4.80/3.26	YES
4.99/3.27	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.99/3.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.99/3.27	
4.99/3.27	
4.99/3.27	Termination w.r.t. Q of the given QTRS could be proven:
4.99/3.27	
4.99/3.27	(0) QTRS
4.99/3.27	(1) QTRSToCSRProof [SOUND, 0 ms]
4.99/3.27	(2) CSR
4.99/3.27	(3) CSRRRRProof [EQUIVALENT, 49 ms]
4.99/3.27	(4) CSR
4.99/3.27	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
4.99/3.27	(6) QCSDP
4.99/3.27	(7) QCSDPInstantiationProcessor [EQUIVALENT, 0 ms]
4.99/3.27	(8) QCSDP
4.99/3.27	(9) QCSUsableRulesProof [EQUIVALENT, 0 ms]
4.99/3.27	(10) QCSDP
4.99/3.27	(11) QCSDPInstantiationProcessor [EQUIVALENT, 0 ms]
4.99/3.27	(12) QCSDP
4.99/3.27	(13) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.99/3.27	(14) TRUE
4.99/3.27	
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(0)
4.99/3.27	Obligation:
4.99/3.27	Q restricted rewrite system:
4.99/3.27	The TRS R consists of the following rules:
4.99/3.27	
4.99/3.27	   active(h(X)) -> mark(g(X, X))
4.99/3.27	   active(g(a, X)) -> mark(f(b, X))
4.99/3.27	   active(f(X, X)) -> mark(h(a))
4.99/3.27	   active(a) -> mark(b)
4.99/3.27	   active(h(X)) -> h(active(X))
4.99/3.27	   active(g(X1, X2)) -> g(active(X1), X2)
4.99/3.27	   active(f(X1, X2)) -> f(active(X1), X2)
4.99/3.27	   h(mark(X)) -> mark(h(X))
4.99/3.27	   g(mark(X1), X2) -> mark(g(X1, X2))
4.99/3.27	   f(mark(X1), X2) -> mark(f(X1, X2))
4.99/3.27	   proper(h(X)) -> h(proper(X))
4.99/3.27	   proper(g(X1, X2)) -> g(proper(X1), proper(X2))
4.99/3.27	   proper(a) -> ok(a)
4.99/3.27	   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
4.99/3.27	   proper(b) -> ok(b)
4.99/3.27	   h(ok(X)) -> ok(h(X))
4.99/3.27	   g(ok(X1), ok(X2)) -> ok(g(X1, X2))
4.99/3.27	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
4.99/3.27	   top(mark(X)) -> top(proper(X))
4.99/3.27	   top(ok(X)) -> top(active(X))
4.99/3.27	
4.99/3.27	The set Q consists of the following terms:
4.99/3.27	
4.99/3.27	   active(h(x0))
4.99/3.27	   active(a)
4.99/3.27	   active(g(x0, x1))
4.99/3.27	   active(f(x0, x1))
4.99/3.27	   h(mark(x0))
4.99/3.27	   g(mark(x0), x1)
4.99/3.27	   f(mark(x0), x1)
4.99/3.27	   proper(h(x0))
4.99/3.27	   proper(g(x0, x1))
4.99/3.27	   proper(a)
4.99/3.27	   proper(f(x0, x1))
4.99/3.27	   proper(b)
4.99/3.27	   h(ok(x0))
4.99/3.27	   g(ok(x0), ok(x1))
4.99/3.27	   f(ok(x0), ok(x1))
4.99/3.27	   top(mark(x0))
4.99/3.27	   top(ok(x0))
4.99/3.27	
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(1) QTRSToCSRProof (SOUND)
4.99/3.27	The following Q TRS is given: Q restricted rewrite system:
4.99/3.27	The TRS R consists of the following rules:
4.99/3.27	
4.99/3.27	   active(h(X)) -> mark(g(X, X))
4.99/3.27	   active(g(a, X)) -> mark(f(b, X))
4.99/3.27	   active(f(X, X)) -> mark(h(a))
4.99/3.27	   active(a) -> mark(b)
4.99/3.27	   active(h(X)) -> h(active(X))
4.99/3.27	   active(g(X1, X2)) -> g(active(X1), X2)
4.99/3.27	   active(f(X1, X2)) -> f(active(X1), X2)
4.99/3.27	   h(mark(X)) -> mark(h(X))
4.99/3.27	   g(mark(X1), X2) -> mark(g(X1, X2))
4.99/3.27	   f(mark(X1), X2) -> mark(f(X1, X2))
4.99/3.27	   proper(h(X)) -> h(proper(X))
4.99/3.27	   proper(g(X1, X2)) -> g(proper(X1), proper(X2))
4.99/3.27	   proper(a) -> ok(a)
4.99/3.27	   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
