openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /BalancedEnumeration.lean
| import OAI.Combinatorics.ContingencyTables.Transport.BalancedFamily | |
| namespace OAI | |
| /-! | |
| # Balanced elimination as a finite assignment sum | |
| The recursive integer marginal equals the finite sum over the choices in | |
| all eliminated large-cell pairs. This identifies the signature marginal | |
| with the completion count. | |
| -/ | |
| namespace ContingencyTables.BalancedEnumeration | |
| open scoped BigOperators Classical | |
| open BalancedFamily IntegerMarginal BalancedBox | |
| @[reducible] def Assignments : List ℕ → Type | |
| | [] => PUnit | |
| | M::Ms => Assignments Ms × Fin (M+1) | |
| @[reducible] instance assignmentsFintype (Ms : List ℕ) : Fintype (Assignments Ms) := by | |
| induction Ms with | |
| | nil => exact inferInstanceAs (Fintype PUnit) | |
| | cons M Ms ih => | |
| letI := ih | |
| exact inferInstanceAs (Fintype (Assignments Ms × Fin (M+1))) | |
| def embedding {I : Type*} : (Ms : List ℕ) → (I → ℕ) → Assignments Ms → Coordinates I Ms → ℕ | |
| | [],z,_ => z | |
| | M::Ms,z,a => profile (embedding Ms z a.1) a.2.val (M-a.2.val) | |
| theorem eliminate_eq_sum {I : Type*} (Ms : List ℕ) | |
| (F : (Coordinates I Ms → ℕ) → ℝ) (z : I → ℕ) : | |
| eliminate Ms F z=∑ a : Assignments Ms, F (embedding Ms z a) := by | |
| induction Ms with | |
| | nil => simp only [eliminate,Assignments,embedding,Fintype.sum_unique] | |
| | cons M Ms ih => | |
| change eliminate Ms (marginal F M) z= | |
| ∑ a : Assignments Ms × Fin (M+1), F (profile (embedding Ms z a.1) a.2.val (M-a.2.val)) | |
| rw [ih,Fintype.sum_prod_type] | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| unfold marginal | |
| exact (Fin.sum_univ_eq_sum_range (fun j => F (profile (embedding Ms z a) j (M-j))) (M+1)).symm | |
| lemma embedding_inBox {I : Type*} (Ms : List ℕ) (m : I → ℕ) (z : I → ℕ) | |
| (hz : InBox m z) (a : Assignments Ms) : InBox (bounds m Ms) (embedding Ms z a) := by | |
| induction Ms with | |
| | nil => exact hz | |
| | cons M Ms ih => | |
| apply profile_inBox (bounds m Ms) (embedding Ms z a.1) M a.2.val (M-a.2.val) | |
| · exact ih a.1 | |
| · have ha := a.2.isLt; omega | |
| · exact Nat.sub_le _ _ | |
| lemma embedding_total {I : Type*} [Fintype I] (Ms : List ℕ) (z : I → ℕ) | |
| (a : Assignments Ms) : | |
| (∑ i, embedding Ms z a i)=(∑ i, z i)+Ms.sum := by | |
| induction Ms with | |
| | nil => simp only [embedding,List.sum_nil,add_zero] | |
| | cons M Ms ih => | |
| change (∑ i, profile (embedding Ms z a.1) a.2.val (M-a.2.val) i)=_ | |
| rw [profile_total,ih] | |
| have ha := a.2.isLt | |
| simp only [List.sum_cons] | |
| omega | |
| end ContingencyTables.BalancedEnumeration | |
| end OAI | |