File size: 4,637 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 85 86 87 88 89 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedScalarCodeBits
import OAI.Combinatorics.EditApproximation.Geometry.SourceRankedEncoding
import OAI.Combinatorics.EditApproximation.Geometry.ChargedRequestSpace
import OAI.Combinatorics.EditApproximation.Geometry.SeedFactorSchedule
namespace OAI
namespace EditApproximation
theorem computed_physicalFamilyCode_bound (N nx ny R : ℕ)
(hN : 0 < N) (hell : 2 ^ 1000 ≤ inputSmallLog N) (hnx : nx ≤ N) (hny : ny ≤ N)
(hR : R ≤ inputHeight N ^ 2)
(hB : Nat.size (localOnlineMassBound (integerParameters N).M (integerParameters N).P
(inputHeight N)) ≤ inputHeight N)
(target : List ℕ) (htarget : target.length = ny)
(key : PhysicalFamilyScalarKey (M := (integerParameters N).M)
(J := intervalTreeDepth (integerParameters N).M nx) (R := R)
target N (integerParameters N).P (inputSmallLog N) (40 * smallLogExponent N)
(inputHeight N) (inputSmallLog N ^ 80) (computedSeedPassCount N) (inputHeight N ^ 2)) :
physicalFamilyCode target N (integerParameters N).P (inputSmallLog N) (40 * smallLogExponent N)
(inputHeight N) (inputSmallLog N ^ 80) (computedSeedPassCount N) (inputHeight N ^ 2) key <
2 ^ (16384 * inputHeight N + 4) := by
let H := inputHeight N
have hH : 2 ≤ H := (by omega : 2 ≤ inputSmallLog N).trans (computed_smallLog_le_height N)
have hlift {e : ℕ} (he : e ≤ 64 * H) : 2 ^ e ≤ 2 ^ (64 * H) :=
Nat.pow_le_pow_right (by decide) he
have hp : computedSeedPassCount N + 1 ≤ 2 ^ (64 * H) :=
((Nat.add_le_add_right (computedSeedPassCount_le_height N hN hell) 1).trans
(Nat.lt_two_pow_self : H < 2 ^ H)).trans (hlift (by omega))
have hc : H ^ 2 + 1 ≤ 2 ^ (64 * H) := by
have hh : H ^ 2 ≤ 2 ^ (2 * H) := by
calc
_ ≤ (2 ^ H) ^ 2 := Nat.pow_le_pow_left Nat.lt_two_pow_self.le 2
_ = _ := by rw [← pow_mul]; congr 1; omega
calc
_ ≤ 2 ^ (2 * H) + 2 ^ (2 * H) := Nat.add_le_add hh Nat.one_le_two_pow
_ = 2 ^ (2 * H + 1) := by rw [pow_succ]; omega
_ ≤ _ := hlift (by omega)
have hscalar (scalar : PhysicalScalarKey (integerParameters N).M
(intervalTreeDepth (integerParameters N).M nx) N target.length (40 * smallLogExponent N)
(inputSmallLog N) (inputSmallLog N ^ 80) R
(localOnlineMassBound (integerParameters N).M (integerParameters N).P H)) :
physicalScalarCode scalar < 2 ^ (64 * (64 * H) + 1) :=
computed_physicalScalarCode_bound N nx target.length R hN (by omega) hnx
(by omega) hR hB scalar
have h := physicalFamilyCode_bound (M := (integerParameters N).M)
(J := intervalTreeDepth (integerParameters N).M nx) (R := R)
target N (integerParameters N).P (inputSmallLog N) (40 * smallLogExponent N) H
(inputSmallLog N ^ 80) (computedSeedPassCount N) (H ^ 2) (64 * H) hp hc hscalar key
convert h using 1
congr 1
omega
theorem mixedCode_lt_two_pow {σ κ : Type*} (sourceCode : σ → ℕ) (tableCode : κ → ℕ)
(b : ℕ) (hs : ∀ source, sourceCode source < 2 ^ b)
(ht : ∀ table, tableCode table < 2 ^ b) (key : Sum σ κ) :
SourceRanked.keyCode sourceCode tableCode key < 2 ^ (b + 1) := by
rw [pow_succ]
cases key with
| inl source => have h := hs source; change 2 * _ < _; omega
| inr table => have h := ht table; change 2 * _ + 1 < _; omega
theorem computed_mixed_cache_nodes {κ : Type*} (N : ℕ)
(rank : κ → ℕ) (program : (key : κ) → FiniteQuery {child : κ // rank child < rank key})
(encode : κ → ℕ) (key : κ) :
(rankedMemoEvaluate rank program (16384 * inputHeight N + 5) encode key .empty).memory.nodes + 1 ≤
(N + 2) ^ 65542 := by
have hcache := rankedMemoEvaluate_nodes_bound rank program (16384 * inputHeight N + 5) encode key
apply hcache.trans
have h4 : 4 ≤ (N + 2) ^ 2 := by nlinarith [Nat.zero_le N]
have hbase : 2 ^ inputHeight N ≤ (N + 2) ^ 4 := by
apply (two_pow_inputHeight_le N).trans
calc
_ ≤ (N + 2) ^ 2 * (N + 2) ^ 2 := Nat.mul_le_mul_right _ h4
_ = _ := by ring
calc
2 ^ (16384 * inputHeight N + 5 + 1) = (2 ^ inputHeight N) ^ 16384 * 2 ^ 6 := by
rw [show 16384 * inputHeight N + 5 + 1 = inputHeight N * 16384 + 6 by omega, pow_add, pow_mul]
_ ≤ ((N + 2) ^ 4) ^ 16384 * (N + 2) ^ 6 :=
Nat.mul_le_mul (Nat.pow_le_pow_left hbase _) (Nat.pow_le_pow_left (by omega) _)
_ = (N + 2) ^ (4 * 16384 + 6) := by
generalize (16384 : ℕ) = exponent
rw [← pow_mul, ← pow_add]
_ = (N + 2) ^ 65542 := congrArg (fun exponent : ℕ => (N + 2) ^ exponent) (by decide)
end EditApproximation
end OAI
|