openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /AllSmallInitialization.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.AllSmallFibres | |
| namespace OAI | |
| /-! # Reading the unique all-small completion directly from its profile -/ | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open scoped BigOperators Classical | |
| open CompletionCounts PhysicalCompletionFibres PaddedCompletions | |
| universe u | |
| variable {I J : Type u} [Fintype I] [Fintype J] [LinearOrder I] [LinearOrder J] | |
| variable (r : I → ℕ) (c : J → ℕ) | |
| /-- The output data consists only of the two stored small views. Existence | |
| of a completion supplies the erased proof that these are valid margins. -/ | |
| noncomputable def allSmallFibre | |
| (hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true) | |
| (z : PaperStates r c) : PaperFibre r c z := by | |
| let p := SmallContextFibres.naturalProfile z.val.val | |
| let views := (fun i j => p (⟨(i,j),hsmall i j⟩,false), | |
| fun i j => stateCapacity r c i j-p (⟨(i,j),hsmall i j⟩,true)) | |
| have hv : ∃ h, (⟨views,h⟩ : RowColumnPair | |
| (fun i => r i+PaddedCompletions.rowLargeCount (FirstPaperPhysicalMarginal.paperSmall r c) i* | |
| ((dimensionAllowance (I := I) (J := J))^12)) | |
| (fun j => c j+PaddedCompletions.columnLargeCount (FirstPaperPhysicalMarginal.paperSmall r c) j* | |
| ((dimensionAllowance (I := I) (J := J))^12))).val=views := by | |
| obtain ⟨Z⟩ := (inferInstance : Nonempty (PaperFibre r c z)) | |
| have he : views=Z.val.val := by | |
| apply Prod.ext | |
| · funext i j | |
| exact (Z.property.2.2.1 i j (hsmall i j)).1.symm | |
| · funext i j | |
| exact (Z.property.2.2.1 i j (hsmall i j)).2.symm | |
| exact ⟨by rw [he]; exact Z.val.property,rfl⟩ | |
| let pair := (⟨views,hv.choose⟩ : RowColumnPair | |
| (fun i => r i+PaddedCompletions.rowLargeCount (FirstPaperPhysicalMarginal.paperSmall r c) i* | |
| ((dimensionAllowance (I := I) (J := J))^12)) | |
| (fun j => c j+PaddedCompletions.columnLargeCount (FirstPaperPhysicalMarginal.paperSmall r c) j* | |
| ((dimensionAllowance (I := I) (J := J))^12))) | |
| refine ⟨pair,?_⟩ | |
| obtain ⟨Z⟩ := (inferInstance : Nonempty (PaperFibre r c z)) | |
| have he : pair=Z.val := by | |
| apply Subtype.ext | |
| apply Prod.ext | |
| · funext i j | |
| exact (Z.property.2.2.1 i j (hsmall i j)).1.symm | |
| · funext i j | |
| exact (Z.property.2.2.1 i j (hsmall i j)).2.symm | |
| exact he.symm ▸ Z.property | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |