/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 0)

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)
Tuples:
   PRIME(0) -> c
   PRIME(s(0)) -> c1
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
   PRIME1(z0, 0) -> c3
   PRIME1(z0, s(0)) -> c4
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
   DIVP(z0, z1) -> c6
S tuples:
   PRIME(0) -> c
   PRIME(s(0)) -> c1
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
   PRIME1(z0, 0) -> c3
   PRIME1(z0, s(0)) -> c4
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
   DIVP(z0, z1) -> c6
K tuples:none
Defined Rule Symbols:   prime_1, prime1_2, divp_2

Defined Pair Symbols:   PRIME_1, PRIME1_2, DIVP_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5_2, c6


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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
Removed 5 trailing nodes:
   PRIME1(z0, s(0)) -> c4
   DIVP(z0, z1) -> c6
   PRIME(s(0)) -> c1
   PRIME(0) -> c
   PRIME1(z0, 0) -> c3

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)
Tuples:
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
S tuples:
   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   prime_1, prime1_2, divp_2

Defined Pair Symbols:   PRIME1_2

Compound Symbols:   c5_2


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)
Tuples:
   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
S tuples:
   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   prime_1, prime1_2, divp_2

Defined Pair Symbols:   PRIME1_2

Compound Symbols:   c5_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
   prime1(z0, 0) -> false
   prime1(z0, s(0)) -> true
   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
   divp(z0, z1) -> =(rem(z0, z1), 0)

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
S tuples:
   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   PRIME1_2

Compound Symbols:   c5_1


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
We considered the (Usable) Rules:none
And the Tuples:
   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(PRIME1(x_1, x_2)) = x_2
   POL(c5(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
S tuples:none
K tuples:
   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
Defined Rule Symbols:none

Defined Pair Symbols:   PRIME1_2

Compound Symbols:   c5_1


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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(12)
BOUNDS(1, 1)

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(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 0)

S is empty.
Rewrite Strategy: INNERMOST
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(15) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

prime1(x, s(s(y))) ->^+ and(not(divp(s(s(y)), x)), prime1(x, s(y)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [y / s(y)].

The result substitution is [ ].




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(16)
Complex Obligation (BEST)

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(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 0)

S is empty.
Rewrite Strategy: INNERMOST
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(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(19)
BOUNDS(n^1, INF)

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(20)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 0)

S is empty.
Rewrite Strategy: INNERMOST