/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 40 ms]
    (4) BOUNDS(1, n^1)
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) SlicingProof [LOWER BOUND(ID), 0 ms]
    (8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 313 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 51 ms]
            (20) typed CpxTrs


----------------------------------------

(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
----------------------------------------

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(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
----------------------------------------

(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

----------------------------------------

(4)
BOUNDS(1, n^1)

----------------------------------------

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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
----------------------------------------

(7) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
::/0
group3#2/1
group3#3/1
group3#3/2
tuple#3/0
tuple#3/1
tuple#3/2
zip3#2/2
zip3#3/1
zip3#3/3

----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
Innermost TRS:
Rules:
group3(@l) -> group3#1(@l)
group3#1(::(@xs)) -> group3#2(@xs)
group3#1(nil) -> nil
group3#2(::(@ys)) -> group3#3(@ys)
group3#2(nil) -> nil
group3#3(::(@zs)) -> ::(group3(@zs))
group3#3(nil) -> nil
zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
zip3#1(::(@xs), @l2, @l3) -> zip3#2(@l2, @l3, @xs)
zip3#1(nil, @l2, @l3) -> nil
zip3#2(::(@ys), @l3, @xs) -> zip3#3(@l3, @xs, @ys)
zip3#2(nil, @l3, @xs) -> nil
zip3#3(::(@zs), @xs, @ys) -> ::(zip3(@xs, @ys, @zs))
zip3#3(nil, @xs, @ys) -> nil

Types:
group3 :: :::nil -> :::nil
group3#1 :: :::nil -> :::nil
:: :: :::nil -> :::nil
group3#2 :: :::nil -> :::nil
nil :: :::nil
group3#3 :: :::nil -> :::nil
zip3 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#1 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#2 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#3 :: :::nil -> :::nil -> :::nil -> :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat -> :::nil

----------------------------------------

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
group3, group3#1, group3#2, group3#3, zip3, zip3#1, zip3#2, zip3#3

They will be analysed ascendingly in the following order:
group3 = group3#1
group3 = group3#2
group3 = group3#3
group3#1 = group3#2
group3#1 = group3#3
group3#2 = group3#3
zip3 = zip3#1
zip3 = zip3#2
zip3 = zip3#3
zip3#1 = zip3#2
zip3#1 = zip3#3
zip3#2 = zip3#3

----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
group3(@l) -> group3#1(@l)
group3#1(::(@xs)) -> group3#2(@xs)
group3#1(nil) -> nil
group3#2(::(@ys)) -> group3#3(@ys)
group3#2(nil) -> nil
group3#3(::(@zs)) -> ::(group3(@zs))
group3#3(nil) -> nil
zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
zip3#1(::(@xs), @l2, @l3) -> zip3#2(@l2, @l3, @xs)
zip3#1(nil, @l2, @l3) -> nil
zip3#2(::(@ys), @l3, @xs) -> zip3#3(@l3, @xs, @ys)
zip3#2(nil, @l3, @xs) -> nil
zip3#3(::(@zs), @xs, @ys) -> ::(zip3(@xs, @ys, @zs))
zip3#3(nil, @xs, @ys) -> nil

Types:
group3 :: :::nil -> :::nil
group3#1 :: :::nil -> :::nil
:: :: :::nil -> :::nil
group3#2 :: :::nil -> :::nil
nil :: :::nil
group3#3 :: :::nil -> :::nil
zip3 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#1 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#2 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#3 :: :::nil -> :::nil -> :::nil -> :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat -> :::nil


Generator Equations:
gen_:::nil2_0(0) <=> nil
gen_:::nil2_0(+(x, 1)) <=> ::(gen_:::nil2_0(x))


The following defined symbols remain to be analysed:
zip3#1, group3, group3#1, group3#2, group3#3, zip3, zip3#2, zip3#3

