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


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YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) CSRInnermostProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(6) QCSDP
(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QCSDP
        (10) QCSDPSubtermProof [EQUIVALENT, 26 ms]
        (11) QCSDP
        (12) PIsEmptyProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QCSDP
        (15) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (16) QCSDP
        (17) PIsEmptyProof [EQUIVALENT, 0 ms]
        (18) YES


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

   active(from(X)) -> mark(cons(X, from(s(X))))
   active(after(0, XS)) -> mark(XS)
   active(after(s(N), cons(X, XS))) -> mark(after(N, XS))
   active(from(X)) -> from(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(after(X1, X2)) -> after(active(X1), X2)
   active(after(X1, X2)) -> after(X1, active(X2))
   from(mark(X)) -> mark(from(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   after(mark(X1), X2) -> mark(after(X1, X2))
   after(X1, mark(X2)) -> mark(after(X1, X2))
   proper(from(X)) -> from(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(after(X1, X2)) -> after(proper(X1), proper(X2))
   proper(0) -> ok(0)
   from(ok(X)) -> ok(from(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   after(ok(X1), ok(X2)) -> ok(after(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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

   active(from(X)) -> mark(cons(X, from(s(X))))
   active(after(0, XS)) -> mark(XS)
   active(after(s(N), cons(X, XS))) -> mark(after(N, XS))
   active(from(X)) -> from(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(after(X1, X2)) -> after(active(X1), X2)
   active(after(X1, X2)) -> after(X1, active(X2))
   from(mark(X)) -> mark(from(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   after(mark(X1), X2) -> mark(after(X1, X2))
   after(X1, mark(X2)) -> mark(after(X1, X2))
   proper(from(X)) -> from(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(after(X1, X2)) -> after(proper(X1), proper(X2))
   proper(0) -> ok(0)
   from(ok(X)) -> ok(from(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   after(ok(X1), ok(X2)) -> ok(after(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:

from: {1}
cons: {1}
s: {1}
after: {1, 2}
0: 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:

   from(X) -> cons(X, from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, XS)

The replacement map contains the following entries:

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

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(3) CSRInnermostProof (EQUIVALENT)
The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
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(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, XS)

The replacement map contains the following entries:

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


Innermost Strategy.

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(5) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
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(6)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, after_2, AFTER_2, FROM_1} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.
The symbols in {U_1} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:
   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)

The collapsing dependency pairs are DP_c:
   AFTER(s(N), cons(X, XS)) -> XS


The hidden terms of R are:

   from(s(x0))

Every hiding context is built from:
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@5d246aef
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@58b92af

Hence, the new unhiding pairs DP_u are :
   AFTER(s(N), cons(X, XS)) -> U(XS)
   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)
   U(from(s(x0))) -> FROM(s(x0))

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, XS)

The set Q consists of the following terms:

   from(x0)
   after(0, x0)
   after(s(x0), cons(x1, x2))


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(7) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 2 less nodes.

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(8)
Complex Obligation (AND)

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(9)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, after_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.
The symbols in {U_1} are not replacing on any position.

The TRS P consists of the following rules:

   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, XS)

The set Q consists of the following terms:

   from(x0)
   after(0, x0)
   after(s(x0), cons(x1, x2))


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(10) QCSDPSubtermProof (EQUIVALENT)
We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.

   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)
The remaining pairs can at least be oriented weakly.
none
Used ordering:  Combined order from the following AFS and order.
U(x1)  =  x1


Subterm Order

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(11)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, after_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, XS)

The set Q consists of the following terms:

   from(x0)
   after(0, x0)
   after(s(x0), cons(x1, x2))


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(12) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
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(13)
YES

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(14)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, after_2, AFTER_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, XS)

The set Q consists of the following terms:

   from(x0)
   after(0, x0)
   after(s(x0), cons(x1, x2))


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(15) QCSDPSubtermProof (EQUIVALENT)
We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.

   AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)
The remaining pairs can at least be oriented weakly.
none
Used ordering:  Combined order from the following AFS and order.
AFTER(x1, x2)  =  x1


Subterm Order

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(16)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, after_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, XS)

The set Q consists of the following terms:

   from(x0)
   after(0, x0)
   after(s(x0), cons(x1, x2))


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(17) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
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(18)
YES