/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 [BOTH BOUNDS(ID, ID), 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)), 61 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:

   sum(0) -> 0
   sum(s(x)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(x))

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:
   sum(0) -> 0
   sum(s(z0)) -> +(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +(*(s(z0), s(z0)), sum(z0))
   sqr(z0) -> *(z0, z0)
Tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SQR(s(z0)), SUM(z0))
   SUM(s(z0)) -> c2(SUM(z0))
   SQR(z0) -> c3
S tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SQR(s(z0)), SUM(z0))
   SUM(s(z0)) -> c2(SUM(z0))
   SQR(z0) -> c3
K tuples:none
Defined Rule Symbols:   sum_1, sqr_1

Defined Pair Symbols:   SUM_1, SQR_1

Compound Symbols:   c, c1_2, c2_1, c3


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   SUM(0) -> c
   SQR(z0) -> c3

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +(*(s(z0), s(z0)), sum(z0))
   sqr(z0) -> *(z0, z0)
Tuples:
   SUM(s(z0)) -> c1(SQR(s(z0)), SUM(z0))
   SUM(s(z0)) -> c2(SUM(z0))
S tuples:
   SUM(s(z0)) -> c1(SQR(s(z0)), SUM(z0))
   SUM(s(z0)) -> c2(SUM(z0))
K tuples:none
Defined Rule Symbols:   sum_1, sqr_1

Defined Pair Symbols:   SUM_1

Compound Symbols:   c1_2, c2_1


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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:
   sum(0) -> 0
   sum(s(z0)) -> +(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +(*(s(z0), s(z0)), sum(z0))
   sqr(z0) -> *(z0, z0)
Tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c1(SUM(z0))
S tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c1(SUM(z0))
K tuples:none
Defined Rule Symbols:   sum_1, sqr_1

Defined Pair Symbols:   SUM_1

Compound Symbols:   c2_1, c1_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   sum(0) -> 0
   sum(s(z0)) -> +(sqr(s(z0)), sum(z0))
   sum(s(z0)) -> +(*(s(z0), s(z0)), sum(z0))
   sqr(z0) -> *(z0, z0)

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c1(SUM(z0))
S tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c1(SUM(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   SUM_1

Compound Symbols:   c2_1, c1_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.
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c1(SUM(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c1(SUM(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(SUM(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c1(SUM(z0))
S tuples:none
K tuples:
   SUM(s(z0)) -> c2(SUM(z0))
   SUM(s(z0)) -> c1(SUM(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   SUM_1

Compound Symbols:   c2_1, c1_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:

   sum(0) -> 0
   sum(s(x)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(x))

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

sum(s(x)) ->^+ +(*(s(x), s(x)), sum(x))

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

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

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:

   sum(0) -> 0
   sum(s(x)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(x))

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:

   sum(0) -> 0
   sum(s(x)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(x))

S is empty.
Rewrite Strategy: INNERMOST