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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 | import OAI.Combinatorics.EditApproximation.Geometry.EarlierInputBudget
import OAI.Combinatorics.EditApproximation.Refinement.ComputedBandSpacing
namespace OAI
/-! The actual earlier-input cover satisfies the polynomial branching
budget at every seed, after substituting the computed time and grid
parameters. -/
namespace EditApproximation
theorem computed_smallLog_le_height (N : ℕ) : inputSmallLog N ≤ inputHeight N := by
have hf : smallLogExponent N ≤ heightExponent N := by
unfold smallLogExponent
apply Nat.clog_le_of_le_pow
exact max_le Nat.one_le_two_pow (heightExponent N).lt_two_pow_self.le
exact pow_le_pow_right₀ (by decide : (1 : ℕ) ≤ 2) hf
theorem computed_history_budget_le_height_sq (N : ℕ) (hH : 4 ≤ inputHeight N) :
(integerParameters N).T + (integerParameters N).S + 1 ≤ inputHeight N ^ 2 := by
have hT : (integerParameters N).T ≤ 2 * inputHeight N := by
change 2 ^ ((3 * heightExponent N + 3) / 4) ≤ 2 * 2 ^ heightExponent N
calc
_ ≤ 2 ^ (heightExponent N + 1) :=
pow_le_pow_right₀ (by decide : (1 : ℕ) ≤ 2) (by omega)
_ = _ := by rw [pow_succ]; ring
have hS : (integerParameters N).S ≤ inputHeight N := Nat.div_le_self _ _
nlinarith only [hT, hS, hH]
theorem computed_window_le_height_cube (N : ℕ) (hN : 0 < N)
(hell : 8 ≤ inputSmallLog N) (hH : 2 ≤ inputHeight N) :
computedSeedInitialExponent N + 8 ≤ inputHeight N ^ 3 := by
calc
_ ≤ 2 * inputSmallLog N ^ 2 := computed_window_exponent_bound N hN hell
_ ≤ inputHeight N * inputHeight N ^ 2 :=
Nat.mul_le_mul hH (Nat.pow_le_pow_left (computed_smallLog_le_height N) 2)
_ = _ := by ring
theorem computed_cell_budget_le_height_pow (N : ℕ) (hH : 64 ≤ inputHeight N) :
((32 * inputSmallLog N + 2) * 2 ^ (40 * smallLogExponent N) + 2) ^ 2 ≤
inputHeight N ^ 84 := by
have hconstant : 1296 ≤ inputHeight N ^ 2 :=
(by norm_num : 1296 ≤ (64 : ℕ) ^ 2).trans (Nat.pow_le_pow_left hH 2)
calc
_ ≤ 1296 * inputSmallLog N ^ 82 :=
computed_local_cell_budget N Nat.one_le_two_pow
_ ≤ inputHeight N ^ 2 * inputHeight N ^ 82 :=
Nat.mul_le_mul hconstant (Nat.pow_le_pow_left (computed_smallLog_le_height N) 82)
_ = _ := by ring
/-- Every literal earlier-input request lies in a cover of cardinality
`H^(256(p+1))`. The only value premise is nonnegativity of the actual
initial query value; no good-seed event is required. -/
theorem computed_queryEarlierInputCover_card {nx ny : ℕ}
(parent : TargetInterval nx) (N P : ℕ) (hN : 0 < N)
(hell : 200 ≤ inputSmallLog N) (hPeq : P = (integerParameters N).P)
(A : ℚ) (hA : A ≤ ((integerParameters N).initialFactor : ℚ))
(initial : TargetInterval ny → ℚ) (q : TargetInterval ny) (hw : 0 ≤ initial q) :
(queryEarlierInputCover (M := (integerParameters N).M) parent N P (inputSmallLog N)
(40 * smallLogExponent N) ((integerParameters N).T + (integerParameters N).S + 1)
(refinementFactor A (inputSmallLog N)) initial q).card ≤
inputHeight N ^ (256 * (coarseExponent (inputSmallLog N) + 1)) := by
let W := computedSeedInitialExponent N + 8
let C := ((32 * inputSmallLog N + 2) * 2 ^ (40 * smallLogExponent N) + 2) ^ 2
have hF : inputSmallLog N ≤ inputHeight N := computed_smallLog_le_height N
have hH : 64 ≤ inputHeight N := (by omega : 64 ≤ inputSmallLog N).trans hF
have hPpos : 1 ≤ P := by
rw [hPeq]
exact one_le_pow₀ Nat.one_le_two_pow
have hPpower : P ≤ inputHeight N ^ (coarseExponent (inputSmallLog N) + 20) := by
rw [hPeq]
exact le_rfl
have hM : (integerParameters N).M ≤ inputHeight N :=
pow_le_pow_right₀ (by decide : (1 : ℕ) ≤ 2) (Nat.div_le_self _ _)
have hbudget := queryEarlierInputCover_le_budget (M := (integerParameters N).M)
parent N P (inputSmallLog N) (40 * smallLogExponent N)
((integerParameters N).T + (integerParameters N).S + 1) W C
(refinementFactor A (inputSmallLog N)) initial q
⟨_, hPeq.trans (computed_P_dyadic N)⟩ (by omega) hw
(computed_refinement_window N hN (by omega) A (initial q) hA) le_rfl
exact hbudget.trans (localEarlierBudget_poly (inputHeight N)
(coarseExponent (inputSmallLog N)) W C (integerParameters N).M P (inputSmallLog N)
((integerParameters N).T + (integerParameters N).S + 1)
hH hPpos hPpower (computed_window_le_height_cube N hN (by omega) (by omega))
(computed_cell_budget_le_height_pow N hH) hM hF
(computed_history_budget_le_height_sq N (by omega)))
end EditApproximation
end OAI
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