File size: 3,116 Bytes
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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseInputStorage
import OAI.Combinatorics.EditApproximation.Coarse.CoarseInitialStorage
import OAI.Combinatorics.EditApproximation.Refinement.ComputedFreshWork
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
open Filter
theorem physicalCoarseInputStorage_polynomial {M J : ℕ} (source target : List ℕ) (N : ℕ)
(hsize : source.length + target.length ≤ N)
(hsymbol : ∀ a ∈ source ++ target, Nat.size a ≤ inputHeight N ^ 3)
(entry : PhysicalEntry M J target.length) :
physicalCoarseInputStorage source target entry ≤ (N + 2) ^ 16 := by
unfold physicalCoarseInputStorage
exact coarsePreparedInputStorage_polynomial source target N
(physicalSourceInterval source.length entry.1).lo
(physicalSourceInterval source.length entry.1).hi entry.2.lo entry.2.hi
(physicalSourceInterval source.length entry.1).ordered
(physicalSourceInterval source.length entry.1).valid entry.2.ordered entry.2.valid hsize hsymbol
theorem computedFamilyCoarseCopyStorage_eq (source target : List ℕ) (N : ℕ)
(input : ComputedFamilyInput source target N) :
computedFamilyCoarseCopyStorage source target N input =
((computedFamilyResult source target N input).freshKeys.map (coarseHistoryKeyCost
(physicalCoarseInputStorage (M := (integerParameters N).M)
(J := intervalTreeDepth (integerParameters N).M source.length) source target))).sum :=
computedFamilyKernelWork_eq_fresh_sum source target N input _
theorem computedFamilyCoarseCopyStorage_bound (source target : List ℕ) (N : ℕ)
(hsize : source.length + target.length ≤ N)
(hsymbol : ∀ a ∈ source ++ target, Nat.size a ≤ inputHeight N ^ 3)
(input : ComputedFamilyInput source target N) :
computedFamilyCoarseCopyStorage source target N input ≤
(N + 2) ^ 16 * (computedFamilyResult source target N input).requests := by
apply computedFamilyKernelWork_le
apply coarseHistoryKeyCost_le
intro entry
exact physicalCoarseInputStorage_polynomial source target N hsize hsymbol entry
theorem computedFamilyCoarseCopyStorage_polynomial :
∀ᶠ N : ℕ in atTop, ∀ source target : List ℕ, source.length + target.length ≤ N →
(∀ a ∈ source ++ target, Nat.size a ≤ inputHeight N ^ 3) →
∀ input : ComputedFamilyInput source target N,
computedFamilyCoarseCopyStorage source target N input ≤ (N + 2) ^ 18 := by
filter_upwards [computedFamilyResult_requests_near_linear 1 (by norm_num)] with N hr
intro source target hsize hsymbol input
have hrequests : (computedFamilyResult source target N input).requests ≤ (N + 2) ^ 2 := by
have h := hr source target hsize input
norm_num at h
exact_mod_cast h
calc
_ ≤ (N + 2) ^ 16 * (computedFamilyResult source target N input).requests :=
computedFamilyCoarseCopyStorage_bound source target N hsize hsymbol input
_ ≤ (N + 2) ^ 16 * (N + 2) ^ 2 := Nat.mul_le_mul_left _ hrequests
_ = _ := by rw [← pow_add]
end EditApproximation
end OAI
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