4.00/1.84	YES
4.00/1.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.00/1.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.00/1.85	
4.00/1.85	
4.00/1.85	Termination w.r.t. Q of the given QTRS could be proven:
4.00/1.85	
4.00/1.85	(0) QTRS
4.00/1.85	(1) QTRSToCSRProof [SOUND, 0 ms]
4.00/1.85	(2) CSR
4.00/1.85	(3) CSRRRRProof [EQUIVALENT, 22 ms]
4.00/1.85	(4) CSR
4.00/1.85	(5) CSRRRRProof [EQUIVALENT, 0 ms]
4.00/1.85	(6) CSR
4.00/1.85	(7) CSRRRRProof [EQUIVALENT, 6 ms]
4.00/1.85	(8) CSR
4.00/1.85	(9) CSRRRRProof [EQUIVALENT, 0 ms]
4.00/1.85	(10) CSR
4.00/1.85	(11) CSRRRRProof [EQUIVALENT, 0 ms]
4.00/1.85	(12) CSR
4.00/1.85	(13) CSRRRRProof [EQUIVALENT, 0 ms]
4.00/1.85	(14) CSR
4.00/1.85	(15) RisEmptyProof [EQUIVALENT, 4 ms]
4.00/1.85	(16) YES
4.00/1.85	
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(0)
4.00/1.85	Obligation:
4.00/1.85	Q restricted rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   active(incr(nil)) -> mark(nil)
4.00/1.85	   active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
4.00/1.85	   active(adx(nil)) -> mark(nil)
4.00/1.85	   active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
4.00/1.85	   active(nats) -> mark(adx(zeros))
4.00/1.85	   active(zeros) -> mark(cons(0, zeros))
4.00/1.85	   active(head(cons(X, L))) -> mark(X)
4.00/1.85	   active(tail(cons(X, L))) -> mark(L)
4.00/1.85	   active(incr(X)) -> incr(active(X))
4.00/1.85	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.00/1.85	   active(s(X)) -> s(active(X))
4.00/1.85	   active(adx(X)) -> adx(active(X))
4.00/1.85	   active(head(X)) -> head(active(X))
4.00/1.85	   active(tail(X)) -> tail(active(X))
4.00/1.85	   incr(mark(X)) -> mark(incr(X))
4.00/1.85	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.00/1.85	   s(mark(X)) -> mark(s(X))
4.00/1.85	   adx(mark(X)) -> mark(adx(X))
4.00/1.85	   head(mark(X)) -> mark(head(X))
4.00/1.85	   tail(mark(X)) -> mark(tail(X))
4.00/1.85	   proper(incr(X)) -> incr(proper(X))
4.00/1.85	   proper(nil) -> ok(nil)
4.00/1.85	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.00/1.85	   proper(s(X)) -> s(proper(X))
4.00/1.85	   proper(adx(X)) -> adx(proper(X))
4.00/1.85	   proper(nats) -> ok(nats)
4.00/1.85	   proper(zeros) -> ok(zeros)
4.00/1.85	   proper(0) -> ok(0)
4.00/1.85	   proper(head(X)) -> head(proper(X))
4.00/1.85	   proper(tail(X)) -> tail(proper(X))
4.00/1.85	   incr(ok(X)) -> ok(incr(X))
4.00/1.85	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.00/1.85	   s(ok(X)) -> ok(s(X))
4.00/1.85	   adx(ok(X)) -> ok(adx(X))
4.00/1.85	   head(ok(X)) -> ok(head(X))
4.00/1.85	   tail(ok(X)) -> ok(tail(X))
4.00/1.85	   top(mark(X)) -> top(proper(X))
4.00/1.85	   top(ok(X)) -> top(active(X))
4.00/1.85	
4.00/1.85	The set Q consists of the following terms:
4.00/1.85	
4.00/1.85	   active(nats)
4.00/1.85	   active(zeros)
4.00/1.85	   active(incr(x0))
4.00/1.85	   active(cons(x0, x1))
4.00/1.85	   active(s(x0))
4.00/1.85	   active(adx(x0))
4.00/1.85	   active(head(x0))
4.00/1.85	   active(tail(x0))
4.00/1.85	   incr(mark(x0))
4.00/1.85	   cons(mark(x0), x1)
4.00/1.85	   s(mark(x0))
4.00/1.85	   adx(mark(x0))
4.00/1.85	   head(mark(x0))
4.00/1.85	   tail(mark(x0))
4.00/1.85	   proper(incr(x0))
4.00/1.85	   proper(nil)
