3.18/1.62	YES
3.18/1.63	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.18/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.18/1.63	
3.18/1.63	
3.18/1.63	Termination w.r.t. Q of the given QTRS could be proven:
3.18/1.63	
3.18/1.63	(0) QTRS
3.18/1.63	(1) QTRSToCSRProof [SOUND, 0 ms]
3.18/1.63	(2) CSR
3.18/1.63	(3) CSRRRRProof [EQUIVALENT, 56 ms]
3.18/1.63	(4) CSR
3.18/1.63	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.18/1.63	(6) CSR
3.18/1.63	(7) CSRInnermostProof [EQUIVALENT, 0 ms]
3.18/1.63	(8) CSR
3.18/1.63	(9) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.18/1.63	(10) QCSDP
3.18/1.63	(11) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.18/1.63	(12) TRUE
3.18/1.63	
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(0)
3.18/1.63	Obligation:
3.18/1.63	Q restricted rewrite system:
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   active(f(a, b, X)) -> mark(f(X, X, X))
3.18/1.63	   active(c) -> mark(a)
3.18/1.63	   active(c) -> mark(b)
3.18/1.63	   active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
3.18/1.63	   f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
3.18/1.63	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
3.18/1.63	   proper(a) -> ok(a)
3.18/1.63	   proper(b) -> ok(b)
3.18/1.63	   proper(c) -> ok(c)
3.18/1.63	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
3.18/1.63	   top(mark(X)) -> top(proper(X))
3.18/1.63	   top(ok(X)) -> top(active(X))
3.18/1.63	
3.18/1.63	The set Q consists of the following terms:
3.18/1.63	
3.18/1.63	   active(c)
3.18/1.63	   active(f(x0, x1, x2))
3.18/1.63	   f(x0, x1, mark(x2))
3.18/1.63	   proper(f(x0, x1, x2))
3.18/1.63	   proper(a)
3.18/1.63	   proper(b)
3.18/1.63	   proper(c)
3.18/1.63	   f(ok(x0), ok(x1), ok(x2))
3.18/1.63	   top(mark(x0))
3.18/1.63	   top(ok(x0))
3.18/1.63	
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(1) QTRSToCSRProof (SOUND)
3.18/1.63	The following Q TRS is given: Q restricted rewrite system:
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   active(f(a, b, X)) -> mark(f(X, X, X))
3.18/1.63	   active(c) -> mark(a)
3.18/1.63	   active(c) -> mark(b)
3.18/1.63	   active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
3.18/1.63	   f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
3.18/1.63	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
3.18/1.63	   proper(a) -> ok(a)
3.18/1.63	   proper(b) -> ok(b)
3.18/1.63	   proper(c) -> ok(c)
3.18/1.63	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
3.18/1.63	   top(mark(X)) -> top(proper(X))
3.18/1.63	   top(ok(X)) -> top(active(X))
3.18/1.63	
3.18/1.63	The set Q consists of the following terms:
3.18/1.63	
3.18/1.63	   active(c)
3.18/1.63	   active(f(x0, x1, x2))
3.18/1.63	   f(x0, x1, mark(x2))
3.18/1.63	   proper(f(x0, x1, x2))
3.18/1.63	   proper(a)
3.18/1.63	   proper(b)
3.18/1.63	   proper(c)
3.18/1.63	   f(ok(x0), ok(x1), ok(x2))
3.18/1.63	   top(mark(x0))
3.18/1.63	   top(ok(x0))
3.18/1.63	
3.18/1.63	Special symbols used for the transformation (see [GM04]):
3.18/1.63	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.18/1.63	The replacement map contains the following entries:
3.18/1.63	
3.18/1.63	f: {3}
3.18/1.63	a: empty set
3.18/1.63	b: empty set
3.18/1.63	c: empty set
3.18/1.63	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.
