File size: 4,032 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.Rooted.LargeMarginProgram
import OAI.Combinatorics.ContingencyTables.Sampling.PaddedResidualSemantics
import OAI.Combinatorics.ContingencyTables.Transport.ReferenceCompletionBlock
namespace OAI
/-! # Exact indexing of the computed large residual margins -/
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms PaddedCompletions SmallContextCoordinates SmallContextFibres
open PhysicalCompletionFibres FirstPaperProfiles CompletionCounts
open scoped BigOperators Classical
noncomputable section
variable {m n q : ℕ}
lemma largeIndices_ofFn_mem (r : Fin m → ℕ) (U : ℕ) (i : Fin m) :
i.val∈largeIndices (U,List.ofFn r) ↔ U≤r i := by
rw [mem_largeIndices]
simp only [List.length_ofFn,i.isLt,true_and,List.getD_eq_getElem?_getD,
List.getElem?_ofFn,dite_true,Option.getD_some,Fin.eta]
private def largeMemberEquiv (r : Fin m → ℕ) (U : ℕ) :
{i // i∈largeIndices (U,List.ofFn r)} ≃ LargeRows r U where
toFun x := by
have hm := (mem_largeIndices U (List.ofFn r) x.val).mp x.property
have hi : x.val < m := by simpa only [List.length_ofFn] using hm.1
exact ⟨⟨x.val,hi⟩,(largeIndices_ofFn_mem r U ⟨x.val,hi⟩).mp x.property⟩
invFun i := ⟨i.val.val,(largeIndices_ofFn_mem r U i.val).mpr i.property⟩
left_inv _ := rfl
right_inv _ := rfl
/-- Increasing list order gives a fixed, computed indexing of every
large row. The same construction is used for columns. -/
def largeIndexEquiv (r : Fin m → ℕ) (U : ℕ) :
Fin (largeIndices (U,List.ofFn r)).length ≃ LargeRows r U :=
(List.Nodup.getEquiv _ (largeIndices_nodup U (List.ofFn r))).trans (largeMemberEquiv r U)
lemma largeIndexEquiv_val (r : Fin m → ℕ) (U : ℕ)
(i : Fin (largeIndices (U,List.ofFn r)).length) :
(largeIndexEquiv r U i).val.val=(largeIndices (U,List.ofFn r))[i.val] := rfl
lemma largePaddedMargins_ofFn (row : Bool) (z : PaddingData) :
largePaddedMargins row z=List.ofFn
(fun i : Fin (largeIndices (z.1.2,marginList row z.1)).length =>
paddedMarginAt row (z,(largeIndices (z.1.2,marginList row z.1))[i.val])) := by
rw [largePaddedMargins_eq]
exact (List.ofFn_getElem_eq_map _ _).symm
theorem largePaddedRows_code (r : Fin m → ℕ) (c : Fin n → ℕ) (U L : ℕ)
(e : Fin q ≃ Cells (firstPaperSmall r c U))
(z : SmallProfile (firstPaperSmall r c U)) :
largePaddedMargins true
(((((cellCatalog (firstPaperSmall r c U) e,viewList (firstPaperSmall r c U) e z false),
(List.ofFn r,List.ofFn c)),U),L))=
List.ofFn (fun i : Fin (largeIndices (U,List.ofFn r)).length =>
r (largeIndexEquiv r U i).val-
smallRow (firstPaperSmall r c U) (smallView (firstPaperSmall r c U) z false)
(largeIndexEquiv r U i).val+
rowLargeCount (firstPaperSmall r c U) (largeIndexEquiv r U i).val*L) := by
rw [largePaddedMargins_ofFn]
apply congrArg List.ofFn
funext i
exact paddedRowAt_code r c U L e z (largeIndexEquiv r U i).val
theorem largePaddedColumns_code (r : Fin m → ℕ) (c : Fin n → ℕ) (U L : ℕ)
(e : Fin q ≃ Cells (firstPaperSmall r c U))
(B : Fin m → Fin n → ℕ) (hB : ∀ a : Cells (firstPaperSmall r c U),B a.val.1 a.val.2=U)
(z : SmallProfile (firstPaperSmall r c U)) :
largePaddedMargins false
(((((cellCatalog (firstPaperSmall r c U) e,
complementValues (U,viewList (firstPaperSmall r c U) e z true)),
(List.ofFn r,List.ofFn c)),U),L))=
List.ofFn (fun j : Fin (largeIndices (U,List.ofFn c)).length =>
c (largeIndexEquiv c U j).val-
smallColumn (firstPaperSmall r c U) B (smallView (firstPaperSmall r c U) z true)
(largeIndexEquiv c U j).val+
columnLargeCount (firstPaperSmall r c U) (largeIndexEquiv c U j).val*L) := by
rw [largePaddedMargins_ofFn]
apply congrArg List.ofFn
funext j
exact paddedColumnAt_code r c U L e B hB z (largeIndexEquiv c U j).val
end
end ContingencyTables.SmallGraphProfiles
end OAI
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