
3.49/1.46	odds: empty set
3.49/1.46	Used ordering:
3.49/1.46	Polynomial interpretation [POLO]:
3.49/1.46	
3.49/1.46	   POL(0) = 0
3.49/1.46	   POL(cons(x_1, x_2)) = x_1 + x_2
3.49/1.46	   POL(incr(x_1)) = x_1
3.49/1.46	   POL(odds) = 0
3.49/1.46	   POL(pairs) = 1
3.49/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.49/1.46	
3.49/1.46	   pairs -> cons(0, incr(odds))
3.49/1.46	
3.49/1.46	
3.49/1.46	
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(8)
3.49/1.46	Obligation:
3.49/1.46	Context-sensitive rewrite system:
3.49/1.46	R is empty.
3.49/1.46	
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(9) RisEmptyProof (EQUIVALENT)
3.49/1.46	The CSR R is empty. Hence, termination is trivially proven.
3.49/1.46	----------------------------------------
3.49/1.46	
3.49/1.46	(10)
3.49/1.46	YES
3.49/1.48	EOF