YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 56 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)

The set Q consists of the following terms:

   terms(x0)
   sqr(0)
   sqr(s(x0))
   dbl(0)
   dbl(s(x0))
   add(0, x0)
   add(s(x0), x1)
   first(0, x0)
   first(s(x0), cons(x1))


----------------------------------------

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
terms_1 > [cons_1, first_2]
terms_1 > recip_1
terms_1 > sqr_1 > 0 > nil
terms_1 > sqr_1 > add_2 > s_1
terms_1 > sqr_1 > dbl_1 > s_1


Status:
terms_1: [1]
cons_1: [1]
recip_1: multiset status
sqr_1: multiset status
0: multiset status
s_1: multiset status
add_2: multiset status
dbl_1: multiset status
first_2: multiset status
nil: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   terms(N) -> cons(recip(sqr(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)




----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
R is empty.
The set Q consists of the following terms:

   terms(x0)
   sqr(0)
   sqr(s(x0))
   dbl(0)
   dbl(s(x0))
   add(0, x0)
   add(s(x0), x1)
   first(0, x0)
   first(s(x0), cons(x1))


----------------------------------------

(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(4)
YES