/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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)), 54 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:

   from(X) -> cons(X, n__from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> 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:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   FROM(z0) -> c
   FROM(z0) -> c1
   AFTER(0, z0) -> c2
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   ACTIVATE(z0) -> c5
S tuples:
   FROM(z0) -> c
   FROM(z0) -> c1
   AFTER(0, z0) -> c2
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   ACTIVATE(z0) -> c5
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   FROM_1, AFTER_2, ACTIVATE_1

Compound Symbols:   c, c1, c2, c3_2, c4_1, c5


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   FROM(z0) -> c1
   ACTIVATE(z0) -> c5
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   AFTER(0, z0) -> c2
   FROM(z0) -> c

----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_2


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
K tuples:none
Defined Rule Symbols:   activate_1, from_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   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.
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
We considered the (Usable) Rules:none
And the Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(AFTER(x_1, x_2)) = x_1
   POL(activate(x_1)) = [3] + [3]x_1
   POL(c3(x_1)) = x_1
   POL(cons(x_1, x_2)) = [3] + x_1 + x_2
   POL(from(x_1)) = [3] + [3]x_1
   POL(n__from(x_1)) = 0
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
S tuples:none
K tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
Defined Rule Symbols:   activate_1, from_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_1


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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(12)
BOUNDS(1, 1)

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(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(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:

   from(X) -> cons(X, n__from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> 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

after(s(N), cons(X, XS)) ->^+ after(N, XS)

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

The pumping substitution is [N / s(N), XS / cons(X, XS)].

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:

   from(X) -> cons(X, n__from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> 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:

   from(X) -> cons(X, n__from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST