File size: 3,019 Bytes
5d5048e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import OAI.Combinatorics.LowSpaceSubsetSum.Storage.OrderedMixerCells
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.BackgroundPreparationCells
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.StoredLeafCells
import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationLeafArrays
namespace OAI
/-! # Live storage while constructing all leaf profiles
The ordered mixer arrays and completed background rows stay shared through
all eight finite-set profile constructions.
-/
noncomputable section
namespace LowSpaceSubsetSum.LeafArraysCells
variable {n : ℕ}
def run (I : Instance n) (data : RepresentationPrefixArrays.Data n) :
(Option (RepresentationLeafArrays.Leaves n) × ℕ) × ℕ :=
let ordered := OrderedMixerCells.run data.layout
match ordered.1.1 with
| none => ((none, ordered.1.2+1), ordered.2+16*n+128)
| some domains =>
let background := BackgroundPreparationCells.run I data.layout.permutation data.guesses
match background.1.1 with
| none => ((none, ordered.1.2+background.1.2+2), ordered.2+background.2+16*n+128)
| some rows =>
let out := StoredLeafCells.run domains data.partitions rows data.guesses
((some out.1.1, ordered.1.2+background.1.2+out.1.2+3),
ordered.2+background.2+out.2+16*n+128)
theorem value (I : Instance n) (data : RepresentationPrefixArrays.Data n) :
(run I data).1 = RepresentationLeafArrays.run I data := by
have ho₁ := congrArg Prod.fst (OrderedMixerCells.value data.layout)
have ho₂ := congrArg Prod.snd (OrderedMixerCells.value data.layout)
have hb₁ := congrArg Prod.fst (BackgroundPreparationCells.value I data.layout.permutation data.guesses)
have hb₂ := congrArg Prod.snd (BackgroundPreparationCells.value I data.layout.permutation data.guesses)
simp only [run, RepresentationLeafArrays.run, ho₁, ho₂, hb₁, hb₂]
split
· simp_all only
· split
· simp_all only
· simp_all only [StoredLeafCells.value]
theorem cells (I : Instance n) (data : RepresentationPrefixArrays.Data n)
(hl : ∀ j : Fin 3, data.layout.domains[j.val].length ≤ n) :
(run I data).2 ≤ 16384*(n+1)^2 := by
have ho := OrderedMixerCells.cells data.layout hl
have hb := BackgroundPreparationCells.cells I data.layout.permutation data.guesses
unfold run
dsimp only
split
· nlinarith only [ho, Nat.zero_le (n^2)]
next domains hd =>
split
· nlinarith only [ho, hb, Nat.zero_le (n^2)]
next rows hr =>
have hs : (RepresentationBackgroundPreparation.run I data.layout.permutation data.guesses).1 =
some rows := by
rw [← BackgroundPreparationCells.value I data.layout.permutation data.guesses]
exact hr
have hw := StoredLeafCells.cells domains data.partitions rows data.guesses
(RepresentationBackgroundPreparation.lengths I data.layout.permutation data.guesses rows hs)
change _+_+_+16*n+128 ≤ _
nlinarith only [ho, hb, hw, Nat.zero_le (n^2)]
end LowSpaceSubsetSum.LeafArraysCells
end
end OAI
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