File size: 2,914 Bytes
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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import OAI.Combinatorics.ContingencyTables.Transport.IntegerMarginal
namespace OAI
/-!
# Fixed-total, finite-box balanced marginal closure
The display and positivity hypotheses of integer marginalization follow
from the box-valid total slice. Positivity of the marginal permits induction
over every finite family of balanced pairs before the equality limit.
-/
namespace ContingencyTables.BalancedBox
open scoped BigOperators Classical Matrix
open CommonBasesFPRAS FactorialSignatures IntegerMarginal
variable {I : Type*} [Fintype I] [DecidableEq I]
def InBox (m z : I → ℕ) : Prop := ∀ i, z i ≤ m i
def PositiveSlice (f : (I → ℕ) → ℝ) (m : I → ℕ) (R : ℕ) : Prop :=
∀ z, InBox m z → (∑ i, z i)=R → 0 < f z
def RemovalSlice (f : (I → ℕ) → ℝ) (m : I → ℕ) (R : ℕ) : Prop :=
∀ z, InBox m z → (∑ i, z i)=R+2 →
OnePositiveDirection (integerRemovalMatrix f z).toBilin'
omit [Fintype I] [DecidableEq I] in
lemma profile_inBox (m z : I → ℕ) (M a b : ℕ) (hz : InBox m z) (ha : a ≤ M) (hb : b ≤ M) :
InBox (profile m M M) (profile z a b) := by
intro i
rcases i with i | (_ | _) <;> simp only [profile,Sum.elim_inl,Sum.elim_inr,ite_true]
· exact hz i
· exact ha
· exact hb
omit [DecidableEq I] in
lemma profile_total (z : I → ℕ) (a b : ℕ) :
(∑ i, profile z a b i)=(∑ i, z i)+a+b := by
simp [profile,Fintype.sum_sum_type]
omega
/-- One balanced pair preserves the exact box-valid integer signature. -/
theorem marginal_removal_slice (F : ((I ⊕ Bool) → ℕ) → ℝ) (m : I → ℕ) (R M : ℕ)
(hM : 2 ≤ M) (hpos : PositiveSlice F (profile m M M) (R+M))
(hsig : RemovalSlice F (profile m M M) (R+M)) :
RemovalSlice (marginal F M) m R := by
intro z hz htotal
apply marginal_removal_onePositive F M z hM
· intro j hj
apply hsig _ (profile_inBox m z M j (M-j) hz hj (Nat.sub_le _ _))
rw [profile_total,htotal]
omega
· intro j hj0 hjM
apply hpos _ (profile_inBox m z M (j-1) (M-1-j) hz (by omega) (by omega))
rw [profile_total,htotal]
omega
omit [DecidableEq I] in
/-- The finite sum stays strictly positive on precisely its required total
slice. This is the induction input for the next balanced pair. -/
theorem marginal_positive_slice (F : ((I ⊕ Bool) → ℕ) → ℝ) (m : I → ℕ) (R M : ℕ)
(hpos : PositiveSlice F (profile m M M) (R+M)) :
PositiveSlice (marginal F M) m R := by
intro z hz htotal
unfold marginal
apply Finset.sum_pos'
· intro j hj
have hjM : j ≤ M := by simpa using Finset.mem_range.mp hj
apply le_of_lt
apply hpos _ (profile_inBox m z M j (M-j) hz hjM (Nat.sub_le _ _))
rw [profile_total,htotal]
omega
· refine ⟨0,by simp,?_⟩
apply hpos _ (profile_inBox m z M 0 (M-0) hz (Nat.zero_le _) (by simp))
rw [profile_total,htotal]
omega
end ContingencyTables.BalancedBox
end OAI
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