2.92/1.42	YES
2.92/1.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.92/1.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.92/1.43	
2.92/1.43	
2.92/1.43	Termination of the given CSR could be proven:
2.92/1.43	
2.92/1.43	(0) CSR
2.92/1.43	(1) CSRInnermostProof [EQUIVALENT, 0 ms]
2.92/1.43	(2) CSR
2.92/1.43	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
2.92/1.43	(4) QCSDP
2.92/1.43	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
2.92/1.43	(6) AND
2.92/1.43	    (7) QCSDP
2.92/1.43	        (8) QCSDPSubtermProof [EQUIVALENT, 10 ms]
2.92/1.43	        (9) QCSDP
2.92/1.43	        (10) PIsEmptyProof [EQUIVALENT, 0 ms]
2.92/1.43	        (11) YES
2.92/1.43	    (12) QCSDP
2.92/1.43	        (13) QCSDPSubtermProof [EQUIVALENT, 0 ms]
2.92/1.43	        (14) QCSDP
2.92/1.43	        (15) PIsEmptyProof [EQUIVALENT, 0 ms]
2.92/1.43	        (16) YES
2.92/1.43	
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(0)
2.92/1.43	Obligation:
2.92/1.43	Context-sensitive rewrite system:
2.92/1.43	The TRS R consists of the following rules:
2.92/1.43	
2.92/1.43	   from(X) -> cons(X, from(s(X)))
2.92/1.43	   head(cons(X, XS)) -> X
2.92/1.43	   2nd(cons(X, XS)) -> head(XS)
2.92/1.43	   take(0, XS) -> nil
2.92/1.43	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
2.92/1.43	   sel(0, cons(X, XS)) -> X
2.92/1.43	   sel(s(N), cons(X, XS)) -> sel(N, XS)
2.92/1.43	
2.92/1.43	The replacement map contains the following entries:
2.92/1.43	
2.92/1.43	from: {1}
2.92/1.43	cons: {1}
2.92/1.43	s: {1}
2.92/1.43	head: {1}
2.92/1.43	2nd: {1}
2.92/1.43	take: {1, 2}
2.92/1.43	0: empty set
2.92/1.43	nil: empty set
2.92/1.43	sel: {1, 2}
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(1) CSRInnermostProof (EQUIVALENT)
2.92/1.43	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(2)
2.92/1.43	Obligation:
2.92/1.43	Context-sensitive rewrite system:
2.92/1.43	The TRS R consists of the following rules:
2.92/1.43	
2.92/1.43	   from(X) -> cons(X, from(s(X)))
2.92/1.43	   head(cons(X, XS)) -> X
2.92/1.43	   2nd(cons(X, XS)) -> head(XS)
2.92/1.43	   take(0, XS) -> nil
2.92/1.43	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
2.92/1.43	   sel(0, cons(X, XS)) -> X
2.92/1.43	   sel(s(N), cons(X, XS)) -> sel(N, XS)
2.92/1.43	
2.92/1.43	The replacement map contains the following entries:
2.92/1.43	
2.92/1.43	from: {1}
2.92/1.43	cons: {1}
2.92/1.43	s: {1}
2.92/1.43	head: {1}
2.92/1.43	2nd: {1}
2.92/1.43	take: {1, 2}
2.92/1.43	0: empty set
2.92/1.43	nil: empty set
2.92/1.43	sel: {1, 2}
2.92/1.43	
2.92/1.43	
2.92/1.43	Innermost Strategy.
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(3) CSDependencyPairsProof (EQUIVALENT)
2.92/1.43	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(4)
2.92/1.43	Obligation:
2.92/1.43	Q-restricted context-sensitive dependency pair problem:
2.92/1.43	The symbols in {from_1, s_1, head_1, 2nd_1, take_2, sel_2, HEAD_1, 2ND_1, SEL_2, FROM_1, TAKE_2} are replacing on all positions.
2.92/1.43	For all symbols f in {cons_2} we have mu(f) = {1}.
2.92/1.43	The symbols in {U_1} are not replacing on any position.
