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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 7 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 1 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 34 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 0 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(18) TRS for Loop Detection
    (19) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (full) 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)) -> *(otimes(x, y), z)
   *(1, y) -> y
   *(+(x, y), z) -> oplus(*(x, z), *(y, z))
   *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
*(x, *(y, z)) -> *(otimes(x, y), z)

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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
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(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: INNERMOST
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(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(6)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   *'_2

Compound Symbols:   c, c1_2, c2_2


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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   *'(1, z0) -> c

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(8)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   *'_2

Compound Symbols:   c1_2, c2_2


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

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
S tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c1_2, c2_2


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(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*'(x_1, x_2)) = x_2 + [2]x_1*x_2
   POL(+(x_1, x_2)) = [2] + x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(oplus(x_1, x_2)) = [2] + x_1 + x_2

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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
S tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
K tuples:
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c1_2, c2_2


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(13) 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) -> c1(*'(z0, z2), *'(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*'(x_1, x_2)) = [1] + [2]x_1 + x_2 + [2]x_1*x_2
   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(oplus(x_1, x_2)) = [1] + x_1 + x_2

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(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
S tuples:none
K tuples:
   *'(z0, oplus(z1, z2)) -> c2(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c1_2, c2_2


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(15) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(16)
BOUNDS(1, 1)

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(17) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(18)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) 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)) -> *(otimes(x, y), z)
   *(1, y) -> y
   *(+(x, y), z) -> oplus(*(x, z), *(y, z))
   *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

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

The rewrite sequence

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

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

The pumping substitution is [y / oplus(y, z)].

The result substitution is [ ].




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(20)
Complex Obligation (BEST)

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(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) 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)) -> *(otimes(x, y), z)
   *(1, y) -> y
   *(+(x, y), z) -> oplus(*(x, z), *(y, z))
   *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

S is empty.
Rewrite Strategy: FULL
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(22) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(23)
BOUNDS(n^1, INF)

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(24)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) 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)) -> *(otimes(x, y), z)
   *(1, y) -> y
   *(+(x, y), z) -> oplus(*(x, z), *(y, z))
   *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

S is empty.
Rewrite Strategy: FULL