/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) UsableRulesProof [EQUIVALENT, 0 ms]
        (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES


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

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

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(s(x)) -> F(x)
   G(x, c(y)) -> G(x, y)
   G(x, c(y)) -> G(x, if(f(x), c(g(s(x), y)), c(y)))
   G(x, c(y)) -> IF(f(x), c(g(s(x), y)), c(y))
   G(x, c(y)) -> F(x)
   G(x, c(y)) -> G(s(x), y)

The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

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 2 SCCs with 3 less nodes.
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(4)
Complex Obligation (AND)

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(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(x)) -> F(x)

The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(6) 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.
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(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(x)) -> F(x)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(8) 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:
*F(s(x)) -> F(x)
The graph contains the following edges 1 > 1


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(9)
YES

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(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   G(x, c(y)) -> G(s(x), y)
   G(x, c(y)) -> G(x, y)

The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(11) 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.
----------------------------------------

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

   G(x, c(y)) -> G(s(x), y)
   G(x, c(y)) -> G(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(13) 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:
*G(x, c(y)) -> G(s(x), y)
The graph contains the following edges 2 > 2


*G(x, c(y)) -> G(x, y)
The graph contains the following edges 1 >= 1, 2 > 2


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(14)
YES