File size: 3,947 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 89 90 91 92 93 94 95 96 | import OAI.Combinatorics.LowSpaceSubsetSum.Storage.PermutationSamplerCells
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.MeasuredFilterCells
import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationLayoutExecution
namespace OAI
/-! # Live stored arrays for the actual permutation and sign draw -/
noncomputable section
namespace LowSpaceSubsetSum.LayoutSamplerCells
variable {n : ℕ}
private theorem mask_ext {a b : IndexedMask n} (h : a.indices = b.indices) : a = b := by
cases a
cases b
cases h
rfl
def signs (n : ℕ) (bits : Fin n → Bool) : (IndexedMask n × ℕ) × ℕ :=
let r := MeasuredFilterCells.run (fun i => ((bits i, 0), 2)) (List.finRange n)
((⟨r.1.1, (MeasuredFilterCells.run_length _ _).trans (by simp only [List.length_finRange]; omega)⟩,
4*n+2), r.2+2*n+8)
theorem signs_value (n : ℕ) (bits : Fin n → Bool) :
(signs n bits).1 = SignMaskExecution.run n bits := by
apply Prod.ext
· apply mask_ext
simp only [signs, SignMaskExecution.run, MeasuredFilterCells.run_value,
MeasuredFilter.runAcc_value, List.reverse_nil, List.nil_append]
· rfl
theorem signs_cells (n : ℕ) (bits : Fin n → Bool) : (signs n bits).2 ≤ 4*n+14 := by
have h := MeasuredFilterCells.run_cells
(fun i : Fin n => ((bits i, 0), 2)) 2 (fun _ => le_rfl) (List.finRange n)
simp only [List.length_finRange] at h
change _+2*n+8 ≤ _
omega
def run (R : RepresentationSetup n) (s : R.layoutCoupling.Seed) :
(Option (Vector (Fin n) n × IndexedMask n) × ℕ) × ℕ :=
let digits := PermutationSamplerCells.run n n s.1
match digits.1.value with
| none => ((none, digits.1.reads+digits.1.tests+1), digits.2+n+16)
| some d =>
let v := PermutationArray.build n d
let mask := signs n s.2
((some (v.1, mask.1.1), digits.1.reads+digits.1.tests+v.2+mask.1.2+3),
digits.2+PermutationArray.peakCells n+mask.2+3*n+32)
theorem run_value (R : RepresentationSetup n) (s : R.layoutCoupling.Seed) :
(run R s).1 = RepresentationLayoutExecution.run R s := by
have h := signs_value n s.2
have h₁ := congrArg Prod.fst h
have h₂ := congrArg Prod.snd h
simp only [run, RepresentationLayoutExecution.run, PermutationSamplerCells.run_value, h₁, h₂]
split <;> simp_all only
theorem run_cells (R : RepresentationSetup n) (s : R.layoutCoupling.Seed) :
(run R s).2 ≤ (n+1)*(3*n^3+128)+32*(n+1)^2 := by
have hd := PermutationSamplerCells.run_cells n n s.1
have hp := PermutationArray.peakCells_bound n
have hs := signs_cells n s.2
simp only [run]
split <;> dsimp only <;> nlinarith only [hd, hp, hs, Nat.zero_le (n^2), Nat.zero_le (n^3)]
theorem run_cells_scale (R : RepresentationSetup n) (hn : 10000 ≤ n)
(s : R.layoutCoupling.Seed) : (run R s).2 ≤ inputScale R.input^6 := by
have hu : n+1 ≤ inputScale R.input := by unfold inputScale; omega
have hn' : n ≤ inputScale R.input := by omega
have hu0 : 10000 ≤ inputScale R.input := by omega
have hp3 := Nat.pow_le_pow_left hn' 3
have hp2 := Nat.pow_le_pow_left hu 2
have hu3 : inputScale R.input ≤ inputScale R.input^3 := by
simpa only [pow_one] using pow_le_pow_right' (by omega : 1 ≤ inputScale R.input)
(by decide : 1 ≤ 3)
have ht : 3*n^3+128 ≤ 4*inputScale R.input^3 := by nlinarith only [hp3, hu3, hu0]
have hm := Nat.mul_le_mul hu ht
have hm' : (n+1)*(3*n^3+128) ≤ 4*inputScale R.input^4 := by
convert hm using 1; ring
have h24 : inputScale R.input^2 ≤ inputScale R.input^4 :=
pow_le_pow_right' (by omega) (by decide)
have h46 : 36*inputScale R.input^4 ≤ inputScale R.input^6 := by
have hcoef : 36 ≤ inputScale R.input^2 := by nlinarith only [hu0]
calc
_ ≤ inputScale R.input^2*inputScale R.input^4 := Nat.mul_le_mul_right _ hcoef
_ = _ := by rw [← pow_add]
have h := run_cells R s
nlinarith only [h, hm', hp2, h24, h46]
end LowSpaceSubsetSum.LayoutSamplerCells
end
end OAI
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