File size: 4,366 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.Geometry.ChargedGlobalQuery
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalRequestBits
import OAI.Combinatorics.EditApproximation.Binary.BinaryMemoSpace
namespace OAI
/-! A polynomial retained-cache bound for every execution. The estimate
uses only fixed-width key insertion and computed parameter sizes, and
therefore does not depend on an approximation success event. -/
namespace EditApproximation
theorem computed_charged_request_bits_le (N nx ny passes copies : ℕ) (hN : 0 < N)
(hell : 200 ≤ inputSmallLog N) (hnx : nx ≤ N) (hny : ny ≤ N)
(hpasses : passes ≤ inputHeight N ^ 2) (hcopies : copies ≤ inputHeight N ^ 2) :
ChargedPhysicalRequest.bits (integerParameters N).M
(intervalTreeDepth (integerParameters N).M nx) ny passes copies
(integerParameters N).T (integerParameters N).S ≤ 11 * inputHeight N + 3 := by
let H := inputHeight N
have hentry := computed_entryCapacity_bound N nx ny hN hell hnx hny
have hbits := computed_request_bits_le N nx ny passes copies hN hell hnx hny hpasses hcopies
have htable : PhysicalTableRequest.capacity (integerParameters N).M
(intervalTreeDepth (integerParameters N).M nx) ny passes copies
(integerParameters N).T (integerParameters N).S + 1 ≤ 2 ^ (11 * H + 1) := by
exact (Nat.clog_le_iff_le_pow (by decide : 1 < (2 : ℕ))).mp hbits
have hentry' : PhysicalTableRequest.entryCapacity (integerParameters N).M
(intervalTreeDepth (integerParameters N).M nx) ny ≤ 2 ^ (11 * H + 1) :=
hentry.trans (pow_le_pow_right₀ (by decide : 1 ≤ (2 : ℕ)) (by omega))
unfold ChargedPhysicalRequest.bits
apply (Nat.clog_le_iff_le_pow (by decide : 1 < (2 : ℕ))).2
unfold ChargedPhysicalRequest.capacity
have hp : 2 ^ (11 * H + 3) = 4 * 2 ^ (11 * H + 1) := by
rw [show 11 * H + 3 = (11 * H + 1) + 2 by omega, pow_add]
norm_num
omega
change 2 * (_ + _) + 1 ≤ 2 ^ (11 * H + 3)
rw [hp]
omega
theorem two_pow_inputHeight_le (N : ℕ) : 2 ^ inputHeight N ≤ 4 * (N + 2) ^ 2 := by
have hc : 0 < Nat.clog 2 (N + 2) :=
Nat.clog_pos (by decide : 1 < (2 : ℕ)) (by omega)
have hH := (rejectionWord_range (Nat.clog 2 (N + 2)) hc).2
have hN := (rejectionWord_range (N + 2) (by omega)).2
change inputHeight N < 2 * Nat.clog 2 (N + 2) at hH
change 2 ^ Nat.clog 2 (N + 2) < 2 * (N + 2) at hN
calc
_ ≤ 2 ^ (2 * Nat.clog 2 (N + 2)) := pow_le_pow_right₀ (by decide : 1 ≤ (2 : ℕ)) hH.le
_ = (2 ^ Nat.clog 2 (N + 2)) ^ 2 := by rw [← pow_mul]; congr 1; omega
_ ≤ (2 * (N + 2)) ^ 2 := Nat.pow_le_pow_left hN.le 2
_ = _ := by ring
theorem computed_charged_cache_nodes {κ : Type*} (N nx ny passes copies : ℕ) (hN : 0 < N)
(hell : 200 ≤ inputSmallLog N) (hnx : nx ≤ N) (hny : ny ≤ N)
(hpasses : passes ≤ inputHeight N ^ 2) (hcopies : copies ≤ inputHeight N ^ 2)
(rank : κ → ℕ) (program : (key : κ) → FiniteQuery {child : κ // rank child < rank key})
(encode : κ → ℕ) (key : κ) :
(rankedMemoEvaluate rank program
(ChargedPhysicalRequest.bits (integerParameters N).M
(intervalTreeDepth (integerParameters N).M nx) ny passes copies
(integerParameters N).T (integerParameters N).S) encode key .empty).memory.nodes + 1 ≤
(N + 2) ^ 48 := by
have hbits := computed_charged_request_bits_le N nx ny passes copies hN hell hnx hny hpasses hcopies
have hcache := rankedMemoEvaluate_nodes_bound rank program
(ChargedPhysicalRequest.bits (integerParameters N).M
(intervalTreeDepth (integerParameters N).M nx) ny passes copies
(integerParameters N).T (integerParameters N).S) encode key
have hpow := Nat.pow_le_pow_right (by decide : 0 < (2 : ℕ)) (Nat.add_le_add_right hbits 1)
apply (hcache.trans hpow).trans
calc
2 ^ (11 * inputHeight N + 3 + 1) = 16 * (2 ^ inputHeight N) ^ 11 := by
rw [show 11 * inputHeight N + 3 + 1 = inputHeight N * 11 + 4 by omega, pow_add, pow_mul]
norm_num
omega
_ ≤ 16 * (4 * (N + 2) ^ 2) ^ 11 := Nat.mul_le_mul_left 16 (Nat.pow_le_pow_left (two_pow_inputHeight_le N) 11)
_ = 2 ^ 26 * (N + 2) ^ 22 := by ring
_ ≤ (N + 2) ^ 26 * (N + 2) ^ 22 :=
Nat.mul_le_mul_right _ (Nat.pow_le_pow_left (by omega) 26)
_ = (N + 2) ^ 48 := by rw [← pow_add]
end EditApproximation
end OAI
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