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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 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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 | import OAI.Combinatorics.EditApproximation.Geometry.CurrentQueryInputs
import OAI.Combinatorics.EditApproximation.Refinement.ComputedGroupNumerical
namespace OAI
/-! Explicit polynomial same-index call budget after substituting the
computed window, cell, scale-label, sampling and rounding parameters. -/
namespace EditApproximation
def localCurrentBudget (window cells labels M Q inverseTheta : ℕ) : ℕ :=
window * cells * labels * (624 * M * Q * inverseTheta ^ 2) + window * cells * M + window * M
theorem localCurrentBudget_poly (ell M window cells labels : ℕ) (hell : 4 ≤ ell)
(hwindow : window ≤ 2 * ell ^ 2) (hcells : cells ≤ 1296 * ell ^ 82)
(hlabels : labels ≤ 2 * ell) :
localCurrentBudget window cells labels M (ell ^ 80) (ell ^ 40) ≤ M * ell ^ 256 := by
have hfirst := Nat.mul_le_mul
(Nat.mul_le_mul (Nat.mul_le_mul hwindow hcells) hlabels)
(le_rfl (a := 624 * M * ell ^ 80 * (ell ^ 40) ^ 2))
have hsecond := Nat.mul_le_mul_right M (Nat.mul_le_mul hwindow hcells)
have hthird := Nat.mul_le_mul_right M hwindow
have h84 : ell ^ 84 ≤ ell ^ 245 := pow_le_pow_right₀ (by omega : 1 ≤ ell) (by decide : 84 ≤ 245)
have h2 : ell ^ 2 ≤ ell ^ 245 := pow_le_pow_right₀ (by omega : 1 ≤ ell) (by decide : 2 ≤ 245)
have hsecond' : 2 * ell ^ 2 * (1296 * ell ^ 82) * M ≤ 2592 * M * ell ^ 245 := by
calc
_ = 2592 * M * ell ^ 84 := by ring
_ ≤ _ := Nat.mul_le_mul_left _ h84
have hthird' : 2 * ell ^ 2 * M ≤ 2 * M * ell ^ 245 := by
calc
_ = 2 * M * ell ^ 2 := by ring
_ ≤ _ := Nat.mul_le_mul_left _ h2
have hsum : localCurrentBudget window cells labels M (ell ^ 80) (ell ^ 40) ≤
3237410 * M * ell ^ 245 := by
apply (Nat.add_le_add (Nat.add_le_add hfirst (hsecond.trans hsecond'))
(hthird.trans hthird')).trans_eq
ring
have hconstant : 3237410 ≤ ell ^ 11 := by
apply (show 3237410 ≤ (4 : ℕ) ^ 11 by norm_num).trans
exact Nat.pow_le_pow_left hell 11
calc
_ ≤ 3237410 * M * ell ^ 245 := hsum
_ ≤ ell ^ 11 * M * ell ^ 245 := Nat.mul_le_mul_right _ (Nat.mul_le_mul_right M hconstant)
_ = M * ell ^ 256 := by ring
theorem computed_window_exponent_bound (N : ℕ) (hN : 0 < N) (hell : 8 ≤ inputSmallLog N) :
computedSeedInitialExponent N + 8 ≤ 2 * inputSmallLog N ^ 2 := by
have hh := (inputSmallLog_bounds N hN).1
have hp : coarseExponent (inputSmallLog N) ≤ inputSmallLog N :=
(Nat.sqrt_le_self _).trans ((Nat.sqrt_le_self _).trans (Nat.log_le_self _ _))
have hmul := Nat.mul_le_mul (Nat.add_le_add_right hp 4) hh
unfold computedSeedInitialExponent
have hslack : 4 * inputSmallLog N + 8 ≤ inputSmallLog N ^ 2 := by nlinarith
nlinarith only [hmul, hslack]
theorem computed_local_cell_budget (N : ℕ) (hell : 1 ≤ inputSmallLog N) :
((32 * inputSmallLog N + 2) * 2 ^ (40 * smallLogExponent N) + 2) ^ 2 ≤
1296 * inputSmallLog N ^ 82 := by
have hpower : 2 ^ (40 * smallLogExponent N) = inputSmallLog N ^ 40 := by
unfold inputSmallLog
rw [← pow_mul, Nat.mul_comm]
rw [hpower]
have hinner : (32 * inputSmallLog N + 2) * inputSmallLog N ^ 40 + 2 ≤
36 * inputSmallLog N ^ 41 := by
