openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /FibreWeightSeparation.lean
| import OAI.NumberTheory.PiExponent.Jets.NormalBasisProducts | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace PiExponent.FibreWeightSeparation | |
| theorem positive_indices {m : ℕ} (w cost : Fin (m+1) → ℝ) (C : ℝ) | |
| (hsep : ∀ A B : Finset (Fin (m+1)), A.card = B.card → | |
| ∀ i, i ≠ 0 → i ∈ A → i ∉ B → | |
| (∀ j, i < j → (j ∈ A ↔ j ∈ B)) → | |
| C * (∏ j ∈ B, cost j) < ∏ j ∈ A, w j) : | |
| ∀ A B : Finset (Fin m), A.card = B.card → | |
| ∀ i, i ∈ A → i ∉ B → (∀ j, i < j → (j ∈ A ↔ j ∈ B)) → | |
| C * (∏ j ∈ B, cost j.succ) < ∏ j ∈ A, w j.succ := by | |
| intro A B hcard i hiA hiB hlarge | |
| have h := hsep (A.map (Fin.succEmb m)) (B.map (Fin.succEmb m)) | |
| (by simpa only [Finset.card_map] using hcard) i.succ (Fin.succ_ne_zero i) | |
| (by simpa using hiA) (by simpa using hiB) (by | |
| intro j hij | |
| cases j using Fin.cases with | |
| | zero => exact (not_lt_of_ge (Fin.zero_le _) hij).elim | |
| | succ j => simpa using hlarge j (Fin.succ_lt_succ_iff.mp hij)) | |
| simpa only [Finset.prod_map, Fin.coe_succEmb] using h | |
| end PiExponent.FibreWeightSeparation | |
| end | |
| end OAI | |