File size: 4,498 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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Combinatorics.ContingencyTables.Dense.DenseTrialSemantics
namespace OAI
/-! # Enumerating tables through their free entries -/
namespace ContingencyTables.BoxCompletion
open DenseAffine DensePrograms
open scoped BigOperators Classical
noncomputable section
abbrev Box (m n N : ℕ) := Fin (m*n) → Fin (N+1)
def freeValues {m n N : ℕ} (z : Box m n N) : Fin m×Fin n → ℕ :=
fun ij => (z (finProdFinEquiv ij)).val
def decode {m n : ℕ} (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ)
(htotal : ∑ i,R i=∑ j,P j) (N : ℕ) (z : Box m n N) : Option (Table R P) :=
if hz : ∀ i j,0 ≤ integerTable R P (freeValues z) i j then
some ⟨naturalTable R P (freeValues z),naturalTable_rows R P _ hz,
naturalTable_columns R P _ hz htotal⟩
else none
def encode {m n N : ℕ} (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ)
(hN : (∑ i,R i)≤N) (X : Table R P) : Box m n N := fun k =>
⟨X.val (some (finProdFinEquiv.symm k).1) (some (finProdFinEquiv.symm k).2),
Nat.lt_succ_of_le ((entry_le_total X.property _ _).trans hN)⟩
lemma freeValues_encode {m n N : ℕ} (R : Option (Fin m) → ℕ)
(P : Option (Fin n) → ℕ) (hN : (∑ i,R i)≤N) (X : Table R P) :
freeValues (encode R P hN X)=fun ij => X.val (some ij.1) (some ij.2) := by
funext ij
simp only [freeValues,encode,Equiv.symm_apply_apply]
lemma decode_encode {m n N : ℕ} (R : Option (Fin m) → ℕ)
(P : Option (Fin n) → ℕ) (htotal : ∑ i,R i=∑ j,P j)
(hN : (∑ i,R i)≤N) (X : Table R P) :
decode R P htotal N (encode R P hN X)=some X := by
have he := integerTable_recover R P X.val X.property.1 X.property.2
have hz : ∀ i j,0 ≤ integerTable R P (freeValues (encode R P hN X)) i j := by
rw [freeValues_encode,he]
exact fun _ _ => Int.natCast_nonneg _
rw [decode,dite_eq_left hz]
congr 1
apply Subtype.ext
funext i j
simp only [naturalTable,freeValues_encode,he,Int.toNat_natCast]
/-- Each feasible table occurs at exactly one enumerated free vector. -/
theorem decode_eq_some {m n N : ℕ} (R : Option (Fin m) → ℕ)
(P : Option (Fin n) → ℕ) (htotal : ∑ i,R i=∑ j,P j)
(hN : (∑ i,R i)≤N) (z : Box m n N) (X : Table R P) :
decode R P htotal N z=some X ↔ z=encode R P hN X := by
constructor
· intro h
unfold decode at h
split at h
next hz =>
have he := Option.some.inj h
funext k
apply Fin.ext
have hv := congrArg (fun Y : Table R P =>
Y.val (some (finProdFinEquiv.symm k).1) (some (finProdFinEquiv.symm k).2)) he
simpa only [naturalTable_free,freeValues,Prod.mk.eta,
Equiv.apply_symm_apply,encode] using hv
next hz => cases h
· rintro rfl
exact decode_encode R P htotal hN X
lemma decode_elim_sum {A : Type*} [Fintype A] (y : Option A) (w : A → ℕ) :
y.elim 0 w=∑ a,if y=some a then w a else 0 := by
cases y <;> simp
/-- Summing over candidate completions counts each feasible table once,
and gives zero weight to every rejected completion. -/
theorem sum_decode {m n N : ℕ} (R : Option (Fin m) → ℕ)
(P : Option (Fin n) → ℕ) (htotal : ∑ i,R i=∑ j,P j)
(hN : (∑ i,R i)≤N) (w : Table R P → ℕ) :
(∑ z : Box m n N,(decode R P htotal N z).elim 0 w)=∑ X,w X := by
simp_rw [decode_elim_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro X _
simp_rw [decode_eq_some R P htotal hN]
simp
/-- The existing executable acceptance test returns the same optional
table as the finite enumeration used in the counting proof. -/
theorem acceptCompletion_eq {m n N : ℕ} (R : Option (Fin m) → ℕ)
(P : Option (Fin n) → ℕ) (htotal : ∑ i,R i=∑ j,P j) (z : Box m n N) :
acceptCompletion ((marginCode R,marginCode P),
matrixCode (fun i j => freeValues z (i,j)))=
(decode R P htotal N z).map tableOutput := by
by_cases hz : ∀ i j,0 ≤ integerTable R P (freeValues z) i j
· have hv := (completionValid_code R P (fun i j => freeValues z (i,j))).mpr hz
simp only [acceptCompletion,hv,ite_true,decode,dite_eq_left hz]
rw [completedMatrix_code]
rfl
· have hv : completionValid ((marginCode R,marginCode P),
matrixCode (fun i j => freeValues z (i,j)))≠true := by
intro h
exact hz ((completionValid_code R P _).mp h)
simp only [acceptCompletion,hv,Bool.false_eq_true,ite_false,decode,dite_eq_right hz,
Option.map_none]
end
end ContingencyTables.BoxCompletion
end OAI
|