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(2)
3.18/1.63	Obligation:
3.18/1.63	Context-sensitive rewrite system:
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   f(a, b, X) -> f(X, X, X)
3.18/1.63	   c -> a
3.18/1.63	   c -> b
3.18/1.63	
3.18/1.63	The replacement map contains the following entries:
3.18/1.63	
3.18/1.63	f: {3}
3.18/1.63	a: empty set
3.18/1.63	b: empty set
3.18/1.63	c: empty set
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(3) CSRRRRProof (EQUIVALENT)
3.18/1.63	The following CSR is given: Context-sensitive rewrite system:
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   f(a, b, X) -> f(X, X, X)
3.18/1.63	   c -> a
3.18/1.63	   c -> b
3.18/1.63	
3.18/1.63	The replacement map contains the following entries:
3.18/1.63	
3.18/1.63	f: {3}
3.18/1.63	a: empty set
3.18/1.63	b: empty set
3.18/1.63	c: empty set
3.18/1.63	Used ordering:
3.18/1.63	Polynomial interpretation [POLO]:
3.18/1.63	
3.18/1.63	   POL(a) = 1
3.18/1.63	   POL(b) = 0
3.18/1.63	   POL(c) = 1
3.18/1.63	   POL(f(x_1, x_2, x_3)) = x_3
3.18/1.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.18/1.63	
3.18/1.63	   c -> b
3.18/1.63	
3.18/1.63	
3.18/1.63	
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(4)
3.18/1.63	Obligation:
3.18/1.63	Context-sensitive rewrite system:
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   f(a, b, X) -> f(X, X, X)
3.18/1.63	   c -> a
3.18/1.63	
3.18/1.63	The replacement map contains the following entries:
3.18/1.63	
3.18/1.63	f: {3}
3.18/1.63	a: empty set
3.18/1.63	b: empty set
3.18/1.63	c: empty set
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(5) CSRRRRProof (EQUIVALENT)
3.18/1.63	The following CSR is given: Context-sensitive rewrite system:
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   f(a, b, X) -> f(X, X, X)
3.18/1.63	   c -> a
3.18/1.63	
3.18/1.63	The replacement map contains the following entries:
3.18/1.63	
3.18/1.63	f: {3}
3.18/1.63	a: empty set
3.18/1.63	b: empty set
3.18/1.63	c: empty set
3.18/1.63	Used ordering:
3.18/1.63	Polynomial interpretation [POLO]:
3.18/1.63	
3.18/1.63	   POL(a) = 0
3.18/1.63	   POL(b) = 0
3.18/1.63	   POL(c) = 1
3.18/1.63	   POL(f(x_1, x_2, x_3)) = x_3
3.18/1.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.18/1.63	
3.18/1.63	   c -> a
3.18/1.63	
3.18/1.63	
3.18/1.63	
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(6)
3.18/1.63	Obligation:
3.18/1.63	Context-sensitive rewrite system:
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   f(a, b, X) -> f(X, X, X)
3.18/1.63	
3.18/1.63	The replacement map contains the following entries:
3.18/1.63	
3.18/1.63	f: {3}
3.18/1.63	a: empty set
3.18/1.63	b: empty set
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(7) CSRInnermostProof (EQUIVALENT)
3.18/1.63	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(8)
3.18/1.63	Obligation:
3.18/1.63	Context-sensitive rewrite system:
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   f(a, b, X) -> f(X, X, X)
3.18/1.63	
3.18/1.63	The replacement map contains the following entries:
3.18/1.63	
3.18/1.63	f: {3}
3.18/1.63	a: empty set
3.18/1.63	b: empty set
3.18/1.63	
3.18/1.63	
3.18/1.63	Innermost Strategy.
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(9) CSDependencyPairsProof (EQUIVALENT)
3.18/1.63	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(10)
3.18/1.63	Obligation:
3.18/1.63	Q-restricted context-sensitive dependency pair problem:
3.18/1.63	For all symbols f in {f_3, F_3} we have mu(f) = {3}.
3.18/1.63	
3.18/1.63	The ordinary context-sensitive dependency pairs DP_o are:
3.18/1.63	   F(a, b, X) -> F(X, X, X)
3.18/1.63	
3.18/1.63	The TRS R consists of the following rules:
3.18/1.63	
3.18/1.63	   f(a, b, X) -> f(X, X, X)
3.18/1.63	
3.18/1.63	The set Q consists of the following terms:
3.18/1.63	
3.18/1.63	   f(a, b, x0)
3.18/1.63	
3.18/1.63	
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(11) QCSDependencyGraphProof (EQUIVALENT)
3.18/1.63	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
3.18/1.63	The rules F(a, b, z0) -> F(z0, z0, z0) and F(a, b, x0) -> F(x0, x0, x0) form no chain, because ECap^mu(F(z0, z0, z0)) = F(z0, z0, z0) does not unify with F(a, b, x0). 
3.18/1.63	----------------------------------------
3.18/1.63	
3.18/1.63	(12)
3.18/1.63	TRUE
3.39/1.65	EOF