4.83/2.15	YES
4.87/2.17	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.87/2.17	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.87/2.17	
4.87/2.17	
4.87/2.17	Termination w.r.t. Q of the given QTRS could be proven:
4.87/2.17	
4.87/2.17	(0) QTRS
4.87/2.17	(1) QTRSToCSRProof [SOUND, 0 ms]
4.87/2.17	(2) CSR
4.87/2.17	(3) CSRRRRProof [EQUIVALENT, 90 ms]
4.87/2.17	(4) CSR
4.87/2.17	(5) CSRRRRProof [EQUIVALENT, 0 ms]
4.87/2.17	(6) CSR
4.87/2.17	(7) CSRRRRProof [EQUIVALENT, 0 ms]
4.87/2.17	(8) CSR
4.87/2.17	(9) CSRRRRProof [EQUIVALENT, 0 ms]
4.87/2.17	(10) CSR
4.87/2.17	(11) CSRRRRProof [EQUIVALENT, 5 ms]
4.87/2.17	(12) CSR
4.87/2.17	(13) CSRRRRProof [EQUIVALENT, 0 ms]
4.87/2.17	(14) CSR
4.87/2.17	(15) CSRRRRProof [EQUIVALENT, 0 ms]
4.87/2.17	(16) CSR
4.87/2.17	(17) CSRRRRProof [EQUIVALENT, 3 ms]
4.87/2.17	(18) CSR
4.87/2.17	(19) CSRRRRProof [EQUIVALENT, 0 ms]
4.87/2.17	(20) CSR
4.87/2.17	(21) RisEmptyProof [EQUIVALENT, 0 ms]
4.87/2.17	(22) YES
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(0)
4.87/2.17	Obligation:
4.87/2.17	Q restricted rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   active(pairNs) -> mark(cons(0, incr(oddNs)))
4.87/2.17	   active(oddNs) -> mark(incr(pairNs))
4.87/2.17	   active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
4.87/2.17	   active(take(0, XS)) -> mark(nil)
4.87/2.17	   active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
4.87/2.17	   active(zip(nil, XS)) -> mark(nil)
4.87/2.17	   active(zip(X, nil)) -> mark(nil)
4.87/2.17	   active(zip(cons(X, XS), cons(Y, YS))) -> mark(cons(pair(X, Y), zip(XS, YS)))
4.87/2.17	   active(tail(cons(X, XS))) -> mark(XS)
4.87/2.17	   active(repItems(nil)) -> mark(nil)
4.87/2.17	   active(repItems(cons(X, XS))) -> mark(cons(X, cons(X, repItems(XS))))
4.87/2.17	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.87/2.17	   active(incr(X)) -> incr(active(X))
4.87/2.17	   active(s(X)) -> s(active(X))
4.87/2.17	   active(take(X1, X2)) -> take(active(X1), X2)
4.87/2.17	   active(take(X1, X2)) -> take(X1, active(X2))
4.87/2.17	   active(zip(X1, X2)) -> zip(active(X1), X2)
4.87/2.17	   active(zip(X1, X2)) -> zip(X1, active(X2))
4.87/2.17	   active(pair(X1, X2)) -> pair(active(X1), X2)
4.87/2.17	   active(pair(X1, X2)) -> pair(X1, active(X2))
4.87/2.17	   active(tail(X)) -> tail(active(X))
4.87/2.17	   active(repItems(X)) -> repItems(active(X))
4.87/2.17	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.87/2.17	   incr(mark(X)) -> mark(incr(X))
4.87/2.17	   s(mark(X)) -> mark(s(X))
4.87/2.17	   take(mark(X1), X2) -> mark(take(X1, X2))
4.87/2.17	   take(X1, mark(X2)) -> mark(take(X1, X2))
4.87/2.17	   zip(mark(X1), X2) -> mark(zip(X1, X2))
4.87/2.17	   zip(X1, mark(X2)) -> mark(zip(X1, X2))
4.87/2.17	   pair(mark(X1), X2) -> mark(pair(X1, X2))
4.87/2.17	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
4.87/2.17	   tail(mark(X)) -> mark(tail(X))
4.87/2.17	   repItems(mark(X)) -> mark(repItems(X))
4.87/2.17	   proper(pairNs) -> ok(pairNs)
4.87/2.17	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.87/2.17	   proper(0) -> ok(0)
4.87/2.17	   proper(incr(X)) -> incr(proper(X))
4.87/2.17	   proper(oddNs) -> ok(oddNs)
4.87/2.17	   proper(s(X)) -> s(proper(X))
4.87/2.17	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
4.87/2.17	   proper(nil) -> ok(nil)
4.87/2.17	   proper(zip(X1, X2)) -> zip(proper(X1), proper(X2))
