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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 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedFamilySampling
import OAI.Combinatorics.EditApproximation.Probability.ExactSampling
namespace OAI
namespace EditApproximation
/-- The stream keeps the current word and one optional accepted value;
failed words are overwritten and the failure count belongs only to the
probability-space description, not to the machine state. -/
def rejectionStateCells (N bits : ℕ) (state : Option (Fin N)) (word : Fin (2 ^ bits)) : ℕ :=
(state.map fun value => value.val.bits.length).getD 0 + word.val.bits.length + 4
theorem rejectionStateCells_bound (N bits : ℕ) (hN : N ≤ 2 ^ bits)
(state : Option (Fin N)) (word : Fin (2 ^ bits)) :
rejectionStateCells N bits state word ≤ 2 * bits + 4 := by
have hw : word.val.bits.length ≤ bits := by
rw [Nat.size_eq_bits_len]
exact Nat.size_le.mpr word.isLt
have hs : (state.map fun value => value.val.bits.length).getD 0 ≤ bits := by
cases state with
| none => exact Nat.zero_le _
| some value =>
change value.val.bits.length ≤ bits
rw [Nat.size_eq_bits_len]
exact Nat.size_le.mpr (value.isLt.trans_le hN)
unfold rejectionStateCells
omega
/-- This bound holds after every transition, regardless of how many
failed trials preceded it. -/
theorem rejectionTransition_cells (N bits : ℕ) (hN : N ≤ 2 ^ bits)
(state : Option (Fin N)) (word nextWord : Fin (2 ^ bits)) :
rejectionStateCells N bits (rejectionTransition N (2 ^ bits) state word) nextWord ≤ 2 * bits + 4 :=
rejectionStateCells_bound N bits hN _ _
theorem computedFamilySampler_cells (source target : List ℕ) (N R : ℕ)
(hMass : Nat.size (localOnlineMassBound (integerParameters N).M (integerParameters N).P
(inputHeight N)) ≤ inputHeight N)
(key : ComputedFamilyScalarKey source target N R)
(state : Option (Fin (physicalFamilyRange target N (integerParameters N).P (inputSmallLog N)
(40 * smallLogExponent N) (inputHeight N) (inputSmallLog N ^ 80)
(computedSeedPassCount N) (inputHeight N ^ 2) key)))
(word : Fin (2 ^ rejectionBitCount (physicalFamilyRange target N (integerParameters N).P (inputSmallLog N)
(40 * smallLogExponent N) (inputHeight N) (inputSmallLog N ^ 80)
(computedSeedPassCount N) (inputHeight N ^ 2) key))) :
rejectionStateCells _ _ state word ≤ 2 * inputHeight N + 4 := by
have h := rejectionStateCells_bound _ _ (Nat.le_pow_clog (by decide : 1 < (2 : ℕ)) _) state word
have hb := computedFamilyRange_bits_le source target N R hMass key
dsimp only [rejectionBitCount] at hb
exact h.trans (by omega)
end EditApproximation
end OAI
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