4.00/1.85	   proper(cons(x0, x1))
4.00/1.85	   proper(s(x0))
4.00/1.85	   proper(adx(x0))
4.00/1.85	   proper(nats)
4.00/1.85	   proper(zeros)
4.00/1.85	   proper(0)
4.00/1.85	   proper(head(x0))
4.00/1.85	   proper(tail(x0))
4.00/1.85	   incr(ok(x0))
4.00/1.85	   cons(ok(x0), ok(x1))
4.00/1.85	   s(ok(x0))
4.00/1.85	   adx(ok(x0))
4.00/1.85	   head(ok(x0))
4.00/1.85	   tail(ok(x0))
4.00/1.85	   top(mark(x0))
4.00/1.85	   top(ok(x0))
4.00/1.85	
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(1) QTRSToCSRProof (SOUND)
4.00/1.85	The following Q TRS is given: Q restricted rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   active(incr(nil)) -> mark(nil)
4.00/1.85	   active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
4.00/1.85	   active(adx(nil)) -> mark(nil)
4.00/1.85	   active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
4.00/1.85	   active(nats) -> mark(adx(zeros))
4.00/1.85	   active(zeros) -> mark(cons(0, zeros))
4.00/1.85	   active(head(cons(X, L))) -> mark(X)
4.00/1.85	   active(tail(cons(X, L))) -> mark(L)
4.00/1.85	   active(incr(X)) -> incr(active(X))
4.00/1.85	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.00/1.85	   active(s(X)) -> s(active(X))
4.00/1.85	   active(adx(X)) -> adx(active(X))
4.00/1.85	   active(head(X)) -> head(active(X))
4.00/1.85	   active(tail(X)) -> tail(active(X))
4.00/1.85	   incr(mark(X)) -> mark(incr(X))
4.00/1.85	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.00/1.85	   s(mark(X)) -> mark(s(X))
4.00/1.85	   adx(mark(X)) -> mark(adx(X))
4.00/1.85	   head(mark(X)) -> mark(head(X))
4.00/1.85	   tail(mark(X)) -> mark(tail(X))
4.00/1.85	   proper(incr(X)) -> incr(proper(X))
4.00/1.85	   proper(nil) -> ok(nil)
4.00/1.85	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.00/1.85	   proper(s(X)) -> s(proper(X))
4.00/1.85	   proper(adx(X)) -> adx(proper(X))
4.00/1.85	   proper(nats) -> ok(nats)
4.00/1.85	   proper(zeros) -> ok(zeros)
4.00/1.85	   proper(0) -> ok(0)
4.00/1.85	   proper(head(X)) -> head(proper(X))
4.00/1.85	   proper(tail(X)) -> tail(proper(X))
4.00/1.85	   incr(ok(X)) -> ok(incr(X))
4.00/1.85	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.00/1.85	   s(ok(X)) -> ok(s(X))
4.00/1.85	   adx(ok(X)) -> ok(adx(X))
4.00/1.85	   head(ok(X)) -> ok(head(X))
4.00/1.85	   tail(ok(X)) -> ok(tail(X))
4.00/1.85	   top(mark(X)) -> top(proper(X))
4.00/1.85	   top(ok(X)) -> top(active(X))
4.00/1.85	
4.00/1.85	The set Q consists of the following terms:
4.00/1.85	
4.00/1.85	   active(nats)
4.00/1.85	   active(zeros)
4.00/1.85	   active(incr(x0))
4.00/1.85	   active(cons(x0, x1))
4.00/1.85	   active(s(x0))
4.00/1.85	   active(adx(x0))
4.00/1.85	   active(head(x0))
4.00/1.85	   active(tail(x0))
4.00/1.85	   incr(mark(x0))
4.00/1.85	   cons(mark(x0), x1)
4.00/1.85	   s(mark(x0))
4.00/1.85	   adx(mark(x0))
4.00/1.85	   head(mark(x0))
4.00/1.85	   tail(mark(x0))
4.00/1.85	   proper(incr(x0))
4.00/1.85	   proper(nil)
4.00/1.85	   proper(cons(x0, x1))
4.00/1.85	   proper(s(x0))
4.00/1.85	   proper(adx(x0))
4.00/1.85	   proper(nats)
4.00/1.85	   proper(zeros)
4.00/1.85	   proper(0)
4.00/1.85	   proper(head(x0))
4.00/1.85	   proper(tail(x0))
4.00/1.85	   incr(ok(x0))
4.00/1.85	   cons(ok(x0), ok(x1))
4.00/1.85	   s(ok(x0))