4.87/2.17	   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
4.87/2.17	   proper(tail(X)) -> tail(proper(X))
4.87/2.17	   proper(repItems(X)) -> repItems(proper(X))
4.87/2.17	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.87/2.17	   incr(ok(X)) -> ok(incr(X))
4.87/2.17	   s(ok(X)) -> ok(s(X))
4.87/2.17	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
4.87/2.17	   zip(ok(X1), ok(X2)) -> ok(zip(X1, X2))
4.87/2.17	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
4.87/2.17	   tail(ok(X)) -> ok(tail(X))
4.87/2.17	   repItems(ok(X)) -> ok(repItems(X))
4.87/2.17	   top(mark(X)) -> top(proper(X))
4.87/2.17	   top(ok(X)) -> top(active(X))
4.87/2.17	
4.87/2.17	The set Q consists of the following terms:
4.87/2.17	
4.87/2.17	   active(pairNs)
4.87/2.17	   active(oddNs)
4.87/2.17	   active(cons(x0, x1))
4.87/2.17	   active(incr(x0))
4.87/2.17	   active(s(x0))
4.87/2.17	   active(take(x0, x1))
4.87/2.17	   active(zip(x0, x1))
4.87/2.17	   active(pair(x0, x1))
4.87/2.17	   active(tail(x0))
4.87/2.17	   active(repItems(x0))
4.87/2.17	   cons(mark(x0), x1)
4.87/2.17	   incr(mark(x0))
4.87/2.17	   s(mark(x0))
4.87/2.17	   take(mark(x0), x1)
4.87/2.17	   take(x0, mark(x1))
4.87/2.17	   zip(mark(x0), x1)
4.87/2.17	   zip(x0, mark(x1))
4.87/2.17	   pair(mark(x0), x1)
4.87/2.17	   pair(x0, mark(x1))
4.87/2.17	   tail(mark(x0))
4.87/2.17	   repItems(mark(x0))
4.87/2.17	   proper(pairNs)
4.87/2.17	   proper(cons(x0, x1))
4.87/2.17	   proper(0)
4.87/2.17	   proper(incr(x0))
4.87/2.17	   proper(oddNs)
4.87/2.17	   proper(s(x0))
4.87/2.17	   proper(take(x0, x1))
4.87/2.17	   proper(nil)
4.87/2.17	   proper(zip(x0, x1))
4.87/2.17	   proper(pair(x0, x1))
4.87/2.17	   proper(tail(x0))
4.87/2.17	   proper(repItems(x0))
4.87/2.17	   cons(ok(x0), ok(x1))
4.87/2.17	   incr(ok(x0))
4.87/2.17	   s(ok(x0))
4.87/2.17	   take(ok(x0), ok(x1))
4.87/2.17	   zip(ok(x0), ok(x1))
4.87/2.17	   pair(ok(x0), ok(x1))
4.87/2.17	   tail(ok(x0))
4.87/2.17	   repItems(ok(x0))
4.87/2.17	   top(mark(x0))
4.87/2.17	   top(ok(x0))
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(1) QTRSToCSRProof (SOUND)
4.87/2.17	The following Q TRS is given: Q restricted rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   active(pairNs) -> mark(cons(0, incr(oddNs)))
4.87/2.17	   active(oddNs) -> mark(incr(pairNs))
4.87/2.17	   active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
4.87/2.17	   active(take(0, XS)) -> mark(nil)
4.87/2.17	   active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
4.87/2.17	   active(zip(nil, XS)) -> mark(nil)
4.87/2.17	   active(zip(X, nil)) -> mark(nil)
4.87/2.17	   active(zip(cons(X, XS), cons(Y, YS))) -> mark(cons(pair(X, Y), zip(XS, YS)))
4.87/2.17	   active(tail(cons(X, XS))) -> mark(XS)
4.87/2.17	   active(repItems(nil)) -> mark(nil)
4.87/2.17	   active(repItems(cons(X, XS))) -> mark(cons(X, cons(X, repItems(XS))))
4.87/2.17	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.87/2.17	   active(incr(X)) -> incr(active(X))
4.87/2.17	   active(s(X)) -> s(active(X))
4.87/2.17	   active(take(X1, X2)) -> take(active(X1), X2)
4.87/2.17	   active(take(X1, X2)) -> take(X1, active(X2))
4.87/2.17	   active(zip(X1, X2)) -> zip(active(X1), X2)
4.87/2.17	   active(zip(X1, X2)) -> zip(X1, active(X2))
4.87/2.17	   active(pair(X1, X2)) -> pair(active(X1), X2)
4.87/2.17	   active(pair(X1, X2)) -> pair(X1, active(X2))
