/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSToCSRProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 45 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 4 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

   active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
   active(sqr(0)) -> mark(0)
   active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
   active(dbl(0)) -> mark(0)
   active(dbl(s(X))) -> mark(s(s(dbl(X))))
   active(add(0, X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(first(0, X)) -> mark(nil)
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(terms(X)) -> terms(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(recip(X)) -> recip(active(X))
   active(sqr(X)) -> sqr(active(X))
   active(add(X1, X2)) -> add(active(X1), X2)
   active(add(X1, X2)) -> add(X1, active(X2))
   active(dbl(X)) -> dbl(active(X))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   terms(mark(X)) -> mark(terms(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   recip(mark(X)) -> mark(recip(X))
   sqr(mark(X)) -> mark(sqr(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   dbl(mark(X)) -> mark(dbl(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(terms(X)) -> terms(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(recip(X)) -> recip(proper(X))
   proper(sqr(X)) -> sqr(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(dbl(X)) -> dbl(proper(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   terms(ok(X)) -> ok(terms(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   recip(ok(X)) -> ok(recip(X))
   sqr(ok(X)) -> ok(sqr(X))
   s(ok(X)) -> ok(s(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   dbl(ok(X)) -> ok(dbl(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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

(1) QTRSToCSRProof (EQUIVALENT)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

   active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
   active(sqr(0)) -> mark(0)
   active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
   active(dbl(0)) -> mark(0)
   active(dbl(s(X))) -> mark(s(s(dbl(X))))
   active(add(0, X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(first(0, X)) -> mark(nil)
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(terms(X)) -> terms(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(recip(X)) -> recip(active(X))
   active(sqr(X)) -> sqr(active(X))
   active(add(X1, X2)) -> add(active(X1), X2)
   active(add(X1, X2)) -> add(X1, active(X2))
   active(dbl(X)) -> dbl(active(X))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   terms(mark(X)) -> mark(terms(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   recip(mark(X)) -> mark(recip(X))
   sqr(mark(X)) -> mark(sqr(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   dbl(mark(X)) -> mark(dbl(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(terms(X)) -> terms(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(recip(X)) -> recip(proper(X))
   proper(sqr(X)) -> sqr(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(dbl(X)) -> dbl(proper(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   terms(ok(X)) -> ok(terms(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   recip(ok(X)) -> ok(recip(X))
   sqr(ok(X)) -> ok(sqr(X))
   s(ok(X)) -> ok(s(X))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   dbl(ok(X)) -> ok(dbl(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set
0: empty set
add: {1, 2}
dbl: {1}
first: {1, 2}
nil: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
----------------------------------------

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), terms(s(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, Z)) -> cons(Y, first(X, Z))

The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set
0: empty set
add: {1, 2}
dbl: {1}
first: {1, 2}
nil: empty set

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

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), terms(s(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, Z)) -> cons(Y, first(X, Z))

The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set
0: empty set
add: {1, 2}
dbl: {1}
first: {1, 2}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(add(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_1
   POL(dbl(x_1)) = x_1
   POL(first(x_1, x_2)) = x_1 + x_2
   POL(nil) = 1
   POL(recip(x_1)) = 1 + x_1
   POL(s(x_1)) = 0
   POL(sqr(x_1)) = x_1
   POL(terms(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   add(0, X) -> X




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

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

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

The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set
0: empty set
add: {1, 2}
dbl: {1}
first: {1, 2}
nil: empty set

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

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

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

The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set
0: empty set
add: {1, 2}
dbl: {1}
first: {1, 2}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(add(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 2 + 2*x_1
   POL(dbl(x_1)) = 1 + x_1
   POL(first(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(nil) = 0
   POL(recip(x_1)) = x_1
   POL(s(x_1)) = 2
   POL(sqr(x_1)) = x_1
   POL(terms(x_1)) = 2 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))




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

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

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

The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set
0: empty set
add: {1, 2}
dbl: {1}

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

(7) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

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

The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set
0: empty set
add: {1, 2}
dbl: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(add(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_1
   POL(dbl(x_1)) = x_1
   POL(recip(x_1)) = x_1
   POL(s(x_1)) = 1
   POL(sqr(x_1)) = 1 + x_1
   POL(terms(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))




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

(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), terms(s(N)))

The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set

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(9) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), terms(s(N)))

The replacement map contains the following entries:

terms: {1}
cons: {1}
recip: {1}
sqr: {1}
s: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons(x_1, x_2)) = x_1
   POL(recip(x_1)) = 2*x_1
   POL(s(x_1)) = 0
   POL(sqr(x_1)) = x_1
   POL(terms(x_1)) = 1 + 2*x_1
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)), terms(s(N)))




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(10)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(11) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(12)
YES