2.92/1.43	
2.92/1.43	The ordinary context-sensitive dependency pairs DP_o are:
2.92/1.43	   2ND(cons(X, XS)) -> HEAD(XS)
2.92/1.43	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
2.92/1.43	
2.92/1.43	The collapsing dependency pairs are DP_c:
2.92/1.43	   2ND(cons(X, XS)) -> XS
2.92/1.43	   SEL(s(N), cons(X, XS)) -> XS
2.92/1.43	
2.92/1.43	
2.92/1.43	The hidden terms of R are:
2.92/1.43	
2.92/1.43	   from(s(x0))
2.92/1.43	   take(x0, x1)
2.92/1.43	
2.92/1.43	Every hiding context is built from:
2.92/1.43	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@77f7c96a
2.92/1.43	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@6168d179
2.92/1.43	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@7bb02f40
2.92/1.43	
2.92/1.43	Hence, the new unhiding pairs DP_u are :
2.92/1.43	   2ND(cons(X, XS)) -> U(XS)
2.92/1.43	   SEL(s(N), cons(X, XS)) -> U(XS)
2.92/1.43	   U(s(x_0)) -> U(x_0)
2.92/1.43	   U(from(x_0)) -> U(x_0)
2.92/1.43	   U(take(x_0, x_1)) -> U(x_0)
2.92/1.43	   U(take(x_0, x_1)) -> U(x_1)
2.92/1.43	   U(from(s(x0))) -> FROM(s(x0))
2.92/1.43	   U(take(x0, x1)) -> TAKE(x0, x1)
2.92/1.43	
2.92/1.43	The TRS R consists of the following rules:
2.92/1.43	
2.92/1.43	   from(X) -> cons(X, from(s(X)))
2.92/1.43	   head(cons(X, XS)) -> X
2.92/1.43	   2nd(cons(X, XS)) -> head(XS)
2.92/1.43	   take(0, XS) -> nil
2.92/1.43	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
2.92/1.43	   sel(0, cons(X, XS)) -> X
2.92/1.43	   sel(s(N), cons(X, XS)) -> sel(N, XS)
2.92/1.43	
2.92/1.43	The set Q consists of the following terms:
2.92/1.43	
2.92/1.43	   from(x0)
2.92/1.43	   head(cons(x0, x1))
2.92/1.43	   2nd(cons(x0, x1))
2.92/1.43	   take(0, x0)
2.92/1.43	   take(s(x0), cons(x1, x2))
2.92/1.43	   sel(0, cons(x0, x1))
2.92/1.43	   sel(s(x0), cons(x1, x2))
2.92/1.43	
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(5) QCSDependencyGraphProof (EQUIVALENT)
2.92/1.43	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 4 less nodes.
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(6)
2.92/1.43	Complex Obligation (AND)
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(7)
2.92/1.43	Obligation:
2.92/1.43	Q-restricted context-sensitive dependency pair problem:
2.92/1.43	The symbols in {from_1, s_1, head_1, 2nd_1, take_2, sel_2} are replacing on all positions.
2.92/1.43	For all symbols f in {cons_2} we have mu(f) = {1}.
2.92/1.43	The symbols in {U_1} are not replacing on any position.
2.92/1.43	
2.92/1.43	The TRS P consists of the following rules:
2.92/1.43	
2.92/1.43	   U(s(x_0)) -> U(x_0)
2.92/1.43	   U(from(x_0)) -> U(x_0)
2.92/1.43	   U(take(x_0, x_1)) -> U(x_0)
2.92/1.43	   U(take(x_0, x_1)) -> U(x_1)
2.92/1.43	
2.92/1.43	The TRS R consists of the following rules:
2.92/1.43	
2.92/1.43	   from(X) -> cons(X, from(s(X)))
2.92/1.43	   head(cons(X, XS)) -> X
2.92/1.43	   2nd(cons(X, XS)) -> head(XS)
2.92/1.43	   take(0, XS) -> nil
2.92/1.43	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
2.92/1.43	   sel(0, cons(X, XS)) -> X
2.92/1.43	   sel(s(N), cons(X, XS)) -> sel(N, XS)
2.92/1.43	
2.92/1.43	The set Q consists of the following terms:
2.92/1.43	
2.92/1.43	   from(x0)
2.92/1.43	   head(cons(x0, x1))
2.92/1.43	   2nd(cons(x0, x1))
2.92/1.43	   take(0, x0)
2.92/1.43	   take(s(x0), cons(x1, x2))
2.92/1.43	   sel(0, cons(x0, x1))
2.92/1.43	   sel(s(x0), cons(x1, x2))
2.92/1.43	
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(8) QCSDPSubtermProof (EQUIVALENT)
2.92/1.43	We use the subterm processor [DA_EMMES].
2.92/1.43	
2.92/1.43	
2.92/1.43	The following pairs can be oriented strictly and are deleted.
2.92/1.43	
2.92/1.43	   U(s(x_0)) -> U(x_0)
2.92/1.43	   U(from(x_0)) -> U(x_0)
2.92/1.43	   U(take(x_0, x_1)) -> U(x_0)
2.92/1.43	   U(take(x_0, x_1)) -> U(x_1)
2.92/1.43	The remaining pairs can at least be oriented weakly.
2.92/1.43	none
2.92/1.43	Used ordering:  Combined order from the following AFS and order.
2.92/1.43	U(x1)  =  x1
2.92/1.43	
2.92/1.43	
2.92/1.43	Subterm Order
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(9)
2.92/1.43	Obligation:
2.92/1.43	Q-restricted context-sensitive dependency pair problem:
2.92/1.43	The symbols in {from_1, s_1, head_1, 2nd_1, take_2, sel_2} are replacing on all positions.
2.92/1.43	For all symbols f in {cons_2} we have mu(f) = {1}.
