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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRSRRRProof [EQUIVALENT, 40 ms]
(2) QTRS
(3) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 0 ms]
(6) QDP
(7) UsableRulesProof [EQUIVALENT, 0 ms]
(8) QDP
(9) QReductionProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(12) YES


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

   b(c(x1)) -> c(b(x1))
   c(b(x1)) -> a(a(a(x1)))
   a(a(a(a(x1)))) -> b(c(x1))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a(x_1)) = 2 + x_1
   POL(b(x_1)) = x_1
   POL(c(x_1)) = 7 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   c(b(x1)) -> a(a(a(x1)))
   a(a(a(a(x1)))) -> b(c(x1))




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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(c(x1)) -> c(b(x1))

Q is empty.

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(3) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(c(x1)) -> c(b(x1))

The set Q consists of the following terms:

   b(c(x0))


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(5) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(c(x1)) -> B(x1)

The TRS R consists of the following rules:

   b(c(x1)) -> c(b(x1))

The set Q consists of the following terms:

   b(c(x0))

We have to consider all minimal (P,Q,R)-chains.
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(7) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(c(x1)) -> B(x1)

R is empty.
The set Q consists of the following terms:

   b(c(x0))

We have to consider all minimal (P,Q,R)-chains.
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(9) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   b(c(x0))


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(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(c(x1)) -> B(x1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(11) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*B(c(x1)) -> B(x1)
The graph contains the following edges 1 > 1


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(12)
YES