3.60/1.75	YES
3.60/1.75	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.60/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.60/1.75	
3.60/1.75	
3.60/1.75	Termination w.r.t. Q of the given QTRS could be proven:
3.60/1.75	
3.60/1.75	(0) QTRS
3.60/1.75	(1) QTRSToCSRProof [SOUND, 0 ms]
3.60/1.75	(2) CSR
3.60/1.75	(3) CSRInnermostProof [EQUIVALENT, 0 ms]
3.60/1.75	(4) CSR
3.60/1.75	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.60/1.75	(6) QCSDP
3.60/1.75	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.60/1.75	(8) AND
3.60/1.75	    (9) QCSDP
3.60/1.75	        (10) QCSDPSubtermProof [EQUIVALENT, 9 ms]
3.60/1.75	        (11) QCSDP
3.60/1.75	        (12) PIsEmptyProof [EQUIVALENT, 0 ms]
3.60/1.75	        (13) YES
3.60/1.75	    (14) QCSDP
3.60/1.75	        (15) QCSDPSubtermProof [EQUIVALENT, 10 ms]
3.60/1.75	        (16) QCSDP
3.60/1.75	        (17) PIsEmptyProof [EQUIVALENT, 0 ms]
3.60/1.75	        (18) YES
3.60/1.75	
3.60/1.75	
3.60/1.75	----------------------------------------
3.60/1.75	
3.60/1.75	(0)
3.60/1.75	Obligation:
3.60/1.75	Q restricted rewrite system:
3.60/1.75	The TRS R consists of the following rules:
3.60/1.75	
3.60/1.75	   active(and(tt, X)) -> mark(X)
3.60/1.75	   active(plus(N, 0)) -> mark(N)
3.60/1.75	   active(plus(N, s(M))) -> mark(s(plus(N, M)))
3.60/1.75	   active(x(N, 0)) -> mark(0)
3.60/1.75	   active(x(N, s(M))) -> mark(plus(x(N, M), N))
3.60/1.75	   active(and(X1, X2)) -> and(active(X1), X2)
3.60/1.75	   active(plus(X1, X2)) -> plus(active(X1), X2)
3.60/1.75	   active(plus(X1, X2)) -> plus(X1, active(X2))
3.60/1.75	   active(s(X)) -> s(active(X))
3.60/1.75	   active(x(X1, X2)) -> x(active(X1), X2)
3.60/1.75	   active(x(X1, X2)) -> x(X1, active(X2))
3.60/1.75	   and(mark(X1), X2) -> mark(and(X1, X2))
3.60/1.75	   plus(mark(X1), X2) -> mark(plus(X1, X2))
3.60/1.75	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
3.60/1.75	   s(mark(X)) -> mark(s(X))
3.60/1.75	   x(mark(X1), X2) -> mark(x(X1, X2))
3.60/1.75	   x(X1, mark(X2)) -> mark(x(X1, X2))
3.60/1.75	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
3.60/1.75	   proper(tt) -> ok(tt)
3.60/1.75	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
3.60/1.75	   proper(0) -> ok(0)
3.60/1.75	   proper(s(X)) -> s(proper(X))
3.60/1.75	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
3.60/1.75	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
3.60/1.75	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
3.60/1.75	   s(ok(X)) -> ok(s(X))
3.60/1.75	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
3.60/1.75	   top(mark(X)) -> top(proper(X))
3.60/1.75	   top(ok(X)) -> top(active(X))
3.60/1.75	
3.60/1.75	The set Q consists of the following terms:
3.60/1.75	
3.60/1.75	   active(and(x0, x1))
3.60/1.75	   active(plus(x0, x1))
3.60/1.75	   active(s(x0))
3.60/1.75	   active(x(x0, x1))
3.60/1.75	   and(mark(x0), x1)
3.60/1.75	   plus(mark(x0), x1)
3.60/1.75	   plus(x0, mark(x1))
3.60/1.75	   s(mark(x0))
3.60/1.75	   x(mark(x0), x1)
3.60/1.75	   x(x0, mark(x1))
3.60/1.75	   proper(and(x0, x1))
3.60/1.75	   proper(tt)
3.60/1.75	   proper(plus(x0, x1))
3.60/1.75	   proper(0)
3.60/1.75	   proper(s(x0))
3.60/1.75	   proper(x(x0, x1))
3.60/1.75	   and(ok(x0), ok(x1))
3.60/1.75	   plus(ok(x0), ok(x1))
3.60/1.75	   s(ok(x0))
3.60/1.75	   x(ok(x0), ok(x1))
3.60/1.75	   top(mark(x0))
3.60/1.75	   top(ok(x0))
3.60/1.75	
3.60/1.75	
3.60/1.75	----------------------------------------
3.60/1.75	
3.60/1.75	(1) QTRSToCSRProof (SOUND)
3.60/1.75	The following Q TRS is given: Q restricted rewrite system:
3.60/1.75	The TRS R consists of the following rules:
3.60/1.75	
3.60/1.75	   active(and(tt, X)) -> mark(X)
3.60/1.75	   active(plus(N, 0)) -> mark(N)
