3.59/1.72	YES
3.59/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.59/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.59/1.73	
3.59/1.73	
3.59/1.73	Termination w.r.t. Q of the given QTRS could be proven:
3.59/1.73	
3.59/1.73	(0) QTRS
3.59/1.73	(1) QTRSToCSRProof [SOUND, 0 ms]
3.59/1.73	(2) CSR
3.59/1.73	(3) CSRRRRProof [EQUIVALENT, 45 ms]
3.59/1.73	(4) CSR
3.59/1.73	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.59/1.73	(6) YES
3.59/1.73	
3.59/1.73	
3.59/1.73	----------------------------------------
3.59/1.73	
3.59/1.73	(0)
3.59/1.73	Obligation:
3.59/1.73	Q restricted rewrite system:
3.59/1.73	The TRS R consists of the following rules:
3.59/1.73	
3.59/1.73	   active(first(0, X)) -> mark(nil)
3.59/1.73	   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
3.59/1.73	   active(from(X)) -> mark(cons(X, from(s(X))))
3.59/1.73	   active(first(X1, X2)) -> first(active(X1), X2)
3.59/1.73	   active(first(X1, X2)) -> first(X1, active(X2))
3.59/1.73	   active(s(X)) -> s(active(X))
3.59/1.73	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.59/1.73	   active(from(X)) -> from(active(X))
3.59/1.73	   first(mark(X1), X2) -> mark(first(X1, X2))
3.59/1.73	   first(X1, mark(X2)) -> mark(first(X1, X2))
3.59/1.73	   s(mark(X)) -> mark(s(X))
3.59/1.73	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.59/1.73	   from(mark(X)) -> mark(from(X))
3.59/1.73	   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
3.59/1.73	   proper(0) -> ok(0)
3.59/1.73	   proper(nil) -> ok(nil)
3.59/1.73	   proper(s(X)) -> s(proper(X))
3.59/1.73	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.59/1.73	   proper(from(X)) -> from(proper(X))
3.59/1.73	   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
3.59/1.73	   s(ok(X)) -> ok(s(X))
3.59/1.73	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.59/1.73	   from(ok(X)) -> ok(from(X))
3.59/1.73	   top(mark(X)) -> top(proper(X))
3.59/1.73	   top(ok(X)) -> top(active(X))
3.59/1.73	
3.59/1.73	The set Q consists of the following terms:
3.59/1.73	
3.59/1.73	   active(from(x0))
3.59/1.73	   active(first(x0, x1))
3.59/1.73	   active(s(x0))
3.59/1.73	   active(cons(x0, x1))
3.59/1.73	   first(mark(x0), x1)
3.59/1.73	   first(x0, mark(x1))
3.59/1.73	   s(mark(x0))
3.59/1.73	   cons(mark(x0), x1)
3.59/1.73	   from(mark(x0))
3.59/1.73	   proper(first(x0, x1))
3.59/1.73	   proper(0)
3.59/1.73	   proper(nil)
3.59/1.73	   proper(s(x0))
3.59/1.73	   proper(cons(x0, x1))
3.59/1.73	   proper(from(x0))
3.59/1.73	   first(ok(x0), ok(x1))
3.59/1.73	   s(ok(x0))
3.59/1.73	   cons(ok(x0), ok(x1))
3.59/1.73	   from(ok(x0))
3.59/1.73	   top(mark(x0))
3.59/1.73	   top(ok(x0))
3.59/1.73	
3.59/1.73	
3.59/1.73	----------------------------------------
3.59/1.73	
3.59/1.73	(1) QTRSToCSRProof (SOUND)
3.59/1.73	The following Q TRS is given: Q restricted rewrite system:
3.59/1.73	The TRS R consists of the following rules:
3.59/1.73	
3.59/1.73	   active(first(0, X)) -> mark(nil)
3.59/1.73	   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
3.59/1.73	   active(from(X)) -> mark(cons(X, from(s(X))))
3.59/1.73	   active(first(X1, X2)) -> first(active(X1), X2)
3.59/1.73	   active(first(X1, X2)) -> first(X1, active(X2))
3.59/1.73	   active(s(X)) -> s(active(X))
3.59/1.73	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.59/1.73	   active(from(X)) -> from(active(X))
3.59/1.73	   first(mark(X1), X2) -> mark(first(X1, X2))
3.59/1.73	   first(X1, mark(X2)) -> mark(first(X1, X2))
3.59/1.73	   s(mark(X)) -> mark(s(X))
3.59/1.73	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.59/1.73	   from(mark(X)) -> mark(from(X))
3.59/1.73	   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
3.59/1.73	   proper(0) -> ok(0)
3.59/1.73	   proper(nil) -> ok(nil)
