3.60/1.68	YES
3.60/1.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.60/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.60/1.69	
3.60/1.69	
3.60/1.69	Termination w.r.t. Q of the given QTRS could be proven:
3.60/1.69	
3.60/1.69	(0) QTRS
3.60/1.69	(1) QTRSToCSRProof [SOUND, 0 ms]
3.60/1.69	(2) CSR
3.60/1.69	(3) CSRInnermostProof [EQUIVALENT, 0 ms]
3.60/1.69	(4) CSR
3.60/1.69	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.60/1.69	(6) QCSDP
3.60/1.69	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.60/1.69	(8) AND
3.60/1.69	    (9) QCSDP
3.60/1.69	        (10) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.60/1.69	        (11) QCSDP
3.60/1.69	        (12) PIsEmptyProof [EQUIVALENT, 0 ms]
3.60/1.69	        (13) YES
3.60/1.69	    (14) QCSDP
3.60/1.69	        (15) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.60/1.69	        (16) QCSDP
3.60/1.69	        (17) PIsEmptyProof [EQUIVALENT, 0 ms]
3.60/1.69	        (18) YES
3.60/1.69	
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(0)
3.60/1.69	Obligation:
3.60/1.69	Q restricted rewrite system:
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   active(from(X)) -> mark(cons(X, from(s(X))))
3.60/1.69	   active(after(0, XS)) -> mark(XS)
3.60/1.69	   active(after(s(N), cons(X, XS))) -> mark(after(N, XS))
3.60/1.69	   active(from(X)) -> from(active(X))
3.60/1.69	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.60/1.69	   active(s(X)) -> s(active(X))
3.60/1.69	   active(after(X1, X2)) -> after(active(X1), X2)
3.60/1.69	   active(after(X1, X2)) -> after(X1, active(X2))
3.60/1.69	   from(mark(X)) -> mark(from(X))
3.60/1.69	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.60/1.69	   s(mark(X)) -> mark(s(X))
3.60/1.69	   after(mark(X1), X2) -> mark(after(X1, X2))
3.60/1.69	   after(X1, mark(X2)) -> mark(after(X1, X2))
3.60/1.69	   proper(from(X)) -> from(proper(X))
3.60/1.69	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.60/1.69	   proper(s(X)) -> s(proper(X))
3.60/1.69	   proper(after(X1, X2)) -> after(proper(X1), proper(X2))
3.60/1.69	   proper(0) -> ok(0)
3.60/1.69	   from(ok(X)) -> ok(from(X))
3.60/1.69	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.60/1.69	   s(ok(X)) -> ok(s(X))
3.60/1.69	   after(ok(X1), ok(X2)) -> ok(after(X1, X2))
3.60/1.69	   top(mark(X)) -> top(proper(X))
3.60/1.69	   top(ok(X)) -> top(active(X))
3.60/1.69	
3.60/1.69	The set Q consists of the following terms:
3.60/1.69	
3.60/1.69	   active(from(x0))
3.60/1.69	   active(cons(x0, x1))
3.60/1.69	   active(s(x0))
3.60/1.69	   active(after(x0, x1))
3.60/1.69	   from(mark(x0))
3.60/1.69	   cons(mark(x0), x1)
3.60/1.69	   s(mark(x0))
3.60/1.69	   after(mark(x0), x1)
3.60/1.69	   after(x0, mark(x1))
3.60/1.69	   proper(from(x0))
3.60/1.69	   proper(cons(x0, x1))
3.60/1.69	   proper(s(x0))
3.60/1.69	   proper(after(x0, x1))
3.60/1.69	   proper(0)
3.60/1.69	   from(ok(x0))
3.60/1.69	   cons(ok(x0), ok(x1))
3.60/1.69	   s(ok(x0))
3.60/1.69	   after(ok(x0), ok(x1))
3.60/1.69	   top(mark(x0))
3.60/1.69	   top(ok(x0))
3.60/1.69	
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(1) QTRSToCSRProof (SOUND)
3.60/1.69	The following Q TRS is given: Q restricted rewrite system:
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   active(from(X)) -> mark(cons(X, from(s(X))))
3.60/1.69	   active(after(0, XS)) -> mark(XS)
