| import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperJointLaw | |
| namespace OAI | |
| /-! # Empty large rectangles have a unique retained completion -/ | |
| namespace ContingencyTables.PhysicalCompletionFibres | |
| open scoped BigOperators Classical | |
| variable {I J : Type*} [Fintype I] [Fintype J] | |
| /-- When every cell is retained, both views are fixed by the profile. -/ | |
| theorem completionFibre_subsingleton (small : I → J → Bool) (B : I → J → ℕ) | |
| (r rbar : I → ℕ) (c cbar : J → ℕ) (z : SmallProfile small) | |
| (hsmall : ∀ i j,small i j=true) : | |
| Subsingleton (CompletionFibre small B r rbar c cbar z) := by | |
| constructor | |
| intro X Y | |
| apply Subtype.ext | |
| apply Subtype.ext | |
| apply Prod.ext | |
| · funext i j | |
| exact (X.property.2.2.1 i j (hsmall i j)).1.trans | |
| (Y.property.2.2.1 i j (hsmall i j)).1.symm | |
| · funext i j | |
| exact (X.property.2.2.1 i j (hsmall i j)).2.trans | |
| (Y.property.2.2.1 i j (hsmall i j)).2.symm | |
| end ContingencyTables.PhysicalCompletionFibres | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open scoped BigOperators Classical | |
| open PhysicalCompletionFibres FirstPaperPhysicalMarginal CompletionCounts SmallContextFibres | |
| universe u | |
| variable {I J : Type u} [Fintype I] [Fintype J] [LinearOrder I] [LinearOrder J] | |
| variable (r : I → ℕ) (c : J → ℕ) | |
| omit [LinearOrder I] [LinearOrder J] in | |
| lemma allSmall_of_no_large_rectangle | |
| (hempty : IsEmpty (LargeRows r ((dimensionAllowance (I := I) (J := J))^20)) ∨ | |
| IsEmpty (LargeColumns c ((dimensionAllowance (I := I) (J := J))^20))) : | |
| ∀ i j,paperSmall r c i j=true := by | |
| intro i j | |
| have he : r i<(dimensionAllowance (I := I) (J := J))^20 ∨ | |
| c j<(dimensionAllowance (I := I) (J := J))^20 := by | |
| rcases hempty with hr|hc | |
| · left | |
| by_contra h | |
| exact hr.false ⟨i,by omega⟩ | |
| · right | |
| by_contra h | |
| exact hc.false ⟨j,by omega⟩ | |
| simpa only [paperSmall,firstPaperSmall,decide_eq_true_eq] using he | |
| lemma paperFibre_card_allSmall (hsmall : ∀ i j,paperSmall r c i j=true) | |
| (z : PaperStates r c) : Fintype.card (PaperFibre r c z)=1 := by | |
| let : Subsingleton (PaperFibre r c z) := completionFibre_subsingleton _ _ _ _ _ _ _ hsmall | |
| let : Unique (PaperFibre r c z) := ⟨⟨Classical.choice inferInstance⟩,fun x => Subsingleton.elim x _⟩ | |
| exact Fintype.card_unique | |
| /-- No dense sampler or numerical integral is needed in this branch. -/ | |
| lemma stateWeight_allSmall (hsmall : ∀ i j,paperSmall r c i j=true) | |
| (z : PaperStates r c) : stateWeight r c z=1 := by | |
| change hardMarginal _ _ _ _ _ _=1 | |
| rw [hardMarginal_eq_fibre_card (paperSmall r c) (stateCapacity r c) r c | |
| ((dimensionAllowance (I := I) (J := J))^12) | |
| (physicalProfile _ (stateCapacity r c) r c _ z) (naturalProfile_bounded z.val.val)] | |
| exact_mod_cast paperFibre_card_allSmall r c hsmall z | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |