openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Rooted /LargeMarginSemantics.lean
| 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 | |