/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(6) YES


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

   f(cons(nil, y)) -> y
   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

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:

   F(cons(f(cons(nil, y)), z)) -> COPY(n, y, z)
   COPY(0, y, z) -> F(z)
   COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))
   COPY(s(x), y, z) -> F(y)

The TRS R consists of the following rules:

   f(cons(nil, y)) -> y
   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

Q is empty.
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 3 less nodes.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))

The TRS R consists of the following rules:

   f(cons(nil, y)) -> y
   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) 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:
*COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))
The graph contains the following edges 1 > 1, 2 >= 2


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(6)
YES