4.00/1.85	   adx(ok(x0))
4.00/1.85	   head(ok(x0))
4.00/1.85	   tail(ok(x0))
4.00/1.85	   top(mark(x0))
4.00/1.85	   top(ok(x0))
4.00/1.85	
4.00/1.85	Special symbols used for the transformation (see [GM04]):
4.00/1.85	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	head: {1}
4.00/1.85	tail: {1}
4.00/1.85	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.
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(2)
4.00/1.85	Obligation:
4.00/1.85	Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(nil) -> nil
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	   head(cons(X, L)) -> X
4.00/1.85	   tail(cons(X, L)) -> L
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	head: {1}
4.00/1.85	tail: {1}
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(3) CSRRRRProof (EQUIVALENT)
4.00/1.85	The following CSR is given: Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(nil) -> nil
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	   head(cons(X, L)) -> X
4.00/1.85	   tail(cons(X, L)) -> L
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	head: {1}
4.00/1.85	tail: {1}
4.00/1.85	Used ordering:
4.00/1.85	Polynomial interpretation [POLO]:
4.00/1.85	
4.00/1.85	   POL(0) = 0
4.00/1.85	   POL(adx(x_1)) = x_1
4.00/1.85	   POL(cons(x_1, x_2)) = x_1 + x_2
4.00/1.85	   POL(head(x_1)) = 1 + x_1
4.00/1.85	   POL(incr(x_1)) = x_1
4.00/1.85	   POL(nats) = 1
4.00/1.85	   POL(nil) = 0
4.00/1.85	   POL(s(x_1)) = x_1
4.00/1.85	   POL(tail(x_1)) = x_1
4.00/1.85	   POL(zeros) = 1
4.00/1.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.00/1.85	
4.00/1.85	   head(cons(X, L)) -> X
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(4)
4.00/1.85	Obligation:
4.00/1.85	Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(nil) -> nil
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	   tail(cons(X, L)) -> L
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	tail: {1}
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(5) CSRRRRProof (EQUIVALENT)
4.00/1.85	The following CSR is given: Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(nil) -> nil
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	   tail(cons(X, L)) -> L
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	tail: {1}
4.00/1.85	Used ordering:
4.00/1.85	Polynomial interpretation [POLO]:
4.00/1.85	
4.00/1.85	   POL(0) = 0
4.00/1.85	   POL(adx(x_1)) = 1 + x_1
4.00/1.85	   POL(cons(x_1, x_2)) = x_1 + x_2
4.00/1.85	   POL(incr(x_1)) = x_1
4.00/1.85	   POL(nats) = 1
4.00/1.85	   POL(nil) = 0
4.00/1.85	   POL(s(x_1)) = x_1
4.00/1.85	   POL(tail(x_1)) = x_1
4.00/1.85	   POL(zeros) = 0
4.00/1.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.00/1.85	
4.00/1.85	   adx(nil) -> nil
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(6)
4.00/1.85	Obligation:
4.00/1.85	Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	   tail(cons(X, L)) -> L
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	tail: {1}
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(7) CSRRRRProof (EQUIVALENT)
4.00/1.85	The following CSR is given: Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	   tail(cons(X, L)) -> L
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	tail: {1}
4.00/1.85	Used ordering:
