WORST_CASE(?,O(n^1))
proof of input_fT3CzJFhSE.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) CpxTrsMatchBoundsTAProof [FINISHED, 58 ms]
(4) BOUNDS(1, n^1)


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(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_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)