They will be analysed ascendingly in the following order:
group3 = group3#1
group3 = group3#2
group3 = group3#3
group3#1 = group3#2
group3#1 = group3#3
group3#2 = group3#3
zip3 = zip3#1
zip3 = zip3#2
zip3 = zip3#3
zip3#1 = zip3#2
zip3#1 = zip3#3
zip3#2 = zip3#3

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
zip3#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0)) -> gen_:::nil2_0(n4_0), rt in Omega(1 + n4_0)

Induction Base:
zip3#1(gen_:::nil2_0(0), gen_:::nil2_0(0), gen_:::nil2_0(0)) ->_R^Omega(1)
nil

Induction Step:
zip3#1(gen_:::nil2_0(+(n4_0, 1)), gen_:::nil2_0(+(n4_0, 1)), gen_:::nil2_0(+(n4_0, 1))) ->_R^Omega(1)
zip3#2(gen_:::nil2_0(+(n4_0, 1)), gen_:::nil2_0(+(n4_0, 1)), gen_:::nil2_0(n4_0)) ->_R^Omega(1)
zip3#3(gen_:::nil2_0(+(1, n4_0)), gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0)) ->_R^Omega(1)
::(zip3(gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0))) ->_R^Omega(1)
::(zip3#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0))) ->_IH
::(gen_:::nil2_0(c5_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Complex Obligation (BEST)

----------------------------------------

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
group3(@l) -> group3#1(@l)
group3#1(::(@xs)) -> group3#2(@xs)
group3#1(nil) -> nil
group3#2(::(@ys)) -> group3#3(@ys)
group3#2(nil) -> nil
group3#3(::(@zs)) -> ::(group3(@zs))
group3#3(nil) -> nil
zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
zip3#1(::(@xs), @l2, @l3) -> zip3#2(@l2, @l3, @xs)
zip3#1(nil, @l2, @l3) -> nil
zip3#2(::(@ys), @l3, @xs) -> zip3#3(@l3, @xs, @ys)
zip3#2(nil, @l3, @xs) -> nil
zip3#3(::(@zs), @xs, @ys) -> ::(zip3(@xs, @ys, @zs))
zip3#3(nil, @xs, @ys) -> nil

Types:
group3 :: :::nil -> :::nil
group3#1 :: :::nil -> :::nil
:: :: :::nil -> :::nil
group3#2 :: :::nil -> :::nil
nil :: :::nil
group3#3 :: :::nil -> :::nil
zip3 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#1 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#2 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#3 :: :::nil -> :::nil -> :::nil -> :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat -> :::nil


Generator Equations:
gen_:::nil2_0(0) <=> nil
gen_:::nil2_0(+(x, 1)) <=> ::(gen_:::nil2_0(x))


The following defined symbols remain to be analysed:
zip3#1, group3, group3#1, group3#2, group3#3, zip3, zip3#2, zip3#3

They will be analysed ascendingly in the following order:
group3 = group3#1
group3 = group3#2
group3 = group3#3
group3#1 = group3#2
group3#1 = group3#3
group3#2 = group3#3
zip3 = zip3#1
zip3 = zip3#2
zip3 = zip3#3
zip3#1 = zip3#2
zip3#1 = zip3#3
zip3#2 = zip3#3

----------------------------------------

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
group3(@l) -> group3#1(@l)
group3#1(::(@xs)) -> group3#2(@xs)
group3#1(nil) -> nil
group3#2(::(@ys)) -> group3#3(@ys)
group3#2(nil) -> nil
group3#3(::(@zs)) -> ::(group3(@zs))
group3#3(nil) -> nil
zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
zip3#1(::(@xs), @l2, @l3) -> zip3#2(@l2, @l3, @xs)
zip3#1(nil, @l2, @l3) -> nil
zip3#2(::(@ys), @l3, @xs) -> zip3#3(@l3, @xs, @ys)
zip3#2(nil, @l3, @xs) -> nil
zip3#3(::(@zs), @xs, @ys) -> ::(zip3(@xs, @ys, @zs))
zip3#3(nil, @xs, @ys) -> nil

