3.24/1.47	YES
3.24/1.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.24/1.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.24/1.48	
3.24/1.48	
3.24/1.48	Termination of the given CSR could be proven:
3.24/1.48	
3.24/1.48	(0) CSR
3.24/1.48	(1) CSRInnermostProof [EQUIVALENT, 0 ms]
3.24/1.48	(2) CSR
3.24/1.48	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.24/1.48	(4) QCSDP
3.24/1.48	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.24/1.48	(6) AND
3.24/1.48	    (7) QCSDP
3.24/1.48	        (8) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.24/1.48	        (9) QCSDP
3.24/1.48	        (10) PIsEmptyProof [EQUIVALENT, 0 ms]
3.24/1.48	        (11) YES
3.24/1.48	    (12) QCSDP
3.24/1.48	        (13) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.24/1.48	        (14) QCSDP
3.24/1.48	        (15) PIsEmptyProof [EQUIVALENT, 0 ms]
3.24/1.48	        (16) YES
3.24/1.48	
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(0)
3.24/1.48	Obligation:
3.24/1.48	Context-sensitive rewrite system:
3.24/1.48	The TRS R consists of the following rules:
3.24/1.48	
3.24/1.48	   and(tt, X) -> X
3.24/1.48	   plus(N, 0) -> N
3.24/1.48	   plus(N, s(M)) -> s(plus(N, M))
3.24/1.48	   x(N, 0) -> 0
3.24/1.48	   x(N, s(M)) -> plus(x(N, M), N)
3.24/1.48	
3.24/1.48	The replacement map contains the following entries:
3.24/1.48	
3.24/1.48	and: {1}
3.24/1.48	tt: empty set
3.24/1.48	plus: {1, 2}
3.24/1.48	0: empty set
3.24/1.48	s: {1}
3.24/1.48	x: {1, 2}
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(1) CSRInnermostProof (EQUIVALENT)
3.24/1.48	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(2)
3.24/1.48	Obligation:
3.24/1.48	Context-sensitive rewrite system:
3.24/1.48	The TRS R consists of the following rules:
3.24/1.48	
3.24/1.48	   and(tt, X) -> X
3.24/1.48	   plus(N, 0) -> N
3.24/1.48	   plus(N, s(M)) -> s(plus(N, M))
3.24/1.48	   x(N, 0) -> 0
3.24/1.48	   x(N, s(M)) -> plus(x(N, M), N)
3.24/1.48	
3.24/1.48	The replacement map contains the following entries:
3.24/1.48	
3.24/1.48	and: {1}
3.24/1.48	tt: empty set
3.24/1.48	plus: {1, 2}
3.24/1.48	0: empty set
3.24/1.48	s: {1}
3.24/1.48	x: {1, 2}
3.24/1.48	
3.24/1.48	
3.24/1.48	Innermost Strategy.
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(3) CSDependencyPairsProof (EQUIVALENT)
3.24/1.48	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(4)
3.24/1.48	Obligation:
3.24/1.48	Q-restricted context-sensitive dependency pair problem:
3.24/1.48	The symbols in {plus_2, s_1, x_2, PLUS_2, X_2} are replacing on all positions.
3.24/1.48	For all symbols f in {and_2, AND_2} we have mu(f) = {1}.
3.24/1.48	The symbols in {U_1} are not replacing on any position.
