/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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)), 42 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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

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:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   OR(true, z0) -> c
   OR(z0, true) -> c1
   OR(false, false) -> c2
   MEM(z0, nil) -> c3
   MEM(z0, set(z1)) -> c4
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
S tuples:
   OR(true, z0) -> c
   OR(z0, true) -> c1
   OR(false, false) -> c2
   MEM(z0, nil) -> c3
   MEM(z0, set(z1)) -> c4
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   OR_2, MEM_2

Compound Symbols:   c, c1, c2, c3, c4, c5_3


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   OR(false, false) -> c2
   OR(true, z0) -> c
   MEM(z0, nil) -> c3
   OR(z0, true) -> c1
   MEM(z0, set(z1)) -> c4

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_3


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_2


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_2


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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.
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(MEM(x_1, x_2)) = x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(union(x_1, x_2)) = [1] + x_1 + x_2

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
S tuples:none
K tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1), MEM(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_2


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


The TRS R consists of the following rules:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

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

mem(x, union(y, z)) ->^+ or(mem(x, y), mem(x, z))

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

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

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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

S is empty.
Rewrite Strategy: INNERMOST