3.60/1.69	   active(after(s(N), cons(X, XS))) -> mark(after(N, XS))
3.60/1.69	   active(from(X)) -> from(active(X))
3.60/1.69	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.60/1.69	   active(s(X)) -> s(active(X))
3.60/1.69	   active(after(X1, X2)) -> after(active(X1), X2)
3.60/1.69	   active(after(X1, X2)) -> after(X1, active(X2))
3.60/1.69	   from(mark(X)) -> mark(from(X))
3.60/1.69	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.60/1.69	   s(mark(X)) -> mark(s(X))
3.60/1.69	   after(mark(X1), X2) -> mark(after(X1, X2))
3.60/1.69	   after(X1, mark(X2)) -> mark(after(X1, X2))
3.60/1.69	   proper(from(X)) -> from(proper(X))
3.60/1.69	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.60/1.69	   proper(s(X)) -> s(proper(X))
3.60/1.69	   proper(after(X1, X2)) -> after(proper(X1), proper(X2))
3.60/1.69	   proper(0) -> ok(0)
3.60/1.69	   from(ok(X)) -> ok(from(X))
3.60/1.69	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.60/1.69	   s(ok(X)) -> ok(s(X))
3.60/1.69	   after(ok(X1), ok(X2)) -> ok(after(X1, X2))
3.60/1.69	   top(mark(X)) -> top(proper(X))
3.60/1.69	   top(ok(X)) -> top(active(X))
3.60/1.69	
3.60/1.69	The set Q consists of the following terms:
3.60/1.69	
3.60/1.69	   active(from(x0))
3.60/1.69	   active(cons(x0, x1))
3.60/1.69	   active(s(x0))
3.60/1.69	   active(after(x0, x1))
3.60/1.69	   from(mark(x0))
3.60/1.69	   cons(mark(x0), x1)
3.60/1.69	   s(mark(x0))
3.60/1.69	   after(mark(x0), x1)
3.60/1.69	   after(x0, mark(x1))
3.60/1.69	   proper(from(x0))
3.60/1.69	   proper(cons(x0, x1))
3.60/1.69	   proper(s(x0))
3.60/1.69	   proper(after(x0, x1))
3.60/1.69	   proper(0)
3.60/1.69	   from(ok(x0))
3.60/1.69	   cons(ok(x0), ok(x1))
3.60/1.69	   s(ok(x0))
3.60/1.69	   after(ok(x0), ok(x1))
3.60/1.69	   top(mark(x0))
3.60/1.69	   top(ok(x0))
3.60/1.69	
3.60/1.69	Special symbols used for the transformation (see [GM04]):
3.60/1.69	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.60/1.69	The replacement map contains the following entries:
3.60/1.69	
3.60/1.69	from: {1}
3.60/1.69	cons: {1}
3.60/1.69	s: {1}
3.60/1.69	after: {1, 2}
3.60/1.69	0: empty set
3.60/1.69	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.
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(2)
3.60/1.69	Obligation:
3.60/1.69	Context-sensitive rewrite system:
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   from(X) -> cons(X, from(s(X)))
3.60/1.69	   after(0, XS) -> XS
3.60/1.69	   after(s(N), cons(X, XS)) -> after(N, XS)
3.60/1.69	
3.60/1.69	The replacement map contains the following entries:
3.60/1.69	
3.60/1.69	from: {1}
3.60/1.69	cons: {1}
3.60/1.69	s: {1}
3.60/1.69	after: {1, 2}
3.60/1.69	0: empty set
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(3) CSRInnermostProof (EQUIVALENT)
3.60/1.69	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(4)
3.60/1.69	Obligation:
3.60/1.69	Context-sensitive rewrite system:
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   from(X) -> cons(X, from(s(X)))
3.60/1.69	   after(0, XS) -> XS
3.60/1.69	   after(s(N), cons(X, XS)) -> after(N, XS)
3.60/1.69	
3.60/1.69	The replacement map contains the following entries:
3.60/1.69	
3.60/1.69	from: {1}
3.60/1.69	cons: {1}
3.60/1.69	s: {1}
3.60/1.69	after: {1, 2}
3.60/1.69	0: empty set
3.60/1.69	
3.60/1.69	
3.60/1.69	Innermost Strategy.