have ha := Nat.mul_le_mul_right (inputSmallLog N ^ 40) (show 32 * inputSmallLog N + 2 ≤
34 * inputSmallLog N by omega)
have hb : 2 ≤ 2 * inputSmallLog N ^ 41 := Nat.mul_le_mul_left 2 (one_le_pow₀ hell)
calc
_ ≤ 34 * inputSmallLog N * inputSmallLog N ^ 40 + 2 := Nat.add_le_add_right ha _
_ ≤ 34 * inputSmallLog N * inputSmallLog N ^ 40 + 2 * inputSmallLog N ^ 41 :=
Nat.add_le_add_left hb _
_ = _ := by ring
calc
_ ≤ (36 * inputSmallLog N ^ 41) ^ 2 := Nat.pow_le_pow_left hinner 2
_ = _ := by ring
theorem computed_refinement_window (N : ℕ) (hN : 0 < N) (hell : 8 ≤ inputSmallLog N)
(A w : ℚ) (hA : A ≤ ((integerParameters N).initialFactor : ℚ)) :
(dyadicQueryWindow N w (refinementFactor A (inputSmallLog N)) (inputSmallLog N)).length ≤
computedSeedInitialExponent N + 8 := by
have hh := inputSmallLog_bounds N hN
have hf : smallLogExponent N ≤ inputSmallLog N := (smallLogExponent N).lt_two_pow_self.le
have he0 : smallLogExponent N ≤ computedSeedInitialExponent N := by
have hfour := Nat.mul_le_mul_right (heightExponent N)
(show 4 ≤ coarseExponent (inputSmallLog N) + 4 by omega)
unfold computedSeedInitialExponent
nlinarith only [hh.2, hf, hfour]
apply refinement_query_window_length N (inputSmallLog N) (computedSeedInitialExponent N) A w (by omega)
· simpa only [inputSmallLog, Nat.cast_pow, Nat.cast_ofNat] using
pow_le_pow_right₀ (by norm_num : (1 : ℚ) ≤ 2) he0
· exact hA.trans_eq (computedSeed_initial_factor N).symm
/-- The actual finite local shared/selected/online input cover is bounded
by M*ell^256 on every seed. The online-loop length premise is the literal
one-action-per-applicable-scale property proved for the physical program. -/
theorem computed_currentQueryInputCover_card {α : Type*} (source target : List α)
(N P : ℕ) (hN : 0 < N) (hell : 200 ≤ inputSmallLog N)
(parent : TargetInterval source.length) (A tau : ℚ)
(hA : A ≤ ((integerParameters N).initialFactor : ℚ))
(parentInitial : TargetInterval target.length → ℚ)
(child initial current : Fin (integerParameters N).M → TargetInterval target.length → ℚ)
(draw : CellGroupDraws (integerParameters N).M (inputSmallLog N) (inputSmallLog N ^ 80))
(q : TargetInterval target.length) (hw : 0 ≤ parentInitial q)
(online : List (PhysicalRationalAction (integerParameters N).M target.length))
(honline : online.length ≤
(dyadicQueryWindow N (parentInitial q) (refinementFactor A (inputSmallLog N)) (inputSmallLog N)).length) :
(currentQueryInputCover source target parent N P (inputSmallLog N) (40 * smallLogExponent N)
(inputSmallLog N ^ 80) tau (refinementFactor A (inputSmallLog N)) parentInitial
child initial current draw q online).card ≤ (integerParameters N).M * inputSmallLog N ^ 256 := by
let ell := inputSmallLog N
let M := (integerParameters N).M
let W := computedSeedInitialExponent N + 8
let C := ((32 * ell + 2) * 2 ^ (40 * smallLogExponent N) + 2) ^ 2
have hell8 : 8 ≤ ell := by omega
have hM : 0 < M := by have h := (computed_branching_bounds N hN hell).1; omega
have hlabels (b : ℕ) (hb : b ∈ dyadicScales N) : (groupDyadicScales M b ell).length ≤ 2 * ell := by
obtain ⟨eb, _, rfl⟩ := List.mem_map.mp hb
exact computed_group_scale_count N eb hN hell8
