/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 242 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:

   revconsapp(C(x1, x2), r) -> revconsapp(x2, C(x1, r))
   deeprevapp(C(x1, x2), rest) -> deeprevapp(x2, C(x1, rest))
   deeprevapp(V(n), rest) -> revconsapp(rest, V(n))
   deeprevapp(N, rest) -> rest
   revconsapp(V(n), r) -> r
   revconsapp(N, r) -> r
   deeprev(C(x1, x2)) -> deeprevapp(C(x1, x2), N)
   deeprev(V(n)) -> V(n)
   deeprev(N) -> N
   second(V(n)) -> N
   second(C(x1, x2)) -> x2
   isVal(C(x1, x2)) -> False
   isVal(V(n)) -> True
   isVal(N) -> False
   isNotEmptyT(C(x1, x2)) -> True
   isNotEmptyT(V(n)) -> False
   isNotEmptyT(N) -> False
   isEmptyT(C(x1, x2)) -> False
   isEmptyT(V(n)) -> False
   isEmptyT(N) -> True
   first(V(n)) -> N
   first(C(x1, x2)) -> x1
   goal(x) -> deeprev(x)

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:

   revconsapp(C(x1, x2), r) -> revconsapp(x2, C(x1, r))
   deeprevapp(C(x1, x2), rest) -> deeprevapp(x2, C(x1, rest))
   deeprevapp(V(n), rest) -> revconsapp(rest, V(n))
   deeprevapp(N, rest) -> rest
   revconsapp(V(n), r) -> r
   revconsapp(N, r) -> r
   deeprev(C(x1, x2)) -> deeprevapp(C(x1, x2), N)
   deeprev(V(n)) -> V(n)
   deeprev(N) -> N
   second(V(n)) -> N
   second(C(x1, x2)) -> x2
   isVal(C(x1, x2)) -> False
   isVal(V(n)) -> True
   isVal(N) -> False
   isNotEmptyT(C(x1, x2)) -> True
   isNotEmptyT(V(n)) -> False
   isNotEmptyT(N) -> False
   isEmptyT(C(x1, x2)) -> False
   isEmptyT(V(n)) -> False
   isEmptyT(N) -> True
   first(V(n)) -> N
   first(C(x1, x2)) -> x1
   goal(x) -> deeprev(x)

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 3. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9]
transitions: 
C0(0, 0) -> 0
V0(0) -> 0
N0() -> 0
False0() -> 0
True0() -> 0
revconsapp0(0, 0) -> 1
deeprevapp0(0, 0) -> 2
deeprev0(0) -> 3
second0(0) -> 4
isVal0(0) -> 5
isNotEmptyT0(0) -> 6
isEmptyT0(0) -> 7
first0(0) -> 8
goal0(0) -> 9
C1(0, 0) -> 10
revconsapp1(0, 10) -> 1
C1(0, 0) -> 11
deeprevapp1(0, 11) -> 2
V1(0) -> 12
revconsapp1(0, 12) -> 2
C1(0, 0) -> 13
N1() -> 14
deeprevapp1(13, 14) -> 3
V1(0) -> 3
N1() -> 3
N1() -> 4
False1() -> 5
True1() -> 5
True1() -> 6
False1() -> 6
False1() -> 7
True1() -> 7
N1() -> 8
deeprev1(0) -> 9
C1(0, 10) -> 10
C1(0, 12) -> 10
revconsapp1(0, 10) -> 2
C1(0, 11) -> 11
C2(0, 14) -> 15
deeprevapp2(0, 15) -> 3
revconsapp1(11, 12) -> 2
deeprevapp1(13, 14) -> 9
V1(0) -> 9
N1() -> 9
C2(0, 12) -> 16
revconsapp2(0, 16) -> 2
revconsapp2(11, 16) -> 2
deeprevapp2(0, 15) -> 9
C1(0, 15) -> 11
deeprevapp1(0, 11) -> 3
revconsapp1(15, 12) -> 3
revconsapp2(15, 16) -> 2
revconsapp1(11, 12) -> 3
revconsapp2(14, 16) -> 3
deeprevapp1(0, 11) -> 9
revconsapp1(15, 12) -> 9
C1(0, 16) -> 10
C2(0, 16) -> 16
revconsapp2(0, 16) -> 3
revconsapp2(11, 16) -> 3
revconsapp2(15, 16) -> 3
revconsapp1(11, 12) -> 9
revconsapp2(14, 16) -> 9
C3(0, 16) -> 17
revconsapp3(14, 17) -> 2
revconsapp2(0, 16) -> 9
revconsapp2(11, 16) -> 9
revconsapp2(15, 16) -> 9
revconsapp1(0, 10) -> 3
revconsapp3(14, 17) -> 3
revconsapp1(0, 10) -> 9
revconsapp3(14, 17) -> 9
0 -> 2
0 -> 1
0 -> 4
0 -> 8
11 -> 2
11 -> 3
11 -> 9
10 -> 1
10 -> 2
10 -> 3
10 -> 9
12 -> 2
15 -> 3
15 -> 9
16 -> 2
16 -> 3
16 -> 9
17 -> 2
17 -> 3
17 -> 9

