3.57/1.86	YES
3.57/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.57/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.57/1.87	
3.57/1.87	
3.57/1.87	Termination w.r.t. Q of the given QTRS could be proven:
3.57/1.87	
3.57/1.87	(0) QTRS
3.57/1.87	(1) QTRSToCSRProof [SOUND, 0 ms]
3.57/1.87	(2) CSR
3.57/1.87	(3) CSRRRRProof [EQUIVALENT, 53 ms]
3.57/1.87	(4) CSR
3.57/1.87	(5) CSRRRRProof [EQUIVALENT, 7 ms]
3.57/1.87	(6) CSR
3.57/1.87	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.57/1.87	(8) CSR
3.57/1.87	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.57/1.87	(10) YES
3.57/1.87	
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(0)
3.57/1.87	Obligation:
3.57/1.87	Q restricted rewrite system:
3.57/1.87	The TRS R consists of the following rules:
3.57/1.87	
3.57/1.87	   active(from(X)) -> mark(cons(X, from(s(X))))
3.57/1.87	   active(length(nil)) -> mark(0)
3.57/1.87	   active(length(cons(X, Y))) -> mark(s(length1(Y)))
3.57/1.87	   active(length1(X)) -> mark(length(X))
3.57/1.87	   active(from(X)) -> from(active(X))
3.57/1.87	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.57/1.87	   active(s(X)) -> s(active(X))
3.57/1.87	   from(mark(X)) -> mark(from(X))
3.57/1.87	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.57/1.87	   s(mark(X)) -> mark(s(X))
3.57/1.87	   proper(from(X)) -> from(proper(X))
3.57/1.87	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.57/1.87	   proper(s(X)) -> s(proper(X))
3.57/1.87	   proper(length(X)) -> length(proper(X))
3.57/1.87	   proper(nil) -> ok(nil)
3.57/1.87	   proper(0) -> ok(0)
3.57/1.87	   proper(length1(X)) -> length1(proper(X))
3.57/1.87	   from(ok(X)) -> ok(from(X))
3.57/1.87	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.57/1.87	   s(ok(X)) -> ok(s(X))
3.57/1.87	   length(ok(X)) -> ok(length(X))
3.57/1.87	   length1(ok(X)) -> ok(length1(X))
3.57/1.87	   top(mark(X)) -> top(proper(X))
3.57/1.87	   top(ok(X)) -> top(active(X))
3.57/1.87	
3.57/1.87	The set Q consists of the following terms:
3.57/1.87	
3.57/1.87	   active(from(x0))
3.57/1.87	   active(length(nil))
3.57/1.87	   active(length(cons(x0, x1)))
3.57/1.87	   active(length1(x0))
3.57/1.87	   active(cons(x0, x1))
3.57/1.87	   active(s(x0))
3.57/1.87	   from(mark(x0))
3.57/1.87	   cons(mark(x0), x1)
3.57/1.87	   s(mark(x0))
3.57/1.87	   proper(from(x0))
3.57/1.87	   proper(cons(x0, x1))
3.57/1.87	   proper(s(x0))
3.57/1.87	   proper(length(x0))
3.57/1.87	   proper(nil)
3.57/1.87	   proper(0)
3.57/1.87	   proper(length1(x0))
3.57/1.87	   from(ok(x0))
3.57/1.87	   cons(ok(x0), ok(x1))
3.57/1.87	   s(ok(x0))
3.57/1.87	   length(ok(x0))
3.57/1.87	   length1(ok(x0))
3.57/1.87	   top(mark(x0))
3.57/1.87	   top(ok(x0))
3.57/1.87	
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(1) QTRSToCSRProof (SOUND)
3.57/1.87	The following Q TRS is given: Q restricted rewrite system:
3.57/1.87	The TRS R consists of the following rules:
3.57/1.87	
3.57/1.87	   active(from(X)) -> mark(cons(X, from(s(X))))
3.57/1.87	   active(length(nil)) -> mark(0)
3.57/1.87	   active(length(cons(X, Y))) -> mark(s(length1(Y)))
3.57/1.87	   active(length1(X)) -> mark(length(X))
3.57/1.87	   active(from(X)) -> from(active(X))
3.57/1.87	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.57/1.87	   active(s(X)) -> s(active(X))
3.57/1.87	   from(mark(X)) -> mark(from(X))
