3.25/1.73	YES
3.25/1.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.25/1.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.25/1.74	
3.25/1.74	
3.25/1.74	Termination w.r.t. Q of the given QTRS could be proven:
3.25/1.74	
3.25/1.74	(0) QTRS
3.25/1.74	(1) QTRSToCSRProof [SOUND, 0 ms]
3.25/1.74	(2) CSR
3.25/1.74	(3) CSRRRRProof [EQUIVALENT, 46 ms]
3.25/1.74	(4) CSR
3.25/1.74	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.25/1.74	(6) QCSDP
3.25/1.74	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.25/1.74	(8) TRUE
3.25/1.74	
3.25/1.74	
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(0)
3.25/1.74	Obligation:
3.25/1.74	Q restricted rewrite system:
3.25/1.74	The TRS R consists of the following rules:
3.25/1.74	
3.25/1.74	   active(f(X, X)) -> mark(f(a, b))
3.25/1.74	   active(b) -> mark(a)
3.25/1.74	   active(f(X1, X2)) -> f(active(X1), X2)
3.25/1.74	   f(mark(X1), X2) -> mark(f(X1, X2))
3.25/1.74	   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
3.25/1.74	   proper(a) -> ok(a)
3.25/1.74	   proper(b) -> ok(b)
3.25/1.74	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
3.25/1.74	   top(mark(X)) -> top(proper(X))
3.25/1.74	   top(ok(X)) -> top(active(X))
3.25/1.74	
3.25/1.74	The set Q consists of the following terms:
3.25/1.74	
3.25/1.74	   active(b)
3.25/1.74	   active(f(x0, x1))
3.25/1.74	   f(mark(x0), x1)
3.25/1.74	   proper(f(x0, x1))
3.25/1.74	   proper(a)
3.25/1.74	   proper(b)
3.25/1.74	   f(ok(x0), ok(x1))
3.25/1.74	   top(mark(x0))
3.25/1.74	   top(ok(x0))
3.25/1.74	
3.25/1.74	
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(1) QTRSToCSRProof (SOUND)
3.25/1.74	The following Q TRS is given: Q restricted rewrite system:
3.25/1.74	The TRS R consists of the following rules:
3.25/1.74	
3.25/1.74	   active(f(X, X)) -> mark(f(a, b))
3.25/1.74	   active(b) -> mark(a)
3.25/1.74	   active(f(X1, X2)) -> f(active(X1), X2)
3.25/1.74	   f(mark(X1), X2) -> mark(f(X1, X2))
3.25/1.74	   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
3.25/1.74	   proper(a) -> ok(a)
3.25/1.74	   proper(b) -> ok(b)
3.25/1.74	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
3.25/1.74	   top(mark(X)) -> top(proper(X))
3.25/1.74	   top(ok(X)) -> top(active(X))
3.25/1.74	
3.25/1.74	The set Q consists of the following terms:
3.25/1.74	
3.25/1.74	   active(b)
3.25/1.74	   active(f(x0, x1))
3.25/1.74	   f(mark(x0), x1)
3.25/1.74	   proper(f(x0, x1))
3.25/1.74	   proper(a)
3.25/1.74	   proper(b)
3.25/1.74	   f(ok(x0), ok(x1))
3.25/1.74	   top(mark(x0))
3.25/1.74	   top(ok(x0))
3.25/1.74	
3.25/1.74	Special symbols used for the transformation (see [GM04]):
3.25/1.74	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.25/1.74	The replacement map contains the following entries:
3.25/1.74	
3.25/1.74	f: {1}
3.25/1.74	a: empty set
3.25/1.74	b: empty set
3.25/1.74	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.
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(2)
3.25/1.74	Obligation:
3.25/1.74	Context-sensitive rewrite system:
3.25/1.74	The TRS R consists of the following rules:
3.25/1.74	
3.25/1.74	   f(X, X) -> f(a, b)
3.25/1.74	   b -> a
3.25/1.74	
3.25/1.74	The replacement map contains the following entries:
3.25/1.74	
3.25/1.74	f: {1}
3.25/1.74	a: empty set
3.25/1.74	b: empty set
3.25/1.74	
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(3) CSRRRRProof (EQUIVALENT)
3.25/1.74	The following CSR is given: Context-sensitive rewrite system:
3.25/1.74	The TRS R consists of the following rules:
3.25/1.74	
3.25/1.74	   f(X, X) -> f(a, b)
3.25/1.74	   b -> a
3.25/1.74	
3.25/1.74	The replacement map contains the following entries:
3.25/1.74	
3.25/1.74	f: {1}
3.25/1.74	a: empty set
3.25/1.74	b: empty set
3.25/1.74	Used ordering:
3.25/1.74	Polynomial interpretation [POLO]:
3.25/1.74	
3.25/1.74	   POL(a) = 0
3.25/1.74	   POL(b) = 2
3.25/1.74	   POL(f(x_1, x_2)) = x_1
3.25/1.74	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.25/1.74	
3.25/1.74	   b -> a
3.25/1.74	
3.25/1.74	
3.25/1.74	
3.25/1.74	
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(4)
3.25/1.74	Obligation:
3.25/1.74	Context-sensitive rewrite system:
3.25/1.74	The TRS R consists of the following rules:
3.25/1.74	
3.25/1.74	   f(X, X) -> f(a, b)
3.25/1.74	
3.25/1.74	The replacement map contains the following entries:
3.25/1.74	
3.25/1.74	f: {1}
3.25/1.74	a: empty set
3.25/1.74	b: empty set
3.25/1.74	
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(5) CSDependencyPairsProof (EQUIVALENT)
3.25/1.74	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(6)
3.25/1.74	Obligation:
3.25/1.74	Q-restricted context-sensitive dependency pair problem:
3.25/1.74	For all symbols f in {f_2, F_2} we have mu(f) = {1}.
3.25/1.74	
3.25/1.74	The ordinary context-sensitive dependency pairs DP_o are:
3.25/1.74	   F(X, X) -> F(a, b)
3.25/1.74	
3.25/1.74	The TRS R consists of the following rules:
3.25/1.74	
3.25/1.74	   f(X, X) -> f(a, b)
3.25/1.74	
3.25/1.74	Q is empty.
3.25/1.74	
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(7) QCSDependencyGraphProof (EQUIVALENT)
3.25/1.74	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
3.25/1.74	The rules F(z0, z0) -> F(a, b) and F(x0, x0) -> F(a, b) form no chain, because ECap^mu(F(a, b)) = F(a, b) does not unify with F(x0, x0). 
3.25/1.74	----------------------------------------
3.25/1.74	
3.25/1.74	(8)
3.25/1.74	TRUE
3.33/1.76	EOF