3.82/1.76	YES
3.82/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.82/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.82/1.77	
3.82/1.77	
3.82/1.77	Termination w.r.t. Q of the given QTRS could be proven:
3.82/1.77	
3.82/1.77	(0) QTRS
3.82/1.77	(1) QTRSToCSRProof [SOUND, 0 ms]
3.82/1.77	(2) CSR
3.82/1.77	(3) CSRRRRProof [EQUIVALENT, 29 ms]
3.82/1.77	(4) CSR
3.82/1.77	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.82/1.77	(6) CSR
3.82/1.77	(7) CSRRRRProof [EQUIVALENT, 4 ms]
3.82/1.77	(8) CSR
3.82/1.77	(9) CSRRRRProof [EQUIVALENT, 0 ms]
3.82/1.77	(10) CSR
3.82/1.77	(11) CSRRRRProof [EQUIVALENT, 0 ms]
3.82/1.77	(12) CSR
3.82/1.77	(13) RisEmptyProof [EQUIVALENT, 0 ms]
3.82/1.77	(14) YES
3.82/1.77	
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(0)
3.82/1.77	Obligation:
3.82/1.77	Q restricted rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   active(nats) -> mark(cons(0, incr(nats)))
3.82/1.77	   active(pairs) -> mark(cons(0, incr(odds)))
3.82/1.77	   active(odds) -> mark(incr(pairs))
3.82/1.77	   active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
3.82/1.77	   active(head(cons(X, XS))) -> mark(X)
3.82/1.77	   active(tail(cons(X, XS))) -> mark(XS)
3.82/1.77	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.82/1.77	   active(incr(X)) -> incr(active(X))
3.82/1.77	   active(s(X)) -> s(active(X))
3.82/1.77	   active(head(X)) -> head(active(X))
3.82/1.77	   active(tail(X)) -> tail(active(X))
3.82/1.77	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.82/1.77	   incr(mark(X)) -> mark(incr(X))
3.82/1.77	   s(mark(X)) -> mark(s(X))
3.82/1.77	   head(mark(X)) -> mark(head(X))
3.82/1.77	   tail(mark(X)) -> mark(tail(X))
3.82/1.77	   proper(nats) -> ok(nats)
3.82/1.77	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.82/1.77	   proper(0) -> ok(0)
3.82/1.77	   proper(incr(X)) -> incr(proper(X))
3.82/1.77	   proper(pairs) -> ok(pairs)
3.82/1.77	   proper(odds) -> ok(odds)
3.82/1.77	   proper(s(X)) -> s(proper(X))
3.82/1.77	   proper(head(X)) -> head(proper(X))
3.82/1.77	   proper(tail(X)) -> tail(proper(X))
3.82/1.77	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.82/1.77	   incr(ok(X)) -> ok(incr(X))
3.82/1.77	   s(ok(X)) -> ok(s(X))
3.82/1.77	   head(ok(X)) -> ok(head(X))
3.82/1.77	   tail(ok(X)) -> ok(tail(X))
3.82/1.77	   top(mark(X)) -> top(proper(X))
3.82/1.77	   top(ok(X)) -> top(active(X))
3.82/1.77	
3.82/1.77	The set Q consists of the following terms:
3.82/1.77	
3.82/1.77	   active(nats)
3.82/1.77	   active(pairs)
3.82/1.77	   active(odds)
3.82/1.77	   active(cons(x0, x1))
3.82/1.77	   active(incr(x0))
3.82/1.77	   active(s(x0))
3.82/1.77	   active(head(x0))
3.82/1.77	   active(tail(x0))
3.82/1.77	   cons(mark(x0), x1)
3.82/1.77	   incr(mark(x0))
3.82/1.77	   s(mark(x0))
3.82/1.77	   head(mark(x0))
3.82/1.77	   tail(mark(x0))
3.82/1.77	   proper(nats)
3.82/1.77	   proper(cons(x0, x1))
3.82/1.77	   proper(0)
3.82/1.77	   proper(incr(x0))
3.82/1.77	   proper(pairs)
3.82/1.77	   proper(odds)
3.82/1.77	   proper(s(x0))
3.82/1.77	   proper(head(x0))
3.82/1.77	   proper(tail(x0))
3.82/1.77	   cons(ok(x0), ok(x1))
3.82/1.77	   incr(ok(x0))
3.82/1.77	   s(ok(x0))
3.82/1.77	   head(ok(x0))
3.82/1.77	   tail(ok(x0))
3.82/1.77	   top(mark(x0))
3.82/1.77	   top(ok(x0))
3.82/1.77	
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(1) QTRSToCSRProof (SOUND)
3.82/1.77	The following Q TRS is given: Q restricted rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   active(nats) -> mark(cons(0, incr(nats)))
3.82/1.77	   active(pairs) -> mark(cons(0, incr(odds)))
3.82/1.77	   active(odds) -> mark(incr(pairs))
3.82/1.77	   active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
3.82/1.77	   active(head(cons(X, XS))) -> mark(X)
3.82/1.77	   active(tail(cons(X, XS))) -> mark(XS)