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(5) CSDependencyPairsProof (EQUIVALENT)
3.60/1.69	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(6)
3.60/1.69	Obligation:
3.60/1.69	Q-restricted context-sensitive dependency pair problem:
3.60/1.69	The symbols in {from_1, s_1, after_2, AFTER_2, FROM_1} are replacing on all positions.
3.60/1.69	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/1.69	The symbols in {U_1} are not replacing on any position.
3.60/1.69	
3.60/1.69	The ordinary context-sensitive dependency pairs DP_o are:
3.60/1.69	   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)
3.60/1.69	
3.60/1.69	The collapsing dependency pairs are DP_c:
3.60/1.69	   AFTER(s(N), cons(X, XS)) -> XS
3.60/1.69	
3.60/1.69	
3.60/1.69	The hidden terms of R are:
3.60/1.69	
3.60/1.69	   from(s(x0))
3.60/1.69	
3.60/1.69	Every hiding context is built from:
3.60/1.69	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@6c7da2cc
3.60/1.69	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@66b48c73
3.60/1.69	
3.60/1.69	Hence, the new unhiding pairs DP_u are :
3.60/1.69	   AFTER(s(N), cons(X, XS)) -> U(XS)
3.60/1.69	   U(s(x_0)) -> U(x_0)
3.60/1.69	   U(from(x_0)) -> U(x_0)
3.60/1.69	   U(from(s(x0))) -> FROM(s(x0))
3.60/1.69	
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   from(X) -> cons(X, from(s(X)))
3.60/1.69	   after(0, XS) -> XS
3.60/1.69	   after(s(N), cons(X, XS)) -> after(N, XS)
3.60/1.69	
3.60/1.69	The set Q consists of the following terms:
3.60/1.69	
3.60/1.69	   from(x0)
3.60/1.69	   after(0, x0)
3.60/1.69	   after(s(x0), cons(x1, x2))
3.60/1.69	
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(7) QCSDependencyGraphProof (EQUIVALENT)
3.60/1.69	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 2 less nodes.
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(8)
3.60/1.69	Complex Obligation (AND)
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(9)
3.60/1.69	Obligation:
3.60/1.69	Q-restricted context-sensitive dependency pair problem:
3.60/1.69	The symbols in {from_1, s_1, after_2} are replacing on all positions.
3.60/1.69	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/1.69	The symbols in {U_1} are not replacing on any position.
3.60/1.69	
3.60/1.69	The TRS P consists of the following rules:
3.60/1.69	
3.60/1.69	   U(s(x_0)) -> U(x_0)
3.60/1.69	   U(from(x_0)) -> U(x_0)
3.60/1.69	
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   from(X) -> cons(X, from(s(X)))
3.60/1.69	   after(0, XS) -> XS
3.60/1.69	   after(s(N), cons(X, XS)) -> after(N, XS)
3.60/1.69	
3.60/1.69	The set Q consists of the following terms:
3.60/1.69	
3.60/1.69	   from(x0)
3.60/1.69	   after(0, x0)
3.60/1.69	   after(s(x0), cons(x1, x2))
3.60/1.69	
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(10) QCSDPSubtermProof (EQUIVALENT)
3.60/1.69	We use the subterm processor [DA_EMMES].
3.60/1.69	
3.60/1.69	
3.60/1.69	The following pairs can be oriented strictly and are deleted.
3.60/1.69	
3.60/1.69	   U(s(x_0)) -> U(x_0)
3.60/1.69	   U(from(x_0)) -> U(x_0)
3.60/1.69	The remaining pairs can at least be oriented weakly.
