4.31/2.03	YES
4.31/2.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.31/2.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.31/2.05	
4.31/2.05	
4.31/2.05	Termination w.r.t. Q of the given QTRS could be proven:
4.31/2.05	
4.31/2.05	(0) QTRS
4.31/2.05	(1) QTRSToCSRProof [SOUND, 0 ms]
4.31/2.05	(2) CSR
4.31/2.05	(3) CSRRRRProof [EQUIVALENT, 67 ms]
4.31/2.05	(4) CSR
4.31/2.05	(5) CSRRRRProof [EQUIVALENT, 12 ms]
4.31/2.05	(6) CSR
4.31/2.05	(7) CSRRRRProof [EQUIVALENT, 0 ms]
4.31/2.05	(8) CSR
4.31/2.05	(9) CSRRRRProof [EQUIVALENT, 0 ms]
4.31/2.05	(10) CSR
4.31/2.05	(11) CSRRRRProof [EQUIVALENT, 3 ms]
4.31/2.05	(12) CSR
4.31/2.05	(13) CSRRRRProof [EQUIVALENT, 0 ms]
4.31/2.05	(14) CSR
4.31/2.05	(15) RisEmptyProof [EQUIVALENT, 0 ms]
4.31/2.05	(16) YES
4.31/2.05	
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(0)
4.31/2.05	Obligation:
4.31/2.05	Q restricted rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
4.31/2.05	   active(sqr(0)) -> mark(0)
4.31/2.05	   active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
4.31/2.05	   active(dbl(0)) -> mark(0)
4.31/2.05	   active(dbl(s(X))) -> mark(s(s(dbl(X))))
4.31/2.05	   active(add(0, X)) -> mark(X)
4.31/2.05	   active(add(s(X), Y)) -> mark(s(add(X, Y)))
4.31/2.05	   active(first(0, X)) -> mark(nil)
4.31/2.05	   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
4.31/2.05	   active(terms(X)) -> terms(active(X))
4.31/2.05	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.31/2.05	   active(recip(X)) -> recip(active(X))
4.31/2.05	   active(sqr(X)) -> sqr(active(X))
4.31/2.05	   active(add(X1, X2)) -> add(active(X1), X2)
4.31/2.05	   active(add(X1, X2)) -> add(X1, active(X2))
4.31/2.05	   active(dbl(X)) -> dbl(active(X))
4.31/2.05	   active(first(X1, X2)) -> first(active(X1), X2)
4.31/2.05	   active(first(X1, X2)) -> first(X1, active(X2))
4.31/2.05	   terms(mark(X)) -> mark(terms(X))
4.31/2.05	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.31/2.05	   recip(mark(X)) -> mark(recip(X))
4.31/2.05	   sqr(mark(X)) -> mark(sqr(X))
4.31/2.05	   add(mark(X1), X2) -> mark(add(X1, X2))
4.31/2.05	   add(X1, mark(X2)) -> mark(add(X1, X2))
4.31/2.05	   dbl(mark(X)) -> mark(dbl(X))
4.31/2.05	   first(mark(X1), X2) -> mark(first(X1, X2))
4.31/2.05	   first(X1, mark(X2)) -> mark(first(X1, X2))
4.31/2.05	   proper(terms(X)) -> terms(proper(X))
4.31/2.05	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.31/2.05	   proper(recip(X)) -> recip(proper(X))
4.31/2.05	   proper(sqr(X)) -> sqr(proper(X))
4.31/2.05	   proper(s(X)) -> s(proper(X))
4.31/2.05	   proper(0) -> ok(0)
4.31/2.05	   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
4.31/2.05	   proper(dbl(X)) -> dbl(proper(X))
4.31/2.05	   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
4.31/2.05	   proper(nil) -> ok(nil)
4.31/2.05	   terms(ok(X)) -> ok(terms(X))
4.31/2.05	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.31/2.05	   recip(ok(X)) -> ok(recip(X))
4.31/2.05	   sqr(ok(X)) -> ok(sqr(X))
4.31/2.05	   s(ok(X)) -> ok(s(X))
4.31/2.05	   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
4.31/2.05	   dbl(ok(X)) -> ok(dbl(X))
4.31/2.05	   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
4.31/2.05	   top(mark(X)) -> top(proper(X))
4.31/2.05	   top(ok(X)) -> top(active(X))
4.31/2.05	
4.31/2.05	The set Q consists of the following terms:
4.31/2.05	
4.31/2.05	   active(terms(x0))
4.31/2.05	   active(cons(x0, x1))
4.31/2.05	   active(recip(x0))
4.31/2.05	   active(sqr(x0))
4.31/2.05	   active(add(x0, x1))
4.31/2.05	   active(dbl(x0))
4.31/2.05	   active(first(x0, x1))
4.31/2.05	   terms(mark(x0))
4.31/2.05	   cons(mark(x0), x1)
4.31/2.05	   recip(mark(x0))
4.31/2.05	   sqr(mark(x0))
4.31/2.05	   add(mark(x0), x1)
4.31/2.05	   add(x0, mark(x1))
4.31/2.05	   dbl(mark(x0))
4.31/2.05	   first(mark(x0), x1)
4.31/2.05	   first(x0, mark(x1))
4.31/2.05	   proper(terms(x0))
4.31/2.05	   proper(cons(x0, x1))
4.31/2.05	   proper(recip(x0))
4.31/2.05	   proper(sqr(x0))
4.31/2.05	   proper(s(x0))
4.31/2.05	   proper(0)
4.31/2.05	   proper(add(x0, x1))
4.31/2.05	   proper(dbl(x0))
4.31/2.05	   proper(first(x0, x1))
4.31/2.05	   proper(nil)
4.31/2.05	   terms(ok(x0))
4.31/2.05	   cons(ok(x0), ok(x1))
4.31/2.05	   recip(ok(x0))
4.31/2.05	   sqr(ok(x0))
4.31/2.05	   s(ok(x0))
4.31/2.05	   add(ok(x0), ok(x1))
4.31/2.05	   dbl(ok(x0))
4.31/2.05	   first(ok(x0), ok(x1))
4.31/2.05	   top(mark(x0))
4.31/2.05	   top(ok(x0))
4.31/2.05	
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(1) QTRSToCSRProof (SOUND)
4.31/2.05	The following Q TRS is given: Q restricted rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
4.31/2.05	   active(sqr(0)) -> mark(0)
4.31/2.05	   active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
4.31/2.05	   active(dbl(0)) -> mark(0)
4.31/2.05	   active(dbl(s(X))) -> mark(s(s(dbl(X))))
4.31/2.05	   active(add(0, X)) -> mark(X)
4.31/2.05	   active(add(s(X), Y)) -> mark(s(add(X, Y)))
4.31/2.05	   active(first(0, X)) -> mark(nil)
4.31/2.05	   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
4.31/2.05	   active(terms(X)) -> terms(active(X))
4.31/2.05	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.31/2.05	   active(recip(X)) -> recip(active(X))
4.31/2.05	   active(sqr(X)) -> sqr(active(X))
4.31/2.05	   active(add(X1, X2)) -> add(active(X1), X2)
4.31/2.05	   active(add(X1, X2)) -> add(X1, active(X2))
4.31/2.05	   active(dbl(X)) -> dbl(active(X))
4.31/2.05	   active(first(X1, X2)) -> first(active(X1), X2)
4.31/2.05	   active(first(X1, X2)) -> first(X1, active(X2))
4.31/2.05	   terms(mark(X)) -> mark(terms(X))
4.31/2.05	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.31/2.05	   recip(mark(X)) -> mark(recip(X))
4.31/2.05	   sqr(mark(X)) -> mark(sqr(X))
4.31/2.05	   add(mark(X1), X2) -> mark(add(X1, X2))
4.31/2.05	   add(X1, mark(X2)) -> mark(add(X1, X2))
4.31/2.05	   dbl(mark(X)) -> mark(dbl(X))
4.31/2.05	   first(mark(X1), X2) -> mark(first(X1, X2))
4.31/2.05	   first(X1, mark(X2)) -> mark(first(X1, X2))
4.31/2.05	   proper(terms(X)) -> terms(proper(X))
4.31/2.05	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.31/2.05	   proper(recip(X)) -> recip(proper(X))
4.31/2.05	   proper(sqr(X)) -> sqr(proper(X))
4.31/2.05	   proper(s(X)) -> s(proper(X))
4.31/2.05	   proper(0) -> ok(0)
4.31/2.05	   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
4.31/2.05	   proper(dbl(X)) -> dbl(proper(X))
4.31/2.05	   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
4.31/2.05	   proper(nil) -> ok(nil)
4.31/2.05	   terms(ok(X)) -> ok(terms(X))
4.31/2.05	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.31/2.05	   recip(ok(X)) -> ok(recip(X))
4.31/2.05	   sqr(ok(X)) -> ok(sqr(X))
4.31/2.05	   s(ok(X)) -> ok(s(X))
4.31/2.05	   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
4.31/2.05	   dbl(ok(X)) -> ok(dbl(X))
4.31/2.05	   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
4.31/2.05	   top(mark(X)) -> top(proper(X))
4.31/2.05	   top(ok(X)) -> top(active(X))
4.31/2.05	
4.31/2.05	The set Q consists of the following terms:
4.31/2.05	
4.31/2.05	   active(terms(x0))
4.31/2.05	   active(cons(x0, x1))
4.31/2.05	   active(recip(x0))
4.31/2.05	   active(sqr(x0))
4.31/2.05	   active(add(x0, x1))
4.31/2.05	   active(dbl(x0))
4.31/2.05	   active(first(x0, x1))
