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d5b7741 | 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 | 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
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