3.61/1.88	YES
3.61/1.88	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.61/1.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.61/1.88	
3.61/1.88	
3.61/1.88	Termination w.r.t. Q of the given QTRS could be proven:
3.61/1.88	
3.61/1.88	(0) QTRS
3.61/1.88	(1) QTRSToCSRProof [SOUND, 0 ms]
3.61/1.88	(2) CSR
3.61/1.88	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.61/1.88	(4) QCSDP
3.61/1.88	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.61/1.88	(6) QCSDP
3.61/1.88	(7) QCSUsableRulesProof [EQUIVALENT, 0 ms]
3.61/1.88	(8) QCSDP
3.61/1.88	(9) QCSDPInstantiationProcessor [EQUIVALENT, 0 ms]
3.61/1.88	(10) QCSDP
3.61/1.88	(11) PIsEmptyProof [EQUIVALENT, 0 ms]
3.61/1.88	(12) YES
3.61/1.88	
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(0)
3.61/1.88	Obligation:
3.61/1.88	Q restricted rewrite system:
3.61/1.88	The TRS R consists of the following rules:
3.61/1.88	
3.61/1.88	   active(f(a, X, X)) -> mark(f(X, b, b))
3.61/1.88	   active(b) -> mark(a)
3.61/1.88	   active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
3.61/1.88	   f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
3.61/1.88	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
3.61/1.88	   proper(a) -> ok(a)
3.61/1.88	   proper(b) -> ok(b)
3.61/1.88	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
3.61/1.88	   top(mark(X)) -> top(proper(X))
3.61/1.88	   top(ok(X)) -> top(active(X))
3.61/1.88	
3.61/1.88	The set Q consists of the following terms:
3.61/1.88	
3.61/1.88	   active(b)
3.61/1.88	   active(f(x0, x1, x2))
3.61/1.88	   f(x0, mark(x1), x2)
3.61/1.88	   proper(f(x0, x1, x2))
3.61/1.88	   proper(a)
3.61/1.88	   proper(b)
3.61/1.88	   f(ok(x0), ok(x1), ok(x2))
3.61/1.88	   top(mark(x0))
3.61/1.88	   top(ok(x0))
3.61/1.88	
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(1) QTRSToCSRProof (SOUND)
3.61/1.88	The following Q TRS is given: Q restricted rewrite system:
3.61/1.88	The TRS R consists of the following rules:
3.61/1.88	
3.61/1.88	   active(f(a, X, X)) -> mark(f(X, b, b))
3.61/1.88	   active(b) -> mark(a)
3.61/1.88	   active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
3.61/1.88	   f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
3.61/1.88	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
3.61/1.88	   proper(a) -> ok(a)
3.61/1.88	   proper(b) -> ok(b)
3.61/1.88	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
3.61/1.88	   top(mark(X)) -> top(proper(X))
3.61/1.88	   top(ok(X)) -> top(active(X))
3.61/1.88	
3.61/1.88	The set Q consists of the following terms:
3.61/1.88	
3.61/1.88	   active(b)
3.61/1.88	   active(f(x0, x1, x2))
3.61/1.88	   f(x0, mark(x1), x2)
3.61/1.88	   proper(f(x0, x1, x2))
3.61/1.88	   proper(a)
3.61/1.88	   proper(b)
3.61/1.88	   f(ok(x0), ok(x1), ok(x2))
3.61/1.88	   top(mark(x0))
3.61/1.88	   top(ok(x0))
3.61/1.88	
3.61/1.88	Special symbols used for the transformation (see [GM04]):
3.61/1.88	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.61/1.88	The replacement map contains the following entries:
3.61/1.88	
3.61/1.88	f: {2}
3.61/1.88	a: empty set
3.61/1.88	b: empty set
3.61/1.88	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.
