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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRSToCSRProof [SOUND, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 49 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   active(first(0, X)) -> mark(nil)
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(from(x0))
   active(first(x0, x1))
   active(s(x0))
   active(cons(x0, x1))
   first(mark(x0), x1)
   first(x0, mark(x1))
   s(mark(x0))
   cons(mark(x0), x1)
   from(mark(x0))
   proper(first(x0, x1))
   proper(0)
   proper(nil)
   proper(s(x0))
   proper(cons(x0, x1))
   proper(from(x0))
   first(ok(x0), ok(x1))
   s(ok(x0))
   cons(ok(x0), ok(x1))
   from(ok(x0))
   top(mark(x0))
   top(ok(x0))


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

   active(first(0, X)) -> mark(nil)
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(from(x0))
   active(first(x0, x1))
   active(s(x0))
   active(cons(x0, x1))
   first(mark(x0), x1)
   first(x0, mark(x1))
   s(mark(x0))
   cons(mark(x0), x1)
   from(mark(x0))
   proper(first(x0, x1))
   proper(0)
   proper(nil)
   proper(s(x0))
   proper(cons(x0, x1))
   proper(from(x0))
   first(ok(x0), ok(x1))
   s(ok(x0))
   cons(ok(x0), ok(x1))
   from(ok(x0))
   top(mark(x0))
   top(ok(x0))

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:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}
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.
----------------------------------------

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

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

The replacement map contains the following entries:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}

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

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

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

The replacement map contains the following entries:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}
Used ordering:
Polynomial interpretation [POLO]:

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

   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))




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

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

   first(0, X) -> nil
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}

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

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

   first(0, X) -> nil
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 2
   POL(cons(x_1, x_2)) = x_1
   POL(first(x_1, x_2)) = 1 + x_1 + x_2
   POL(from(x_1)) = 2*x_1
   POL(nil) = 1
   POL(s(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:

   first(0, X) -> nil




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

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

   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

s: {1}
cons: {1}
from: {1}

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

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

   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

s: {1}
cons: {1}
from: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons(x_1, x_2)) = x_1
   POL(from(x_1)) = 1 + 2*x_1
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   from(X) -> cons(X, from(s(X)))




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

(8)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(9) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(10)
YES