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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:

   revconsapp(C(x1, x2), r) -> revconsapp(x2, C(x1, r))
   deeprevapp(C(x1, x2), rest) -> deeprevapp(x2, C(x1, rest))
   deeprevapp(V(n), rest) -> revconsapp(rest, V(n))
   deeprevapp(N, rest) -> rest
   revconsapp(V(n), r) -> r
   revconsapp(N, r) -> r
   deeprev(C(x1, x2)) -> deeprevapp(C(x1, x2), N)
   deeprev(V(n)) -> V(n)
   deeprev(N) -> N
   second(V(n)) -> N
   second(C(x1, x2)) -> x2
   isVal(C(x1, x2)) -> False
   isVal(V(n)) -> True
   isVal(N) -> False
   isNotEmptyT(C(x1, x2)) -> True
   isNotEmptyT(V(n)) -> False
   isNotEmptyT(N) -> False
   isEmptyT(C(x1, x2)) -> False
   isEmptyT(V(n)) -> False
   isEmptyT(N) -> True
   first(V(n)) -> N
   first(C(x1, x2)) -> x1
   goal(x) -> deeprev(x)

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

revconsapp(C(x1, x2), r) ->^+ revconsapp(x2, C(x1, r))

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

The pumping substitution is [x2 / C(x1, x2)].

The result substitution is [r / C(x1, r)].




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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:

   revconsapp(C(x1, x2), r) -> revconsapp(x2, C(x1, r))
   deeprevapp(C(x1, x2), rest) -> deeprevapp(x2, C(x1, rest))
   deeprevapp(V(n), rest) -> revconsapp(rest, V(n))
   deeprevapp(N, rest) -> rest
   revconsapp(V(n), r) -> r
   revconsapp(N, r) -> r
   deeprev(C(x1, x2)) -> deeprevapp(C(x1, x2), N)
   deeprev(V(n)) -> V(n)
   deeprev(N) -> N
   second(V(n)) -> N
   second(C(x1, x2)) -> x2
   isVal(C(x1, x2)) -> False
   isVal(V(n)) -> True
   isVal(N) -> False
   isNotEmptyT(C(x1, x2)) -> True
   isNotEmptyT(V(n)) -> False
   isNotEmptyT(N) -> False
   isEmptyT(C(x1, x2)) -> False
   isEmptyT(V(n)) -> False
   isEmptyT(N) -> True
   first(V(n)) -> N
   first(C(x1, x2)) -> x1
   goal(x) -> deeprev(x)

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:

   revconsapp(C(x1, x2), r) -> revconsapp(x2, C(x1, r))
   deeprevapp(C(x1, x2), rest) -> deeprevapp(x2, C(x1, rest))
   deeprevapp(V(n), rest) -> revconsapp(rest, V(n))
   deeprevapp(N, rest) -> rest
   revconsapp(V(n), r) -> r
   revconsapp(N, r) -> r
   deeprev(C(x1, x2)) -> deeprevapp(C(x1, x2), N)
   deeprev(V(n)) -> V(n)
   deeprev(N) -> N
   second(V(n)) -> N
   second(C(x1, x2)) -> x2
   isVal(C(x1, x2)) -> False
   isVal(V(n)) -> True
   isVal(N) -> False
   isNotEmptyT(C(x1, x2)) -> True
   isNotEmptyT(V(n)) -> False
   isNotEmptyT(N) -> False
   isEmptyT(C(x1, x2)) -> False
   isEmptyT(V(n)) -> False
   isEmptyT(N) -> True
   first(V(n)) -> N
   first(C(x1, x2)) -> x1
   goal(x) -> deeprev(x)

S is empty.
Rewrite Strategy: INNERMOST