have hwindow := computed_refinement_window N hN hell8 A (parentInitial q) hA
have hkeys : (localGroupQueryKeys N ell (40 * smallLogExponent N)
(refinementFactor A ell) parentInitial q).length ≤ W * C :=
(localGroupQueryKeys_length N ell (40 * smallLogExponent N) (refinementFactor A ell)
parentInitial q (by omega) hw).trans (Nat.mul_le_mul_right C hwindow)
have hgroups := localCellGroupActions_length_refinement source target parent N P ell
(40 * smallLogExponent N) (ell ^ 80) (computedSeedInitialExponent N) tau A
parentInitial child initial current draw q (by omega) hw
have hraw := currentQueryInputCover_card source target parent N P ell (40 * smallLogExponent N)
(ell ^ 80) (2 * ell) tau (refinementFactor A ell) parentInitial child initial current draw q online
hM (by omega) (one_le_pow₀ (by omega)) hlabels
have hrawN : (currentQueryInputCover source target parent N P ell (40 * smallLogExponent N)
(ell ^ 80) tau (refinementFactor A ell) parentInitial child initial current draw q online).card ≤
(localGroupQueryKeys N ell (40 * smallLogExponent N) (refinementFactor A ell) parentInitial q).length *
(2 * ell) * (624 * M * ell ^ 80 * (2 ^ (40 * smallLogExponent N)) ^ 2) +
(localCellGroupActions source target parent N P ell (40 * smallLogExponent N) (ell ^ 80) tau
(refinementFactor A ell) parentInitial child initial current draw q).length * M + online.length * M := by
exact_mod_cast hraw
have hFpower : (ell : ℚ) ≤ (2 : ℚ) ^ computedSeedInitialExponent N := by
have hh := inputSmallLog_bounds N hN
have hf : smallLogExponent N ≤ ell := (smallLogExponent N).lt_two_pow_self.le
have hfour := Nat.mul_le_mul_right (heightExponent N)
(show 4 ≤ coarseExponent ell + 4 by omega)
have he : smallLogExponent N ≤ computedSeedInitialExponent N := by
unfold computedSeedInitialExponent
nlinarith only [hh.2, hf, hfour]
simpa only [ell, inputSmallLog, Nat.cast_pow, Nat.cast_ofNat] using
pow_le_pow_right₀ (by norm_num : (1 : ℚ) ≤ 2) he
have hAbound : A ≤ (2 : ℚ) ^ computedSeedInitialExponent N := hA.trans_eq (computedSeed_initial_factor N).symm
have hg : (localCellGroupActions source target parent N P ell (40 * smallLogExponent N) (ell ^ 80) tau
(refinementFactor A ell) parentInitial child initial current draw q).length ≤ W * C :=
hgroups hFpower hAbound
have hbudget : (currentQueryInputCover source target parent N P ell (40 * smallLogExponent N)
(ell ^ 80) tau (refinementFactor A ell) parentInitial child initial current draw q online).card ≤
localCurrentBudget W C (2 * ell) M (ell ^ 80) (2 ^ (40 * smallLogExponent N)) := by
exact hrawN.trans (Nat.add_le_add (Nat.add_le_add
(Nat.mul_le_mul_right _ (Nat.mul_le_mul_right (2 * ell) hkeys))
(Nat.mul_le_mul_right M hg)) (Nat.mul_le_mul_right M (honline.trans hwindow)))
have hpower : 2 ^ (40 * smallLogExponent N) = ell ^ 40 := by
dsimp only [ell]
unfold inputSmallLog
rw [← pow_mul, Nat.mul_comm]
rw [hpower] at hbudget
exact hbudget.trans (localCurrentBudget_poly ell M W C (2 * ell)
((by decide : 4 ≤ (8 : ℕ)).trans hell8)
(computed_window_exponent_bound N hN hell8)
(computed_local_cell_budget N ((by decide : 1 ≤ (8 : ℕ)).trans hell8)) le_rfl)
end EditApproximation
end OAI
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