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e63eb7f | 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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Combinatorics.EditApproximation.Binary.BinaryActualBandOptimizer
import OAI.Combinatorics.EditApproximation.Numerics.OptimizerBitEnvelope
namespace OAI
/-! A fixed polynomial bounds the complete envelope construction and band
optimizer. In particular the child masses are computed from the initial
table, rather than supplied as already constructed binary fractions. -/
namespace EditApproximation
private theorem square_work_cap (S coefficient width : ℕ) (hc : coefficient ≤ S ^ 2)
(hw : width ≤ S ^ 2) : coefficient * width ^ 2 ≤ S ^ 6 := by
calc
_ ≤ S ^ 2 * (S ^ 2) ^ 2 := Nat.mul_le_mul hc (Nat.pow_le_pow_left hw 2)
_ = _ := by ring
private theorem square_work_with_constant_cap (S coefficient width extra : ℕ)
(hS : 64 ≤ S) (hc : coefficient ≤ S ^ 2) (hw : width ≤ S ^ 2)
(he : extra ≤ S ^ 6) : coefficient * width ^ 2 + extra ≤ S ^ 7 := by
have h := square_work_cap S coefficient width hc hw
calc
_ ≤ 2 * S ^ 6 := by omega
_ ≤ S * S ^ 6 := Nat.mul_le_mul_right _ (by omega)
_ = _ := by ring
theorem bandEnvelopeBitBudget_le_power (b M vertices B S : ℕ) (hS : 64 ≤ S)
(hb : Nat.size b ≤ S) (hM : Nat.size M ≤ S) (hv : vertices ≤ S) (hB : B ≤ S) :
BinaryFraction.bandEnvelopeBitBudget b M vertices B ≤ S ^ 10 := by
let W := BinaryFraction.bandEnvelopeWordBudget b M B
have hW : W ≤ 3 * S + 8 := by
dsimp only [W, BinaryFraction.bandEnvelopeWordBudget]
omega
have hW1 : W + 1 ≤ S ^ 2 := by nlinarith
have hW3 : 3 * W + 2 ≤ S ^ 2 := by nlinarith
have hW2 : 2 * W ≤ S ^ 2 := by nlinarith
have htail : 3 * W + 6 ≤ S ^ 2 := by nlinarith
have hfloor : Nat.size b + Nat.size M + 6 ≤ S ^ 2 := by nlinarith
have h1000 : 1000 ≤ S ^ 2 := by nlinarith
have h2100 : 2100 ≤ S ^ 2 := by nlinarith
have hpos : 0 < S := by omega
have hpow6 : 2 ≤ S ^ 6 := (by omega : 2 ≤ S).trans (Nat.le_pow (by omega))
have hfc := square_work_cap S 2100 (Nat.size b + Nat.size M + 6) h2100 hfloor
have hcmp := square_work_with_constant_cap S 1000 (W + 1) 2 hS h1000 hW1 hpow6
have hbig := square_work_with_constant_cap S 1000 (3 * W + 2) 2 hS h1000 hW3 hpow6
have hmember : vertices * (1000 * (W + 1) ^ 2 + 2) ≤ S ^ 8 := by
calc
_ ≤ S * S ^ 7 := Nat.mul_le_mul hv hcmp
_ = _ := by ring
have hscan : 2 * W * (1000 * (3 * W + 2) ^ 2 + 2) ≤ S ^ 9 := by
calc
_ ≤ S ^ 2 * S ^ 7 := Nat.mul_le_mul hW2 hbig
_ = _ := by ring
have h6 : S ^ 6 ≤ S ^ 9 := Nat.pow_le_pow_right hpos (by omega)
have h7 : S ^ 7 ≤ S ^ 9 := Nat.pow_le_pow_right hpos (by omega)
have h8 : S ^ 8 ≤ S ^ 9 := Nat.pow_le_pow_right hpos (by omega)
have h2 : S ^ 2 ≤ S ^ 9 := Nat.pow_le_pow_right hpos (by omega)
have hsmall : vertices + 6 ≤ S ^ 9 := by
have hsq : S + 6 ≤ S ^ 2 := by nlinarith
omega
