/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) QTRSRRRProof [EQUIVALENT, 96 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) TRUE


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

   a__f(X, X) -> a__f(a, b)
   a__b -> a
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2) -> f(X1, X2)
   a__b -> b

The set Q consists of the following terms:

   a__b
   mark(f(x0, x1))
   mark(b)
   mark(a)
   a__f(x0, x1)


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

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[a__b, mark_1] > a__f_2 > [a, b, f_2]


Status:
a__f_2: [1,2]
a: multiset status
b: multiset status
a__b: multiset status
mark_1: [1]
f_2: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a__b -> a
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2) -> f(X1, X2)
   a__b -> b




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

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

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

The set Q consists of the following terms:

   a__b
   mark(f(x0, x1))
   mark(b)
   mark(a)
   a__f(x0, x1)


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

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   a__b
   mark(f(x0, x1))
   mark(b)
   mark(a)
   a__f(x0, x1)

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
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(6)
TRUE