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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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, 86 ms]
    (4) BOUNDS(1, n^1)
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
    (7) DecreasingLoopProof [LOWER BOUND(ID), 44 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:

   group3(@l) -> group3#1(@l)
   group3#1(::(@x, @xs)) -> group3#2(@xs, @x)
   group3#1(nil) -> nil
   group3#2(::(@y, @ys), @x) -> group3#3(@ys, @x, @y)
   group3#2(nil, @x) -> nil
   group3#3(::(@z, @zs), @x, @y) -> ::(tuple#3(@x, @y, @z), group3(@zs))
   group3#3(nil, @x, @y) -> nil
   zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
   zip3#1(::(@x, @xs), @l2, @l3) -> zip3#2(@l2, @l3, @x, @xs)
   zip3#1(nil, @l2, @l3) -> nil
   zip3#2(::(@y, @ys), @l3, @x, @xs) -> zip3#3(@l3, @x, @xs, @y, @ys)
   zip3#2(nil, @l3, @x, @xs) -> nil
   zip3#3(::(@z, @zs), @x, @xs, @y, @ys) -> ::(tuple#3(@x, @y, @z), zip3(@xs, @ys, @zs))
   zip3#3(nil, @x, @xs, @y, @ys) -> nil

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:

   group3(@l) -> group3#1(@l)
   group3#1(::(@x, @xs)) -> group3#2(@xs, @x)
   group3#1(nil) -> nil
   group3#2(::(@y, @ys), @x) -> group3#3(@ys, @x, @y)
   group3#2(nil, @x) -> nil
   group3#3(::(@z, @zs), @x, @y) -> ::(tuple#3(@x, @y, @z), group3(@zs))
   group3#3(nil, @x, @y) -> nil
   zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
   zip3#1(::(@x, @xs), @l2, @l3) -> zip3#2(@l2, @l3, @x, @xs)
   zip3#1(nil, @l2, @l3) -> nil
   zip3#2(::(@y, @ys), @l3, @x, @xs) -> zip3#3(@l3, @x, @xs, @y, @ys)
   zip3#2(nil, @l3, @x, @xs) -> nil
   zip3#3(::(@z, @zs), @x, @xs, @y, @ys) -> ::(tuple#3(@x, @y, @z), zip3(@xs, @ys, @zs))
   zip3#3(nil, @x, @xs, @y, @ys) -> nil

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, 4, 5, 6, 7, 8]
transitions: 
::0(0, 0) -> 0
nil0() -> 0
tuple#30(0, 0, 0) -> 0
group30(0) -> 1
group3#10(0) -> 2
group3#20(0, 0) -> 3
group3#30(0, 0, 0) -> 4
zip30(0, 0, 0) -> 5
zip3#10(0, 0, 0) -> 6
zip3#20(0, 0, 0, 0) -> 7
zip3#30(0, 0, 0, 0, 0) -> 8
group3#11(0) -> 1
group3#21(0, 0) -> 2
nil1() -> 2
group3#31(0, 0, 0) -> 3
nil1() -> 3
tuple#31(0, 0, 0) -> 9
group31(0) -> 10
::1(9, 10) -> 4
nil1() -> 4
zip3#11(0, 0, 0) -> 5
zip3#21(0, 0, 0, 0) -> 6
nil1() -> 6
zip3#31(0, 0, 0, 0, 0) -> 7
nil1() -> 7
zip31(0, 0, 0) -> 11
::1(9, 11) -> 8
nil1() -> 8
group3#12(0) -> 10
group3#21(0, 0) -> 1
nil1() -> 1
group3#31(0, 0, 0) -> 2
::1(9, 10) -> 3
zip3#12(0, 0, 0) -> 11
zip3#21(0, 0, 0, 0) -> 5
nil1() -> 5
zip3#31(0, 0, 0, 0, 0) -> 6
::1(9, 11) -> 7
group3#31(0, 0, 0) -> 1
::1(9, 10) -> 2
zip3#31(0, 0, 0, 0, 0) -> 5
::1(9, 11) -> 6
group3#21(0, 0) -> 10
nil1() -> 10
zip3#21(0, 0, 0, 0) -> 11
nil1() -> 11
group3#31(0, 0, 0) -> 10
::1(9, 10) -> 1
zip3#31(0, 0, 0, 0, 0) -> 11
::1(9, 11) -> 5
::1(9, 10) -> 10
::1(9, 11) -> 11

