/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 64 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) 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^1).


The TRS R consists of the following rules:

   h(e(x), y) -> h(d(x, y), s(y))
   d(g(g(0, x), y), s(z)) -> g(e(x), d(g(g(0, x), y), z))
   d(g(g(0, x), y), 0) -> e(y)
   d(g(0, x), y) -> e(x)
   d(g(x, y), z) -> g(d(x, z), e(y))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of h: d
The following defined symbols can occur below the 0th argument of d: d

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
d(g(g(0, x), y), s(z)) -> g(e(x), d(g(g(0, x), y), z))
d(g(g(0, x), y), 0) -> e(y)
d(g(0, x), y) -> e(x)
d(g(x, y), z) -> g(d(x, z), e(y))

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


The TRS R consists of the following rules:

   h(e(x), y) -> h(d(x, y), s(y))
   g(e(x), e(y)) -> e(g(x, y))

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

As the TRS does not nest defined symbols, we have rc = irc.
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   h(e(x), y) -> h(d(x, y), s(y))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   h(e(z0), z1) -> h(d(z0, z1), s(z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))
Tuples:
   H(e(z0), z1) -> c(H(d(z0, z1), s(z1)))
   G(e(z0), e(z1)) -> c1(G(z0, z1))
S tuples:
   H(e(z0), z1) -> c(H(d(z0, z1), s(z1)))
   G(e(z0), e(z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, g_2

Defined Pair Symbols:   H_2, G_2

Compound Symbols:   c_1, c1_1


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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   H(e(z0), z1) -> c(H(d(z0, z1), s(z1)))

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   h(e(z0), z1) -> h(d(z0, z1), s(z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))
Tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
S tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, g_2

Defined Pair Symbols:   G_2

Compound Symbols:   c1_1


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(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   h(e(z0), z1) -> h(d(z0, z1), s(z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
S tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c1_1


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(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(e(z0), e(z1)) -> c1(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(G(x_1, x_2)) = x_2
   POL(c1(x_1)) = x_1
   POL(e(x_1)) = [1] + x_1

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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
S tuples:none
K tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c1_1


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(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(14)
BOUNDS(1, 1)

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(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(16)
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^1).


The TRS R consists of the following rules:

   h(e(x), y) -> h(d(x, y), s(y))
   d(g(g(0, x), y), s(z)) -> g(e(x), d(g(g(0, x), y), z))
   d(g(g(0, x), y), 0) -> e(y)
   d(g(0, x), y) -> e(x)
   d(g(x, y), z) -> g(d(x, z), e(y))
   g(e(x), e(y)) -> e(g(x, y))

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

The rewrite sequence

g(e(x), e(y)) ->^+ e(g(x, y))

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

The pumping substitution is [x / e(x), y / e(y)].

The result substitution is [ ].




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(18)
Complex Obligation (BEST)

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(19)
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^1).


The TRS R consists of the following rules:

   h(e(x), y) -> h(d(x, y), s(y))
   d(g(g(0, x), y), s(z)) -> g(e(x), d(g(g(0, x), y), z))
   d(g(g(0, x), y), 0) -> e(y)
   d(g(0, x), y) -> e(x)
   d(g(x, y), z) -> g(d(x, z), e(y))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: FULL
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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
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^1).


The TRS R consists of the following rules:

   h(e(x), y) -> h(d(x, y), s(y))
   d(g(g(0, x), y), s(z)) -> g(e(x), d(g(g(0, x), y), z))
   d(g(g(0, x), y), 0) -> e(y)
   d(g(0, x), y) -> e(x)
   d(g(x, y), z) -> g(d(x, z), e(y))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: FULL