4.99/3.27	   proper(b) -> ok(b)
4.99/3.27	   h(ok(X)) -> ok(h(X))
4.99/3.27	   g(ok(X1), ok(X2)) -> ok(g(X1, X2))
4.99/3.27	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
4.99/3.27	   top(mark(X)) -> top(proper(X))
4.99/3.27	   top(ok(X)) -> top(active(X))
4.99/3.27	
4.99/3.27	The set Q consists of the following terms:
4.99/3.27	
4.99/3.27	   active(h(x0))
4.99/3.27	   active(a)
4.99/3.27	   active(g(x0, x1))
4.99/3.27	   active(f(x0, x1))
4.99/3.27	   h(mark(x0))
4.99/3.27	   g(mark(x0), x1)
4.99/3.27	   f(mark(x0), x1)
4.99/3.27	   proper(h(x0))
4.99/3.27	   proper(g(x0, x1))
4.99/3.27	   proper(a)
4.99/3.27	   proper(f(x0, x1))
4.99/3.27	   proper(b)
4.99/3.27	   h(ok(x0))
4.99/3.27	   g(ok(x0), ok(x1))
4.99/3.27	   f(ok(x0), ok(x1))
4.99/3.27	   top(mark(x0))
4.99/3.27	   top(ok(x0))
4.99/3.27	
4.99/3.27	Special symbols used for the transformation (see [GM04]):
4.99/3.27	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.99/3.27	The replacement map contains the following entries:
4.99/3.27	
4.99/3.27	h: {1}
4.99/3.27	g: {1}
4.99/3.27	a: empty set
4.99/3.27	f: {1}
4.99/3.27	b: empty set
4.99/3.27	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.
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(2)
4.99/3.27	Obligation:
4.99/3.27	Context-sensitive rewrite system:
4.99/3.27	The TRS R consists of the following rules:
4.99/3.27	
4.99/3.27	   h(X) -> g(X, X)
4.99/3.27	   g(a, X) -> f(b, X)
4.99/3.27	   f(X, X) -> h(a)
4.99/3.27	   a -> b
4.99/3.27	
4.99/3.27	The replacement map contains the following entries:
4.99/3.27	
4.99/3.27	h: {1}
4.99/3.27	g: {1}
4.99/3.27	a: empty set
4.99/3.27	f: {1}
4.99/3.27	b: empty set
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(3) CSRRRRProof (EQUIVALENT)
4.99/3.27	The following CSR is given: Context-sensitive rewrite system:
4.99/3.27	The TRS R consists of the following rules:
4.99/3.27	
4.99/3.27	   h(X) -> g(X, X)
4.99/3.27	   g(a, X) -> f(b, X)
4.99/3.27	   f(X, X) -> h(a)
4.99/3.27	   a -> b
4.99/3.27	
4.99/3.27	The replacement map contains the following entries:
4.99/3.27	
4.99/3.27	h: {1}
4.99/3.27	g: {1}
4.99/3.27	a: empty set
4.99/3.27	f: {1}
4.99/3.27	b: empty set
4.99/3.27	Used ordering:
4.99/3.27	Polynomial interpretation [POLO]:
4.99/3.27	
4.99/3.27	   POL(a) = 1
4.99/3.27	   POL(b) = 0
4.99/3.27	   POL(f(x_1, x_2)) = 2 + 2*x_1
4.99/3.27	   POL(g(x_1, x_2)) = 2*x_1
4.99/3.27	   POL(h(x_1)) = 2*x_1
4.99/3.27	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.99/3.27	
4.99/3.27	   a -> b
4.99/3.27	
4.99/3.27	
4.99/3.27	
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(4)
4.99/3.27	Obligation:
4.99/3.27	Context-sensitive rewrite system:
4.99/3.27	The TRS R consists of the following rules:
4.99/3.27	
4.99/3.27	   h(X) -> g(X, X)
4.99/3.27	   g(a, X) -> f(b, X)
4.99/3.27	   f(X, X) -> h(a)
4.99/3.27	
4.99/3.27	The replacement map contains the following entries:
4.99/3.27	
4.99/3.27	h: {1}
4.99/3.27	g: {1}
4.99/3.27	a: empty set
4.99/3.27	f: {1}
4.99/3.27	b: empty set
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(5) CSDependencyPairsProof (EQUIVALENT)
4.99/3.27	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(6)
4.99/3.27	Obligation:
4.99/3.27	Q-restricted context-sensitive dependency pair problem:
4.99/3.27	The symbols in {h_1, H_1} are replacing on all positions.