4.87/2.17	   active(tail(X)) -> tail(active(X))
4.87/2.17	   active(repItems(X)) -> repItems(active(X))
4.87/2.17	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.87/2.17	   incr(mark(X)) -> mark(incr(X))
4.87/2.17	   s(mark(X)) -> mark(s(X))
4.87/2.17	   take(mark(X1), X2) -> mark(take(X1, X2))
4.87/2.17	   take(X1, mark(X2)) -> mark(take(X1, X2))
4.87/2.17	   zip(mark(X1), X2) -> mark(zip(X1, X2))
4.87/2.17	   zip(X1, mark(X2)) -> mark(zip(X1, X2))
4.87/2.17	   pair(mark(X1), X2) -> mark(pair(X1, X2))
4.87/2.17	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
4.87/2.17	   tail(mark(X)) -> mark(tail(X))
4.87/2.17	   repItems(mark(X)) -> mark(repItems(X))
4.87/2.17	   proper(pairNs) -> ok(pairNs)
4.87/2.17	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.87/2.17	   proper(0) -> ok(0)
4.87/2.17	   proper(incr(X)) -> incr(proper(X))
4.87/2.17	   proper(oddNs) -> ok(oddNs)
4.87/2.17	   proper(s(X)) -> s(proper(X))
4.87/2.17	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
4.87/2.17	   proper(nil) -> ok(nil)
4.87/2.17	   proper(zip(X1, X2)) -> zip(proper(X1), proper(X2))
4.87/2.17	   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
4.87/2.17	   proper(tail(X)) -> tail(proper(X))
4.87/2.17	   proper(repItems(X)) -> repItems(proper(X))
4.87/2.17	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.87/2.17	   incr(ok(X)) -> ok(incr(X))
4.87/2.17	   s(ok(X)) -> ok(s(X))
4.87/2.17	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
4.87/2.17	   zip(ok(X1), ok(X2)) -> ok(zip(X1, X2))
4.87/2.17	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
4.87/2.17	   tail(ok(X)) -> ok(tail(X))
4.87/2.17	   repItems(ok(X)) -> ok(repItems(X))
4.87/2.17	   top(mark(X)) -> top(proper(X))
4.87/2.17	   top(ok(X)) -> top(active(X))
4.87/2.17	
4.87/2.17	The set Q consists of the following terms:
4.87/2.17	
4.87/2.17	   active(pairNs)
4.87/2.17	   active(oddNs)
4.87/2.17	   active(cons(x0, x1))
4.87/2.17	   active(incr(x0))
4.87/2.17	   active(s(x0))
4.87/2.17	   active(take(x0, x1))
4.87/2.17	   active(zip(x0, x1))
4.87/2.17	   active(pair(x0, x1))
4.87/2.17	   active(tail(x0))
4.87/2.17	   active(repItems(x0))
4.87/2.17	   cons(mark(x0), x1)
4.87/2.17	   incr(mark(x0))
4.87/2.17	   s(mark(x0))
4.87/2.17	   take(mark(x0), x1)
4.87/2.17	   take(x0, mark(x1))
4.87/2.17	   zip(mark(x0), x1)
4.87/2.17	   zip(x0, mark(x1))
4.87/2.17	   pair(mark(x0), x1)
4.87/2.17	   pair(x0, mark(x1))
4.87/2.17	   tail(mark(x0))
4.87/2.17	   repItems(mark(x0))
4.87/2.17	   proper(pairNs)
4.87/2.17	   proper(cons(x0, x1))
4.87/2.17	   proper(0)
4.87/2.17	   proper(incr(x0))
4.87/2.17	   proper(oddNs)
4.87/2.17	   proper(s(x0))
4.87/2.17	   proper(take(x0, x1))
4.87/2.17	   proper(nil)
4.87/2.17	   proper(zip(x0, x1))
4.87/2.17	   proper(pair(x0, x1))
4.87/2.17	   proper(tail(x0))
4.87/2.17	   proper(repItems(x0))
4.87/2.17	   cons(ok(x0), ok(x1))
4.87/2.17	   incr(ok(x0))
4.87/2.17	   s(ok(x0))
4.87/2.17	   take(ok(x0), ok(x1))
4.87/2.17	   zip(ok(x0), ok(x1))
4.87/2.17	   pair(ok(x0), ok(x1))
4.87/2.17	   tail(ok(x0))
4.87/2.17	   repItems(ok(x0))
4.87/2.17	   top(mark(x0))
4.87/2.17	   top(ok(x0))
4.87/2.17	
4.87/2.17	Special symbols used for the transformation (see [GM04]):
4.87/2.17	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	tail: {1}
4.87/2.17	repItems: {1}
4.87/2.17	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.
