3.98/1.86	YES
3.98/1.87	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.98/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.98/1.87	
3.98/1.87	
3.98/1.87	Termination w.r.t. Q of the given QTRS could be proven:
3.98/1.87	
3.98/1.87	(0) QTRS
3.98/1.87	(1) QTRSToCSRProof [SOUND, 0 ms]
3.98/1.87	(2) CSR
3.98/1.87	(3) CSRRRRProof [EQUIVALENT, 32 ms]
3.98/1.87	(4) CSR
3.98/1.87	(5) CSRRRRProof [EQUIVALENT, 18 ms]
3.98/1.87	(6) CSR
3.98/1.87	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.98/1.87	(8) CSR
3.98/1.87	(9) CSRRRRProof [EQUIVALENT, 0 ms]
3.98/1.87	(10) CSR
3.98/1.87	(11) CSRRRRProof [EQUIVALENT, 0 ms]
3.98/1.87	(12) CSR
3.98/1.87	(13) RisEmptyProof [EQUIVALENT, 0 ms]
3.98/1.87	(14) YES
3.98/1.87	
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(0)
3.98/1.87	Obligation:
3.98/1.87	Q restricted rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   active(minus(0, Y)) -> mark(0)
3.98/1.87	   active(minus(s(X), s(Y))) -> mark(minus(X, Y))
3.98/1.87	   active(geq(X, 0)) -> mark(true)
3.98/1.87	   active(geq(0, s(Y))) -> mark(false)
3.98/1.87	   active(geq(s(X), s(Y))) -> mark(geq(X, Y))
3.98/1.87	   active(div(0, s(Y))) -> mark(0)
3.98/1.87	   active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
3.98/1.87	   active(if(true, X, Y)) -> mark(X)
3.98/1.87	   active(if(false, X, Y)) -> mark(Y)
3.98/1.87	   active(s(X)) -> s(active(X))
3.98/1.87	   active(div(X1, X2)) -> div(active(X1), X2)
3.98/1.87	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
3.98/1.87	   s(mark(X)) -> mark(s(X))
3.98/1.87	   div(mark(X1), X2) -> mark(div(X1, X2))
3.98/1.87	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
3.98/1.87	   proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
3.98/1.87	   proper(0) -> ok(0)
3.98/1.87	   proper(s(X)) -> s(proper(X))
3.98/1.87	   proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
3.98/1.87	   proper(true) -> ok(true)
3.98/1.87	   proper(false) -> ok(false)
3.98/1.87	   proper(div(X1, X2)) -> div(proper(X1), proper(X2))
3.98/1.87	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
3.98/1.87	   minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
3.98/1.87	   s(ok(X)) -> ok(s(X))
3.98/1.87	   geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
3.98/1.87	   div(ok(X1), ok(X2)) -> ok(div(X1, X2))
3.98/1.87	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
3.98/1.87	   top(mark(X)) -> top(proper(X))
3.98/1.87	   top(ok(X)) -> top(active(X))
3.98/1.87	
3.98/1.87	The set Q consists of the following terms:
3.98/1.87	
3.98/1.87	   active(minus(0, x0))
3.98/1.87	   active(minus(s(x0), s(x1)))
3.98/1.87	   active(geq(x0, 0))
3.98/1.87	   active(geq(0, s(x0)))
3.98/1.87	   active(geq(s(x0), s(x1)))
3.98/1.87	   active(s(x0))
3.98/1.87	   active(div(x0, x1))
3.98/1.87	   active(if(x0, x1, x2))
3.98/1.87	   s(mark(x0))
3.98/1.87	   div(mark(x0), x1)
3.98/1.87	   if(mark(x0), x1, x2)
3.98/1.87	   proper(minus(x0, x1))
3.98/1.87	   proper(0)
3.98/1.87	   proper(s(x0))
3.98/1.87	   proper(geq(x0, x1))
3.98/1.87	   proper(true)
3.98/1.87	   proper(false)
3.98/1.87	   proper(div(x0, x1))
3.98/1.87	   proper(if(x0, x1, x2))
