/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms]
    (8) CdtProblem
    (9) CdtKnowledgeProof [FINISHED, 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:

   f(empty, l) -> l
   f(cons(x, k), l) -> g(k, l, cons(x, k))
   g(a, b, c) -> f(a, cons(b, c))

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(empty, z0) -> z0
   f(cons(z0, z1), z2) -> g(z1, z2, cons(z0, z1))
   g(z0, z1, z2) -> f(z0, cons(z1, z2))
Tuples:
   F(empty, z0) -> c
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
S tuples:
   F(empty, z0) -> c
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   f_2, g_3

Defined Pair Symbols:   F_2, G_3

Compound Symbols:   c, c1_1, c2_1


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

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(empty, z0) -> z0
   f(cons(z0, z1), z2) -> g(z1, z2, cons(z0, z1))
   g(z0, z1, z2) -> f(z0, cons(z1, z2))
Tuples:
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
S tuples:
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   f_2, g_3

Defined Pair Symbols:   F_2, G_3

Compound Symbols:   c1_1, c2_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(empty, z0) -> z0
   f(cons(z0, z1), z2) -> g(z1, z2, cons(z0, z1))
   g(z0, z1, z2) -> f(z0, cons(z1, z2))

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
S tuples:
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2, G_3

Compound Symbols:   c1_1, c2_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.
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
We considered the (Usable) Rules:none
And the Tuples:
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2, x_3)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
S tuples:
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
K tuples:
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
Defined Rule Symbols:none

Defined Pair Symbols:   F_2, G_3

Compound Symbols:   c1_1, c2_1


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(9) CdtKnowledgeProof (FINISHED)
The following tuples could be moved from S to K by knowledge propagation:
   G(z0, z1, z2) -> c2(F(z0, cons(z1, z2)))
   F(cons(z0, z1), z2) -> c1(G(z1, z2, cons(z0, z1)))
Now 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:

   f(empty, l) -> l
   f(cons(x, k), l) -> g(k, l, cons(x, k))
   g(a, b, c) -> f(a, cons(b, c))

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

g(cons(x1_0, k2_0), b, c) ->^+ g(k2_0, cons(b, c), cons(x1_0, k2_0))

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

The pumping substitution is [k2_0 / cons(x1_0, k2_0)].

The result substitution is [b / cons(b, c), c / cons(x1_0, k2_0)].




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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:

   f(empty, l) -> l
   f(cons(x, k), l) -> g(k, l, cons(x, k))
   g(a, b, c) -> f(a, cons(b, c))

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:

   f(empty, l) -> l
   f(cons(x, k), l) -> g(k, l, cons(x, k))
   g(a, b, c) -> f(a, cons(b, c))

S is empty.
Rewrite Strategy: INNERMOST