3.57/1.87	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.57/1.87	   s(mark(X)) -> mark(s(X))
3.57/1.87	   proper(from(X)) -> from(proper(X))
3.57/1.87	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.57/1.87	   proper(s(X)) -> s(proper(X))
3.57/1.87	   proper(length(X)) -> length(proper(X))
3.57/1.87	   proper(nil) -> ok(nil)
3.57/1.87	   proper(0) -> ok(0)
3.57/1.87	   proper(length1(X)) -> length1(proper(X))
3.57/1.87	   from(ok(X)) -> ok(from(X))
3.57/1.87	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.57/1.87	   s(ok(X)) -> ok(s(X))
3.57/1.87	   length(ok(X)) -> ok(length(X))
3.57/1.87	   length1(ok(X)) -> ok(length1(X))
3.57/1.87	   top(mark(X)) -> top(proper(X))
3.57/1.87	   top(ok(X)) -> top(active(X))
3.57/1.87	
3.57/1.87	The set Q consists of the following terms:
3.57/1.87	
3.57/1.87	   active(from(x0))
3.57/1.87	   active(length(nil))
3.57/1.87	   active(length(cons(x0, x1)))
3.57/1.87	   active(length1(x0))
3.57/1.87	   active(cons(x0, x1))
3.57/1.87	   active(s(x0))
3.57/1.87	   from(mark(x0))
3.57/1.87	   cons(mark(x0), x1)
3.57/1.87	   s(mark(x0))
3.57/1.87	   proper(from(x0))
3.57/1.87	   proper(cons(x0, x1))
3.57/1.87	   proper(s(x0))
3.57/1.87	   proper(length(x0))
3.57/1.87	   proper(nil)
3.57/1.87	   proper(0)
3.57/1.87	   proper(length1(x0))
3.57/1.87	   from(ok(x0))
3.57/1.87	   cons(ok(x0), ok(x1))
3.57/1.87	   s(ok(x0))
3.57/1.87	   length(ok(x0))
3.57/1.87	   length1(ok(x0))
3.57/1.87	   top(mark(x0))
3.57/1.87	   top(ok(x0))
3.57/1.87	
3.57/1.87	Special symbols used for the transformation (see [GM04]):
3.57/1.87	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.57/1.87	The replacement map contains the following entries:
3.57/1.87	
3.57/1.87	from: {1}
3.57/1.87	cons: {1}
3.57/1.87	s: {1}
3.57/1.87	length: empty set
3.57/1.87	nil: empty set
3.57/1.87	0: empty set
3.57/1.87	length1: empty set
3.57/1.87	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.
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(2)
3.57/1.87	Obligation:
3.57/1.87	Context-sensitive rewrite system:
3.57/1.87	The TRS R consists of the following rules:
3.57/1.87	
3.57/1.87	   from(X) -> cons(X, from(s(X)))
3.57/1.87	   length(nil) -> 0
3.57/1.87	   length(cons(X, Y)) -> s(length1(Y))
3.57/1.87	   length1(X) -> length(X)
3.57/1.87	
3.57/1.87	The replacement map contains the following entries:
3.57/1.87	
3.57/1.87	from: {1}
3.57/1.87	cons: {1}
3.57/1.87	s: {1}
3.57/1.87	length: empty set
3.57/1.87	nil: empty set
3.57/1.87	0: empty set
3.57/1.87	length1: empty set
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(3) CSRRRRProof (EQUIVALENT)
3.57/1.87	The following CSR is given: Context-sensitive rewrite system:
3.57/1.87	The TRS R consists of the following rules:
3.57/1.87	
3.57/1.87	   from(X) -> cons(X, from(s(X)))
3.57/1.87	   length(nil) -> 0
3.57/1.87	   length(cons(X, Y)) -> s(length1(Y))
3.57/1.87	   length1(X) -> length(X)
3.57/1.87	
3.57/1.87	The replacement map contains the following entries:
3.57/1.87	
3.57/1.87	from: {1}
3.57/1.87	cons: {1}
3.57/1.87	s: {1}
3.57/1.87	length: empty set
3.57/1.87	nil: empty set
3.57/1.87	0: empty set
3.57/1.87	length1: empty set
3.57/1.87	Used ordering:
3.57/1.87	Polynomial interpretation [POLO]:
3.57/1.87	
3.57/1.87	   POL(0) = 0
3.57/1.87	   POL(cons(x_1, x_2)) = x_1
3.57/1.87	   POL(from(x_1)) = 1 + 2*x_1
3.57/1.87	   POL(length(x_1)) = 0
3.57/1.87	   POL(length1(x_1)) = 0
3.57/1.87	   POL(nil) = 2
3.57/1.87	   POL(s(x_1)) = 2*x_1
3.57/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.57/1.87	
3.57/1.87	   from(X) -> cons(X, from(s(X)))
3.57/1.87	
3.57/1.87	
3.57/1.87	
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(4)
3.57/1.87	Obligation:
3.57/1.87	Context-sensitive rewrite system:
3.57/1.87	The TRS R consists of the following rules:
3.57/1.87	
3.57/1.87	   length(nil) -> 0
3.57/1.87	   length(cons(X, Y)) -> s(length1(Y))
3.57/1.87	   length1(X) -> length(X)
3.57/1.87	
3.57/1.87	The replacement map contains the following entries:
3.57/1.87	
3.57/1.87	cons: {1}
3.57/1.87	s: {1}
3.57/1.87	length: empty set
3.57/1.87	nil: empty set
3.57/1.87	0: empty set
3.57/1.87	length1: empty set
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(5) CSRRRRProof (EQUIVALENT)
3.57/1.87	The following CSR is given: Context-sensitive rewrite system:
3.57/1.87	The TRS R consists of the following rules:
3.57/1.87	
3.57/1.87	   length(nil) -> 0
3.57/1.87	   length(cons(X, Y)) -> s(length1(Y))
3.57/1.87	   length1(X) -> length(X)
3.57/1.87	
3.57/1.87	The replacement map contains the following entries:
3.57/1.87	
3.57/1.87	cons: {1}
3.57/1.87	s: {1}
3.57/1.87	length: empty set
3.57/1.87	nil: empty set
3.57/1.87	0: empty set
3.57/1.87	length1: empty set
3.57/1.87	Used ordering:
3.57/1.87	Polynomial interpretation [POLO]:
3.57/1.87	
3.57/1.87	   POL(0) = 1
3.57/1.87	   POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2
3.57/1.87	   POL(length(x_1)) = 1 + 2*x_1
3.57/1.87	   POL(length1(x_1)) = 2 + 2*x_1
3.57/1.87	   POL(nil) = 0
3.57/1.87	   POL(s(x_1)) = x_1
3.57/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.57/1.87	
3.57/1.87	   length(cons(X, Y)) -> s(length1(Y))
3.57/1.87	   length1(X) -> length(X)
3.57/1.87	
3.57/1.87	
3.57/1.87	
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(6)
3.57/1.87	Obligation:
3.57/1.87	Context-sensitive rewrite system:
3.57/1.87	The TRS R consists of the following rules:
3.57/1.87	
3.57/1.87	   length(nil) -> 0
3.57/1.87	
3.57/1.87	The replacement map contains the following entries:
3.57/1.87	
3.57/1.87	length: empty set
3.57/1.87	nil: empty set
3.57/1.87	0: empty set
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(7) CSRRRRProof (EQUIVALENT)
3.57/1.87	The following CSR is given: Context-sensitive rewrite system:
3.57/1.87	The TRS R consists of the following rules:
3.57/1.87	
3.57/1.87	   length(nil) -> 0
3.57/1.87	
3.57/1.87	The replacement map contains the following entries:
3.57/1.87	
3.57/1.87	length: empty set
3.57/1.87	nil: empty set
3.57/1.87	0: empty set
3.57/1.87	Used ordering:
3.57/1.87	Polynomial interpretation [POLO]:
3.57/1.87	
3.57/1.87	   POL(0) = 0
3.57/1.87	   POL(length(x_1)) = 2 + 2*x_1
3.57/1.87	   POL(nil) = 1
3.57/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.57/1.87	
3.57/1.87	   length(nil) -> 0
3.57/1.87	
3.57/1.87	
3.57/1.87	
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(8)
3.57/1.87	Obligation:
3.57/1.87	Context-sensitive rewrite system:
3.57/1.87	R is empty.
3.57/1.87	
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(9) RisEmptyProof (EQUIVALENT)
3.57/1.87	The CSR R is empty. Hence, termination is trivially proven.
3.57/1.87	----------------------------------------
3.57/1.87	
3.57/1.87	(10)
3.57/1.87	YES
3.68/1.88	EOF