| import OAI.Combinatorics.EditApproximation.Coarse.CoarseGridInvariant | |
| namespace OAI | |
| namespace EditApproximation | |
| theorem coarseScale_root_grid (B q L j d : ℕ) (allowance : ℚ) (hq : 0 < q) | |
| (ha : allowance.den ∣ (L * B * q) ^ j) : | |
| (allowance / q).den ∣ coarseCommonDenominator 1 q L B (j + d) := by | |
| apply (coarse_rat_div_nat_den_dvd allowance q _ hq ha).trans | |
| refine ⟨(L * B * q) ^ d, ?_⟩ | |
| simp only [coarseCommonDenominator, one_mul, pow_add] | |
| ring | |
| theorem coarseThreshold_root_grid (B q L j d : ℕ) (allowance : ℚ) (draw : Fin B) | |
| (hB : 0 < B) (hq : 0 < q) | |
| (ha : allowance.den ∣ (L * B * q) ^ j) : | |
| (coarseChildThreshold B q allowance draw).den ∣ | |
| coarseCommonDenominator 1 q L B (j + (d + 1)) := by | |
| have ht : (coarseChildThreshold B q allowance draw).den ∣ (L * B * q) ^ j * (B * q) := by | |
| simpa only [coarseChildThreshold, Nat.cast_mul] using | |
| coarse_rat_div_nat_den_dvd (allowance * ((draw.val + 1 : ℕ) : ℚ)) (B * q) ((L * B * q) ^ j) | |
| (Nat.mul_pos hB hq) (coarse_rat_mul_nat_den_dvd allowance (draw.val + 1) _ ha) | |
| apply ht.trans | |
| refine ⟨q * L * (L * B * q) ^ d, ?_⟩ | |
| simp only [coarseCommonDenominator, one_mul, pow_add, pow_succ] | |
| ring | |
| theorem coarseChildAllowance_root_grid (B q L j d : ℕ) (allowance : ℚ) (draw : Fin B) | |
| (hB : 0 < B) (hq : 0 < q) (hL : 0 < L) | |
| (ha : allowance.den ∣ (L * B * q) ^ j) : | |
| ((1 / (L : ℚ)) * coarseChildThreshold B q allowance draw).den ∣ | |
| coarseCommonDenominator 1 q L B (j + (d + 1)) := by | |
| apply (coarseChildAllowance_grid B q L j allowance draw hB hq hL ha).trans | |
| refine ⟨q * (L * B * q) ^ d, ?_⟩ | |
| simp only [coarseCommonDenominator, one_mul, pow_add, pow_succ] | |
| ring | |
| end EditApproximation | |
| end OAI | |