3.23/1.45	YES
3.23/1.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.23/1.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.23/1.46	
3.23/1.46	
3.23/1.46	Termination of the given CSR could be proven:
3.23/1.46	
3.23/1.46	(0) CSR
3.23/1.46	(1) CSRInnermostProof [EQUIVALENT, 0 ms]
3.23/1.46	(2) CSR
3.23/1.46	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.23/1.46	(4) QCSDP
3.23/1.46	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.23/1.46	(6) QCSDP
3.23/1.46	(7) QCSDPSubtermProof [EQUIVALENT, 16 ms]
3.23/1.46	(8) QCSDP
3.23/1.46	(9) PIsEmptyProof [EQUIVALENT, 0 ms]
3.23/1.46	(10) YES
3.23/1.46	
3.23/1.46	
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(0)
3.23/1.46	Obligation:
3.23/1.46	Context-sensitive rewrite system:
3.23/1.46	The TRS R consists of the following rules:
3.23/1.46	
3.23/1.46	   2nd(cons1(X, cons(Y, Z))) -> Y
3.23/1.46	   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
3.23/1.46	   from(X) -> cons(X, from(s(X)))
3.23/1.46	
3.23/1.46	The replacement map contains the following entries:
3.23/1.46	
3.23/1.46	2nd: {1}
3.23/1.46	cons1: {1, 2}
3.23/1.46	cons: {1}
3.23/1.46	from: {1}
3.23/1.46	s: {1}
3.23/1.46	
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(1) CSRInnermostProof (EQUIVALENT)
3.23/1.46	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(2)
3.23/1.46	Obligation:
3.23/1.46	Context-sensitive rewrite system:
3.23/1.46	The TRS R consists of the following rules:
3.23/1.46	
3.23/1.46	   2nd(cons1(X, cons(Y, Z))) -> Y
3.23/1.46	   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
3.23/1.46	   from(X) -> cons(X, from(s(X)))
3.23/1.46	
3.23/1.46	The replacement map contains the following entries:
3.23/1.46	
3.23/1.46	2nd: {1}
3.23/1.46	cons1: {1, 2}
3.23/1.46	cons: {1}
3.23/1.46	from: {1}
3.23/1.46	s: {1}
3.23/1.46	
3.23/1.46	
3.23/1.46	Innermost Strategy.
3.23/1.46	
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(3) CSDependencyPairsProof (EQUIVALENT)
3.23/1.46	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(4)
3.23/1.46	Obligation:
3.23/1.46	Q-restricted context-sensitive dependency pair problem:
3.23/1.46	The symbols in {2nd_1, cons1_2, from_1, s_1, 2ND_1, FROM_1} are replacing on all positions.
3.23/1.46	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.46	The symbols in {U_1} are not replacing on any position.
3.23/1.46	
3.23/1.46	The ordinary context-sensitive dependency pairs DP_o are:
3.23/1.46	   2ND(cons(X, X1)) -> 2ND(cons1(X, X1))
3.23/1.46	
3.23/1.46	The collapsing dependency pairs are DP_c:
3.23/1.46	   2ND(cons(X, X1)) -> X1
3.23/1.46	
3.23/1.46	
3.23/1.46	The hidden terms of R are:
3.23/1.46	
3.23/1.46	   from(s(x0))
3.23/1.46	
3.23/1.46	Every hiding context is built from:
3.23/1.46	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@3e32ccfe
3.23/1.46	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@333a3111
3.23/1.46	
3.23/1.46	Hence, the new unhiding pairs DP_u are :
3.23/1.46	   2ND(cons(X, X1)) -> U(X1)
3.23/1.46	   U(s(x_0)) -> U(x_0)
3.23/1.46	   U(from(x_0)) -> U(x_0)
3.23/1.46	   U(from(s(x0))) -> FROM(s(x0))
3.23/1.46	
3.23/1.46	The TRS R consists of the following rules:
3.23/1.46	
3.23/1.46	   2nd(cons1(X, cons(Y, Z))) -> Y
3.23/1.46	   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
3.23/1.46	   from(X) -> cons(X, from(s(X)))
3.23/1.46	
3.23/1.46	The set Q consists of the following terms:
3.23/1.46	
3.23/1.46	   2nd(cons1(x0, cons(x1, x2)))
3.23/1.46	   2nd(cons(x0, x1))
3.23/1.46	   from(x0)
3.23/1.46	
3.23/1.46	
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(5) QCSDependencyGraphProof (EQUIVALENT)
3.23/1.46	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 2 less nodes.
