File size: 3,838 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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import OAI.Combinatorics.LowSpaceSubsetSum.Storage.IntervalDomainCells
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.AllocationCells
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.BackgroundBlockCells
import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationBackgroundPreparation
namespace OAI
/-! # Live storage for the allocated background construction
The scanned outside list is shared by all eight row copies. The six integer
width computations run before the size guard and are charged on both branches.
-/
noncomputable section
namespace LowSpaceSubsetSum.BackgroundPreparationCells
open scoped BigOperators
variable {n : ℕ}
private def assemble (outside : List (Fin n) × ℕ) (counts : Vector ℕ 8 × ℕ) :
Option (Vector (List (Fin n)) 8) × ℕ :=
if h : (∑ i : Fin 8, counts.1[i.val]) = outside.1.length then
let xs : Vector (Fin n) outside.1.length := ⟨outside.1.toArray, by simp only [List.size_toArray]⟩
let rows := StoredBackgroundBlocks.run xs counts.1 h
(some rows.1, outside.2+counts.2+rows.2+20)
else (none, outside.2+counts.2+20)
private def core (outside : (List (Fin n) × ℕ) × ℕ)
(counts : (Vector ℕ 8 × ℕ) × ℕ) :
(Option (Vector (List (Fin n)) 8) × ℕ) × ℕ :=
if h : (∑ i : Fin 8, counts.1.1[i.val]) = outside.1.1.length then
let xs : Vector (Fin n) outside.1.1.length := ⟨outside.1.1.toArray, by simp only [List.size_toArray]⟩
let rows := BackgroundBlockCells.run xs counts.1.1 h
((some rows.1.1, outside.1.2+counts.1.2+rows.1.2+20),
outside.2+counts.2+rows.2+12*n+128)
else ((none, outside.1.2+counts.1.2+20), outside.2+counts.2+12*n+128)
private theorem core_value (outside : (List (Fin n) × ℕ) × ℕ)
(counts : (Vector ℕ 8 × ℕ) × ℕ) :
(core outside counts).1 = assemble outside.1 counts.1 := by
unfold core assemble
dsimp only
split <;> simp only [BackgroundBlockCells.value]
def run (_I : Instance n) (v : Vector (Fin n) n) (guesses : Vector ℕ 3) :
(Option (Vector (List (Fin n)) 8) × ℕ) × ℕ :=
core (IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n))
(AllocationCells.run n guesses)
theorem value (I : Instance n) (v : Vector (Fin n) n) (guesses : Vector ℕ 3) :
(run I v guesses).1 = RepresentationBackgroundPreparation.run I v guesses := by
have ho : (IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)).1 =
CanonicalBackground.run v := IntervalDomainCells.run_value v _ _
calc
(run I v guesses).1 = assemble
(IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)).1
(AllocationCells.run n guesses).1 := core_value _ _
_ = assemble (CanonicalBackground.run v)
(RepresentationAllocationExecution.run n guesses) :=
congrArg₂ assemble ho (AllocationCells.value n guesses)
_ = RepresentationBackgroundPreparation.run I v guesses := rfl
theorem cells (I : Instance n) (v : Vector (Fin n) n) (guesses : Vector ℕ 3) :
(run I v guesses).2 ≤ 2048*(n+1)^2 := by
have ho := IntervalDomainCells.run_cells v (3*mixerSize n) (n-3*mixerSize n)
have hc := AllocationCells.cells n guesses
have hl : (IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)).1.1.length ≤ n := by
rw [IntervalDomainCells.run_value]
exact CanonicalBackground.length_le v
unfold run core
dsimp only
split
next h =>
have hr := BackgroundBlockCells.cells
(⟨(IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)).1.1.toArray,
by simp only [List.size_toArray]⟩)
(AllocationCells.run n guesses).1.1 h
change _+_+_+12*n+128 ≤ _
nlinarith only [ho, hc, hl, hr, Nat.zero_le (n^2)]
· change _+_+12*n+128 ≤ _
nlinarith only [ho, hc, Nat.zero_le (n^2)]
end LowSpaceSubsetSum.BackgroundPreparationCells
end
end OAI
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