3.82/1.77	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.82/1.77	   active(incr(X)) -> incr(active(X))
3.82/1.77	   active(s(X)) -> s(active(X))
3.82/1.77	   active(head(X)) -> head(active(X))
3.82/1.77	   active(tail(X)) -> tail(active(X))
3.82/1.77	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.82/1.77	   incr(mark(X)) -> mark(incr(X))
3.82/1.77	   s(mark(X)) -> mark(s(X))
3.82/1.77	   head(mark(X)) -> mark(head(X))
3.82/1.77	   tail(mark(X)) -> mark(tail(X))
3.82/1.77	   proper(nats) -> ok(nats)
3.82/1.77	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.82/1.77	   proper(0) -> ok(0)
3.82/1.77	   proper(incr(X)) -> incr(proper(X))
3.82/1.77	   proper(pairs) -> ok(pairs)
3.82/1.77	   proper(odds) -> ok(odds)
3.82/1.77	   proper(s(X)) -> s(proper(X))
3.82/1.77	   proper(head(X)) -> head(proper(X))
3.82/1.77	   proper(tail(X)) -> tail(proper(X))
3.82/1.77	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.82/1.77	   incr(ok(X)) -> ok(incr(X))
3.82/1.77	   s(ok(X)) -> ok(s(X))
3.82/1.77	   head(ok(X)) -> ok(head(X))
3.82/1.77	   tail(ok(X)) -> ok(tail(X))
3.82/1.77	   top(mark(X)) -> top(proper(X))
3.82/1.77	   top(ok(X)) -> top(active(X))
3.82/1.77	
3.82/1.77	The set Q consists of the following terms:
3.82/1.77	
3.82/1.77	   active(nats)
3.82/1.77	   active(pairs)
3.82/1.77	   active(odds)
3.82/1.77	   active(cons(x0, x1))
3.82/1.77	   active(incr(x0))
3.82/1.77	   active(s(x0))
3.82/1.77	   active(head(x0))
3.82/1.77	   active(tail(x0))
3.82/1.77	   cons(mark(x0), x1)
3.82/1.77	   incr(mark(x0))
3.82/1.77	   s(mark(x0))
3.82/1.77	   head(mark(x0))
3.82/1.77	   tail(mark(x0))
3.82/1.77	   proper(nats)
3.82/1.77	   proper(cons(x0, x1))
3.82/1.77	   proper(0)
3.82/1.77	   proper(incr(x0))
3.82/1.77	   proper(pairs)
3.82/1.77	   proper(odds)
3.82/1.77	   proper(s(x0))
3.82/1.77	   proper(head(x0))
3.82/1.77	   proper(tail(x0))
3.82/1.77	   cons(ok(x0), ok(x1))
3.82/1.77	   incr(ok(x0))
3.82/1.77	   s(ok(x0))
3.82/1.77	   head(ok(x0))
3.82/1.77	   tail(ok(x0))
3.82/1.77	   top(mark(x0))
3.82/1.77	   top(ok(x0))
3.82/1.77	
3.82/1.77	Special symbols used for the transformation (see [GM04]):
3.82/1.77	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	s: {1}
3.82/1.77	head: {1}
3.82/1.77	tail: {1}
3.82/1.77	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(2)
3.82/1.77	Obligation:
3.82/1.77	Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.82/1.77	   head(cons(X, XS)) -> X
3.82/1.77	   tail(cons(X, XS)) -> XS
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	s: {1}
3.82/1.77	head: {1}
3.82/1.77	tail: {1}
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(3) CSRRRRProof (EQUIVALENT)
3.82/1.77	The following CSR is given: Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.82/1.77	   head(cons(X, XS)) -> X
3.82/1.77	   tail(cons(X, XS)) -> XS
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	s: {1}
3.82/1.77	head: {1}
3.82/1.77	tail: {1}
3.82/1.77	Used ordering:
3.82/1.77	Polynomial interpretation [POLO]:
3.82/1.77	
3.82/1.77	   POL(0) = 0
3.82/1.77	   POL(cons(x_1, x_2)) = 2*x_1 + x_2
3.82/1.77	   POL(head(x_1)) = 1 + 2*x_1
3.82/1.77	   POL(incr(x_1)) = x_1
3.82/1.77	   POL(nats) = 0
3.82/1.77	   POL(odds) = 0
3.82/1.77	   POL(pairs) = 0
3.82/1.77	   POL(s(x_1)) = x_1
3.82/1.77	   POL(tail(x_1)) = 2*x_1
3.82/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.82/1.77	
3.82/1.77	   head(cons(X, XS)) -> X
3.82/1.77	
3.82/1.77	
3.82/1.77	
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(4)
3.82/1.77	Obligation:
3.82/1.77	Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.82/1.77	   tail(cons(X, XS)) -> XS
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	s: {1}
3.82/1.77	tail: {1}
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(5) CSRRRRProof (EQUIVALENT)