4.99/3.27	For all symbols f in {g_2, f_2, G_2, F_2} we have mu(f) = {1}.
4.99/3.27	
4.99/3.27	The ordinary context-sensitive dependency pairs DP_o are:
4.99/3.27	   H(X) -> G(X, X)
4.99/3.27	   G(a, X) -> F(b, X)
4.99/3.27	   F(X, X) -> H(a)
4.99/3.27	
4.99/3.27	The TRS R consists of the following rules:
4.99/3.27	
4.99/3.27	   h(X) -> g(X, X)
4.99/3.27	   g(a, X) -> f(b, X)
4.99/3.27	   f(X, X) -> h(a)
4.99/3.27	
4.99/3.27	Q is empty.
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(7) QCSDPInstantiationProcessor (EQUIVALENT)
4.99/3.27	Using the Context-Sensitive Instantiation[LPAR08,DA_EMMES] Processor
4.99/3.27	
4.99/3.27	the pair H(X) -> G(X, X)
4.99/3.27	
4.99/3.27	was transformed to the following new pairs: 
4.99/3.27	   H(a) -> G(a, a)
4.99/3.27	
4.99/3.27	
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(8)
4.99/3.27	Obligation:
4.99/3.27	Q-restricted context-sensitive dependency pair problem:
4.99/3.27	The symbols in {h_1, H_1} are replacing on all positions.
4.99/3.27	For all symbols f in {g_2, f_2, F_2, G_2} we have mu(f) = {1}.
4.99/3.27	
4.99/3.27	The TRS P consists of the following rules:
4.99/3.27	
4.99/3.27	   G(a, X) -> F(b, X)
4.99/3.27	   F(X, X) -> H(a)
4.99/3.27	   H(a) -> G(a, a)
4.99/3.27	
4.99/3.27	The TRS R consists of the following rules:
4.99/3.27	
4.99/3.27	   h(X) -> g(X, X)
4.99/3.27	   g(a, X) -> f(b, X)
4.99/3.27	   f(X, X) -> h(a)
4.99/3.27	
4.99/3.27	Q is empty.
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(9) QCSUsableRulesProof (EQUIVALENT)
4.99/3.27	The following rules are not useable [DA_EMMES] and can be deleted:
4.99/3.27	
4.99/3.27	   h(x0) -> g(x0, x0)
4.99/3.27	   g(a, x0) -> f(b, x0)
4.99/3.27	   f(x0, x0) -> h(a)
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(10)
4.99/3.27	Obligation:
4.99/3.27	Q-restricted context-sensitive dependency pair problem:
4.99/3.27	The symbols in {H_1} are replacing on all positions.
4.99/3.27	For all symbols f in {F_2, G_2} we have mu(f) = {1}.
4.99/3.27	
4.99/3.27	The TRS P consists of the following rules:
4.99/3.27	
4.99/3.27	   G(a, X) -> F(b, X)
4.99/3.27	   F(X, X) -> H(a)
4.99/3.27	   H(a) -> G(a, a)
4.99/3.27	
4.99/3.27	R is empty.
4.99/3.27	Q is empty.
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(11) QCSDPInstantiationProcessor (EQUIVALENT)
4.99/3.27	Using the Context-Sensitive Instantiation[LPAR08,DA_EMMES] Processor
4.99/3.27	
4.99/3.27	the pair G(a, X) -> F(b, X)
4.99/3.27	
4.99/3.27	was transformed to the following new pairs: 
4.99/3.27	   G(a, a) -> F(b, a)
4.99/3.27	
4.99/3.27	
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(12)
4.99/3.27	Obligation:
4.99/3.27	Q-restricted context-sensitive dependency pair problem:
4.99/3.27	The symbols in {H_1} are replacing on all positions.
4.99/3.27	For all symbols f in {F_2, G_2} we have mu(f) = {1}.
4.99/3.27	
4.99/3.27	The TRS P consists of the following rules:
4.99/3.27	
4.99/3.27	   F(X, X) -> H(a)
4.99/3.27	   H(a) -> G(a, a)
4.99/3.27	   G(a, a) -> F(b, a)
4.99/3.27	
4.99/3.27	R is empty.
4.99/3.27	Q is empty.
4.99/3.27	
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(13) QCSDependencyGraphProof (EQUIVALENT)
4.99/3.27	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes.
4.99/3.27	The rules G(a, a) -> F(b, a) and F(x0, x0) -> H(a) form no chain, because ECap^mu(F(b, a)) = F(b, a) does not unify with F(x0, x0). 
4.99/3.27	----------------------------------------
4.99/3.27	
4.99/3.27	(14)
4.99/3.27	TRUE
4.99/3.30	EOF