3.23/1.46	The rules 2ND(cons(z0, z1)) -> 2ND(cons1(z0, z1)) and 2ND(cons(x0, x1)) -> 2ND(cons1(x0, x1)) form no chain, because ECap^mu(2ND(cons1(z0, z1))) = 2ND(cons1(z0, x_1)) does not unify with 2ND(cons(x0, x1)). The rules 2ND(cons(z0, z1)) -> 2ND(cons1(z0, z1)) and 2ND(cons(x0, x1)) -> U(x1) form no chain, because ECap^mu(2ND(cons1(z0, z1))) = 2ND(cons1(z0, x_1)) does not unify with 2ND(cons(x0, x1)). 
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(6)
3.23/1.46	Obligation:
3.23/1.46	Q-restricted context-sensitive dependency pair problem:
3.23/1.46	The symbols in {2nd_1, cons1_2, from_1, s_1} are replacing on all positions.
3.23/1.46	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.46	The symbols in {U_1} are not replacing on any position.
3.23/1.46	
3.23/1.46	The TRS P consists of the following rules:
3.23/1.46	
3.23/1.46	   U(s(x_0)) -> U(x_0)
3.23/1.46	   U(from(x_0)) -> U(x_0)
3.23/1.46	
3.23/1.46	The TRS R consists of the following rules:
3.23/1.46	
3.23/1.46	   2nd(cons1(X, cons(Y, Z))) -> Y
3.23/1.46	   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
3.23/1.46	   from(X) -> cons(X, from(s(X)))
3.23/1.46	
3.23/1.46	The set Q consists of the following terms:
3.23/1.46	
3.23/1.46	   2nd(cons1(x0, cons(x1, x2)))
3.23/1.46	   2nd(cons(x0, x1))
3.23/1.46	   from(x0)
3.23/1.46	
3.23/1.46	
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(7) QCSDPSubtermProof (EQUIVALENT)
3.23/1.46	We use the subterm processor [DA_EMMES].
3.23/1.46	
3.23/1.46	
3.23/1.46	The following pairs can be oriented strictly and are deleted.
3.23/1.46	
3.23/1.46	   U(s(x_0)) -> U(x_0)
3.23/1.46	   U(from(x_0)) -> U(x_0)
3.23/1.46	The remaining pairs can at least be oriented weakly.
3.23/1.46	none
3.23/1.46	Used ordering:  Combined order from the following AFS and order.
3.23/1.46	U(x1)  =  x1
3.23/1.46	
3.23/1.46	
3.23/1.46	Subterm Order
3.23/1.46	
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(8)
3.23/1.46	Obligation:
3.23/1.46	Q-restricted context-sensitive dependency pair problem:
3.23/1.46	The symbols in {2nd_1, cons1_2, from_1, s_1} are replacing on all positions.
3.23/1.46	For all symbols f in {cons_2} we have mu(f) = {1}.
3.23/1.46	
3.23/1.46	The TRS P consists of the following rules:
3.23/1.46	none
3.23/1.46	
3.23/1.46	The TRS R consists of the following rules:
3.23/1.46	
3.23/1.46	   2nd(cons1(X, cons(Y, Z))) -> Y
3.23/1.46	   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
3.23/1.46	   from(X) -> cons(X, from(s(X)))
3.23/1.46	
3.23/1.46	The set Q consists of the following terms:
3.23/1.46	
3.23/1.46	   2nd(cons1(x0, cons(x1, x2)))
3.23/1.46	   2nd(cons(x0, x1))
3.23/1.46	   from(x0)
3.23/1.46	
3.23/1.46	
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(9) PIsEmptyProof (EQUIVALENT)
3.23/1.46	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.23/1.46	----------------------------------------
3.23/1.46	
3.23/1.46	(10)
3.23/1.46	YES
3.23/1.49	EOF