3.24/1.48	
3.24/1.48	The ordinary context-sensitive dependency pairs DP_o are:
3.24/1.48	   PLUS(N, s(M)) -> PLUS(N, M)
3.24/1.48	   X(N, s(M)) -> PLUS(x(N, M), N)
3.24/1.48	   X(N, s(M)) -> X(N, M)
3.24/1.48	
3.24/1.48	The collapsing dependency pairs are DP_c:
3.24/1.48	   AND(tt, X) -> X
3.24/1.48	
3.24/1.48	
3.24/1.48	The hidden terms of R are:
3.24/1.48	none
3.24/1.48	
3.24/1.48	Every hiding context is built from:none
3.24/1.48	
3.24/1.48	Hence, the new unhiding pairs DP_u are :
3.24/1.48	   AND(tt, X) -> U(X)
3.24/1.48	
3.24/1.48	The TRS R consists of the following rules:
3.24/1.48	
3.24/1.48	   and(tt, X) -> X
3.24/1.48	   plus(N, 0) -> N
3.24/1.48	   plus(N, s(M)) -> s(plus(N, M))
3.24/1.48	   x(N, 0) -> 0
3.24/1.48	   x(N, s(M)) -> plus(x(N, M), N)
3.24/1.48	
3.24/1.48	The set Q consists of the following terms:
3.24/1.48	
3.24/1.48	   and(tt, x0)
3.24/1.48	   plus(x0, 0)
3.24/1.48	   plus(x0, s(x1))
3.24/1.48	   x(x0, 0)
3.24/1.48	   x(x0, s(x1))
3.24/1.48	
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(5) QCSDependencyGraphProof (EQUIVALENT)
3.24/1.48	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 1 less node.
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(6)
3.24/1.48	Complex Obligation (AND)
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(7)
3.24/1.48	Obligation:
3.24/1.48	Q-restricted context-sensitive dependency pair problem:
3.24/1.48	The symbols in {plus_2, s_1, x_2, PLUS_2} are replacing on all positions.
3.24/1.48	For all symbols f in {and_2} we have mu(f) = {1}.
3.24/1.48	
3.24/1.48	The TRS P consists of the following rules:
3.24/1.48	
3.24/1.48	   PLUS(N, s(M)) -> PLUS(N, M)
3.24/1.48	
3.24/1.48	The TRS R consists of the following rules:
3.24/1.48	
3.24/1.48	   and(tt, X) -> X
3.24/1.48	   plus(N, 0) -> N
3.24/1.48	   plus(N, s(M)) -> s(plus(N, M))
3.24/1.48	   x(N, 0) -> 0
3.24/1.48	   x(N, s(M)) -> plus(x(N, M), N)
3.24/1.48	
3.24/1.48	The set Q consists of the following terms:
3.24/1.48	
3.24/1.48	   and(tt, x0)
3.24/1.48	   plus(x0, 0)
3.24/1.48	   plus(x0, s(x1))
3.24/1.48	   x(x0, 0)
3.24/1.48	   x(x0, s(x1))
3.24/1.48	
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(8) QCSDPSubtermProof (EQUIVALENT)
3.24/1.48	We use the subterm processor [DA_EMMES].
3.24/1.48	
3.24/1.48	
3.24/1.48	The following pairs can be oriented strictly and are deleted.
3.24/1.48	
3.24/1.48	   PLUS(N, s(M)) -> PLUS(N, M)
3.24/1.48	The remaining pairs can at least be oriented weakly.
3.24/1.48	none
3.24/1.48	Used ordering:  Combined order from the following AFS and order.
3.24/1.48	PLUS(x1, x2)  =  x2
3.24/1.48	
3.24/1.48	
3.24/1.48	Subterm Order
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(9)
3.24/1.48	Obligation:
3.24/1.48	Q-restricted context-sensitive dependency pair problem:
3.24/1.48	The symbols in {plus_2, s_1, x_2} are replacing on all positions.
3.24/1.48	For all symbols f in {and_2} we have mu(f) = {1}.