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(2)
3.61/1.88	Obligation:
3.61/1.88	Context-sensitive rewrite system:
3.61/1.88	The TRS R consists of the following rules:
3.61/1.88	
3.61/1.88	   f(a, X, X) -> f(X, b, b)
3.61/1.88	   b -> a
3.61/1.88	
3.61/1.88	The replacement map contains the following entries:
3.61/1.88	
3.61/1.88	f: {2}
3.61/1.88	a: empty set
3.61/1.88	b: empty set
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(3) CSDependencyPairsProof (EQUIVALENT)
3.61/1.88	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(4)
3.61/1.88	Obligation:
3.61/1.88	Q-restricted context-sensitive dependency pair problem:
3.61/1.88	For all symbols f in {f_3, F_3} we have mu(f) = {2}.
3.61/1.88	
3.61/1.88	The ordinary context-sensitive dependency pairs DP_o are:
3.61/1.88	   F(a, X, X) -> F(X, b, b)
3.61/1.88	   F(a, X, X) -> B
3.61/1.88	
3.61/1.88	The TRS R consists of the following rules:
3.61/1.88	
3.61/1.88	   f(a, X, X) -> f(X, b, b)
3.61/1.88	   b -> a
3.61/1.88	
3.61/1.88	Q is empty.
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(5) QCSDependencyGraphProof (EQUIVALENT)
3.61/1.88	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 1 less node.
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(6)
3.61/1.88	Obligation:
3.61/1.88	Q-restricted context-sensitive dependency pair problem:
3.61/1.88	For all symbols f in {f_3, F_3} we have mu(f) = {2}.
3.61/1.88	
3.61/1.88	The TRS P consists of the following rules:
3.61/1.88	
3.61/1.88	   F(a, X, X) -> F(X, b, b)
3.61/1.88	
3.61/1.88	The TRS R consists of the following rules:
3.61/1.88	
3.61/1.88	   f(a, X, X) -> f(X, b, b)
3.61/1.88	   b -> a
3.61/1.88	
3.61/1.88	Q is empty.
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(7) QCSUsableRulesProof (EQUIVALENT)
3.61/1.88	The following rules are not useable [DA_EMMES] and can be deleted:
3.61/1.88	
3.61/1.88	   f(a, x0, x0) -> f(x0, b, b)
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(8)
3.61/1.88	Obligation:
3.61/1.88	Q-restricted context-sensitive dependency pair problem:
3.61/1.88	For all symbols f in {F_3} we have mu(f) = {2}.
3.61/1.88	
3.61/1.88	The TRS P consists of the following rules:
3.61/1.88	
3.61/1.88	   F(a, X, X) -> F(X, b, b)
3.61/1.88	
3.61/1.88	The TRS R consists of the following rules:
3.61/1.88	
3.61/1.88	   b -> a
3.61/1.88	
3.61/1.88	Q is empty.
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(9) QCSDPInstantiationProcessor (EQUIVALENT)
3.61/1.88	Using the Context-Sensitive Instantiation[LPAR08,DA_EMMES] Processor
3.61/1.88	
3.61/1.88	the pair F(a, X, X) -> F(X, b, b)
3.61/1.88	
3.61/1.88	was transformed to the following new pairs: 
3.61/1.88	   F(a, b, b) -> F(b, b, b)
3.61/1.88	
3.61/1.88	
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(10)
3.61/1.88	Obligation:
3.61/1.88	Q-restricted context-sensitive dependency pair problem:
3.61/1.88	
3.61/1.88	The TRS P consists of the following rules:
3.61/1.88	none
3.61/1.88	
3.61/1.88	The TRS R consists of the following rules:
3.61/1.88	
3.61/1.88	   b -> a
3.61/1.88	
3.61/1.88	Q is empty.
3.61/1.88	
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(11) PIsEmptyProof (EQUIVALENT)
3.61/1.88	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.61/1.88	----------------------------------------
3.61/1.88	
3.61/1.88	(12)
3.61/1.88	YES
3.61/1.91	EOF