/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 49 ms]
    (6) CdtProblem
    (7) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) BOUNDS(1, 1)
(9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(10) TRS for Loop Detection
    (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (12) BEST
        (13) proven lower bound
            (14) LowerBoundPropagationProof [FINISHED, 0 ms]
            (15) BOUNDS(n^1, INF)
        (16) 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:

   f(0, y) -> 0
   f(s(x), y) -> f(f(x, y), y)

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:
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   F(0, z0) -> c
   F(s(z0), z1) -> c1(F(f(z0, z1), z1), F(z0, z1))
S tuples:
   F(0, z0) -> c
   F(s(z0), z1) -> c1(F(f(z0, z1), z1), F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c, c1_2


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(0, z0) -> c

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   F(s(z0), z1) -> c1(F(f(z0, z1), z1), F(z0, z1))
S tuples:
   F(s(z0), z1) -> c1(F(f(z0, z1), z1), F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c1_2


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(5) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0), z1) -> c1(F(f(z0, z1), z1), F(z0, z1))
We considered the (Usable) Rules:
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
And the Tuples:
   F(s(z0), z1) -> c1(F(f(z0, z1), z1), F(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(f(x_1, x_2)) = 0
   POL(s(x_1)) = [1] + x_1

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   F(s(z0), z1) -> c1(F(f(z0, z1), z1), F(z0, z1))
S tuples:none
K tuples:
   F(s(z0), z1) -> c1(F(f(z0, z1), z1), F(z0, z1))
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c1_2


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(7) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(8)
BOUNDS(1, 1)

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(9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(10)
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:

   f(0, y) -> 0
   f(s(x), y) -> f(f(x, y), y)

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

The rewrite sequence

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

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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(12)
Complex Obligation (BEST)

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(13)
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:

   f(0, y) -> 0
   f(s(x), y) -> f(f(x, y), y)

S is empty.
Rewrite Strategy: INNERMOST
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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
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:

   f(0, y) -> 0
   f(s(x), y) -> f(f(x, y), y)

S is empty.
Rewrite Strategy: INNERMOST