lean/OAI/Combinatorics/EditApproximation/Refinement/ComputedNumericalParameters.lean

import OAI.Combinatorics.EditApproximation.Geometry.LocalOnlineMassBound
import OAI.Combinatorics.EditApproximation.Refinement.ComputedEarlierInputBudget
import OAI.Combinatorics.EditApproximation.Numerics.NumericalWork

namespace OAI

/-! All-seed numerical dimension and source-range bounds after substituting
the computed grid size. These bounds do not use an accuracy event. -/

namespace EditApproximation

theorem wide_coordinate_budget_le (H M P p : ℕ) (hH : 64 ≤ H) (hM : M ≤ H)
    (hP : 1 ≤ P) (hPcap : P ≤ H ^ (p + 20)) :
    M * (40 * P + 2) ^ 2 ≤ H ^ (2 * p + 45) := by
  have hbox : 40 * P + 2 ≤ H ^ (p + 22) := by
    have hconstant : 42 ≤ H ^ 2 := by nlinarith
    calc
      _ ≤ 42 * P := by omega
      _ ≤ H ^ 2 * H ^ (p + 20) := Nat.mul_le_mul hconstant hPcap
      _ = H ^ (2 + (p + 20)) := (pow_add H 2 (p + 20)).symm
      _ = _ := by congr 1; omega
  calc
    _ ≤ H * (H ^ (p + 22)) ^ 2 := Nat.mul_le_mul hM (Nat.pow_le_pow_left hbox 2)
    _ = H * H ^ ((p + 22) * 2) := by rw [pow_mul]
    _ = H ^ ((p + 22) * 2 + 1) := (pow_succ' H _).symm
    _ = _ := by congr 1; omega

theorem optimizer_parameter_budget_le (H M P p : ℕ) (hH : 64 ≤ H) (hM : M ≤ H)
    (hP : 1 ≤ P) (hPcap : P ≤ H ^ (p + 20)) :
    H + M * (40 * P + 2) ^ 2 + 4 ≤ H ^ (2 * p + 46) := by
  have hdimension := wide_coordinate_budget_le H M P p hH hM hP hPcap
  have hlarge : H ≤ H ^ (2 * p + 45) := Nat.le_pow (by omega)
  have hfour : 4 ≤ H ^ (2 * p + 45) := (by omega : 4 ≤ H).trans hlarge
  calc
    _ ≤ 3 * H ^ (2 * p + 45) := by omega
    _ ≤ H * H ^ (2 * p + 45) := Nat.mul_le_mul_right _ (by omega)
    _ = H ^ (2 * p + 45 + 1) := (pow_succ' H _).symm
    _ = _ := rfl

theorem optimizerTriangularMass_le_square (K : ℕ) : optimizerTriangularMass K ≤ K ^ 2 := by
  have htwice := optimizerTriangularMass_twice K
  have hpower : K ≤ K ^ 2 := Nat.le_pow (by decide)
  nlinarith only [htwice, hpower]

theorem localOnlineMassBound_polynomial (H M P p : ℕ) (hH : 64 ≤ H) (hM : M ≤ H)
    (hP : 1 ≤ P) (hPcap : P ≤ H ^ (p + 20)) :
    localOnlineMassBound M P H ≤ H ^ (8000 * (p + 1)) := by
  have hG : H + M * (40 * P + 2) ^ 2 + 2 ≤ H ^ (2 * p + 46) :=
    (by omega : H + M * (40 * P + 2) ^ 2 + 2 ≤ H + M * (40 * P + 2) ^ 2 + 4).trans
      (optimizer_parameter_budget_le H M P p hH hM hP hPcap)
  calc
    _ ≤ ((H + M * (40 * P + 2) ^ 2 + 2) ^ 80) ^ 2 := optimizerTriangularMass_le_square _
    _ = (H + M * (40 * P + 2) ^ 2 + 2) ^ (80 * 2) := (pow_mul _ 80 2).symm
    _ ≤ (H ^ (2 * p + 46)) ^ (80 * 2) := Nat.pow_le_pow_left hG _
    _ = H ^ ((2 * p + 46) * (80 * 2)) := (pow_mul H _ _).symm
    _ ≤ _ := Nat.pow_le_pow_right (by omega) (by omega)

theorem computed_localOnlineMassBound_polynomial (N : ℕ) (hH : 64 ≤ inputHeight N) :
    localOnlineMassBound (integerParameters N).M (integerParameters N).P (inputHeight N) ≤
      inputHeight N ^ (8000 * (coarseExponent (inputSmallLog N) + 1)) := by
  apply localOnlineMassBound_polynomial _ _ _ _ hH
  · exact Nat.pow_le_pow_right (by decide : 0 < (2 : ℕ)) (Nat.div_le_self _ _)
  · exact Nat.one_le_pow _ _ (by omega)
  · exact le_rfl

theorem computed_localOnlineMassBound_bits (N : ℕ) (hH : 64 ≤ inputHeight N) :
    Nat.size (localOnlineMassBound (integerParameters N).M (integerParameters N).P (inputHeight N)) ≤
      8000 * (coarseExponent (inputSmallLog N) + 1) * Nat.size (inputHeight N) + 1 :=
  (Nat.size_le_size (computed_localOnlineMassBound_polynomial N hH)).trans
    (natural_power_bit_length _ _)

theorem rejectionBitCount_le_size (B : ℕ) : rejectionBitCount B ≤ Nat.size B := by
  apply Nat.clog_le_of_le_pow
  exact (Nat.lt_size_self B).le

theorem computed_localOnlineMassBound_rejection_bits (N : ℕ) (hH : 64 ≤ inputHeight N) :
    rejectionBitCount (localOnlineMassBound (integerParameters N).M (integerParameters N).P (inputHeight N)) ≤
      8000 * (coarseExponent (inputSmallLog N) + 1) * Nat.size (inputHeight N) + 1 :=
  (rejectionBitCount_le_size _).trans (computed_localOnlineMassBound_bits N hH)

end EditApproximation

end OAI