2.70/1.41	YES
2.87/1.41	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.87/1.41	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.87/1.41	
2.87/1.41	
2.87/1.41	Termination of the given CSR could be proven:
2.87/1.41	
2.87/1.41	(0) CSR
2.87/1.41	(1) CSRInnermostProof [EQUIVALENT, 0 ms]
2.87/1.41	(2) CSR
2.87/1.41	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
2.87/1.41	(4) QCSDP
2.87/1.41	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
2.87/1.41	(6) QCSDP
2.87/1.41	(7) QCSDPSubtermProof [EQUIVALENT, 0 ms]
2.87/1.41	(8) QCSDP
2.87/1.41	(9) PIsEmptyProof [EQUIVALENT, 0 ms]
2.87/1.41	(10) YES
2.87/1.41	
2.87/1.41	
2.87/1.41	----------------------------------------
2.87/1.41	
2.87/1.41	(0)
2.87/1.41	Obligation:
2.87/1.41	Context-sensitive rewrite system:
2.87/1.41	The TRS R consists of the following rules:
2.87/1.41	
2.87/1.41	   2nd(cons(X, cons(Y, Z))) -> Y
2.87/1.41	   from(X) -> cons(X, from(s(X)))
2.87/1.41	
2.87/1.41	The replacement map contains the following entries:
2.87/1.41	
2.87/1.41	2nd: {1}
2.87/1.41	cons: {1}
2.87/1.41	from: {1}
2.87/1.41	s: {1}
2.87/1.41	
2.87/1.41	----------------------------------------
2.87/1.41	
2.87/1.41	(1) CSRInnermostProof (EQUIVALENT)
2.87/1.41	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
2.87/1.41	----------------------------------------
2.87/1.41	
2.87/1.41	(2)
2.87/1.41	Obligation:
2.87/1.41	Context-sensitive rewrite system:
2.87/1.41	The TRS R consists of the following rules:
2.87/1.41	
2.87/1.41	   2nd(cons(X, cons(Y, Z))) -> Y
2.87/1.41	   from(X) -> cons(X, from(s(X)))
2.87/1.41	
2.87/1.41	The replacement map contains the following entries:
2.87/1.41	
2.87/1.41	2nd: {1}
2.87/1.41	cons: {1}
2.87/1.41	from: {1}
2.87/1.41	s: {1}
2.87/1.41	
2.87/1.41	
2.87/1.41	Innermost Strategy.
2.87/1.41	
2.87/1.41	----------------------------------------
2.87/1.41	
2.87/1.41	(3) CSDependencyPairsProof (EQUIVALENT)
2.87/1.41	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
2.87/1.41	----------------------------------------
2.87/1.41	
2.87/1.41	(4)
2.87/1.41	Obligation:
2.87/1.41	Q-restricted context-sensitive dependency pair problem:
2.87/1.41	The symbols in {2nd_1, from_1, s_1, 2ND_1, FROM_1} are replacing on all positions.
2.87/1.41	For all symbols f in {cons_2} we have mu(f) = {1}.
2.87/1.41	The symbols in {U_1} are not replacing on any position.
2.87/1.41	
2.87/1.41	The collapsing dependency pairs are DP_c:
2.87/1.41	   2ND(cons(X, cons(Y, Z))) -> Y
2.87/1.41	
2.87/1.41	
2.87/1.41	The hidden terms of R are:
2.87/1.41	
2.87/1.41	   from(s(x0))
2.87/1.41	
2.87/1.41	Every hiding context is built from:
2.87/1.41	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@6654e7f5
2.87/1.41	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@5ed0b466
2.87/1.41	
2.87/1.41	Hence, the new unhiding pairs DP_u are :
2.87/1.41	   2ND(cons(X, cons(Y, Z))) -> U(Y)
2.87/1.41	   U(s(x_0)) -> U(x_0)
2.87/1.41	   U(from(x_0)) -> U(x_0)
2.87/1.41	   U(from(s(x0))) -> FROM(s(x0))
2.87/1.41	
2.87/1.41	The TRS R consists of the following rules:
2.87/1.41	
2.87/1.41	   2nd(cons(X, cons(Y, Z))) -> Y
2.87/1.41	   from(X) -> cons(X, from(s(X)))
2.87/1.41	
2.87/1.41	The set Q consists of the following terms:
2.87/1.41	
2.87/1.41	   2nd(cons(x0, cons(x1, x2)))
2.87/1.41	   from(x0)
2.87/1.41	
2.87/1.41	
2.87/1.41	----------------------------------------
2.87/1.41	
2.87/1.41	(5) QCSDependencyGraphProof (EQUIVALENT)
2.87/1.41	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 2 less nodes.
2.87/1.41	
2.87/1.41	----------------------------------------
2.87/1.41	
2.87/1.41	(6)
2.87/1.41	Obligation:
2.87/1.41	Q-restricted context-sensitive dependency pair problem:
2.87/1.41	The symbols in {2nd_1, from_1, s_1} are replacing on all positions.
2.87/1.41	For all symbols f in {cons_2} we have mu(f) = {1}.
2.87/1.41	The symbols in {U_1} are not replacing on any position.
2.87/1.41	
2.87/1.41	The TRS P consists of the following rules:
2.87/1.41	
2.87/1.41	   U(s(x_0)) -> U(x_0)
2.87/1.41	   U(from(x_0)) -> U(x_0)
2.87/1.41	
2.87/1.41	The TRS R consists of the following rules:
2.87/1.41	
2.87/1.41	   2nd(cons(X, cons(Y, Z))) -> Y
2.87/1.41	   from(X) -> cons(X, from(s(X)))
2.87/1.41	
2.87/1.41	The set Q consists of the following terms:
2.87/1.41	
2.87/1.41	   2nd(cons(x0, cons(x1, x2)))
2.87/1.42	   from(x0)
2.87/1.42	
2.87/1.42	
2.87/1.42	----------------------------------------
2.87/1.42	
2.87/1.42	(7) QCSDPSubtermProof (EQUIVALENT)
2.87/1.42	We use the subterm processor [DA_EMMES].
2.87/1.42	
2.87/1.42	
2.87/1.42	The following pairs can be oriented strictly and are deleted.
2.87/1.42	
2.87/1.42	   U(s(x_0)) -> U(x_0)
2.87/1.42	   U(from(x_0)) -> U(x_0)
2.87/1.42	The remaining pairs can at least be oriented weakly.
2.87/1.42	none
2.87/1.42	Used ordering:  Combined order from the following AFS and order.
2.87/1.42	U(x1)  =  x1
2.87/1.42	
2.87/1.42	
2.87/1.42	Subterm Order
2.87/1.42	
2.87/1.42	----------------------------------------
2.87/1.42	
2.87/1.42	(8)
2.87/1.42	Obligation:
2.87/1.42	Q-restricted context-sensitive dependency pair problem:
2.87/1.42	The symbols in {2nd_1, from_1, s_1} are replacing on all positions.
2.87/1.42	For all symbols f in {cons_2} we have mu(f) = {1}.
2.87/1.42	
2.87/1.42	The TRS P consists of the following rules:
2.87/1.42	none
2.87/1.42	
2.87/1.42	The TRS R consists of the following rules:
2.87/1.42	
2.87/1.42	   2nd(cons(X, cons(Y, Z))) -> Y
2.87/1.42	   from(X) -> cons(X, from(s(X)))
2.87/1.42	
2.87/1.42	The set Q consists of the following terms:
2.87/1.42	
2.87/1.42	   2nd(cons(x0, cons(x1, x2)))
2.87/1.42	   from(x0)
2.87/1.42	
2.87/1.42	
2.87/1.42	----------------------------------------
2.87/1.42	
2.87/1.42	(9) PIsEmptyProof (EQUIVALENT)
2.87/1.42	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
2.87/1.42	----------------------------------------
2.87/1.42	
2.87/1.42	(10)
2.87/1.42	YES
2.87/1.45	EOF