3.98/1.87	   minus(ok(x0), ok(x1))
3.98/1.87	   s(ok(x0))
3.98/1.87	   geq(ok(x0), ok(x1))
3.98/1.87	   div(ok(x0), ok(x1))
3.98/1.87	   if(ok(x0), ok(x1), ok(x2))
3.98/1.87	   top(mark(x0))
3.98/1.87	   top(ok(x0))
3.98/1.87	
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(1) QTRSToCSRProof (SOUND)
3.98/1.87	The following Q TRS is given: Q restricted rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   active(minus(0, Y)) -> mark(0)
3.98/1.87	   active(minus(s(X), s(Y))) -> mark(minus(X, Y))
3.98/1.87	   active(geq(X, 0)) -> mark(true)
3.98/1.87	   active(geq(0, s(Y))) -> mark(false)
3.98/1.87	   active(geq(s(X), s(Y))) -> mark(geq(X, Y))
3.98/1.87	   active(div(0, s(Y))) -> mark(0)
3.98/1.87	   active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
3.98/1.87	   active(if(true, X, Y)) -> mark(X)
3.98/1.87	   active(if(false, X, Y)) -> mark(Y)
3.98/1.87	   active(s(X)) -> s(active(X))
3.98/1.87	   active(div(X1, X2)) -> div(active(X1), X2)
3.98/1.87	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
3.98/1.87	   s(mark(X)) -> mark(s(X))
3.98/1.87	   div(mark(X1), X2) -> mark(div(X1, X2))
3.98/1.87	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
3.98/1.87	   proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
3.98/1.87	   proper(0) -> ok(0)
3.98/1.87	   proper(s(X)) -> s(proper(X))
3.98/1.87	   proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
3.98/1.87	   proper(true) -> ok(true)
3.98/1.87	   proper(false) -> ok(false)
3.98/1.87	   proper(div(X1, X2)) -> div(proper(X1), proper(X2))
3.98/1.87	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
3.98/1.87	   minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
3.98/1.87	   s(ok(X)) -> ok(s(X))
3.98/1.87	   geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
3.98/1.87	   div(ok(X1), ok(X2)) -> ok(div(X1, X2))
3.98/1.87	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
3.98/1.87	   top(mark(X)) -> top(proper(X))
3.98/1.87	   top(ok(X)) -> top(active(X))
3.98/1.87	
3.98/1.87	The set Q consists of the following terms:
3.98/1.87	
3.98/1.87	   active(minus(0, x0))
3.98/1.87	   active(minus(s(x0), s(x1)))
3.98/1.87	   active(geq(x0, 0))
3.98/1.87	   active(geq(0, s(x0)))
3.98/1.87	   active(geq(s(x0), s(x1)))
3.98/1.87	   active(s(x0))
3.98/1.87	   active(div(x0, x1))
3.98/1.87	   active(if(x0, x1, x2))
3.98/1.87	   s(mark(x0))
3.98/1.87	   div(mark(x0), x1)
3.98/1.87	   if(mark(x0), x1, x2)
3.98/1.87	   proper(minus(x0, x1))
3.98/1.87	   proper(0)
3.98/1.87	   proper(s(x0))
3.98/1.87	   proper(geq(x0, x1))
3.98/1.87	   proper(true)
3.98/1.87	   proper(false)
3.98/1.87	   proper(div(x0, x1))
3.98/1.87	   proper(if(x0, x1, x2))
3.98/1.87	   minus(ok(x0), ok(x1))
3.98/1.87	   s(ok(x0))
3.98/1.87	   geq(ok(x0), ok(x1))
3.98/1.87	   div(ok(x0), ok(x1))
3.98/1.87	   if(ok(x0), ok(x1), ok(x2))
3.98/1.87	   top(mark(x0))
3.98/1.87	   top(ok(x0))
3.98/1.87	
3.98/1.87	Special symbols used for the transformation (see [GM04]):
3.98/1.87	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	div: {1}
3.98/1.87	if: {1}
3.98/1.87	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.
