File size: 4,510 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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseInitialAllocation
import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseHistorySpace
import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseBitWork
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
open Filter
def physicalCoarseAllocationEntry {M J B : ℕ} [NeZero B] (source target : List ℕ) (N : ℕ)
(depth : PhysicalEntry M J target.length → ℕ) (draw : PhysicalCoarseDraw source target B depth)
(entry : PhysicalEntry M J target.length) : ℕ :=
BinaryFraction.coarseIndexedInitialAllocation (inputHeight N ^ 4) (64 * inputHeight N) N
(Nat.pos_of_neZero B) (pow_pos (Nat.two_pow_pos _) _)
(Nat.mul_pos (by decide) (Nat.two_pow_pos _))
(physicalSource source entry.1) (substring target entry.2.lo entry.2.hi) (draw entry)
theorem physicalCoarseAllocationEntry_fits {M J : ℕ} :
∀ᶠ N : ℕ in atTop, ∀ B : ℕ, ∀ (_hB : NeZero B),
∀ (source target : List ℕ) (depth : PhysicalEntry M J target.length → ℕ),
B = (integerParameters N).B →
(∀ entry, depth entry = computedCoarseDepth N
(coarseEntryWidth (physicalSource source entry.1) (substring target entry.2.lo entry.2.hi))) →
source.length + target.length ≤ N →
(∀ a ∈ source ++ target, Nat.size a ≤ inputHeight N ^ 3) →
∀ (draw : PhysicalCoarseDraw source target B depth) (entry : PhysicalEntry M J target.length),
physicalCoarseAllocationEntry source target N depth draw entry ≤ (N + 2) ^ 181 := by
filter_upwards [BinaryFraction.computedCoarseIndexedInitialAllocation_polynomial] with N hN
intro B inst source target depth hB hd hsize hsymbols draw entry
have hx := physicalSource_length_le source entry.1
have hy : (substring target entry.2.lo entry.2.hi).length ≤ target.length := by
simp only [substring, List.length_drop, List.length_take]
exact (Nat.sub_le _ _).trans (min_le_right _ _)
have hs : ∀ a ∈ physicalSource source entry.1 ++ substring target entry.2.lo entry.2.hi,
Nat.size a ≤ inputHeight N ^ 3 := by
intro a ha
rcases List.mem_append.mp ha with ha | ha
· exact hsymbols a (List.mem_append_left _ (physicalSource_mem source entry.1 a ha))
· exact hsymbols a (List.mem_append_right _ (List.mem_of_mem_take (List.mem_of_mem_drop ha)))
exact hN B (depth entry) (physicalSource source entry.1)
(substring target entry.2.lo entry.2.hi) hB (hd entry)
(max_le (by omega) (by omega)) hs (Nat.pos_of_neZero B)
(pow_pos (Nat.two_pow_pos _) _) (Nat.mul_pos (by decide) (Nat.two_pow_pos _)) (draw entry)
/-- Reserve every fresh cell allocated by every coarse cache miss until
termination. This deliberately bounds more than simultaneously live scratch. -/
theorem computedFamilyCoarseScratchCapacity_eq (source target : List ℕ) (N : ℕ)
(input : ComputedFamilyInput source target N) :
computedFamilyCoarseScratchCapacity source target N input =
((computedFamilyResult source target N input).freshKeys.map
(coarseHistoryKeyCost (fun _ => (N + 2) ^ 181))).sum :=
computedFamilyKernelWork_eq_fresh_sum source target N input _
theorem computedFamilyCoarseScratchCapacity_bound (source target : List ℕ) (N : ℕ)
(input : ComputedFamilyInput source target N) :
computedFamilyCoarseScratchCapacity source target N input ≤
(N + 2) ^ 181 * (computedFamilyResult source target N input).requests := by
apply computedFamilyKernelWork_le
exact coarseHistoryKeyCost_le _ _ (fun _ => le_rfl)
theorem computedFamilyCoarseScratchCapacity_polynomial :
∀ᶠ N : ℕ in atTop, ∀ source target : List ℕ, source.length + target.length ≤ N →
∀ input : ComputedFamilyInput source target N,
computedFamilyCoarseScratchCapacity source target N input ≤ (N + 2) ^ 183 := by
filter_upwards [computedFamilyResult_requests_near_linear 1 (by norm_num)] with N hr
intro source target hsize 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) ^ 181 * (computedFamilyResult source target N input).requests :=
computedFamilyCoarseScratchCapacity_bound source target N input
_ ≤ (N + 2) ^ 181 * (N + 2) ^ 2 := Nat.mul_le_mul_left _ hrequests
_ = _ := by rw [← pow_add]
end EditApproximation
end OAI
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