4.31/2.05	   terms(mark(x0))
4.31/2.05	   cons(mark(x0), x1)
4.31/2.05	   recip(mark(x0))
4.31/2.05	   sqr(mark(x0))
4.31/2.05	   add(mark(x0), x1)
4.31/2.05	   add(x0, mark(x1))
4.31/2.05	   dbl(mark(x0))
4.31/2.05	   first(mark(x0), x1)
4.31/2.05	   first(x0, mark(x1))
4.31/2.05	   proper(terms(x0))
4.31/2.05	   proper(cons(x0, x1))
4.31/2.05	   proper(recip(x0))
4.31/2.05	   proper(sqr(x0))
4.31/2.05	   proper(s(x0))
4.31/2.05	   proper(0)
4.31/2.05	   proper(add(x0, x1))
4.31/2.05	   proper(dbl(x0))
4.31/2.05	   proper(first(x0, x1))
4.31/2.05	   proper(nil)
4.31/2.05	   terms(ok(x0))
4.31/2.05	   cons(ok(x0), ok(x1))
4.31/2.05	   recip(ok(x0))
4.31/2.05	   sqr(ok(x0))
4.31/2.05	   s(ok(x0))
4.31/2.05	   add(ok(x0), ok(x1))
4.31/2.05	   dbl(ok(x0))
4.31/2.05	   first(ok(x0), ok(x1))
4.31/2.05	   top(mark(x0))
4.31/2.05	   top(ok(x0))
4.31/2.05	
4.31/2.05	Special symbols used for the transformation (see [GM04]):
4.31/2.05	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	first: {1, 2}
4.31/2.05	nil: empty set
4.31/2.05	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.
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(2)
4.31/2.05	Obligation:
4.31/2.05	Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	   add(0, X) -> X
4.31/2.05	   add(s(X), Y) -> s(add(X, Y))
4.31/2.05	   first(0, X) -> nil
4.31/2.05	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	first: {1, 2}
4.31/2.05	nil: empty set
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(3) CSRRRRProof (EQUIVALENT)
4.31/2.05	The following CSR is given: Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	   add(0, X) -> X
4.31/2.05	   add(s(X), Y) -> s(add(X, Y))
4.31/2.05	   first(0, X) -> nil
4.31/2.05	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	first: {1, 2}
4.31/2.05	nil: empty set
4.31/2.05	Used ordering:
4.31/2.05	Polynomial interpretation [POLO]:
4.31/2.05	
4.31/2.05	   POL(0) = 1
4.31/2.05	   POL(add(x_1, x_2)) = x_1 + x_2
4.31/2.05	   POL(cons(x_1, x_2)) = x_1
4.31/2.05	   POL(dbl(x_1)) = x_1
4.31/2.05	   POL(first(x_1, x_2)) = x_1 + x_2
4.31/2.05	   POL(nil) = 1
4.31/2.05	   POL(recip(x_1)) = 1 + x_1
4.31/2.05	   POL(s(x_1)) = 0
4.31/2.05	   POL(sqr(x_1)) = x_1
4.31/2.05	   POL(terms(x_1)) = 1 + x_1
4.31/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.31/2.05	
4.31/2.05	   add(0, X) -> X
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(4)
4.31/2.05	Obligation:
4.31/2.05	Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	   add(s(X), Y) -> s(add(X, Y))
4.31/2.05	   first(0, X) -> nil
4.31/2.05	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	first: {1, 2}
4.31/2.05	nil: empty set
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(5) CSRRRRProof (EQUIVALENT)
4.31/2.05	The following CSR is given: Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	   add(s(X), Y) -> s(add(X, Y))
4.31/2.05	   first(0, X) -> nil
4.31/2.05	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	first: {1, 2}
4.31/2.05	nil: empty set
4.31/2.05	Used ordering:
4.31/2.05	Polynomial interpretation [POLO]:
4.31/2.05	
4.31/2.05	   POL(0) = 1
4.31/2.05	   POL(add(x_1, x_2)) = x_1 + x_2
4.31/2.05	   POL(cons(x_1, x_2)) = x_1
4.31/2.05	   POL(dbl(x_1)) = x_1
4.31/2.05	   POL(first(x_1, x_2)) = 1 + x_1 + x_2
4.31/2.05	   POL(nil) = 0
4.31/2.05	   POL(recip(x_1)) = 1 + x_1
4.31/2.05	   POL(s(x_1)) = 1
4.31/2.05	   POL(sqr(x_1)) = x_1
4.31/2.05	   POL(terms(x_1)) = 1 + x_1
4.31/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.31/2.05	
4.31/2.05	   first(0, X) -> nil
4.31/2.05	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(6)
4.31/2.05	Obligation:
