/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 34 ms]
(4) CSR
(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(6) QCSDP
(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(8) TRUE


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

   active(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
   active(g(b)) -> mark(c)
   active(b) -> mark(c)
   active(g(X)) -> g(active(X))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
   proper(g(X)) -> g(proper(X))
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   g(ok(X)) -> ok(g(X))
   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(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
   active(g(b)) -> mark(c)
   active(b) -> mark(c)
   active(g(X)) -> g(active(X))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
   proper(g(X)) -> g(proper(X))
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   g(ok(X)) -> ok(g(X))
   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:

f: empty set
g: {1}
b: empty set
c: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
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(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(X, g(X), Y) -> f(Y, Y, Y)
   g(b) -> c
   b -> c

The replacement map contains the following entries:

f: empty set
g: {1}
b: empty set
c: empty set

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

   f(X, g(X), Y) -> f(Y, Y, Y)
   g(b) -> c
   b -> c

The replacement map contains the following entries:

f: empty set
g: {1}
b: empty set
c: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(b) = 2
   POL(c) = 1
   POL(f(x_1, x_2, x_3)) = 0
   POL(g(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:

   g(b) -> c
   b -> c




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

   f(X, g(X), Y) -> f(Y, Y, Y)

The replacement map contains the following entries:

f: empty set
g: {1}

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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 {g_1} are replacing on all positions.
The symbols in {f_3, F_3} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:
   F(X, g(X), Y) -> F(Y, Y, Y)

The TRS R consists of the following rules:

   f(X, g(X), Y) -> f(Y, Y, Y)

Q is empty.

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(7) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F(z0, g(z0), z1) -> F(z1, z1, z1) and F(x0, g(x0), x1) -> F(x1, x1, x1) form no chain, because ECap^mu(F(z1, z1, z1)) = F(z1, z1, z1) does not unify with F(x0, g(x0), x1). 
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(8)
TRUE