/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 13 ms]
(4) CSR
(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(6) QCSDP
(7) QCSDPInstantiationProcessor [EQUIVALENT, 0 ms]
(8) QCSDP
(9) QCSUsableRulesProof [EQUIVALENT, 0 ms]
(10) QCSDP
(11) QCSDPInstantiationProcessor [EQUIVALENT, 0 ms]
(12) QCSDP
(13) QCSDependencyGraphProof [EQUIVALENT, 4 ms]
(14) TRUE


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

   active(h(X)) -> mark(g(X, X))
   active(g(a, X)) -> mark(f(b, X))
   active(f(X, X)) -> mark(h(a))
   active(a) -> mark(b)
   active(h(X)) -> h(active(X))
   active(g(X1, X2)) -> g(active(X1), X2)
   active(f(X1, X2)) -> f(active(X1), X2)
   h(mark(X)) -> mark(h(X))
   g(mark(X1), X2) -> mark(g(X1, X2))
   f(mark(X1), X2) -> mark(f(X1, X2))
   proper(h(X)) -> h(proper(X))
   proper(g(X1, X2)) -> g(proper(X1), proper(X2))
   proper(a) -> ok(a)
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(b) -> ok(b)
   h(ok(X)) -> ok(h(X))
   g(ok(X1), ok(X2)) -> ok(g(X1, X2))
   f(ok(X1), ok(X2)) -> ok(f(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(h(X)) -> mark(g(X, X))
   active(g(a, X)) -> mark(f(b, X))
   active(f(X, X)) -> mark(h(a))
   active(a) -> mark(b)
   active(h(X)) -> h(active(X))
   active(g(X1, X2)) -> g(active(X1), X2)
   active(f(X1, X2)) -> f(active(X1), X2)
   h(mark(X)) -> mark(h(X))
   g(mark(X1), X2) -> mark(g(X1, X2))
   f(mark(X1), X2) -> mark(f(X1, X2))
   proper(h(X)) -> h(proper(X))
   proper(g(X1, X2)) -> g(proper(X1), proper(X2))
   proper(a) -> ok(a)
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(b) -> ok(b)
   h(ok(X)) -> ok(h(X))
   g(ok(X1), ok(X2)) -> ok(g(X1, X2))
   f(ok(X1), ok(X2)) -> ok(f(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:

h: {1}
g: {1}
a: empty set
f: {1}
b: 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:

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

The replacement map contains the following entries:

h: {1}
g: {1}
a: empty set
f: {1}
b: 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:

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

The replacement map contains the following entries:

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

   POL(a) = 1
   POL(b) = 0
   POL(f(x_1, x_2)) = 2 + 2*x_1
   POL(g(x_1, x_2)) = 2*x_1
   POL(h(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a -> b




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

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

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

The replacement map contains the following entries:

h: {1}
g: {1}
a: empty set
f: {1}
b: empty set

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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 {h_1, H_1} are replacing on all positions.
For all symbols f in {g_2, f_2, G_2, F_2} we have mu(f) = {1}.

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

The TRS R consists of the following rules:

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

Q is empty.

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(7) QCSDPInstantiationProcessor (EQUIVALENT)
Using the Context-Sensitive Instantiation[LPAR08,DA_EMMES] Processor

the pair H(X) -> G(X, X)

was transformed to the following new pairs: 
   H(a) -> G(a, a)



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

(8)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {h_1, H_1} are replacing on all positions.
For all symbols f in {g_2, f_2, F_2, G_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   G(a, X) -> F(b, X)
   F(X, X) -> H(a)
   H(a) -> G(a, a)

The TRS R consists of the following rules:

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

Q is empty.

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(9) QCSUsableRulesProof (EQUIVALENT)
The following rules are not useable [DA_EMMES] and can be deleted:

   h(x0) -> g(x0, x0)
   g(a, x0) -> f(b, x0)
   f(x0, x0) -> h(a)

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

(10)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {H_1} are replacing on all positions.
For all symbols f in {F_2, G_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   G(a, X) -> F(b, X)
   F(X, X) -> H(a)
   H(a) -> G(a, a)

R is empty.
Q is empty.

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(11) QCSDPInstantiationProcessor (EQUIVALENT)
Using the Context-Sensitive Instantiation[LPAR08,DA_EMMES] Processor

the pair G(a, X) -> F(b, X)

was transformed to the following new pairs: 
   G(a, a) -> F(b, a)



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

(12)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {H_1} are replacing on all positions.
For all symbols f in {F_2, G_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   F(X, X) -> H(a)
   H(a) -> G(a, a)
   G(a, a) -> F(b, a)

R is empty.
Q is empty.

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(13) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes.
The rules G(a, a) -> F(b, a) and F(x0, x0) -> H(a) form no chain, because ECap^mu(F(b, a)) = F(b, a) does not unify with F(x0, x0). 
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(14)
TRUE