File size: 3,135 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 84 85 86 87 88 | import OAI.Combinatorics.ContingencyTables.Model
namespace OAI
/-! # Complementing a bounded table
The map X ↦ b-X transforms cell minima into maxima. All margin
subtractions are justified by an actual feasible table.
-/
namespace ContingencyTables.BoundedTableComplement
open scoped BigOperators
variable {m n : ℕ}
def complement (b X : Fin m→Fin n→ℕ) (i : Fin m) (j : Fin n) : ℕ := b i j-X i j
def rowTotal (b : Fin m→Fin n→ℕ) (i : Fin m) : ℕ := ∑ j,b i j
def columnTotal (b : Fin m→Fin n→ℕ) (j : Fin n) : ℕ := ∑ i,b i j
def rows (b : Fin m→Fin n→ℕ) (r : Fin m→ℕ) (i : Fin m) : ℕ := rowTotal b i-r i
def columns (b : Fin m→Fin n→ℕ) (c : Fin n→ℕ) (j : Fin n) : ℕ := columnTotal b j-c j
lemma complement_bounds (b X : Fin m→Fin n→ℕ) :
∀ i j,complement b X i j≤b i j := fun _ _=>Nat.sub_le _ _
lemma complement_twice (b X : Fin m→Fin n→ℕ) (hb : ∀ i j,X i j≤b i j) :
complement b (complement b X)=X := by
funext i j
exact Nat.sub_sub_self (hb i j)
lemma complement_margins (b X : Fin m→Fin n→ℕ) (r : Fin m→ℕ) (c : Fin n→ℕ)
(hm : HasMargins X r c) (hb : ∀ i j,X i j≤b i j) :
HasMargins (complement b X) (rows b r) (columns b c) := by
constructor
· intro i
have hs : (∑ j,complement b X i j)+(∑ j,X i j)=rowTotal b i := by
rw [←Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro j _
exact Nat.sub_add_cancel (hb i j)
rw [hm.1 i] at hs
unfold rows
omega
· intro j
have hs : (∑ i,complement b X i j)+(∑ i,X i j)=columnTotal b j := by
rw [←Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro i _
exact Nat.sub_add_cancel (hb i j)
rw [hm.2 j] at hs
unfold columns
omega
lemma row_total_ge (b X : Fin m→Fin n→ℕ) (r : Fin m→ℕ) (c : Fin n→ℕ)
(hm : HasMargins X r c) (hb : ∀ i j,X i j≤b i j) (i : Fin m) :
r i≤rowTotal b i := by
rw [←hm.1 i]
exact Finset.sum_le_sum (fun j _=>hb i j)
lemma column_total_ge (b X : Fin m→Fin n→ℕ) (r : Fin m→ℕ) (c : Fin n→ℕ)
(hm : HasMargins X r c) (hb : ∀ i j,X i j≤b i j) (j : Fin n) :
c j≤columnTotal b j := by
rw [←hm.2 j]
exact Finset.sum_le_sum (fun i _=>hb i j)
lemma margins_twice (b X : Fin m→Fin n→ℕ) (r : Fin m→ℕ) (c : Fin n→ℕ)
(hm : HasMargins X r c) (hb : ∀ i j,X i j≤b i j) :
rows b (rows b r)=r ∧ columns b (columns b c)=c := by
constructor
· funext i
exact Nat.sub_sub_self (row_total_ge b X r c hm hb i)
· funext j
exact Nat.sub_sub_self (column_total_ge b X r c hm hb j)
lemma complement_reverse_margins (b X Y : Fin m→Fin n→ℕ)
(r : Fin m→ℕ) (c : Fin n→ℕ) (hx : HasMargins X r c) (hxb : ∀ i j,X i j≤b i j)
(hy : HasMargins Y (rows b r) (columns b c)) (hyb : ∀ i j,Y i j≤b i j) :
HasMargins (complement b Y) r c := by
have h := complement_margins b Y (rows b r) (columns b c) hy hyb
obtain ⟨hr,hc⟩ := margins_twice b X r c hx hxb
rwa [hr,hc] at h
end ContingencyTables.BoundedTableComplement
end OAI
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