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(2)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	   tail(cons(X, XS)) -> XS
4.87/2.17	   repItems(nil) -> nil
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	tail: {1}
4.87/2.17	repItems: {1}
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(3) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	   tail(cons(X, XS)) -> XS
4.87/2.17	   repItems(nil) -> nil
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	tail: {1}
4.87/2.17	repItems: {1}
4.87/2.17	Used ordering:
4.87/2.17	Polynomial interpretation [POLO]:
4.87/2.17	
4.87/2.17	   POL(0) = 0
4.87/2.17	   POL(cons(x_1, x_2)) = 2*x_1 + x_2
4.87/2.17	   POL(incr(x_1)) = 2*x_1
4.87/2.17	   POL(nil) = 0
4.87/2.17	   POL(oddNs) = 0
4.87/2.17	   POL(pair(x_1, x_2)) = x_1 + x_2
4.87/2.17	   POL(pairNs) = 0
4.87/2.17	   POL(repItems(x_1)) = 2 + 2*x_1
4.87/2.17	   POL(s(x_1)) = 2*x_1
4.87/2.17	   POL(tail(x_1)) = 2*x_1
4.87/2.17	   POL(take(x_1, x_2)) = x_1 + 2*x_2
4.87/2.17	   POL(zip(x_1, x_2)) = 2*x_1 + 2*x_2
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   repItems(nil) -> nil
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(4)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	   tail(cons(X, XS)) -> XS
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	tail: {1}
4.87/2.17	repItems: {1}
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(5) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	   tail(cons(X, XS)) -> XS
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	tail: {1}
4.87/2.17	repItems: {1}
4.87/2.17	Used ordering:
4.87/2.17	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(pairNs) =  	[[0], [0]]
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 1]] 	* 	x_2
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(0) =  	[[0], [1]]
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(incr(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(oddNs) =  	[[0], [1]]
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(take(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(nil) =  	[[0], [0]]
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(zip(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(pair(x_1, x_2)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(tail(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(repItems(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   tail(cons(X, XS)) -> XS
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(6)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	repItems: {1}
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(7) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	repItems: {1}
4.87/2.17	Used ordering:
4.87/2.17	Polynomial interpretation [POLO]:
4.87/2.17	
4.87/2.17	   POL(0) = 0
4.87/2.17	   POL(cons(x_1, x_2)) = 1 + x_1
4.87/2.17	   POL(incr(x_1)) = x_1
4.87/2.17	   POL(nil) = 1
4.87/2.17	   POL(oddNs) = 1
4.87/2.17	   POL(pair(x_1, x_2)) = 1 + x_1 + x_2
4.87/2.17	   POL(pairNs) = 1
4.87/2.17	   POL(repItems(x_1)) = x_1
4.87/2.17	   POL(s(x_1)) = x_1
4.87/2.17	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
4.87/2.17	   POL(zip(x_1, x_2)) = x_1 + x_2
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(8)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	repItems: {1}
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(9) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	repItems: {1}
4.87/2.17	Used ordering:
4.87/2.17	Polynomial interpretation [POLO]:
4.87/2.17	
4.87/2.17	   POL(0) = 0
4.87/2.17	   POL(cons(x_1, x_2)) = 1 + x_1
4.87/2.17	   POL(incr(x_1)) = x_1
4.87/2.17	   POL(nil) = 1
4.87/2.17	   POL(oddNs) = 1
4.87/2.17	   POL(pair(x_1, x_2)) = 1 + x_1 + x_2
4.87/2.17	   POL(pairNs) = 1
4.87/2.17	   POL(repItems(x_1)) = 1 + x_1
4.87/2.17	   POL(s(x_1)) = x_1
4.87/2.17	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
4.87/2.17	   POL(zip(x_1, x_2)) = x_1 + x_2
