/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 20 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 6 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

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

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

The duplicating contexts are:
sum(s([]))


The defined contexts are:
+([], s(x1))
+([], x1)


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, 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^2).


The TRS R consists of the following rules:

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

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))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, 0) -> c2
   +'(z0, s(z1)) -> c3(+'(z0, z1))
S tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, 0) -> c2
   +'(z0, s(z1)) -> c3(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   sum_1, +_2

Defined Pair Symbols:   SUM_1, +'_2

Compound Symbols:   c, c1_2, c2, c3_1


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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   SUM(0) -> c
   +'(z0, 0) -> c2

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
S tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   sum_1, +_2

Defined Pair Symbols:   SUM_1, +'_2

Compound Symbols:   c1_2, c3_1


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(7) 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), s(z0)), SUM(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [3] + [3]x_2
   POL(+'(x_1, x_2)) = 0
   POL(0) = [3]
   POL(SUM(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(s(x_1)) = [2] + x_1
   POL(sum(x_1)) = [3]

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
S tuples:
   +'(z0, s(z1)) -> c3(+'(z0, z1))
K tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
Defined Rule Symbols:   sum_1, +_2

Defined Pair Symbols:   SUM_1, +'_2

Compound Symbols:   c1_2, c3_1


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(z0, s(z1)) -> c3(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1] + x_2 + [2]x_2^2
   POL(+'(x_1, x_2)) = x_2
   POL(0) = 0
   POL(SUM(x_1)) = x_1^2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1
   POL(sum(x_1)) = [2] + [2]x_1 + [2]x_1^2

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
S tuples:none
K tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
Defined Rule Symbols:   sum_1, +_2

Defined Pair Symbols:   SUM_1, +'_2

Compound Symbols:   c1_2, c3_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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
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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

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

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

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


The TRS R consists of the following rules:

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

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


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL