| import OAI.Combinatorics.ContingencyTables.Sampling.SmallBalancedMarginal |
| import OAI.Combinatorics.ContingencyTables.Transport.EqualityReindex |
|
|
| namespace OAI |
|
|
| /-! # The actual balanced finite family allows every small-cell capacity |
|
|
| The proved zero/one boundary cases complete the finite balanced |
| elimination theorem. The physical equality-softened tree marginal can |
| therefore eliminate the original small cells directly. |
| -/ |
|
|
| namespace ContingencyTables.BalancedBox |
|
|
| open scoped BigOperators Classical |
| open IntegerMarginal |
|
|
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| theorem marginal_removal_slice_all (F : ((I⊕Bool) → ℕ) → ℝ) |
| (m : I → ℕ) (R M : ℕ) |
| (hpos : PositiveSlice F (profile m M M) (R+M)) |
| (hsig : RemovalSlice F (profile m M M) (R+M)) : |
| RemovalSlice (marginal F M) m R := by |
| rcases M with _|M |
| · exact marginal_zero_removal_slice F m R (by simpa only [add_zero] using hsig) |
| rcases M with _|M |
| · exact marginal_one_removal_slice F m R hpos hsig |
| · exact marginal_removal_slice F m R _ (by omega) hpos hsig |
|
|
| end ContingencyTables.BalancedBox |
|
|
| namespace ContingencyTables.BalancedFamily |
|
|
| open scoped BigOperators Classical |
| open IntegerMarginal BalancedBox |
|
|
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| /-- Every original small-cell width, including zero and one, may be |
| eliminated on its precise box-valid total slice. -/ |
| theorem eliminate_slices_all (Ms : List ℕ) (m : I → ℕ) (R : ℕ) |
| (F : (Coordinates I Ms → ℕ) → ℝ) |
| (hpos : PositiveSlice F (bounds m Ms) (R+Ms.sum)) |
| (hsig : RemovalSlice F (bounds m Ms) (R+Ms.sum)) : |
| PositiveSlice (eliminate Ms F) m R ∧ RemovalSlice (eliminate Ms F) m R := by |
| induction Ms with |
| | nil => simpa only [List.sum_nil,add_zero,eliminate,bounds] using And.intro hpos hsig |
| | cons M Ms ih => |
| have htotal : R+(M::Ms).sum=(R+Ms.sum)+M := by simp only [List.sum_cons]; omega |
| rw [htotal] at hpos hsig |
| exact ih (marginal F M) |
| (marginal_positive_slice F (bounds m Ms) (R+Ms.sum) M hpos) |
| (marginal_removal_slice_all F (bounds m Ms) (R+Ms.sum) M hpos hsig) |
|
|
| end ContingencyTables.BalancedFamily |
|
|
| namespace ContingencyTables.EqualityReindex |
|
|
| open scoped BigOperators Classical |
| open EqualitySoftening BalancedFamily BalancedBox |
|
|
| variable {S D G : Type*} [Fintype S] [DecidableEq S] [Fintype D] [Fintype G] |
|
|
| omit [DecidableEq S] in |
| lemma softenedWeight_reindex (Ms : List ℕ) (e : Coordinates S Ms ≃ D) |
| (line : D → G) (root : G → Bool) (rate offset slope : G → ℝ) |
| (urate ell uslope cap : D → ℝ) (L : ℝ) (z : Coordinates S Ms → ℕ) : |
| softenedWeight (line ∘ e) root rate offset slope (urate ∘ e) (ell ∘ e) |
| (uslope ∘ e) (cap ∘ e) L z= |
| softenedWeight line root rate offset slope urate ell uslope cap L (z ∘ e.symm) := by |
| unfold softenedWeight |
| exact productWeight_reindex e line (lineFactor root rate offset slope L) |
| (unaryFactor urate ell uslope cap) z |
|
|
| /-- The literal physical positive factor family can retain one tree |
| pair while summing all other original pairs, at every capacity. -/ |
| theorem physical_softened_slices_all (Ms : List ℕ) (m : S → ℕ) (R : ℕ) |
| (e : Coordinates S Ms ≃ D) (line : D → G) (root : G → Bool) |
| (rate offset slope : G → ℝ) (urate ell uslope cap : D → ℝ) (L : ℝ) |
| (hrate : ∀ g,0 ≤ rate g) (hu : ∀ d,0 ≤ urate d) (hL : 0 ≤ L) : |
| PositiveSlice (eliminate Ms (fun z => |
| softenedWeight line root rate offset slope urate ell uslope cap L (z ∘ e.symm))) m R ∧ |
| RemovalSlice (eliminate Ms (fun z => |
| softenedWeight line root rate offset slope urate ell uslope cap L (z ∘ e.symm))) m R := by |
| have heq : (fun z => softenedWeight line root rate offset slope urate ell uslope cap L |
| (z ∘ e.symm))=softenedWeight (line ∘ e) root rate offset slope |
| (urate ∘ e) (ell ∘ e) (uslope ∘ e) (cap ∘ e) L := by |
| funext z |
| exact (softenedWeight_reindex Ms e line root rate offset slope urate ell uslope cap L z).symm |
| rw [heq] |
| apply eliminate_slices_all Ms m R |
| · intro z _ _ |
| exact softenedWeight_positive (line ∘ e) root rate offset slope |
| (urate ∘ e) (ell ∘ e) (uslope ∘ e) (cap ∘ e) L z |
| · intro z _ _ |
| convert softenedWeight_removal (line ∘ e) root rate offset slope |
| (urate ∘ e) (ell ∘ e) (uslope ∘ e) (cap ∘ e) L hrate (fun i => hu (e i)) hL z using 1 |
| congr 2 <;> exact Subsingleton.elim _ _ |
|
|
| end ContingencyTables.EqualityReindex |
|
|
| end OAI |
|
|
