/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^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (4) CdtProblem
    (5) CdtLeafRemovalProof [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)), 49 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) 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^1).


The TRS R consists of the following rules:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

S is empty.
Rewrite Strategy: FULL
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(1) 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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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

S is empty.
Rewrite Strategy: INNERMOST
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(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   sum1(0) -> 0
   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))
Tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(0) -> c2
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(0) -> c2
   SUM1(s(z0)) -> c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:   sum_1, sum1_1

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c, c1_1, c2, c3_1


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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   SUM1(0) -> c2
   SUM(0) -> c

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   sum1(0) -> 0
   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))
Tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:   sum_1, sum1_1

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c1_1, c3_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)) -> +(sum(z0), s(z0))
   sum1(0) -> 0
   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c1_1, c3_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.
   SUM1(s(z0)) -> c3(SUM1(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(SUM(x_1)) = 0
   POL(SUM1(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c3(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)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:
   SUM(s(z0)) -> c1(SUM(z0))
K tuples:
   SUM1(s(z0)) -> c3(SUM1(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c1_1, c3_1


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(11) 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)) -> c1(SUM(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(SUM(x_1)) = x_1
   POL(SUM1(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   SUM(s(z0)) -> c1(SUM(z0))
   SUM1(s(z0)) -> c3(SUM1(z0))
S tuples:none
K tuples:
   SUM1(s(z0)) -> c3(SUM1(z0))
   SUM(s(z0)) -> c1(SUM(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   SUM_1, SUM1_1

Compound Symbols:   c1_1, c3_1


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(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(14)
BOUNDS(1, 1)

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(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(16)
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^1).


The TRS R consists of the following rules:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

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

The rewrite sequence

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

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

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

The result substitution is [ ].




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(18)
Complex Obligation (BEST)

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(19)
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^1).


The TRS R consists of the following rules:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

S is empty.
Rewrite Strategy: FULL
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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
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^1).


The TRS R consists of the following rules:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

S is empty.
Rewrite Strategy: FULL