3.60/1.69	none
3.60/1.69	Used ordering:  Combined order from the following AFS and order.
3.60/1.69	U(x1)  =  x1
3.60/1.69	
3.60/1.69	
3.60/1.69	Subterm Order
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(11)
3.60/1.69	Obligation:
3.60/1.69	Q-restricted context-sensitive dependency pair problem:
3.60/1.69	The symbols in {from_1, s_1, after_2} are replacing on all positions.
3.60/1.69	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/1.69	
3.60/1.69	The TRS P consists of the following rules:
3.60/1.69	none
3.60/1.69	
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   from(X) -> cons(X, from(s(X)))
3.60/1.69	   after(0, XS) -> XS
3.60/1.69	   after(s(N), cons(X, XS)) -> after(N, XS)
3.60/1.69	
3.60/1.69	The set Q consists of the following terms:
3.60/1.69	
3.60/1.69	   from(x0)
3.60/1.69	   after(0, x0)
3.60/1.69	   after(s(x0), cons(x1, x2))
3.60/1.69	
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(12) PIsEmptyProof (EQUIVALENT)
3.60/1.69	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(13)
3.60/1.69	YES
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(14)
3.60/1.69	Obligation:
3.60/1.69	Q-restricted context-sensitive dependency pair problem:
3.60/1.69	The symbols in {from_1, s_1, after_2, AFTER_2} are replacing on all positions.
3.60/1.69	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/1.69	
3.60/1.69	The TRS P consists of the following rules:
3.60/1.69	
3.60/1.69	   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)
3.60/1.69	
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   from(X) -> cons(X, from(s(X)))
3.60/1.69	   after(0, XS) -> XS
3.60/1.69	   after(s(N), cons(X, XS)) -> after(N, XS)
3.60/1.69	
3.60/1.69	The set Q consists of the following terms:
3.60/1.69	
3.60/1.69	   from(x0)
3.60/1.69	   after(0, x0)
3.60/1.69	   after(s(x0), cons(x1, x2))
3.60/1.69	
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(15) QCSDPSubtermProof (EQUIVALENT)
3.60/1.69	We use the subterm processor [DA_EMMES].
3.60/1.69	
3.60/1.69	
3.60/1.69	The following pairs can be oriented strictly and are deleted.
3.60/1.69	
3.60/1.69	   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)
3.60/1.69	The remaining pairs can at least be oriented weakly.
3.60/1.69	none
3.60/1.69	Used ordering:  Combined order from the following AFS and order.
3.60/1.69	AFTER(x1, x2)  =  x1
3.60/1.69	
3.60/1.69	
3.60/1.69	Subterm Order
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(16)
3.60/1.69	Obligation:
3.60/1.69	Q-restricted context-sensitive dependency pair problem:
3.60/1.69	The symbols in {from_1, s_1, after_2} are replacing on all positions.
3.60/1.69	For all symbols f in {cons_2} we have mu(f) = {1}.
3.60/1.69	
3.60/1.69	The TRS P consists of the following rules:
3.60/1.69	none
3.60/1.69	
3.60/1.69	The TRS R consists of the following rules:
3.60/1.69	
3.60/1.69	   from(X) -> cons(X, from(s(X)))
3.60/1.69	   after(0, XS) -> XS
3.60/1.69	   after(s(N), cons(X, XS)) -> after(N, XS)
3.60/1.69	
3.60/1.69	The set Q consists of the following terms:
3.60/1.69	
3.60/1.69	   from(x0)
3.60/1.69	   after(0, x0)
3.60/1.69	   after(s(x0), cons(x1, x2))
3.60/1.69	
3.60/1.69	
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(17) PIsEmptyProof (EQUIVALENT)
3.60/1.69	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.60/1.69	----------------------------------------
3.60/1.69	
3.60/1.69	(18)
3.60/1.69	YES
3.60/1.72	EOF