Types:
group3 :: :::nil -> :::nil
group3#1 :: :::nil -> :::nil
:: :: :::nil -> :::nil
group3#2 :: :::nil -> :::nil
nil :: :::nil
group3#3 :: :::nil -> :::nil
zip3 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#1 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#2 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#3 :: :::nil -> :::nil -> :::nil -> :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat -> :::nil


Lemmas:
zip3#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0)) -> gen_:::nil2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_:::nil2_0(0) <=> nil
gen_:::nil2_0(+(x, 1)) <=> ::(gen_:::nil2_0(x))


The following defined symbols remain to be analysed:
zip3#2, group3, group3#1, group3#2, group3#3, zip3, zip3#3

They will be analysed ascendingly in the following order:
group3 = group3#1
group3 = group3#2
group3 = group3#3
group3#1 = group3#2
group3#1 = group3#3
group3#2 = group3#3
zip3 = zip3#1
zip3 = zip3#2
zip3 = zip3#3
zip3#1 = zip3#2
zip3#1 = zip3#3
zip3#2 = zip3#3

----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
group3#1(gen_:::nil2_0(*(3, n929_0))) -> gen_:::nil2_0(n929_0), rt in Omega(1 + n929_0)

Induction Base:
group3#1(gen_:::nil2_0(*(3, 0))) ->_R^Omega(1)
nil

Induction Step:
group3#1(gen_:::nil2_0(*(3, +(n929_0, 1)))) ->_R^Omega(1)
group3#2(gen_:::nil2_0(+(2, *(3, n929_0)))) ->_R^Omega(1)
group3#3(gen_:::nil2_0(+(1, *(3, n929_0)))) ->_R^Omega(1)
::(group3(gen_:::nil2_0(*(3, n929_0)))) ->_R^Omega(1)
::(group3#1(gen_:::nil2_0(*(3, n929_0)))) ->_IH
::(gen_:::nil2_0(c930_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
group3(@l) -> group3#1(@l)
group3#1(::(@xs)) -> group3#2(@xs)
group3#1(nil) -> nil
group3#2(::(@ys)) -> group3#3(@ys)
group3#2(nil) -> nil
group3#3(::(@zs)) -> ::(group3(@zs))
group3#3(nil) -> nil
zip3(@l1, @l2, @l3) -> zip3#1(@l1, @l2, @l3)
zip3#1(::(@xs), @l2, @l3) -> zip3#2(@l2, @l3, @xs)
zip3#1(nil, @l2, @l3) -> nil
zip3#2(::(@ys), @l3, @xs) -> zip3#3(@l3, @xs, @ys)
zip3#2(nil, @l3, @xs) -> nil
zip3#3(::(@zs), @xs, @ys) -> ::(zip3(@xs, @ys, @zs))
zip3#3(nil, @xs, @ys) -> nil

Types:
group3 :: :::nil -> :::nil
group3#1 :: :::nil -> :::nil
:: :: :::nil -> :::nil
group3#2 :: :::nil -> :::nil
nil :: :::nil
group3#3 :: :::nil -> :::nil
zip3 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#1 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#2 :: :::nil -> :::nil -> :::nil -> :::nil
zip3#3 :: :::nil -> :::nil -> :::nil -> :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat -> :::nil


Lemmas:
zip3#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0), gen_:::nil2_0(n4_0)) -> gen_:::nil2_0(n4_0), rt in Omega(1 + n4_0)
group3#1(gen_:::nil2_0(*(3, n929_0))) -> gen_:::nil2_0(n929_0), rt in Omega(1 + n929_0)


Generator Equations:
gen_:::nil2_0(0) <=> nil
gen_:::nil2_0(+(x, 1)) <=> ::(gen_:::nil2_0(x))


The following defined symbols remain to be analysed:
group3#2, group3, group3#3

They will be analysed ascendingly in the following order:
group3 = group3#1
group3 = group3#2
group3 = group3#3
group3#1 = group3#2
group3#1 = group3#3
group3#2 = group3#3