3.90/1.88	YES
3.90/1.89	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.90/1.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.90/1.89	
3.90/1.89	
3.90/1.89	Termination w.r.t. Q of the given QTRS could be proven:
3.90/1.89	
3.90/1.89	(0) QTRS
3.90/1.89	(1) QTRSToCSRProof [SOUND, 0 ms]
3.90/1.89	(2) CSR
3.90/1.89	(3) CSRRRRProof [EQUIVALENT, 50 ms]
3.90/1.89	(4) CSR
3.90/1.89	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.90/1.89	(6) CSR
3.90/1.89	(7) CSRRRRProof [EQUIVALENT, 9 ms]
3.90/1.89	(8) CSR
3.90/1.89	(9) CSRRRRProof [EQUIVALENT, 0 ms]
3.90/1.89	(10) CSR
3.90/1.89	(11) CSRRRRProof [EQUIVALENT, 0 ms]
3.90/1.89	(12) CSR
3.90/1.89	(13) CSRRRRProof [EQUIVALENT, 0 ms]
3.90/1.89	(14) CSR
3.90/1.89	(15) RisEmptyProof [EQUIVALENT, 0 ms]
3.90/1.89	(16) YES
3.90/1.89	
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(0)
3.90/1.89	Obligation:
3.90/1.89	Q restricted rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   active(nats) -> mark(adx(zeros))
3.90/1.89	   active(zeros) -> mark(cons(0, zeros))
3.90/1.89	   active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
3.90/1.89	   active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
3.90/1.89	   active(hd(cons(X, Y))) -> mark(X)
3.90/1.89	   active(tl(cons(X, Y))) -> mark(Y)
3.90/1.89	   active(adx(X)) -> adx(active(X))
3.90/1.89	   active(incr(X)) -> incr(active(X))
3.90/1.89	   active(hd(X)) -> hd(active(X))
3.90/1.89	   active(tl(X)) -> tl(active(X))
3.90/1.89	   adx(mark(X)) -> mark(adx(X))
3.90/1.89	   incr(mark(X)) -> mark(incr(X))
3.90/1.89	   hd(mark(X)) -> mark(hd(X))
3.90/1.89	   tl(mark(X)) -> mark(tl(X))
3.90/1.89	   proper(nats) -> ok(nats)
3.90/1.89	   proper(adx(X)) -> adx(proper(X))
3.90/1.89	   proper(zeros) -> ok(zeros)
3.90/1.89	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.90/1.89	   proper(0) -> ok(0)
3.90/1.89	   proper(incr(X)) -> incr(proper(X))
3.90/1.89	   proper(s(X)) -> s(proper(X))
3.90/1.89	   proper(hd(X)) -> hd(proper(X))
3.90/1.89	   proper(tl(X)) -> tl(proper(X))
3.90/1.89	   adx(ok(X)) -> ok(adx(X))
3.90/1.89	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.90/1.89	   incr(ok(X)) -> ok(incr(X))
3.90/1.89	   s(ok(X)) -> ok(s(X))
3.90/1.89	   hd(ok(X)) -> ok(hd(X))
3.90/1.89	   tl(ok(X)) -> ok(tl(X))
3.90/1.89	   top(mark(X)) -> top(proper(X))
3.90/1.89	   top(ok(X)) -> top(active(X))
3.90/1.89	
3.90/1.89	The set Q consists of the following terms:
3.90/1.89	
3.90/1.89	   active(nats)
3.90/1.89	   active(zeros)
3.90/1.89	   active(adx(x0))
3.90/1.89	   active(incr(x0))
3.90/1.89	   active(hd(x0))
3.90/1.89	   active(tl(x0))
3.90/1.89	   adx(mark(x0))
3.90/1.89	   incr(mark(x0))
3.90/1.89	   hd(mark(x0))
3.90/1.89	   tl(mark(x0))
3.90/1.89	   proper(nats)
3.90/1.89	   proper(adx(x0))
3.90/1.89	   proper(zeros)
3.90/1.89	   proper(cons(x0, x1))
3.90/1.89	   proper(0)
3.90/1.89	   proper(incr(x0))
3.90/1.89	   proper(s(x0))