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(10)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(11) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	zip: {1, 2}
4.87/2.17	pair: {1, 2}
4.87/2.17	Used ordering:
4.87/2.17	Polynomial interpretation [POLO]:
4.87/2.17	
4.87/2.17	   POL(0) = 0
4.87/2.17	   POL(cons(x_1, x_2)) = 1 + x_1
4.87/2.17	   POL(incr(x_1)) = x_1
4.87/2.17	   POL(nil) = 1
4.87/2.17	   POL(oddNs) = 1
4.87/2.17	   POL(pair(x_1, x_2)) = 1 + x_1 + x_2
4.87/2.17	   POL(pairNs) = 1
4.87/2.17	   POL(s(x_1)) = x_1
4.87/2.17	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
4.87/2.17	   POL(zip(x_1, x_2)) = 1 + x_1 + x_2
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   zip(nil, XS) -> nil
4.87/2.17	   zip(X, nil) -> nil
4.87/2.17	   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(12)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(13) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	take: {1, 2}
4.87/2.17	nil: empty set
4.87/2.17	Used ordering:
4.87/2.17	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(pairNs) =  	[[0], [1]]
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(0) =  	[[0], [0]]
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(incr(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(oddNs) =  	[[0], [1]]
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(take(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	   <<<
4.87/2.17	 POL(nil) =  	[[0], [1]]
4.87/2.17	>>>
4.87/2.17	
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   take(0, XS) -> nil
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(14)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(15) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	0: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	Used ordering:
4.87/2.17	Polynomial interpretation [POLO]:
4.87/2.17	
4.87/2.17	   POL(0) = 0
4.87/2.17	   POL(cons(x_1, x_2)) = x_1
4.87/2.17	   POL(incr(x_1)) = x_1
4.87/2.17	   POL(oddNs) = 1
4.87/2.17	   POL(pairNs) = 1
4.87/2.17	   POL(s(x_1)) = x_1
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   pairNs -> cons(0, incr(oddNs))
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(16)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(17) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	cons: {1}
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	s: {1}
4.87/2.17	Used ordering:
4.87/2.17	Polynomial interpretation [POLO]:
4.87/2.17	
4.87/2.17	   POL(cons(x_1, x_2)) = 2 + x_1
4.87/2.17	   POL(incr(x_1)) = 1 + 2*x_1
4.87/2.17	   POL(oddNs) = 1
4.87/2.17	   POL(pairNs) = 0
4.87/2.17	   POL(s(x_1)) = x_1
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   incr(cons(X, XS)) -> cons(s(X), incr(XS))
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(18)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(19) CSRRRRProof (EQUIVALENT)
4.87/2.17	The following CSR is given: Context-sensitive rewrite system:
4.87/2.17	The TRS R consists of the following rules:
4.87/2.17	
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	
4.87/2.17	The replacement map contains the following entries:
4.87/2.17	
4.87/2.17	pairNs: empty set
4.87/2.17	incr: {1}
4.87/2.17	oddNs: empty set
4.87/2.17	Used ordering:
4.87/2.17	Polynomial interpretation [POLO]:
4.87/2.17	
4.87/2.17	   POL(incr(x_1)) = 1 + 2*x_1
4.87/2.17	   POL(oddNs) = 2
4.87/2.17	   POL(pairNs) = 0
4.87/2.17	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.87/2.17	
4.87/2.17	   oddNs -> incr(pairNs)
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(20)
4.87/2.17	Obligation:
4.87/2.17	Context-sensitive rewrite system:
4.87/2.17	R is empty.
4.87/2.17	
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(21) RisEmptyProof (EQUIVALENT)
4.87/2.17	The CSR R is empty. Hence, termination is trivially proven.
4.87/2.17	----------------------------------------
4.87/2.17	
4.87/2.17	(22)
4.87/2.17	YES
4.87/2.19	EOF