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97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Dense.DenseExponentListProgram
namespace OAI
/-! # The binary list exponent equals the geometric dense exponent -/
namespace ContingencyTables.DensePrograms
open scoped BigOperators Classical
open GridBoundary ProfilePrograms
variable {m n : ℕ}
def exponentDataCode (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ)
(d : ℕ) (z : Fin m → Fin n → ℕ) : ExponentData :=
(((marginCode R,marginCode P),matrixCode z),d)
lemma optionIndex_lt (i : Option (Fin m)) : optionIndex i < m + 1 := by
cases i with
| none => simp [optionIndex]
| some i => simpa only [optionIndex,Nat.add_lt_add_iff_right] using i.isLt
lemma exists_optionIndex (a : ℕ) (ha : a < m + 1) :
∃ i : Option (Fin m),optionIndex i=a := by
cases a with
| zero => exact ⟨none,rfl⟩
| succ a => exact ⟨some ⟨a,by omega⟩,rfl⟩
lemma exponentCell_code (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ)
(d : ℕ) (z : Fin m → Fin n → ℕ) (i : Option (Fin m)) (j : Option (Fin n)) :
exponentCell (exponentDataCode R P d z,(optionIndex i,optionIndex j))=
componentExponent d (min (R i) (P j))
(integerParts R P (fun p => z p.1 p.2) i j).1
(integerParts R P (fun p => z p.1 p.2) i j).2 := by
simp only [exponentCell,exponentCellData,exponentDataCode,componentValue,
marginCode_getD,completionParts_code]
lemma mem_exponentCells_code (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ)
(d : ℕ) (z : Fin m → Fin n → ℕ) (i : Option (Fin m)) (j : Option (Fin n)) :
(optionIndex i,optionIndex j)∈exponentCells (exponentDataCode R P d z) := by
apply (mem_cellPairs _ _ _ _).mpr
simpa only [marginCode,List.length_cons,List.length_ofFn] using
And.intro (optionIndex_lt i) (optionIndex_lt j)
/-- The list maximum contains every cell once; its initial zero accounts
for the extra zero component in the nonnegative rational penalty. -/
theorem fullExponent_code {q K : ℕ} (R : Option (Fin m) → ℕ)
(P : Option (Fin n) → ℕ) (d : ℕ) (e : Fin m×Fin n ≃ Fin q)
(a : Grid (I := Fin q) K) :
fullExponent (exponentDataCode R P d (fun i j => cornerFree R P d e a (i,j)))=
denseExponent R P d e a := by
let z := fun i j => cornerFree R P d e a (i,j)
let data := exponentDataCode R P d z
rw [denseExponent_eq_arithmetic]
apply le_antisymm
· apply (listMaximum_le (exponentValues data) 0 _).mpr
refine ⟨Nat.zero_le _,?_⟩
intro b hb
obtain ⟨ij,hij,rfl⟩ := List.mem_map.mp hb
have hbound := (mem_cellPairs _ _ ij.1 ij.2).mp hij
have hi : ij.1 < m + 1 := by
simpa only [data,exponentDataCode,marginCode,List.length_cons,List.length_ofFn] using hbound.1
have hj : ij.2 < n + 1 := by
simpa only [data,exponentDataCode,marginCode,List.length_cons,List.length_ofFn] using hbound.2
obtain ⟨i,hi⟩ := exists_optionIndex ij.1 hi
obtain ⟨j,hj⟩ := exists_optionIndex ij.2 hj
have he : ij=(optionIndex i,optionIndex j) := Prod.ext hi.symm hj.symm
rw [he]
change exponentCell (exponentDataCode R P d z,(optionIndex i,optionIndex j))≤_
rw [exponentCell_code]
exact Finset.le_sup' (arithmeticExponentAt R P d e a) (Finset.mem_univ (some (i,j)))
· apply Finset.sup'_le
intro g _
cases g with
| none => exact Nat.zero_le _
| some ij =>
have hmem : exponentCell (data,(optionIndex ij.1,optionIndex ij.2))∈exponentValues data :=
List.mem_map.mpr ⟨(optionIndex ij.1,optionIndex ij.2),
mem_exponentCells_code R P d z ij.1 ij.2,rfl⟩
have hle := le_listMaximum_member (exponentValues data) 0 _ hmem
rw [show data=exponentDataCode R P d z from rfl,exponentCell_code] at hle
exact hle
end ContingencyTables.DensePrograms
end OAI
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