openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Rooted /ForestBoxCardinality.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionFactorialApplication | |
| namespace OAI | |
| /-! # The expanded physical forest box still has a short cardinality bound | |
| The tree interval width 20db has at most (b+1)^2 integer positions | |
| when every large capacity is at least 20d. This provides a uniform bound | |
| for the positive equality-softened balanced sums before taking limits. | |
| -/ | |
| namespace ContingencyTables.ForestBoxCardinality | |
| open scoped BigOperators Classical | |
| open CompanionFactorialApplication ForestLiteralCount | |
| variable {I J : Type*} [Fintype I] [Fintype J] | |
| lemma width_add_one_le_square (A : Instance I J) (hW : 20*A.d ≤ A.W) (i : I) (j : J) : | |
| width A.T A.d A.b i j+1 ≤ (A.b i j+1)^2 := by | |
| unfold width | |
| split_ifs with ht | |
| · have hs : A.small i j=false := A.hTG ht | |
| have hb : 20*A.d ≤ A.b i j := hW.trans (A.hlarge i j hs).1 | |
| nlinarith | |
| · nlinarith | |
| abbrev Assignments (A : Instance I J) := | |
| (a : I×J) → Fin (width A.T A.d A.b a.1 a.2+1) | |
| theorem cardinality_le_square (A : Instance I J) (hW : 20*A.d ≤ A.W) : | |
| Fintype.card (Assignments A) ≤ (∏ a : I×J,(A.b a.1 a.2+1))^2 := by | |
| simp only [Assignments,Fintype.card_pi,Fintype.card_fin] | |
| calc | |
| _ ≤ ∏ a : I×J,(A.b a.1 a.2+1)^2 := by | |
| apply Finset.prod_le_prod | |
| intro a _ | |
| exact width_add_one_le_square A hW a.1 a.2 | |
| _ = _ := Finset.prod_pow Finset.univ 2 (fun a : I×J => A.b a.1 a.2+1) | |
| /-- The full balanced physical box, including every expanded tree | |
| interval, has at most 2^K profiles under the source's bit-size allowance. -/ | |
| theorem cardinality_le_two_pow (A : Instance I J) (hW : 20*A.d ≤ A.W) | |
| (L K : ℕ) (hL : (∏ a : I×J,(A.b a.1 a.2+1)) ≤ 2^L) (hK : 2*L ≤ K) : | |
| Fintype.card (Assignments A) ≤ 2^K := by | |
| calc | |
| _ ≤ (∏ a : I×J,(A.b a.1 a.2+1))^2 := cardinality_le_square A hW | |
| _ ≤ (2^L)^2 := Nat.pow_le_pow_left hL 2 | |
| _ = 2^(2*L) := by rw [← pow_mul]; congr 1; omega | |
| _ ≤ _ := Nat.pow_le_pow_right (by omega) hK | |
| end ContingencyTables.ForestBoxCardinality | |
| end OAI | |