3.90/1.89	   proper(hd(x0))
3.90/1.89	   proper(tl(x0))
3.90/1.89	   adx(ok(x0))
3.90/1.89	   cons(ok(x0), ok(x1))
3.90/1.89	   incr(ok(x0))
3.90/1.89	   s(ok(x0))
3.90/1.89	   hd(ok(x0))
3.90/1.89	   tl(ok(x0))
3.90/1.89	   top(mark(x0))
3.90/1.89	   top(ok(x0))
3.90/1.89	
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(1) QTRSToCSRProof (SOUND)
3.90/1.89	The following Q TRS is given: Q restricted rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   active(nats) -> mark(adx(zeros))
3.90/1.89	   active(zeros) -> mark(cons(0, zeros))
3.90/1.89	   active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
3.90/1.89	   active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
3.90/1.89	   active(hd(cons(X, Y))) -> mark(X)
3.90/1.89	   active(tl(cons(X, Y))) -> mark(Y)
3.90/1.89	   active(adx(X)) -> adx(active(X))
3.90/1.89	   active(incr(X)) -> incr(active(X))
3.90/1.89	   active(hd(X)) -> hd(active(X))
3.90/1.89	   active(tl(X)) -> tl(active(X))
3.90/1.89	   adx(mark(X)) -> mark(adx(X))
3.90/1.89	   incr(mark(X)) -> mark(incr(X))
3.90/1.89	   hd(mark(X)) -> mark(hd(X))
3.90/1.89	   tl(mark(X)) -> mark(tl(X))
3.90/1.89	   proper(nats) -> ok(nats)
3.90/1.89	   proper(adx(X)) -> adx(proper(X))
3.90/1.89	   proper(zeros) -> ok(zeros)
3.90/1.89	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.90/1.89	   proper(0) -> ok(0)
3.90/1.89	   proper(incr(X)) -> incr(proper(X))
3.90/1.89	   proper(s(X)) -> s(proper(X))
3.90/1.89	   proper(hd(X)) -> hd(proper(X))
3.90/1.89	   proper(tl(X)) -> tl(proper(X))
3.90/1.89	   adx(ok(X)) -> ok(adx(X))
3.90/1.89	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.90/1.89	   incr(ok(X)) -> ok(incr(X))
3.90/1.89	   s(ok(X)) -> ok(s(X))
3.90/1.89	   hd(ok(X)) -> ok(hd(X))
3.90/1.89	   tl(ok(X)) -> ok(tl(X))
3.90/1.89	   top(mark(X)) -> top(proper(X))
3.90/1.89	   top(ok(X)) -> top(active(X))
3.90/1.89	
3.90/1.89	The set Q consists of the following terms:
3.90/1.89	
3.90/1.89	   active(nats)
3.90/1.89	   active(zeros)
3.90/1.89	   active(adx(x0))
3.90/1.89	   active(incr(x0))
3.90/1.89	   active(hd(x0))
3.90/1.89	   active(tl(x0))
3.90/1.89	   adx(mark(x0))
3.90/1.89	   incr(mark(x0))
3.90/1.89	   hd(mark(x0))
3.90/1.89	   tl(mark(x0))
3.90/1.89	   proper(nats)
3.90/1.89	   proper(adx(x0))
3.90/1.89	   proper(zeros)
3.90/1.89	   proper(cons(x0, x1))
3.90/1.89	   proper(0)
3.90/1.89	   proper(incr(x0))
3.90/1.89	   proper(s(x0))
3.90/1.89	   proper(hd(x0))
3.90/1.89	   proper(tl(x0))
3.90/1.89	   adx(ok(x0))
3.90/1.89	   cons(ok(x0), ok(x1))
3.90/1.89	   incr(ok(x0))
3.90/1.89	   s(ok(x0))
3.90/1.89	   hd(ok(x0))
3.90/1.89	   tl(ok(x0))
3.90/1.89	   top(mark(x0))
3.90/1.89	   top(ok(x0))
3.90/1.89	
3.90/1.89	Special symbols used for the transformation (see [GM04]):
3.90/1.89	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	hd: {1}
3.90/1.89	tl: {1}
3.90/1.89	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.