2.92/1.43	
2.92/1.43	The TRS P consists of the following rules:
2.92/1.43	none
2.92/1.43	
2.92/1.43	The TRS R consists of the following rules:
2.92/1.43	
2.92/1.43	   from(X) -> cons(X, from(s(X)))
2.92/1.43	   head(cons(X, XS)) -> X
2.92/1.43	   2nd(cons(X, XS)) -> head(XS)
2.92/1.43	   take(0, XS) -> nil
2.92/1.43	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
2.92/1.43	   sel(0, cons(X, XS)) -> X
2.92/1.43	   sel(s(N), cons(X, XS)) -> sel(N, XS)
2.92/1.43	
2.92/1.43	The set Q consists of the following terms:
2.92/1.43	
2.92/1.43	   from(x0)
2.92/1.43	   head(cons(x0, x1))
2.92/1.43	   2nd(cons(x0, x1))
2.92/1.43	   take(0, x0)
2.92/1.43	   take(s(x0), cons(x1, x2))
2.92/1.43	   sel(0, cons(x0, x1))
2.92/1.43	   sel(s(x0), cons(x1, x2))
2.92/1.43	
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(10) PIsEmptyProof (EQUIVALENT)
2.92/1.43	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(11)
2.92/1.43	YES
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(12)
2.92/1.43	Obligation:
2.92/1.43	Q-restricted context-sensitive dependency pair problem:
2.92/1.43	The symbols in {from_1, s_1, head_1, 2nd_1, take_2, sel_2, SEL_2} are replacing on all positions.
2.92/1.43	For all symbols f in {cons_2} we have mu(f) = {1}.
2.92/1.43	
2.92/1.43	The TRS P consists of the following rules:
2.92/1.43	
2.92/1.43	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
2.92/1.43	
2.92/1.43	The TRS R consists of the following rules:
2.92/1.43	
2.92/1.43	   from(X) -> cons(X, from(s(X)))
2.92/1.43	   head(cons(X, XS)) -> X
2.92/1.43	   2nd(cons(X, XS)) -> head(XS)
2.92/1.43	   take(0, XS) -> nil
2.92/1.43	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
2.92/1.43	   sel(0, cons(X, XS)) -> X
2.92/1.43	   sel(s(N), cons(X, XS)) -> sel(N, XS)
2.92/1.43	
2.92/1.43	The set Q consists of the following terms:
2.92/1.43	
2.92/1.43	   from(x0)
2.92/1.43	   head(cons(x0, x1))
2.92/1.43	   2nd(cons(x0, x1))
2.92/1.43	   take(0, x0)
2.92/1.43	   take(s(x0), cons(x1, x2))
2.92/1.43	   sel(0, cons(x0, x1))
2.92/1.43	   sel(s(x0), cons(x1, x2))
2.92/1.43	
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(13) QCSDPSubtermProof (EQUIVALENT)
2.92/1.43	We use the subterm processor [DA_EMMES].
2.92/1.43	
2.92/1.43	
2.92/1.43	The following pairs can be oriented strictly and are deleted.
2.92/1.43	
2.92/1.43	   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
2.92/1.43	The remaining pairs can at least be oriented weakly.
2.92/1.43	none
2.92/1.43	Used ordering:  Combined order from the following AFS and order.
2.92/1.43	SEL(x1, x2)  =  x1
2.92/1.43	
2.92/1.43	
2.92/1.43	Subterm Order
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(14)
2.92/1.43	Obligation:
2.92/1.43	Q-restricted context-sensitive dependency pair problem:
2.92/1.43	The symbols in {from_1, s_1, head_1, 2nd_1, take_2, sel_2} are replacing on all positions.
2.92/1.43	For all symbols f in {cons_2} we have mu(f) = {1}.
2.92/1.43	
2.92/1.43	The TRS P consists of the following rules:
2.92/1.43	none
2.92/1.43	
2.92/1.43	The TRS R consists of the following rules:
2.92/1.43	
2.92/1.43	   from(X) -> cons(X, from(s(X)))
2.92/1.43	   head(cons(X, XS)) -> X
2.92/1.43	   2nd(cons(X, XS)) -> head(XS)
2.92/1.43	   take(0, XS) -> nil
2.92/1.43	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
2.92/1.43	   sel(0, cons(X, XS)) -> X
2.92/1.43	   sel(s(N), cons(X, XS)) -> sel(N, XS)
2.92/1.43	
2.92/1.43	The set Q consists of the following terms:
2.92/1.43	
2.92/1.43	   from(x0)
2.92/1.43	   head(cons(x0, x1))
2.92/1.43	   2nd(cons(x0, x1))
2.92/1.43	   take(0, x0)
2.92/1.43	   take(s(x0), cons(x1, x2))
2.92/1.43	   sel(0, cons(x0, x1))
2.92/1.43	   sel(s(x0), cons(x1, x2))
2.92/1.43	
2.92/1.43	
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(15) PIsEmptyProof (EQUIVALENT)
2.92/1.43	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
2.92/1.43	----------------------------------------
2.92/1.43	
2.92/1.43	(16)
2.92/1.43	YES
3.16/1.45	EOF