3.60/1.75	   active(plus(N, s(M))) -> mark(s(plus(N, M)))
3.60/1.75	   active(x(N, 0)) -> mark(0)
3.60/1.75	   active(x(N, s(M))) -> mark(plus(x(N, M), N))
3.60/1.75	   active(and(X1, X2)) -> and(active(X1), X2)
3.60/1.75	   active(plus(X1, X2)) -> plus(active(X1), X2)
3.60/1.75	   active(plus(X1, X2)) -> plus(X1, active(X2))
3.60/1.75	   active(s(X)) -> s(active(X))
3.60/1.75	   active(x(X1, X2)) -> x(active(X1), X2)
3.60/1.75	   active(x(X1, X2)) -> x(X1, active(X2))
3.60/1.76	   and(mark(X1), X2) -> mark(and(X1, X2))
3.60/1.76	   plus(mark(X1), X2) -> mark(plus(X1, X2))
3.60/1.76	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
3.60/1.76	   s(mark(X)) -> mark(s(X))
3.60/1.76	   x(mark(X1), X2) -> mark(x(X1, X2))
3.60/1.76	   x(X1, mark(X2)) -> mark(x(X1, X2))
3.60/1.76	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
3.60/1.76	   proper(tt) -> ok(tt)
3.60/1.76	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
3.60/1.76	   proper(0) -> ok(0)
3.60/1.76	   proper(s(X)) -> s(proper(X))
3.60/1.76	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
3.60/1.76	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
3.60/1.76	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
3.60/1.76	   s(ok(X)) -> ok(s(X))
3.60/1.76	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
3.60/1.76	   top(mark(X)) -> top(proper(X))
3.60/1.76	   top(ok(X)) -> top(active(X))
3.60/1.76	
3.60/1.76	The set Q consists of the following terms:
3.60/1.76	
3.60/1.76	   active(and(x0, x1))
3.60/1.76	   active(plus(x0, x1))
3.60/1.76	   active(s(x0))
3.60/1.76	   active(x(x0, x1))
3.60/1.76	   and(mark(x0), x1)
3.60/1.76	   plus(mark(x0), x1)
3.60/1.76	   plus(x0, mark(x1))
3.60/1.76	   s(mark(x0))
3.60/1.76	   x(mark(x0), x1)
3.60/1.76	   x(x0, mark(x1))
3.60/1.76	   proper(and(x0, x1))
3.60/1.76	   proper(tt)
3.60/1.76	   proper(plus(x0, x1))
3.60/1.76	   proper(0)
3.60/1.76	   proper(s(x0))
3.60/1.76	   proper(x(x0, x1))
3.60/1.76	   and(ok(x0), ok(x1))
3.60/1.76	   plus(ok(x0), ok(x1))
3.60/1.76	   s(ok(x0))
3.60/1.76	   x(ok(x0), ok(x1))
3.60/1.76	   top(mark(x0))
3.60/1.76	   top(ok(x0))
3.60/1.76	
3.60/1.76	Special symbols used for the transformation (see [GM04]):
3.60/1.76	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.60/1.76	The replacement map contains the following entries:
3.60/1.76	
3.60/1.76	and: {1}
3.60/1.76	tt: empty set
3.60/1.76	plus: {1, 2}
3.60/1.76	0: empty set
3.60/1.76	s: {1}
3.60/1.76	x: {1, 2}
3.60/1.76	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(2)
3.60/1.76	Obligation:
3.60/1.76	Context-sensitive rewrite system:
3.60/1.76	The TRS R consists of the following rules:
3.60/1.76	
3.60/1.76	   and(tt, X) -> X
3.60/1.76	   plus(N, 0) -> N
3.60/1.76	   plus(N, s(M)) -> s(plus(N, M))
3.60/1.76	   x(N, 0) -> 0
3.60/1.76	   x(N, s(M)) -> plus(x(N, M), N)
3.60/1.76	
3.60/1.76	The replacement map contains the following entries:
3.60/1.76	
3.60/1.76	and: {1}
3.60/1.76	tt: empty set
3.60/1.76	plus: {1, 2}
3.60/1.76	0: empty set
3.60/1.76	s: {1}
3.60/1.76	x: {1, 2}
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(3) CSRInnermostProof (EQUIVALENT)
3.60/1.76	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(4)
3.60/1.76	Obligation:
3.60/1.76	Context-sensitive rewrite system:
3.60/1.76	The TRS R consists of the following rules:
3.60/1.76	
3.60/1.76	   and(tt, X) -> X
3.60/1.76	   plus(N, 0) -> N
3.60/1.76	   plus(N, s(M)) -> s(plus(N, M))
3.60/1.76	   x(N, 0) -> 0
3.60/1.76	   x(N, s(M)) -> plus(x(N, M), N)
3.60/1.76	
3.60/1.76	The replacement map contains the following entries:
3.60/1.76	
3.60/1.76	and: {1}
3.60/1.76	tt: empty set
3.60/1.76	plus: {1, 2}
3.60/1.76	0: empty set
3.60/1.76	s: {1}
3.60/1.76	x: {1, 2}
3.60/1.76	
3.60/1.76	
3.60/1.76	Innermost Strategy.
