2.78/1.54	YES
2.78/1.55	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.78/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.78/1.55	
2.78/1.55	
2.78/1.55	Termination of the given CSR could be proven:
2.78/1.55	
2.78/1.55	(0) CSR
2.78/1.55	(1) CSRInnermostProof [EQUIVALENT, 0 ms]
2.78/1.55	(2) CSR
2.78/1.55	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
2.78/1.55	(4) QCSDP
2.78/1.55	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
2.78/1.55	(6) AND
2.78/1.55	    (7) QCSDP
2.78/1.55	        (8) QCSDPSubtermProof [EQUIVALENT, 22 ms]
2.78/1.55	        (9) QCSDP
2.78/1.55	        (10) PIsEmptyProof [EQUIVALENT, 0 ms]
2.78/1.55	        (11) YES
2.78/1.55	    (12) QCSDP
2.78/1.55	        (13) QCSDPSubtermProof [EQUIVALENT, 0 ms]
2.78/1.55	        (14) QCSDP
2.78/1.55	        (15) PIsEmptyProof [EQUIVALENT, 0 ms]
2.78/1.55	        (16) YES
2.78/1.55	
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(0)
2.78/1.55	Obligation:
2.78/1.55	Context-sensitive rewrite system:
2.78/1.55	The TRS R consists of the following rules:
2.78/1.55	
2.78/1.55	   from(X) -> cons(X, from(s(X)))
2.78/1.55	   after(0, XS) -> XS
2.78/1.55	   after(s(N), cons(X, XS)) -> after(N, XS)
2.78/1.55	
2.78/1.55	The replacement map contains the following entries:
2.78/1.55	
2.78/1.55	from: {1}
2.78/1.55	cons: {1}
2.78/1.55	s: {1}
2.78/1.55	after: {1, 2}
2.78/1.55	0: empty set
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(1) CSRInnermostProof (EQUIVALENT)
2.78/1.55	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(2)
2.78/1.55	Obligation:
2.78/1.55	Context-sensitive rewrite system:
2.78/1.55	The TRS R consists of the following rules:
2.78/1.55	
2.78/1.55	   from(X) -> cons(X, from(s(X)))
2.78/1.55	   after(0, XS) -> XS
2.78/1.55	   after(s(N), cons(X, XS)) -> after(N, XS)
2.78/1.55	
2.78/1.55	The replacement map contains the following entries:
2.78/1.55	
2.78/1.55	from: {1}
2.78/1.55	cons: {1}
2.78/1.55	s: {1}
2.78/1.55	after: {1, 2}
2.78/1.55	0: empty set
2.78/1.55	
2.78/1.55	
2.78/1.55	Innermost Strategy.
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(3) CSDependencyPairsProof (EQUIVALENT)
2.78/1.55	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(4)
2.78/1.55	Obligation:
2.78/1.55	Q-restricted context-sensitive dependency pair problem:
2.78/1.55	The symbols in {from_1, s_1, after_2, AFTER_2, FROM_1} are replacing on all positions.
2.78/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
2.78/1.55	The symbols in {U_1} are not replacing on any position.
2.78/1.55	
2.78/1.55	The ordinary context-sensitive dependency pairs DP_o are:
2.78/1.55	   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)
2.78/1.55	
2.78/1.55	The collapsing dependency pairs are DP_c:
2.78/1.55	   AFTER(s(N), cons(X, XS)) -> XS
2.78/1.55	
2.78/1.55	
2.78/1.55	The hidden terms of R are:
2.78/1.55	
2.78/1.55	   from(s(x0))
2.78/1.55	
2.78/1.55	Every hiding context is built from:
2.78/1.55	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@1b878b36
2.78/1.55	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@46e2a944
2.78/1.55	
2.78/1.55	Hence, the new unhiding pairs DP_u are :
2.78/1.55	   AFTER(s(N), cons(X, XS)) -> U(XS)
2.78/1.55	   U(s(x_0)) -> U(x_0)
2.78/1.55	   U(from(x_0)) -> U(x_0)
2.78/1.55	   U(from(s(x0))) -> FROM(s(x0))
2.78/1.55	
2.78/1.55	The TRS R consists of the following rules:
2.78/1.55	
2.78/1.55	   from(X) -> cons(X, from(s(X)))
2.78/1.55	   after(0, XS) -> XS
2.78/1.55	   after(s(N), cons(X, XS)) -> after(N, XS)
2.78/1.55	
2.78/1.55	The set Q consists of the following terms:
2.78/1.55	
2.78/1.55	   from(x0)
2.78/1.55	   after(0, x0)
2.78/1.55	   after(s(x0), cons(x1, x2))
2.78/1.55	
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(5) QCSDependencyGraphProof (EQUIVALENT)
2.78/1.55	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 2 less nodes.
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(6)
2.78/1.55	Complex Obligation (AND)
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(7)
2.78/1.55	Obligation:
2.78/1.55	Q-restricted context-sensitive dependency pair problem:
2.78/1.55	The symbols in {from_1, s_1, after_2} are replacing on all positions.
2.78/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
2.78/1.55	The symbols in {U_1} are not replacing on any position.
