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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 34 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRInnermostProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(10) QCSDP
(11) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(12) TRUE


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

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
   f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
   proper(a) -> ok(a)
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(c)
   active(f(x0, x1, x2))
   f(x0, x1, mark(x2))
   proper(f(x0, x1, x2))
   proper(a)
   proper(b)
   proper(c)
   f(ok(x0), ok(x1), ok(x2))
   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(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
   f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
   proper(a) -> ok(a)
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(c)
   active(f(x0, x1, x2))
   f(x0, x1, mark(x2))
   proper(f(x0, x1, x2))
   proper(a)
   proper(b)
   proper(c)
   f(ok(x0), ok(x1), ok(x2))
   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:

f: {3}
a: empty set
b: empty set
c: empty set
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.
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(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(a, b, X) -> f(X, X, X)
   c -> a
   c -> b

The replacement map contains the following entries:

f: {3}
a: empty set
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(a, b, X) -> f(X, X, X)
   c -> a
   c -> b

The replacement map contains the following entries:

f: {3}
a: empty set
b: empty set
c: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(a) = 1
   POL(b) = 0
   POL(c) = 1
   POL(f(x_1, x_2, x_3)) = x_3
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   c -> b




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

   f(a, b, X) -> f(X, X, X)
   c -> a

The replacement map contains the following entries:

f: {3}
a: empty set
b: empty set
c: empty set

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

   f(a, b, X) -> f(X, X, X)
   c -> a

The replacement map contains the following entries:

f: {3}
a: empty set
b: empty set
c: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(a) = 0
   POL(b) = 0
   POL(c) = 1
   POL(f(x_1, x_2, x_3)) = x_3
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   c -> a




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

   f(a, b, X) -> f(X, X, X)

The replacement map contains the following entries:

f: {3}
a: empty set
b: empty set

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(7) CSRInnermostProof (EQUIVALENT)
The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
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(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(a, b, X) -> f(X, X, X)

The replacement map contains the following entries:

f: {3}
a: empty set
b: empty set


Innermost Strategy.

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(9) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
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(10)
Obligation:
Q-restricted context-sensitive dependency pair problem:
For all symbols f in {f_3, F_3} we have mu(f) = {3}.

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

The TRS R consists of the following rules:

   f(a, b, X) -> f(X, X, X)

The set Q consists of the following terms:

   f(a, b, x0)


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(11) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F(a, b, z0) -> F(z0, z0, z0) and F(a, b, x0) -> F(x0, x0, x0) form no chain, because ECap^mu(F(z0, z0, z0)) = F(z0, z0, z0) does not unify with F(a, b, x0). 
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(12)
TRUE