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(5) CSDependencyPairsProof (EQUIVALENT)
3.60/1.76	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(6)
3.60/1.76	Obligation:
3.60/1.76	Q-restricted context-sensitive dependency pair problem:
3.60/1.76	The symbols in {plus_2, s_1, x_2, PLUS_2, X_2} are replacing on all positions.
3.60/1.76	For all symbols f in {and_2, AND_2} we have mu(f) = {1}.
3.60/1.76	The symbols in {U_1} are not replacing on any position.
3.60/1.76	
3.60/1.76	The ordinary context-sensitive dependency pairs DP_o are:
3.60/1.76	   PLUS(N, s(M)) -> PLUS(N, M)
3.60/1.76	   X(N, s(M)) -> PLUS(x(N, M), N)
3.60/1.76	   X(N, s(M)) -> X(N, M)
3.60/1.76	
3.60/1.76	The collapsing dependency pairs are DP_c:
3.60/1.76	   AND(tt, X) -> X
3.60/1.76	
3.60/1.76	
3.60/1.76	The hidden terms of R are:
3.60/1.76	none
3.60/1.76	
3.60/1.76	Every hiding context is built from:none
3.60/1.76	
3.60/1.76	Hence, the new unhiding pairs DP_u are :
3.60/1.76	   AND(tt, X) -> U(X)
3.60/1.76	
3.60/1.76	The TRS R consists of the following rules:
3.60/1.76	
3.60/1.76	   and(tt, X) -> X
3.60/1.76	   plus(N, 0) -> N
3.60/1.76	   plus(N, s(M)) -> s(plus(N, M))
3.60/1.76	   x(N, 0) -> 0
3.60/1.76	   x(N, s(M)) -> plus(x(N, M), N)
3.60/1.76	
3.60/1.76	The set Q consists of the following terms:
3.60/1.76	
3.60/1.76	   and(tt, x0)
3.60/1.76	   plus(x0, 0)
3.60/1.76	   plus(x0, s(x1))
3.60/1.76	   x(x0, 0)
3.60/1.76	   x(x0, s(x1))
3.60/1.76	
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(7) QCSDependencyGraphProof (EQUIVALENT)
3.60/1.76	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 1 less node.
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(8)
3.60/1.76	Complex Obligation (AND)
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(9)
3.60/1.76	Obligation:
3.60/1.76	Q-restricted context-sensitive dependency pair problem:
3.60/1.76	The symbols in {plus_2, s_1, x_2, PLUS_2} are replacing on all positions.
3.60/1.76	For all symbols f in {and_2} we have mu(f) = {1}.
3.60/1.76	
3.60/1.76	The TRS P consists of the following rules:
3.60/1.76	
3.60/1.76	   PLUS(N, s(M)) -> PLUS(N, M)
3.60/1.76	
3.60/1.76	The TRS R consists of the following rules:
3.60/1.76	
3.60/1.76	   and(tt, X) -> X
3.60/1.76	   plus(N, 0) -> N
3.60/1.76	   plus(N, s(M)) -> s(plus(N, M))
3.60/1.76	   x(N, 0) -> 0
3.60/1.76	   x(N, s(M)) -> plus(x(N, M), N)
3.60/1.76	
3.60/1.76	The set Q consists of the following terms:
3.60/1.76	
3.60/1.76	   and(tt, x0)
3.60/1.76	   plus(x0, 0)
3.60/1.76	   plus(x0, s(x1))
3.60/1.76	   x(x0, 0)
3.60/1.76	   x(x0, s(x1))
3.60/1.76	
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(10) QCSDPSubtermProof (EQUIVALENT)
3.60/1.76	We use the subterm processor [DA_EMMES].
3.60/1.76	
3.60/1.76	
3.60/1.76	The following pairs can be oriented strictly and are deleted.
3.60/1.76	
3.60/1.76	   PLUS(N, s(M)) -> PLUS(N, M)
3.60/1.76	The remaining pairs can at least be oriented weakly.
3.60/1.76	none
3.60/1.76	Used ordering:  Combined order from the following AFS and order.
