WORST_CASE(Omega(n^1), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 7 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 40 ms]
    (8) CdtProblem
    (9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) BOUNDS(1, 1)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) 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^2).


The TRS R consists of the following rules:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, 1) -> x
   *(1, y) -> y

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:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, 1) -> c2
   *'(1, z0) -> c3
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, 1) -> c2
   *'(1, z0) -> c3
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1_2, c2, c3


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(1, z0) -> c3
   *'(z0, 1) -> c2

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1_2


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1_2


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(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*'(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_1*x_2
   POL(+(x_1, x_2)) = [2] + x_1 + x_2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
S tuples:none
K tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1_2


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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(10)
BOUNDS(1, 1)

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(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(12)
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^2).


The TRS R consists of the following rules:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, 1) -> x
   *(1, y) -> y

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

The rewrite sequence

*(+(x, y), z) ->^+ +(*(x, z), *(y, z))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x / +(x, y)].

The result substitution is [ ].




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(14)
Complex Obligation (BEST)

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(15)
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^2).


The TRS R consists of the following rules:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, 1) -> x
   *(1, y) -> y

S is empty.
Rewrite Strategy: INNERMOST
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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
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^2).


The TRS R consists of the following rules:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, 1) -> x
   *(1, y) -> y

S is empty.
Rewrite Strategy: INNERMOST