/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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 [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 10 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^1).


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(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:
   duplicate(Cons(z0, z1)) -> Cons(z0, Cons(z0, duplicate(z1)))
   duplicate(Nil) -> Nil
   goal(z0) -> duplicate(z0)
Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
   DUPLICATE(Nil) -> c1
   GOAL(z0) -> c2(DUPLICATE(z0))
S tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
   DUPLICATE(Nil) -> c1
   GOAL(z0) -> c2(DUPLICATE(z0))
K tuples:none
Defined Rule Symbols:   duplicate_1, goal_1

Defined Pair Symbols:   DUPLICATE_1, GOAL_1

Compound Symbols:   c_1, c1, c2_1


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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c2(DUPLICATE(z0))
Removed 1 trailing nodes:
   DUPLICATE(Nil) -> c1

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   duplicate(Cons(z0, z1)) -> Cons(z0, Cons(z0, duplicate(z1)))
   duplicate(Nil) -> Nil
   goal(z0) -> duplicate(z0)
Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
S tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
K tuples:none
Defined Rule Symbols:   duplicate_1, goal_1

Defined Pair Symbols:   DUPLICATE_1

Compound Symbols:   c_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   duplicate(Cons(z0, z1)) -> Cons(z0, Cons(z0, duplicate(z1)))
   duplicate(Nil) -> Nil
   goal(z0) -> duplicate(z0)

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
S tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   DUPLICATE_1

Compound Symbols:   c_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.
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
We considered the (Usable) Rules:none
And the Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(DUPLICATE(x_1)) = x_1
   POL(c(x_1)) = x_1

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
S tuples:none
K tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   DUPLICATE_1

Compound Symbols:   c_1


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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^1).


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

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

duplicate(Cons(x, xs)) ->^+ Cons(x, Cons(x, duplicate(xs)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].

The pumping substitution is [xs / Cons(x, xs)].

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^1).


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

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^1).


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

S is empty.
Rewrite Strategy: INNERMOST