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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 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedFamilyEncoding
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
theorem computedFamilyMemoCode_injective (source target : List ℕ) (N R : ℕ)
(hN : 0 < N) (hell : 200 ≤ inputSmallLog N) :
Function.Injective (computedFamilyMemoCode source target N R) := by
have hM : 1 ≤ (integerParameters N).M := (by omega : 1 ≤ 2).trans
(computed_branching_bounds N hN hell).1
exact SourceRanked.keyCode_injective _ _
(physicalFamilyCode_injective target N (integerParameters N).P (inputSmallLog N)
(40 * smallLogExponent N) (inputHeight N) (inputSmallLog N ^ 80)
(computedSeedPassCount N) (inputHeight N ^ 2) hM)
(ChargedPhysicalRequest.code_injective hM)
theorem computedFamilyMemoCode_fits (source target : List ℕ) (N R : ℕ)
(hN : 0 < N) (hell : 2 ^ 1000 ≤ inputSmallLog N) (hsize : source.length + target.length ≤ N)
(hR : R ≤ inputHeight N ^ 2)
(hB : Nat.size (localOnlineMassBound (integerParameters N).M (integerParameters N).P
(inputHeight N)) ≤ inputHeight N)
(key : Sum (ComputedFamilyScalarKey source target N R) (ComputedFamilyTableKey source target N)) :
computedFamilyMemoCode source target N R key < 2 ^ (16384 * inputHeight N + 5) := by
have hM : 1 ≤ (integerParameters N).M := (by omega : 1 ≤ 2).trans
(computed_branching_bounds N hN (by omega)).1
have hH : 2 ≤ inputHeight N := (by omega : 2 ≤ inputSmallLog N).trans (computed_smallLog_le_height N)
have hpass : computedSeedPassCount N + 1 ≤ inputHeight N ^ 2 := by
have h := computedSeedPassCount_le_height N hN hell
nlinarith only [h, hH]
apply mixedCode_lt_two_pow _ _ (16384 * inputHeight N + 4)
· intro scalar
exact computed_physicalFamilyCode_bound N source.length target.length R hN hell
(by omega) (by omega) hR hB target rfl scalar
· intro table
have hb := computed_charged_request_bits_le N source.length target.length
(computedSeedPassCount N + 1) (inputHeight N ^ 2) hN (by omega) (by omega) (by omega)
hpass le_rfl
exact (ChargedPhysicalRequest.code_fits hM table).trans_le
(Nat.pow_le_pow_right (by decide) (hb.trans (by omega)))
end EditApproximation
end OAI
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