3.59/1.73	   proper(s(X)) -> s(proper(X))
3.59/1.73	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.59/1.73	   proper(from(X)) -> from(proper(X))
3.59/1.73	   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
3.59/1.73	   s(ok(X)) -> ok(s(X))
3.59/1.73	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.59/1.73	   from(ok(X)) -> ok(from(X))
3.59/1.73	   top(mark(X)) -> top(proper(X))
3.59/1.73	   top(ok(X)) -> top(active(X))
3.59/1.73	
3.59/1.73	The set Q consists of the following terms:
3.59/1.73	
3.59/1.73	   active(from(x0))
3.59/1.73	   active(first(x0, x1))
3.59/1.73	   active(s(x0))
3.59/1.73	   active(cons(x0, x1))
3.59/1.73	   first(mark(x0), x1)
3.59/1.73	   first(x0, mark(x1))
3.59/1.73	   s(mark(x0))
3.59/1.73	   cons(mark(x0), x1)
3.59/1.73	   from(mark(x0))
3.59/1.73	   proper(first(x0, x1))
3.59/1.73	   proper(0)
3.59/1.73	   proper(nil)
3.59/1.73	   proper(s(x0))
3.59/1.73	   proper(cons(x0, x1))
3.59/1.73	   proper(from(x0))
3.59/1.73	   first(ok(x0), ok(x1))
3.59/1.73	   s(ok(x0))
3.59/1.73	   cons(ok(x0), ok(x1))
3.59/1.73	   from(ok(x0))
3.59/1.73	   top(mark(x0))
3.59/1.73	   top(ok(x0))
3.59/1.73	
3.59/1.73	Special symbols used for the transformation (see [GM04]):
3.59/1.73	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.59/1.73	The replacement map contains the following entries:
3.59/1.73	
3.59/1.73	first: {1, 2}
3.59/1.73	0: empty set
3.59/1.73	nil: empty set
3.59/1.73	s: {1}
3.59/1.73	cons: {1}
3.59/1.73	from: {1}
3.59/1.73	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.
3.59/1.73	----------------------------------------
3.59/1.73	
3.59/1.73	(2)
3.59/1.73	Obligation:
3.59/1.73	Context-sensitive rewrite system:
3.59/1.73	The TRS R consists of the following rules:
3.59/1.73	
3.59/1.73	   first(0, X) -> nil
3.59/1.73	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.59/1.73	   from(X) -> cons(X, from(s(X)))
3.59/1.73	
3.59/1.73	The replacement map contains the following entries:
3.59/1.73	
3.59/1.73	first: {1, 2}
3.59/1.73	0: empty set
3.59/1.73	nil: empty set
3.59/1.73	s: {1}
3.59/1.73	cons: {1}
3.59/1.73	from: {1}
3.59/1.73	
3.59/1.73	----------------------------------------
3.59/1.73	
3.59/1.73	(3) CSRRRRProof (EQUIVALENT)
3.59/1.73	The following CSR is given: Context-sensitive rewrite system:
3.59/1.73	The TRS R consists of the following rules:
3.59/1.73	
3.59/1.73	   first(0, X) -> nil
3.59/1.73	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.59/1.73	   from(X) -> cons(X, from(s(X)))
3.59/1.73	
3.59/1.73	The replacement map contains the following entries:
3.59/1.73	
3.59/1.73	first: {1, 2}
3.59/1.73	0: empty set
3.59/1.73	nil: empty set
3.59/1.73	s: {1}
3.59/1.73	cons: {1}
3.59/1.73	from: {1}
3.59/1.73	Used ordering:
3.59/1.73	Polynomial interpretation [POLO]:
3.59/1.73	
3.59/1.73	   POL(0) = 1
3.59/1.73	   POL(cons(x_1, x_2)) = x_1
3.59/1.73	   POL(first(x_1, x_2)) = 2 + 2*x_1 + x_2
3.59/1.73	   POL(from(x_1)) = 1 + x_1
3.59/1.73	   POL(nil) = 0
3.59/1.73	   POL(s(x_1)) = 2 + x_1
3.59/1.73	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.59/1.73	
3.59/1.73	   first(0, X) -> nil
3.59/1.73	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.59/1.73	   from(X) -> cons(X, from(s(X)))
3.59/1.73	
3.59/1.73	
3.59/1.73	
3.59/1.73	
3.59/1.73	----------------------------------------
3.59/1.73	
3.59/1.73	(4)
3.59/1.73	Obligation:
3.59/1.73	Context-sensitive rewrite system:
3.59/1.73	R is empty.
3.59/1.73	
3.59/1.73	----------------------------------------
3.59/1.73	
3.59/1.73	(5) RisEmptyProof (EQUIVALENT)
3.59/1.73	The CSR R is empty. Hence, termination is trivially proven.
3.59/1.73	----------------------------------------
3.59/1.73	
3.59/1.73	(6)
3.59/1.73	YES
3.59/1.75	EOF