2.78/1.55	
2.78/1.55	The TRS P consists of the following rules:
2.78/1.55	
2.78/1.55	   U(s(x_0)) -> U(x_0)
2.78/1.55	   U(from(x_0)) -> U(x_0)
2.78/1.55	
2.78/1.55	The TRS R consists of the following rules:
2.78/1.55	
2.78/1.55	   from(X) -> cons(X, from(s(X)))
2.78/1.55	   after(0, XS) -> XS
2.78/1.55	   after(s(N), cons(X, XS)) -> after(N, XS)
2.78/1.55	
2.78/1.55	The set Q consists of the following terms:
2.78/1.55	
2.78/1.55	   from(x0)
2.78/1.55	   after(0, x0)
2.78/1.55	   after(s(x0), cons(x1, x2))
2.78/1.55	
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(8) QCSDPSubtermProof (EQUIVALENT)
2.78/1.55	We use the subterm processor [DA_EMMES].
2.78/1.55	
2.78/1.55	
2.78/1.55	The following pairs can be oriented strictly and are deleted.
2.78/1.55	
2.78/1.55	   U(s(x_0)) -> U(x_0)
2.78/1.55	   U(from(x_0)) -> U(x_0)
2.78/1.55	The remaining pairs can at least be oriented weakly.
2.78/1.55	none
2.78/1.55	Used ordering:  Combined order from the following AFS and order.
2.78/1.55	U(x1)  =  x1
2.78/1.55	
2.78/1.55	
2.78/1.55	Subterm Order
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(9)
2.78/1.55	Obligation:
2.78/1.55	Q-restricted context-sensitive dependency pair problem:
2.78/1.55	The symbols in {from_1, s_1, after_2} are replacing on all positions.
2.78/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
2.78/1.55	
2.78/1.55	The TRS P consists of the following rules:
2.78/1.55	none
2.78/1.55	
2.78/1.55	The TRS R consists of the following rules:
2.78/1.55	
2.78/1.55	   from(X) -> cons(X, from(s(X)))
2.78/1.55	   after(0, XS) -> XS
2.78/1.55	   after(s(N), cons(X, XS)) -> after(N, XS)
2.78/1.55	
2.78/1.55	The set Q consists of the following terms:
2.78/1.55	
2.78/1.55	   from(x0)
2.78/1.55	   after(0, x0)
2.78/1.55	   after(s(x0), cons(x1, x2))
2.78/1.55	
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(10) PIsEmptyProof (EQUIVALENT)
2.78/1.55	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(11)
2.78/1.55	YES
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(12)
2.78/1.55	Obligation:
2.78/1.55	Q-restricted context-sensitive dependency pair problem:
2.78/1.55	The symbols in {from_1, s_1, after_2, AFTER_2} are replacing on all positions.
2.78/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
2.78/1.55	
2.78/1.55	The TRS P consists of the following rules:
2.78/1.55	
2.78/1.55	   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)
2.78/1.55	
2.78/1.55	The TRS R consists of the following rules:
2.78/1.55	
2.78/1.55	   from(X) -> cons(X, from(s(X)))
2.78/1.55	   after(0, XS) -> XS
2.78/1.55	   after(s(N), cons(X, XS)) -> after(N, XS)
2.78/1.55	
2.78/1.55	The set Q consists of the following terms:
2.78/1.55	
2.78/1.55	   from(x0)
2.78/1.55	   after(0, x0)
2.78/1.55	   after(s(x0), cons(x1, x2))
2.78/1.55	
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(13) QCSDPSubtermProof (EQUIVALENT)
2.78/1.55	We use the subterm processor [DA_EMMES].
2.78/1.55	
2.78/1.55	
2.78/1.55	The following pairs can be oriented strictly and are deleted.
2.78/1.55	
2.78/1.55	   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)
2.78/1.55	The remaining pairs can at least be oriented weakly.
2.78/1.55	none
2.78/1.55	Used ordering:  Combined order from the following AFS and order.
2.78/1.55	AFTER(x1, x2)  =  x1
2.78/1.55	
2.78/1.55	
2.78/1.55	Subterm Order
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(14)
2.78/1.55	Obligation:
2.78/1.55	Q-restricted context-sensitive dependency pair problem:
2.78/1.55	The symbols in {from_1, s_1, after_2} are replacing on all positions.
2.78/1.55	For all symbols f in {cons_2} we have mu(f) = {1}.
2.78/1.55	
2.78/1.55	The TRS P consists of the following rules:
2.78/1.55	none
2.78/1.55	
2.78/1.55	The TRS R consists of the following rules:
2.78/1.55	
2.78/1.55	   from(X) -> cons(X, from(s(X)))
2.78/1.55	   after(0, XS) -> XS
2.78/1.55	   after(s(N), cons(X, XS)) -> after(N, XS)
2.78/1.55	
2.78/1.55	The set Q consists of the following terms:
2.78/1.55	
2.78/1.55	   from(x0)
2.78/1.55	   after(0, x0)
2.78/1.55	   after(s(x0), cons(x1, x2))
2.78/1.55	
2.78/1.55	
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(15) PIsEmptyProof (EQUIVALENT)
2.78/1.55	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
2.78/1.55	----------------------------------------
2.78/1.55	
2.78/1.55	(16)
2.78/1.55	YES
3.05/1.58	EOF