change 2100 * (Nat.size b + Nat.size M + 6) ^ 2 +
(1 + vertices * (1000 * (W + 1) ^ 2 + 2)) + (1000 * (W + 1) ^ 2 + 1) +
(1 + 2 * W * (1000 * (3 * W + 2) ^ 2 + 2) + 3 * W + 5) + vertices + 4 ≤ _
have hten : S ^ 10 = S * S ^ 9 := by ring
calc
_ ≤ 6 * S ^ 9 := by
nlinarith only [hfc, hcmp, hmember, hscan, htail, h6, h7, h8, h2, hsmall]
_ ≤ S * S ^ 9 := Nat.mul_le_mul_right _ (by omega)
_ = _ := hten.symm
def actualBandOptimizerBitDegree : ℕ := max 12 (2 * preparedOptimizerBitPolynomialDegree) + 2
theorem actualBandOptimizerBitBudget_le_fixed_power
(M d b vertices G K t Be Bk Bi Bt E S : ℕ) (hS : 64 ≤ S)
(hM : M ≤ S) (hd : d ≤ S) (hb : Nat.size b ≤ S) (hv : vertices ≤ S)
(hG : G ≤ S) (hK : K ≤ S) (ht : t ≤ S)
(hBe : Be ≤ S) (hBk : Bk ≤ S) (hBi : Bi ≤ S) (hBt : Bt ≤ S) (hE : E ≤ S) :
BinaryFraction.actualBandOptimizerBitBudget M d b vertices G K t Be Bk Bi Bt E ≤
S ^ actualBandOptimizerBitDegree := by
have hpos : 0 < S := by omega
have hSsq : S ≤ S ^ 2 := Nat.le_pow (by omega)
have hMbits : Nat.size M ≤ S := (Nat.size_le.mpr Nat.lt_two_pow_self).trans hM
have hmass : 3 * BinaryFraction.bandEnvelopeWordBudget b M Bi + 1 ≤ S ^ 2 := by
unfold BinaryFraction.bandEnvelopeWordBudget
nlinarith
have hprepared := preparedBandOptimizerBitBudget_le_fixed_power
M d vertices G K t Be Bk (3 * BinaryFraction.bandEnvelopeWordBudget b M Bi + 1) Bt E
(S ^ 2) ((by omega : 2 ≤ S).trans hSsq)
(hM.trans hSsq) (hd.trans hSsq) (hv.trans hSsq) (hG.trans hSsq) (hK.trans hSsq)
(ht.trans hSsq) (hBe.trans hSsq) (hBk.trans hSsq) hmass (hBt.trans hSsq) (hE.trans hSsq)
rw [← pow_mul] at hprepared
have henv := bandEnvelopeBitBudget_le_power b M vertices Bi S hS hb hMbits hv hBi
have h6 : 6 ≤ S ^ 10 := (by omega : 6 ≤ S).trans (Nat.le_pow (by omega))
have henvelope : M * (BinaryFraction.bandEnvelopeBitBudget b M vertices Bi + 6) ≤ S ^ 12 := by
calc
_ ≤ S * (S ^ 10 + 6) := Nat.mul_le_mul hM (by omega)
_ ≤ S * (S * S ^ 10) := Nat.mul_le_mul_left S (by
calc
_ ≤ 2 * S ^ 10 := by omega
_ ≤ S * S ^ 10 := Nat.mul_le_mul_right _ (by omega))
_ = _ := by ring
let D := max 12 (2 * preparedOptimizerBitPolynomialDegree)
have hleft : S ^ 12 ≤ S ^ D := Nat.pow_le_pow_right hpos (Nat.le_max_left _ _)
have hright : S ^ (2 * preparedOptimizerBitPolynomialDegree) ≤ S ^ D :=
Nat.pow_le_pow_right hpos (Nat.le_max_right _ _)
have hone : 1 ≤ S ^ D := Nat.one_le_pow _ _ hpos
change M * (BinaryFraction.bandEnvelopeBitBudget b M vertices Bi + 6) +
BinaryFraction.preparedBandOptimizerBitBudget M d vertices G K t Be Bk
(3 * BinaryFraction.bandEnvelopeWordBudget b M Bi + 1) Bt E + 1 ≤ S ^ (D + 2)
calc
_ ≤ 3 * S ^ D := by omega
_ ≤ S ^ D * S ^ 2 := by
simpa only [Nat.mul_comm] using
Nat.mul_le_mul_right (S ^ D) (show 3 ≤ S ^ 2 by nlinarith)
_ = _ := (pow_add S D 2).symm
end EditApproximation
end OAI
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