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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:

   group3(@l) -> group3#1(@l)
   group3#1(::(@x, @xs)) -> group3#2(@xs, @x)
   group3#1(nil) -> nil
   group3#2(::(@y, @ys), @x) -> group3#3(@ys, @x, @y)
   group3#2(nil, @x) -> nil
   group3#3(::(@z, @zs), @x, @y) -> ::(tuple#3(@x, @y, @z), group3(@zs))
   group3#3(nil, @x, @y) -> nil
   zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
   zip3#1(::(@x, @xs), @l2, @l3) -> zip3#2(@l2, @l3, @x, @xs)
   zip3#1(nil, @l2, @l3) -> nil
   zip3#2(::(@y, @ys), @l3, @x, @xs) -> zip3#3(@l3, @x, @xs, @y, @ys)
   zip3#2(nil, @l3, @x, @xs) -> nil
   zip3#3(::(@z, @zs), @x, @xs, @y, @ys) -> ::(tuple#3(@x, @y, @z), zip3(@xs, @ys, @zs))
   zip3#3(nil, @x, @xs, @y, @ys) -> nil

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

group3#2(::(@y, ::(@z1_0, ::(@x1_1, @xs2_1))), @x) ->^+ ::(tuple#3(@x, @y, @z1_0), group3#2(@xs2_1, @x1_1))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [@xs2_1 / ::(@y, ::(@z1_0, ::(@x1_1, @xs2_1)))].

The result substitution is [@x / @x1_1].




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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:

   group3(@l) -> group3#1(@l)
   group3#1(::(@x, @xs)) -> group3#2(@xs, @x)
   group3#1(nil) -> nil
   group3#2(::(@y, @ys), @x) -> group3#3(@ys, @x, @y)
   group3#2(nil, @x) -> nil
   group3#3(::(@z, @zs), @x, @y) -> ::(tuple#3(@x, @y, @z), group3(@zs))
   group3#3(nil, @x, @y) -> nil
   zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
   zip3#1(::(@x, @xs), @l2, @l3) -> zip3#2(@l2, @l3, @x, @xs)
   zip3#1(nil, @l2, @l3) -> nil
   zip3#2(::(@y, @ys), @l3, @x, @xs) -> zip3#3(@l3, @x, @xs, @y, @ys)
   zip3#2(nil, @l3, @x, @xs) -> nil
   zip3#3(::(@z, @zs), @x, @xs, @y, @ys) -> ::(tuple#3(@x, @y, @z), zip3(@xs, @ys, @zs))
   zip3#3(nil, @x, @xs, @y, @ys) -> nil

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:

   group3(@l) -> group3#1(@l)
   group3#1(::(@x, @xs)) -> group3#2(@xs, @x)
   group3#1(nil) -> nil
   group3#2(::(@y, @ys), @x) -> group3#3(@ys, @x, @y)
   group3#2(nil, @x) -> nil
   group3#3(::(@z, @zs), @x, @y) -> ::(tuple#3(@x, @y, @z), group3(@zs))
   group3#3(nil, @x, @y) -> nil
   zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
   zip3#1(::(@x, @xs), @l2, @l3) -> zip3#2(@l2, @l3, @x, @xs)
   zip3#1(nil, @l2, @l3) -> nil
   zip3#2(::(@y, @ys), @l3, @x, @xs) -> zip3#3(@l3, @x, @xs, @y, @ys)
   zip3#2(nil, @l3, @x, @xs) -> nil
   zip3#3(::(@z, @zs), @x, @xs, @y, @ys) -> ::(tuple#3(@x, @y, @z), zip3(@xs, @ys, @zs))
   zip3#3(nil, @x, @xs, @y, @ys) -> nil

S is empty.
Rewrite Strategy: INNERMOST