/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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 of the given CSR could be proven:

(0) CSR
(1) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(2) QCSDP
(3) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(4) QCSDP
(5) QCSUsableRulesProof [EQUIVALENT, 0 ms]
(6) QCSDP
(7) QCSDPInstantiationProcessor [EQUIVALENT, 0 ms]
(8) QCSDP
(9) PIsEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

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

The replacement map contains the following entries:

f: {2}
a: empty set
b: empty set

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

(1) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
----------------------------------------

(2)
Obligation:
Q-restricted context-sensitive dependency pair problem:
For all symbols f in {f_3, F_3} we have mu(f) = {2}.

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

The TRS R consists of the following rules:

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

Q is empty.

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

(3) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 1 less node.

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

(4)
Obligation:
Q-restricted context-sensitive dependency pair problem:
For all symbols f in {f_3, F_3} we have mu(f) = {2}.

The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

Q is empty.

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

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

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

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

(6)
Obligation:
Q-restricted context-sensitive dependency pair problem:
For all symbols f in {F_3} we have mu(f) = {2}.

The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   b -> a

Q is empty.

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

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

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

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



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

(8)
Obligation:
Q-restricted context-sensitive dependency pair problem:

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   b -> a

Q is empty.

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(9) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
----------------------------------------

(10)
YES