/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 17 ms]
(2) QDP
(3) QDPOrderProof [EQUIVALENT, 4 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) UsableRulesProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES


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

   a(b(x1)) -> b(r(x1))
   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)

Q is empty.

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(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(b(x1)) -> R(x1)
   R(a(x1)) -> D(r(x1))
   R(a(x1)) -> R(x1)
   R(x1) -> D(x1)
   D(a(x1)) -> A(a(d(x1)))
   D(a(x1)) -> A(d(x1))
   D(a(x1)) -> D(x1)
   D(x1) -> A(x1)

The TRS R consists of the following rules:

   a(b(x1)) -> b(r(x1))
   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(3) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   R(a(x1)) -> D(r(x1))
   R(x1) -> D(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( D_1(x_1) ) = 2x_1
POL( A_1(x_1) ) = max{0, 2x_1 - 1}
POL( b_1(x_1) ) = 2x_1 + 1
POL( r_1(x_1) ) = x_1
POL( a_1(x_1) ) = x_1
POL( d_1(x_1) ) = x_1
POL( R_1(x_1) ) = 2x_1 + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)
   a(b(x1)) -> b(r(x1))


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

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(b(x1)) -> R(x1)
   R(a(x1)) -> R(x1)
   D(a(x1)) -> A(a(d(x1)))
   D(a(x1)) -> A(d(x1))
   D(a(x1)) -> D(x1)
   D(x1) -> A(x1)

The TRS R consists of the following rules:

   a(b(x1)) -> b(r(x1))
   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
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(6)
Complex Obligation (AND)

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(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   R(a(x1)) -> R(x1)

The TRS R consists of the following rules:

   a(b(x1)) -> b(r(x1))
   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(8) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   R(a(x1)) -> R(x1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(10) 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:
*R(a(x1)) -> R(x1)
The graph contains the following edges 1 > 1


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(11)
YES

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(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   D(a(x1)) -> D(x1)

The TRS R consists of the following rules:

   a(b(x1)) -> b(r(x1))
   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   D(a(x1)) -> D(x1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) 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:
*D(a(x1)) -> D(x1)
The graph contains the following edges 1 > 1


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(16)
YES