/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 20 ms]
(6) QDP
(7) PisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   a__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

The set Q consists of the following terms:

   a__c
   mark(f(x0, x1, x2))
   mark(c)
   mark(a)
   mark(b)
   a__f(x0, x1, x2)


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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__F(a, b, X) -> A__F(X, X, mark(X))
   A__F(a, b, X) -> MARK(X)
   MARK(f(X1, X2, X3)) -> A__F(X1, X2, mark(X3))
   MARK(f(X1, X2, X3)) -> MARK(X3)
   MARK(c) -> A__C

The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

The set Q consists of the following terms:

   a__c
   mark(f(x0, x1, x2))
   mark(c)
   mark(a)
   mark(b)
   a__f(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
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(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__F(a, b, X) -> MARK(X)
   MARK(f(X1, X2, X3)) -> A__F(X1, X2, mark(X3))
   MARK(f(X1, X2, X3)) -> MARK(X3)

The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

The set Q consists of the following terms:

   a__c
   mark(f(x0, x1, x2))
   mark(c)
   mark(a)
   mark(b)
   a__f(x0, x1, x2)

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


The following pairs can be oriented strictly and are deleted.

   A__F(a, b, X) -> MARK(X)
   MARK(f(X1, X2, X3)) -> A__F(X1, X2, mark(X3))
   MARK(f(X1, X2, X3)) -> MARK(X3)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( A__F_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( mark_1(x_1) ) = x_1
POL( f_3(x_1, ..., x_3) ) = x_3 + 2
POL( a__f_3(x_1, ..., x_3) ) = x_3 + 2
POL( c ) = 0
POL( a__c ) = 0
POL( a ) = 0
POL( b ) = 0
POL( MARK_1(x_1) ) = 2x_1

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

   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   a__c -> c


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(6)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

The set Q consists of the following terms:

   a__c
   mark(f(x0, x1, x2))
   mark(c)
   mark(a)
   mark(b)
   a__f(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
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(7) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(8)
YES