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(2)
3.98/1.87	Obligation:
3.98/1.87	Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   minus(s(X), s(Y)) -> minus(X, Y)
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   geq(s(X), s(Y)) -> geq(X, Y)
3.98/1.87	   div(0, s(Y)) -> 0
3.98/1.87	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	div: {1}
3.98/1.87	if: {1}
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(3) CSRRRRProof (EQUIVALENT)
3.98/1.87	The following CSR is given: Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   minus(s(X), s(Y)) -> minus(X, Y)
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   geq(s(X), s(Y)) -> geq(X, Y)
3.98/1.87	   div(0, s(Y)) -> 0
3.98/1.87	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	div: {1}
3.98/1.87	if: {1}
3.98/1.87	Used ordering:
3.98/1.87	Polynomial interpretation [POLO]:
3.98/1.87	
3.98/1.87	   POL(0) = 0
3.98/1.87	   POL(div(x_1, x_2)) = x_1 + x_2
3.98/1.87	   POL(false) = 0
3.98/1.87	   POL(geq(x_1, x_2)) = 0
3.98/1.87	   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.98/1.87	   POL(minus(x_1, x_2)) = x_1
3.98/1.87	   POL(s(x_1)) = 1 + x_1
3.98/1.87	   POL(true) = 0
3.98/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.98/1.87	
3.98/1.87	   minus(s(X), s(Y)) -> minus(X, Y)
3.98/1.87	   div(0, s(Y)) -> 0
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(4)
3.98/1.87	Obligation:
3.98/1.87	Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   geq(s(X), s(Y)) -> geq(X, Y)
3.98/1.87	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	div: {1}
3.98/1.87	if: {1}
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(5) CSRRRRProof (EQUIVALENT)
3.98/1.87	The following CSR is given: Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   geq(s(X), s(Y)) -> geq(X, Y)
3.98/1.87	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	div: {1}
3.98/1.87	if: {1}
3.98/1.87	Used ordering:
3.98/1.87	Polynomial interpretation [POLO]:
3.98/1.87	
3.98/1.87	   POL(0) = 0
3.98/1.87	   POL(div(x_1, x_2)) = 2*x_1
3.98/1.87	   POL(false) = 0
3.98/1.87	   POL(geq(x_1, x_2)) = x_1
3.98/1.87	   POL(if(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3
3.98/1.87	   POL(minus(x_1, x_2)) = 0
3.98/1.87	   POL(s(x_1)) = 1 + 2*x_1
3.98/1.87	   POL(true) = 0
3.98/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.98/1.87	
3.98/1.87	   geq(s(X), s(Y)) -> geq(X, Y)
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(6)
3.98/1.87	Obligation:
3.98/1.87	Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	div: {1}
3.98/1.87	if: {1}
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(7) CSRRRRProof (EQUIVALENT)
3.98/1.87	The following CSR is given: Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	div: {1}
3.98/1.87	if: {1}
3.98/1.87	Used ordering:
3.98/1.87	Polynomial interpretation [POLO]:
3.98/1.87	
3.98/1.87	   POL(0) = 0
3.98/1.87	   POL(div(x_1, x_2)) = 1 + 2*x_1
3.98/1.87	   POL(false) = 0
3.98/1.87	   POL(geq(x_1, x_2)) = 2*x_1
3.98/1.87	   POL(if(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
3.98/1.87	   POL(minus(x_1, x_2)) = 0
3.98/1.87	   POL(s(x_1)) = 2 + 2*x_1
3.98/1.87	   POL(true) = 0
3.98/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.98/1.87	
3.98/1.87	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(8)
3.98/1.87	Obligation:
3.98/1.87	Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	if: {1}
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(9) CSRRRRProof (EQUIVALENT)
3.98/1.87	The following CSR is given: Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	minus: empty set
3.98/1.87	0: empty set
3.98/1.87	s: {1}
3.98/1.87	geq: empty set
3.98/1.87	true: empty set
3.98/1.87	false: empty set
3.98/1.87	if: {1}
3.98/1.87	Used ordering:
3.98/1.87	Polynomial interpretation [POLO]:
3.98/1.87	
3.98/1.87	   POL(0) = 2
3.98/1.87	   POL(false) = 0
3.98/1.87	   POL(geq(x_1, x_2)) = 1 + x_1 + x_2
3.98/1.87	   POL(if(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
3.98/1.87	   POL(minus(x_1, x_2)) = 2 + x_1
3.98/1.87	   POL(s(x_1)) = 2 + x_1
3.98/1.87	   POL(true) = 2
3.98/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.98/1.87	
3.98/1.87	   minus(0, Y) -> 0
3.98/1.87	   geq(X, 0) -> true
3.98/1.87	   geq(0, s(Y)) -> false
3.98/1.87	   if(true, X, Y) -> X
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(10)
3.98/1.87	Obligation:
3.98/1.87	Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	false: empty set
3.98/1.87	if: {1}
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(11) CSRRRRProof (EQUIVALENT)
3.98/1.87	The following CSR is given: Context-sensitive rewrite system:
3.98/1.87	The TRS R consists of the following rules:
3.98/1.87	
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	The replacement map contains the following entries:
3.98/1.87	
3.98/1.87	false: empty set
3.98/1.87	if: {1}
3.98/1.87	Used ordering:
3.98/1.87	Polynomial interpretation [POLO]:
3.98/1.87	
3.98/1.87	   POL(false) = 1
3.98/1.87	   POL(if(x_1, x_2, x_3)) = x_1 + x_3
3.98/1.87	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.98/1.87	
3.98/1.87	   if(false, X, Y) -> Y
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(12)
3.98/1.87	Obligation:
3.98/1.87	Context-sensitive rewrite system:
3.98/1.87	R is empty.
3.98/1.87	
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(13) RisEmptyProof (EQUIVALENT)
3.98/1.87	The CSR R is empty. Hence, termination is trivially proven.
3.98/1.87	----------------------------------------
3.98/1.87	
3.98/1.87	(14)
3.98/1.87	YES
3.98/1.90	EOF