3.13/1.60	YES
3.13/1.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.13/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.13/1.61	
3.13/1.61	
3.13/1.61	Termination w.r.t. Q of the given QTRS could be proven:
3.13/1.61	
3.13/1.61	(0) QTRS
3.13/1.61	(1) QTRSToCSRProof [SOUND, 0 ms]
3.13/1.61	(2) CSR
3.13/1.61	(3) CSRRRRProof [EQUIVALENT, 60 ms]
3.13/1.61	(4) CSR
3.13/1.61	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.13/1.61	(6) QCSDP
3.13/1.61	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.13/1.61	(8) TRUE
3.13/1.61	
3.13/1.61	
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(0)
3.13/1.61	Obligation:
3.13/1.61	Q restricted rewrite system:
3.13/1.61	The TRS R consists of the following rules:
3.13/1.61	
3.13/1.61	   active(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
3.13/1.61	   active(g(b)) -> mark(c)
3.13/1.61	   active(b) -> mark(c)
3.13/1.61	   active(g(X)) -> g(active(X))
3.13/1.61	   g(mark(X)) -> mark(g(X))
3.13/1.61	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
3.13/1.61	   proper(g(X)) -> g(proper(X))
3.13/1.61	   proper(b) -> ok(b)
3.13/1.61	   proper(c) -> ok(c)
3.13/1.61	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
3.13/1.61	   g(ok(X)) -> ok(g(X))
3.13/1.61	   top(mark(X)) -> top(proper(X))
3.13/1.61	   top(ok(X)) -> top(active(X))
3.13/1.61	
3.13/1.61	The set Q consists of the following terms:
3.13/1.61	
3.13/1.61	   active(f(x0, g(x0), x1))
3.13/1.61	   active(b)
3.13/1.61	   active(g(x0))
3.13/1.61	   g(mark(x0))
3.13/1.61	   proper(f(x0, x1, x2))
3.13/1.61	   proper(g(x0))
3.13/1.61	   proper(b)
3.13/1.61	   proper(c)
3.13/1.61	   f(ok(x0), ok(x1), ok(x2))
3.13/1.61	   g(ok(x0))
3.13/1.61	   top(mark(x0))
3.13/1.61	   top(ok(x0))
3.13/1.61	
3.13/1.61	
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(1) QTRSToCSRProof (SOUND)
3.13/1.61	The following Q TRS is given: Q restricted rewrite system:
3.13/1.61	The TRS R consists of the following rules:
3.13/1.61	
3.13/1.61	   active(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
3.13/1.61	   active(g(b)) -> mark(c)
3.13/1.61	   active(b) -> mark(c)
3.13/1.61	   active(g(X)) -> g(active(X))
3.13/1.61	   g(mark(X)) -> mark(g(X))
3.13/1.61	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
3.13/1.61	   proper(g(X)) -> g(proper(X))
3.13/1.61	   proper(b) -> ok(b)
3.13/1.61	   proper(c) -> ok(c)
3.13/1.61	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
3.13/1.61	   g(ok(X)) -> ok(g(X))
3.13/1.61	   top(mark(X)) -> top(proper(X))
3.13/1.61	   top(ok(X)) -> top(active(X))
3.13/1.61	
3.13/1.61	The set Q consists of the following terms:
3.13/1.61	
3.13/1.61	   active(f(x0, g(x0), x1))
3.13/1.61	   active(b)
3.13/1.61	   active(g(x0))
3.13/1.61	   g(mark(x0))
3.13/1.61	   proper(f(x0, x1, x2))
3.13/1.61	   proper(g(x0))
3.13/1.61	   proper(b)
3.13/1.61	   proper(c)
3.13/1.61	   f(ok(x0), ok(x1), ok(x2))
3.13/1.61	   g(ok(x0))
3.13/1.61	   top(mark(x0))
3.13/1.61	   top(ok(x0))
3.13/1.61	
3.13/1.61	Special symbols used for the transformation (see [GM04]):
3.13/1.61	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.13/1.61	The replacement map contains the following entries:
3.13/1.61	
3.13/1.61	f: empty set
3.13/1.61	g: {1}
3.13/1.61	b: empty set
3.13/1.61	c: empty set
3.13/1.61	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.