3.82/1.77	The following CSR is given: Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.82/1.77	   tail(cons(X, XS)) -> XS
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	s: {1}
3.82/1.77	tail: {1}
3.82/1.77	Used ordering:
3.82/1.77	Polynomial interpretation [POLO]:
3.82/1.77	
3.82/1.77	   POL(0) = 0
3.82/1.77	   POL(cons(x_1, x_2)) = x_1 + x_2
3.82/1.77	   POL(incr(x_1)) = x_1
3.82/1.77	   POL(nats) = 0
3.82/1.77	   POL(odds) = 0
3.82/1.77	   POL(pairs) = 0
3.82/1.77	   POL(s(x_1)) = x_1
3.82/1.77	   POL(tail(x_1)) = 1 + x_1
3.82/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.82/1.77	
3.82/1.77	   tail(cons(X, XS)) -> XS
3.82/1.77	
3.82/1.77	
3.82/1.77	
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(6)
3.82/1.77	Obligation:
3.82/1.77	Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	s: {1}
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(7) CSRRRRProof (EQUIVALENT)
3.82/1.77	The following CSR is given: Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	s: {1}
3.82/1.77	Used ordering:
3.82/1.77	Polynomial interpretation [POLO]:
3.82/1.77	
3.82/1.77	   POL(0) = 0
3.82/1.77	   POL(cons(x_1, x_2)) = x_1
3.82/1.77	   POL(incr(x_1)) = 1 + x_1
3.82/1.77	   POL(nats) = 0
3.82/1.77	   POL(odds) = 1
3.82/1.77	   POL(pairs) = 0
3.82/1.77	   POL(s(x_1)) = x_1
3.82/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.82/1.77	
3.82/1.77	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
3.82/1.77	
3.82/1.77	
3.82/1.77	
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(8)
3.82/1.77	Obligation:
3.82/1.77	Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(9) CSRRRRProof (EQUIVALENT)
3.82/1.77	The following CSR is given: Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	nats: empty set
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	Used ordering:
3.82/1.77	Polynomial interpretation [POLO]:
3.82/1.77	
3.82/1.77	   POL(0) = 0
3.82/1.77	   POL(cons(x_1, x_2)) = x_1
3.82/1.77	   POL(incr(x_1)) = 2*x_1
3.82/1.77	   POL(nats) = 1
3.82/1.77	   POL(odds) = 1
3.82/1.77	   POL(pairs) = 0
3.82/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.82/1.77	
3.82/1.77	   nats -> cons(0, incr(nats))
3.82/1.77	   odds -> incr(pairs)
3.82/1.77	
3.82/1.77	
3.82/1.77	
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(10)
3.82/1.77	Obligation:
3.82/1.77	Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
3.82/1.77	
3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	
3.82/1.77	----------------------------------------
3.82/1.77	
3.82/1.77	(11) CSRRRRProof (EQUIVALENT)
3.82/1.77	The following CSR is given: Context-sensitive rewrite system:
3.82/1.77	The TRS R consists of the following rules:
3.82/1.77	
3.82/1.77	   pairs -> cons(0, incr(odds))
3.82/1.77	
3.82/1.77	The replacement map contains the following entries:
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3.82/1.77	cons: {1}
3.82/1.77	0: empty set
3.82/1.77	incr: {1}
3.82/1.77	pairs: empty set
3.82/1.77	odds: empty set
3.82/1.77	Used ordering:
3.82/1.77	Polynomial interpretation [POLO]:
3.82/1.77	
3.82/1.77	   POL(0) = 0
3.82/1.77	   POL(cons(x_1, x_2)) = 1 + x_1
3.82/1.77	   POL(incr(x_1)) = 1 + 2*x_1
3.82/1.77	   POL(odds) = 0
3.82/1.77	   POL(pairs) = 2
3.82/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.82/1.77	
3.82/1.77	   pairs -> cons(0, incr(odds))
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3.82/1.77	
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3.82/1.77	
3.82/1.77	(12)
3.82/1.77	Obligation:
3.82/1.77	Context-sensitive rewrite system:
3.82/1.77	R is empty.
3.82/1.77	
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3.82/1.77	
3.82/1.77	(13) RisEmptyProof (EQUIVALENT)
3.82/1.77	The CSR R is empty. Hence, termination is trivially proven.
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3.82/1.77	
3.82/1.77	(14)
3.82/1.77	YES
3.85/1.79	EOF