4.31/2.05	Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	   add(s(X), Y) -> s(add(X, Y))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(7) CSRRRRProof (EQUIVALENT)
4.31/2.05	The following CSR is given: Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	   add(s(X), Y) -> s(add(X, Y))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	Used ordering:
4.31/2.05	Polynomial interpretation [POLO]:
4.31/2.05	
4.31/2.05	   POL(0) = 0
4.31/2.05	   POL(add(x_1, x_2)) = 1 + x_1 + x_2
4.31/2.05	   POL(cons(x_1, x_2)) = x_1 + x_2
4.31/2.05	   POL(dbl(x_1)) = x_1
4.31/2.05	   POL(recip(x_1)) = x_1
4.31/2.05	   POL(s(x_1)) = 0
4.31/2.05	   POL(sqr(x_1)) = x_1
4.31/2.05	   POL(terms(x_1)) = 1 + x_1
4.31/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.31/2.05	
4.31/2.05	   add(s(X), Y) -> s(add(X, Y))
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(8)
4.31/2.05	Obligation:
4.31/2.05	Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(9) CSRRRRProof (EQUIVALENT)
4.31/2.05	The following CSR is given: Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	Used ordering:
4.31/2.05	Polynomial interpretation [POLO]:
4.31/2.05	
4.31/2.05	   POL(0) = 0
4.31/2.05	   POL(add(x_1, x_2)) = x_1 + x_2
4.31/2.05	   POL(cons(x_1, x_2)) = x_1 + x_2
4.31/2.05	   POL(dbl(x_1)) = 1 + x_1
4.31/2.05	   POL(recip(x_1)) = x_1
4.31/2.05	   POL(s(x_1)) = 0
4.31/2.05	   POL(sqr(x_1)) = x_1
4.31/2.05	   POL(terms(x_1)) = 1 + x_1
4.31/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.31/2.05	
4.31/2.05	   dbl(0) -> 0
4.31/2.05	   dbl(s(X)) -> s(s(dbl(X)))
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(10)
4.31/2.05	Obligation:
4.31/2.05	Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(11) CSRRRRProof (EQUIVALENT)
4.31/2.05	The following CSR is given: Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	0: empty set
4.31/2.05	add: {1, 2}
4.31/2.05	dbl: {1}
4.31/2.05	Used ordering:
4.31/2.05	Polynomial interpretation [POLO]:
4.31/2.05	
4.31/2.05	   POL(0) = 0
4.31/2.05	   POL(add(x_1, x_2)) = x_1 + x_2
4.31/2.05	   POL(cons(x_1, x_2)) = x_1
4.31/2.05	   POL(dbl(x_1)) = x_1
4.31/2.05	   POL(recip(x_1)) = x_1
4.31/2.05	   POL(s(x_1)) = 1
4.31/2.05	   POL(sqr(x_1)) = 1 + x_1
4.31/2.05	   POL(terms(x_1)) = 1 + x_1
4.31/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.31/2.05	
4.31/2.05	   sqr(0) -> 0
4.31/2.05	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(12)
4.31/2.05	Obligation:
4.31/2.05	Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(13) CSRRRRProof (EQUIVALENT)
4.31/2.05	The following CSR is given: Context-sensitive rewrite system:
4.31/2.05	The TRS R consists of the following rules:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	
4.31/2.05	The replacement map contains the following entries:
4.31/2.05	
4.31/2.05	terms: {1}
4.31/2.05	cons: {1}
4.31/2.05	recip: {1}
4.31/2.05	sqr: {1}
4.31/2.05	s: empty set
4.31/2.05	Used ordering:
4.31/2.05	Polynomial interpretation [POLO]:
4.31/2.05	
4.31/2.05	   POL(cons(x_1, x_2)) = x_1
4.31/2.05	   POL(recip(x_1)) = 2*x_1
4.31/2.05	   POL(s(x_1)) = 0
4.31/2.05	   POL(sqr(x_1)) = x_1
4.31/2.05	   POL(terms(x_1)) = 1 + 2*x_1
4.31/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.31/2.05	
4.31/2.05	   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(14)
4.31/2.05	Obligation:
4.31/2.05	Context-sensitive rewrite system:
4.31/2.05	R is empty.
4.31/2.05	
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(15) RisEmptyProof (EQUIVALENT)
4.31/2.05	The CSR R is empty. Hence, termination is trivially proven.
4.31/2.05	----------------------------------------
4.31/2.05	
4.31/2.05	(16)
4.31/2.05	YES
4.50/2.09	EOF