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(2)
3.90/1.89	Obligation:
3.90/1.89	Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.90/1.89	   hd(cons(X, Y)) -> X
3.90/1.89	   tl(cons(X, Y)) -> Y
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	hd: {1}
3.90/1.89	tl: {1}
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(3) CSRRRRProof (EQUIVALENT)
3.90/1.89	The following CSR is given: Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.90/1.89	   hd(cons(X, Y)) -> X
3.90/1.89	   tl(cons(X, Y)) -> Y
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	hd: {1}
3.90/1.89	tl: {1}
3.90/1.89	Used ordering:
3.90/1.89	Polynomial interpretation [POLO]:
3.90/1.89	
3.90/1.89	   POL(0) = 0
3.90/1.89	   POL(adx(x_1)) = 1 + x_1
3.90/1.89	   POL(cons(x_1, x_2)) = x_1 + x_2
3.90/1.89	   POL(hd(x_1)) = 1 + x_1
3.90/1.89	   POL(incr(x_1)) = x_1
3.90/1.89	   POL(nats) = 1
3.90/1.89	   POL(s(x_1)) = 0
3.90/1.89	   POL(tl(x_1)) = x_1
3.90/1.89	   POL(zeros) = 0
3.90/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.90/1.89	
3.90/1.89	   hd(cons(X, Y)) -> X
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(4)
3.90/1.89	Obligation:
3.90/1.89	Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.90/1.89	   tl(cons(X, Y)) -> Y
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	tl: {1}
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(5) CSRRRRProof (EQUIVALENT)
3.90/1.89	The following CSR is given: Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.90/1.89	   tl(cons(X, Y)) -> Y
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	tl: {1}
3.90/1.89	Used ordering:
3.90/1.89	Polynomial interpretation [POLO]:
3.90/1.89	
3.90/1.89	   POL(0) = 0
3.90/1.89	   POL(adx(x_1)) = 1 + x_1
3.90/1.89	   POL(cons(x_1, x_2)) = x_1 + x_2
3.90/1.89	   POL(incr(x_1)) = x_1
3.90/1.89	   POL(nats) = 1
3.90/1.89	   POL(s(x_1)) = x_1
3.90/1.89	   POL(tl(x_1)) = 1 + x_1
3.90/1.89	   POL(zeros) = 0
3.90/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.90/1.89	
3.90/1.89	   tl(cons(X, Y)) -> Y
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(6)
3.90/1.89	Obligation:
3.90/1.89	Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(7) CSRRRRProof (EQUIVALENT)
3.90/1.89	The following CSR is given: Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	Used ordering:
3.90/1.89	Polynomial interpretation [POLO]:
3.90/1.89	
3.90/1.89	   POL(0) = 2
3.90/1.89	   POL(adx(x_1)) = 1 + 2*x_1
3.90/1.89	   POL(cons(x_1, x_2)) = 0
3.90/1.89	   POL(incr(x_1)) = 2*x_1
3.90/1.89	   POL(nats) = 1
3.90/1.89	   POL(s(x_1)) = 2 + x_1
3.90/1.89	   POL(zeros) = 0
3.90/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.90/1.89	
3.90/1.89	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(8)
3.90/1.89	Obligation:
3.90/1.89	Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(9) CSRRRRProof (EQUIVALENT)
3.90/1.89	The following CSR is given: Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	incr: {1}
3.90/1.89	s: empty set
3.90/1.89	Used ordering:
3.90/1.89	Polynomial interpretation [POLO]:
3.90/1.89	
3.90/1.89	   POL(0) = 2
3.90/1.89	   POL(adx(x_1)) = x_1
3.90/1.89	   POL(cons(x_1, x_2)) = 2
3.90/1.89	   POL(incr(x_1)) = 1 + 2*x_1
3.90/1.89	   POL(nats) = 2
3.90/1.89	   POL(s(x_1)) = 2 + x_1
3.90/1.89	   POL(zeros) = 2
3.90/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.90/1.89	
3.90/1.89	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(10)
3.90/1.89	Obligation:
3.90/1.89	Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(11) CSRRRRProof (EQUIVALENT)
3.90/1.89	The following CSR is given: Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	cons: empty set
3.90/1.89	0: empty set
3.90/1.89	Used ordering:
3.90/1.89	Polynomial interpretation [POLO]:
3.90/1.89	
3.90/1.89	   POL(0) = 1
3.90/1.89	   POL(adx(x_1)) = x_1
3.90/1.89	   POL(cons(x_1, x_2)) = 0
3.90/1.89	   POL(nats) = 1
3.90/1.89	   POL(zeros) = 1
3.90/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.90/1.89	
3.90/1.89	   zeros -> cons(0, zeros)
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(12)
3.90/1.89	Obligation:
3.90/1.89	Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(13) CSRRRRProof (EQUIVALENT)
3.90/1.89	The following CSR is given: Context-sensitive rewrite system:
3.90/1.89	The TRS R consists of the following rules:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	
3.90/1.89	The replacement map contains the following entries:
3.90/1.89	
3.90/1.89	nats: empty set
3.90/1.89	adx: {1}
3.90/1.89	zeros: empty set
3.90/1.89	Used ordering:
3.90/1.89	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.90/1.89	
3.90/1.89	   <<<
3.90/1.89	 POL(nats) =  	[[1]]
3.90/1.89	>>>
3.90/1.89	
3.90/1.89	   <<<
3.90/1.89	 POL(adx(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
3.90/1.89	>>>
3.90/1.89	
3.90/1.89	   <<<
3.90/1.89	 POL(zeros) =  	[[0], [0]]
3.90/1.89	>>>
3.90/1.89	
3.90/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.90/1.89	
3.90/1.89	   nats -> adx(zeros)
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(14)
3.90/1.89	Obligation:
3.90/1.89	Context-sensitive rewrite system:
3.90/1.89	R is empty.
3.90/1.89	
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(15) RisEmptyProof (EQUIVALENT)
3.90/1.89	The CSR R is empty. Hence, termination is trivially proven.
3.90/1.89	----------------------------------------
3.90/1.89	
3.90/1.89	(16)
3.90/1.89	YES
3.95/1.92	EOF