3.24/1.48	
3.24/1.48	The TRS P consists of the following rules:
3.24/1.48	none
3.24/1.48	
3.24/1.48	The TRS R consists of the following rules:
3.24/1.48	
3.24/1.48	   and(tt, X) -> X
3.24/1.48	   plus(N, 0) -> N
3.24/1.48	   plus(N, s(M)) -> s(plus(N, M))
3.24/1.48	   x(N, 0) -> 0
3.24/1.48	   x(N, s(M)) -> plus(x(N, M), N)
3.24/1.48	
3.24/1.48	The set Q consists of the following terms:
3.24/1.48	
3.24/1.48	   and(tt, x0)
3.24/1.48	   plus(x0, 0)
3.24/1.48	   plus(x0, s(x1))
3.24/1.48	   x(x0, 0)
3.24/1.48	   x(x0, s(x1))
3.24/1.48	
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(10) PIsEmptyProof (EQUIVALENT)
3.24/1.48	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(11)
3.24/1.48	YES
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(12)
3.24/1.48	Obligation:
3.24/1.48	Q-restricted context-sensitive dependency pair problem:
3.24/1.48	The symbols in {plus_2, s_1, x_2, X_2} are replacing on all positions.
3.24/1.48	For all symbols f in {and_2} we have mu(f) = {1}.
3.24/1.48	
3.24/1.48	The TRS P consists of the following rules:
3.24/1.48	
3.24/1.48	   X(N, s(M)) -> X(N, M)
3.24/1.48	
3.24/1.48	The TRS R consists of the following rules:
3.24/1.48	
3.24/1.48	   and(tt, X) -> X
3.24/1.48	   plus(N, 0) -> N
3.24/1.48	   plus(N, s(M)) -> s(plus(N, M))
3.24/1.48	   x(N, 0) -> 0
3.24/1.48	   x(N, s(M)) -> plus(x(N, M), N)
3.24/1.48	
3.24/1.48	The set Q consists of the following terms:
3.24/1.48	
3.24/1.48	   and(tt, x0)
3.24/1.48	   plus(x0, 0)
3.24/1.48	   plus(x0, s(x1))
3.24/1.48	   x(x0, 0)
3.24/1.48	   x(x0, s(x1))
3.24/1.48	
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(13) QCSDPSubtermProof (EQUIVALENT)
3.24/1.48	We use the subterm processor [DA_EMMES].
3.24/1.48	
3.24/1.48	
3.24/1.48	The following pairs can be oriented strictly and are deleted.
3.24/1.48	
3.24/1.48	   X(N, s(M)) -> X(N, M)
3.24/1.48	The remaining pairs can at least be oriented weakly.
3.24/1.48	none
3.24/1.48	Used ordering:  Combined order from the following AFS and order.
3.24/1.48	X(x1, x2)  =  x2
3.24/1.48	
3.24/1.48	
3.24/1.48	Subterm Order
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(14)
3.24/1.48	Obligation:
3.24/1.48	Q-restricted context-sensitive dependency pair problem:
3.24/1.48	The symbols in {plus_2, s_1, x_2} are replacing on all positions.
3.24/1.48	For all symbols f in {and_2} we have mu(f) = {1}.
3.24/1.48	
3.24/1.48	The TRS P consists of the following rules:
3.24/1.48	none
3.24/1.48	
3.24/1.48	The TRS R consists of the following rules:
3.24/1.48	
3.24/1.48	   and(tt, X) -> X
3.24/1.48	   plus(N, 0) -> N
3.24/1.48	   plus(N, s(M)) -> s(plus(N, M))
3.24/1.48	   x(N, 0) -> 0
3.24/1.48	   x(N, s(M)) -> plus(x(N, M), N)
3.24/1.48	
3.24/1.48	The set Q consists of the following terms:
3.24/1.48	
3.24/1.48	   and(tt, x0)
3.24/1.48	   plus(x0, 0)
3.24/1.48	   plus(x0, s(x1))
3.24/1.48	   x(x0, 0)
3.24/1.48	   x(x0, s(x1))
3.24/1.48	
3.24/1.48	
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(15) PIsEmptyProof (EQUIVALENT)
3.24/1.48	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.24/1.48	----------------------------------------
3.24/1.48	
3.24/1.48	(16)
3.24/1.48	YES
3.24/1.50	EOF