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(2)
3.13/1.61	Obligation:
3.13/1.61	Context-sensitive rewrite system:
3.13/1.61	The TRS R consists of the following rules:
3.13/1.61	
3.13/1.61	   f(X, g(X), Y) -> f(Y, Y, Y)
3.13/1.61	   g(b) -> c
3.13/1.61	   b -> c
3.13/1.61	
3.13/1.61	The replacement map contains the following entries:
3.13/1.61	
3.13/1.61	f: empty set
3.13/1.61	g: {1}
3.13/1.61	b: empty set
3.13/1.61	c: empty set
3.13/1.61	
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(3) CSRRRRProof (EQUIVALENT)
3.13/1.61	The following CSR is given: Context-sensitive rewrite system:
3.13/1.61	The TRS R consists of the following rules:
3.13/1.61	
3.13/1.61	   f(X, g(X), Y) -> f(Y, Y, Y)
3.13/1.61	   g(b) -> c
3.13/1.61	   b -> c
3.13/1.61	
3.13/1.61	The replacement map contains the following entries:
3.13/1.61	
3.13/1.61	f: empty set
3.13/1.61	g: {1}
3.13/1.61	b: empty set
3.13/1.61	c: empty set
3.13/1.61	Used ordering:
3.13/1.61	Polynomial interpretation [POLO]:
3.13/1.61	
3.13/1.61	   POL(b) = 2
3.13/1.61	   POL(c) = 1
3.13/1.61	   POL(f(x_1, x_2, x_3)) = 0
3.13/1.61	   POL(g(x_1)) = 2 + 2*x_1
3.13/1.61	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.13/1.61	
3.13/1.61	   g(b) -> c
3.13/1.61	   b -> c
3.13/1.61	
3.13/1.61	
3.13/1.61	
3.13/1.61	
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(4)
3.13/1.61	Obligation:
3.13/1.61	Context-sensitive rewrite system:
3.13/1.61	The TRS R consists of the following rules:
3.13/1.61	
3.13/1.61	   f(X, g(X), Y) -> f(Y, Y, Y)
3.13/1.61	
3.13/1.61	The replacement map contains the following entries:
3.13/1.61	
3.13/1.61	f: empty set
3.13/1.61	g: {1}
3.13/1.61	
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(5) CSDependencyPairsProof (EQUIVALENT)
3.13/1.61	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(6)
3.13/1.61	Obligation:
3.13/1.61	Q-restricted context-sensitive dependency pair problem:
3.13/1.61	The symbols in {g_1} are replacing on all positions.
3.13/1.61	The symbols in {f_3, F_3} are not replacing on any position.
3.13/1.61	
3.13/1.61	The ordinary context-sensitive dependency pairs DP_o are:
3.13/1.61	   F(X, g(X), Y) -> F(Y, Y, Y)
3.13/1.61	
3.13/1.61	The TRS R consists of the following rules:
3.13/1.61	
3.13/1.61	   f(X, g(X), Y) -> f(Y, Y, Y)
3.13/1.61	
3.13/1.61	Q is empty.
3.13/1.61	
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(7) QCSDependencyGraphProof (EQUIVALENT)
3.13/1.61	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
3.13/1.61	The rules F(z0, g(z0), z1) -> F(z1, z1, z1) and F(x0, g(x0), x1) -> F(x1, x1, x1) form no chain, because ECap^mu(F(z1, z1, z1)) = F(z1, z1, z1) does not unify with F(x0, g(x0), x1). 
3.13/1.61	----------------------------------------
3.13/1.61	
3.13/1.61	(8)
3.13/1.61	TRUE
3.31/1.64	EOF