4.00/1.85	Polynomial interpretation [POLO]:
4.00/1.85	
4.00/1.85	   POL(0) = 0
4.00/1.85	   POL(adx(x_1)) = 1 + x_1
4.00/1.85	   POL(cons(x_1, x_2)) = x_1 + x_2
4.00/1.85	   POL(incr(x_1)) = x_1
4.00/1.85	   POL(nats) = 1
4.00/1.85	   POL(nil) = 0
4.00/1.85	   POL(s(x_1)) = x_1
4.00/1.85	   POL(tail(x_1)) = 1 + x_1
4.00/1.85	   POL(zeros) = 0
4.00/1.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.00/1.85	
4.00/1.85	   tail(cons(X, L)) -> L
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(8)
4.00/1.85	Obligation:
4.00/1.85	Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(9) CSRRRRProof (EQUIVALENT)
4.00/1.85	The following CSR is given: Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	0: empty set
4.00/1.85	Used ordering:
4.00/1.85	Polynomial interpretation [POLO]:
4.00/1.85	
4.00/1.85	   POL(0) = 0
4.00/1.85	   POL(adx(x_1)) = x_1
4.00/1.85	   POL(cons(x_1, x_2)) = x_1
4.00/1.85	   POL(incr(x_1)) = x_1
4.00/1.85	   POL(nats) = 1
4.00/1.85	   POL(nil) = 0
4.00/1.85	   POL(s(x_1)) = x_1
4.00/1.85	   POL(zeros) = 1
4.00/1.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.00/1.85	
4.00/1.85	   zeros -> cons(0, zeros)
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(10)
4.00/1.85	Obligation:
4.00/1.85	Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(11) CSRRRRProof (EQUIVALENT)
4.00/1.85	The following CSR is given: Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	incr: {1}
4.00/1.85	nil: empty set
4.00/1.85	cons: {1}
4.00/1.85	s: {1}
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	Used ordering:
4.00/1.85	Polynomial interpretation [POLO]:
4.00/1.85	
4.00/1.85	   POL(adx(x_1)) = 2 + 2*x_1
4.00/1.85	   POL(cons(x_1, x_2)) = 1 + x_1
4.00/1.85	   POL(incr(x_1)) = 1 + 2*x_1
4.00/1.85	   POL(nats) = 2
4.00/1.85	   POL(nil) = 0
4.00/1.85	   POL(s(x_1)) = 2*x_1
4.00/1.85	   POL(zeros) = 0
4.00/1.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.00/1.85	
4.00/1.85	   incr(nil) -> nil
4.00/1.85	   incr(cons(X, L)) -> cons(s(X), incr(L))
4.00/1.85	   adx(cons(X, L)) -> incr(cons(X, adx(L)))
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(12)
4.00/1.85	Obligation:
4.00/1.85	Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(13) CSRRRRProof (EQUIVALENT)
4.00/1.85	The following CSR is given: Context-sensitive rewrite system:
4.00/1.85	The TRS R consists of the following rules:
4.00/1.85	
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	
4.00/1.85	The replacement map contains the following entries:
4.00/1.85	
4.00/1.85	adx: {1}
4.00/1.85	nats: empty set
4.00/1.85	zeros: empty set
4.00/1.85	Used ordering:
4.00/1.85	Polynomial interpretation [POLO]:
4.00/1.85	
4.00/1.85	   POL(adx(x_1)) = x_1
4.00/1.85	   POL(nats) = 1
4.00/1.85	   POL(zeros) = 0
4.00/1.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.00/1.85	
4.00/1.85	   nats -> adx(zeros)
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(14)
4.00/1.85	Obligation:
4.00/1.85	Context-sensitive rewrite system:
4.00/1.85	R is empty.
4.00/1.85	
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(15) RisEmptyProof (EQUIVALENT)
4.00/1.85	The CSR R is empty. Hence, termination is trivially proven.
4.00/1.85	----------------------------------------
4.00/1.85	
4.00/1.85	(16)
4.00/1.85	YES
4.00/1.88	EOF