/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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, 51 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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
   compS_f#1(id, x3) -> S(x3)
   iter#3(0) -> id
   iter#3(S(x6)) -> compS_f(iter#3(x6))
   main(0) -> 0
   main(S(x9)) -> compS_f#1(iter#3(x9), 0)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
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:

   compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
   compS_f#1(id, x3) -> S(x3)
   iter#3(0) -> id
   iter#3(S(x6)) -> compS_f(iter#3(x6))
   main(0) -> 0
   main(S(x9)) -> compS_f#1(iter#3(x9), 0)

S is empty.
Rewrite Strategy: INNERMOST
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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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
compS_f0(0) -> 0
S0(0) -> 0
id0() -> 0
00() -> 0
compS_f#10(0, 0) -> 1
iter#30(0) -> 2
main0(0) -> 3
S1(0) -> 4
compS_f#11(0, 4) -> 1
S1(0) -> 1
id1() -> 2
iter#31(0) -> 5
compS_f1(5) -> 2
01() -> 3
iter#31(0) -> 6
01() -> 7
compS_f#11(6, 7) -> 3
S1(4) -> 4
S1(4) -> 1
id1() -> 5
id1() -> 6
compS_f1(5) -> 5
compS_f1(5) -> 6
S2(7) -> 8
compS_f#12(5, 8) -> 3
S2(7) -> 3
S2(8) -> 8
S2(8) -> 3

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


The TRS R consists of the following rules:

   compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
   compS_f#1(id, x3) -> S(x3)
   iter#3(0) -> id
   iter#3(S(x6)) -> compS_f(iter#3(x6))
   main(0) -> 0
   main(S(x9)) -> compS_f#1(iter#3(x9), 0)

S is empty.
Rewrite Strategy: INNERMOST
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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

compS_f#1(compS_f(x2), x1) ->^+ compS_f#1(x2, S(x1))

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

The pumping substitution is [x2 / compS_f(x2)].

The result substitution is [x1 / S(x1)].




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


The TRS R consists of the following rules:

   compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
   compS_f#1(id, x3) -> S(x3)
   iter#3(0) -> id
   iter#3(S(x6)) -> compS_f(iter#3(x6))
   main(0) -> 0
   main(S(x9)) -> compS_f#1(iter#3(x9), 0)

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
   compS_f#1(id, x3) -> S(x3)
   iter#3(0) -> id
   iter#3(S(x6)) -> compS_f(iter#3(x6))
   main(0) -> 0
   main(S(x9)) -> compS_f#1(iter#3(x9), 0)

S is empty.
Rewrite Strategy: INNERMOST