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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 39 ms]
    (4) BOUNDS(1, n^1)
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (8) BEST
        (9) proven lower bound
            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
            (11) BOUNDS(n^1, INF)
        (12) 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:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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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:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

S is empty.
Rewrite Strategy: FULL
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(3) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
nil0() -> 0
00() -> 0
g0(0, 0) -> 0
s0(0) -> 0
norm0(0) -> 1
f0(0, 0) -> 2
rem0(0, 0) -> 3
01() -> 1
norm1(0) -> 4
s1(4) -> 1
nil1() -> 5
g1(5, 0) -> 2
f1(0, 0) -> 6
g1(6, 0) -> 2
nil1() -> 3
g1(0, 0) -> 3
rem1(0, 0) -> 3
01() -> 4
s1(4) -> 4
g1(5, 0) -> 6
g1(6, 0) -> 6

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(4)
BOUNDS(1, n^1)

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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(6)
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:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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

The rewrite sequence

rem(g(x, y), s(z)) ->^+ rem(x, z)

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

The pumping substitution is [x / g(x, y), z / s(z)].

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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(9)
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:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

S is empty.
Rewrite Strategy: FULL
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
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:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

S is empty.
Rewrite Strategy: FULL