3.60/1.76	PLUS(x1, x2)  =  x2
3.60/1.76	
3.60/1.76	
3.60/1.76	Subterm Order
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(11)
3.60/1.76	Obligation:
3.60/1.76	Q-restricted context-sensitive dependency pair problem:
3.60/1.76	The symbols in {plus_2, s_1, x_2} are replacing on all positions.
3.60/1.76	For all symbols f in {and_2} we have mu(f) = {1}.
3.60/1.76	
3.60/1.76	The TRS P consists of the following rules:
3.60/1.76	none
3.60/1.76	
3.60/1.76	The TRS R consists of the following rules:
3.60/1.76	
3.60/1.76	   and(tt, X) -> X
3.60/1.76	   plus(N, 0) -> N
3.60/1.76	   plus(N, s(M)) -> s(plus(N, M))
3.60/1.76	   x(N, 0) -> 0
3.60/1.76	   x(N, s(M)) -> plus(x(N, M), N)
3.60/1.76	
3.60/1.76	The set Q consists of the following terms:
3.60/1.76	
3.60/1.76	   and(tt, x0)
3.60/1.76	   plus(x0, 0)
3.60/1.76	   plus(x0, s(x1))
3.60/1.76	   x(x0, 0)
3.60/1.76	   x(x0, s(x1))
3.60/1.76	
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(12) PIsEmptyProof (EQUIVALENT)
3.60/1.76	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(13)
3.60/1.76	YES
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(14)
3.60/1.76	Obligation:
3.60/1.76	Q-restricted context-sensitive dependency pair problem:
3.60/1.76	The symbols in {plus_2, s_1, x_2, X_2} are replacing on all positions.
3.60/1.76	For all symbols f in {and_2} we have mu(f) = {1}.
3.60/1.76	
3.60/1.76	The TRS P consists of the following rules:
3.60/1.76	
3.60/1.76	   X(N, s(M)) -> X(N, M)
3.60/1.76	
3.60/1.76	The TRS R consists of the following rules:
3.60/1.76	
3.60/1.76	   and(tt, X) -> X
3.60/1.76	   plus(N, 0) -> N
3.60/1.76	   plus(N, s(M)) -> s(plus(N, M))
3.60/1.76	   x(N, 0) -> 0
3.60/1.76	   x(N, s(M)) -> plus(x(N, M), N)
3.60/1.76	
3.60/1.76	The set Q consists of the following terms:
3.60/1.76	
3.60/1.76	   and(tt, x0)
3.60/1.76	   plus(x0, 0)
3.60/1.76	   plus(x0, s(x1))
3.60/1.76	   x(x0, 0)
3.60/1.76	   x(x0, s(x1))
3.60/1.76	
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(15) QCSDPSubtermProof (EQUIVALENT)
3.60/1.76	We use the subterm processor [DA_EMMES].
3.60/1.76	
3.60/1.76	
3.60/1.76	The following pairs can be oriented strictly and are deleted.
3.60/1.76	
3.60/1.76	   X(N, s(M)) -> X(N, M)
3.60/1.76	The remaining pairs can at least be oriented weakly.
3.60/1.76	none
3.60/1.76	Used ordering:  Combined order from the following AFS and order.
3.60/1.76	X(x1, x2)  =  x2
3.60/1.76	
3.60/1.76	
3.60/1.76	Subterm Order
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(16)
3.60/1.76	Obligation:
3.60/1.76	Q-restricted context-sensitive dependency pair problem:
3.60/1.76	The symbols in {plus_2, s_1, x_2} are replacing on all positions.
3.60/1.76	For all symbols f in {and_2} we have mu(f) = {1}.
3.60/1.76	
3.60/1.76	The TRS P consists of the following rules:
3.60/1.76	none
3.60/1.76	
3.60/1.76	The TRS R consists of the following rules:
3.60/1.76	
3.60/1.76	   and(tt, X) -> X
3.60/1.76	   plus(N, 0) -> N
3.60/1.76	   plus(N, s(M)) -> s(plus(N, M))
3.60/1.76	   x(N, 0) -> 0
3.60/1.76	   x(N, s(M)) -> plus(x(N, M), N)
3.60/1.76	
3.60/1.76	The set Q consists of the following terms:
3.60/1.76	
3.60/1.76	   and(tt, x0)
3.60/1.76	   plus(x0, 0)
3.60/1.76	   plus(x0, s(x1))
3.60/1.76	   x(x0, 0)
3.60/1.76	   x(x0, s(x1))
3.60/1.76	
3.60/1.76	
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(17) PIsEmptyProof (EQUIVALENT)
3.60/1.76	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.60/1.76	----------------------------------------
3.60/1.76	
3.60/1.76	(18)
3.60/1.76	YES
3.79/1.77	EOF