/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 1 ms]
(4) QDP
(5) UsableRulesProof [EQUIVALENT, 0 ms]
(6) QDP
(7) MNOCProof [EQUIVALENT, 0 ms]
(8) QDP
(9) MRRProof [EQUIVALENT, 0 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) TRUE


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

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

Q is empty.

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

(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:

   -^1(x, s(y)) -> -^1(x, p(s(y)))
   -^1(x, s(y)) -> P(s(y))

The TRS R consists of the following rules:

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

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

   -^1(x, s(y)) -> -^1(x, p(s(y)))

The TRS R consists of the following rules:

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

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

(5) 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.
----------------------------------------

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

   -^1(x, s(y)) -> -^1(x, p(s(y)))

The TRS R consists of the following rules:

   p(s(x)) -> x

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

(7) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   -^1(x, s(y)) -> -^1(x, p(s(y)))

The TRS R consists of the following rules:

   p(s(x)) -> x

The set Q consists of the following terms:

   p(s(x0))

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

(9) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.


Strictly oriented rules of the TRS R:

   p(s(x)) -> x

Used ordering: Polynomial interpretation [POLO]:

   POL(-^1(x_1, x_2)) = x_1 + 2*x_2
   POL(p(x_1)) = x_1
   POL(s(x_1)) = 2 + x_1


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

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

   -^1(x, s(y)) -> -^1(x, p(s(y)))

R is empty.
The set Q consists of the following terms:

   p(s(x0))

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

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

(12)
TRUE