openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /BalancedPhysicalPartition.lean
| import OAI.Combinatorics.ContingencyTables.Transport.BalancedCellFibres | |
| namespace OAI | |
| /-! # Partitioning actual balanced profiles into small and large cells -/ | |
| namespace ContingencyTables.BalancedCellFibres | |
| open scoped BigOperators Classical | |
| open PhysicalLargeProfiles LargePairEnumeration | |
| variable {C : Type*} | |
| abbrev SmallProfiles (small : C → Bool) (M : C → ℕ) := | |
| (s : {c : C // small c=true}) → Fin (M s.val+1) | |
| noncomputable def partition (small : C → Bool) (M : C → ℕ) : | |
| Profiles M ≃ SmallProfiles small M×PhysicalAssignments small M where | |
| toFun a := (fun s => a s.val,fun l => a l.val) | |
| invFun a c := if hs : small c=true then a.1 ⟨c,hs⟩ | |
| else a.2 ⟨c,Bool.eq_false_iff.mpr hs⟩ | |
| left_inv a := by funext c; dsimp only; split_ifs <;> rfl | |
| right_inv a := by | |
| apply Prod.ext | |
| · funext s | |
| simp only [s.property,dite_eq_left] | |
| · funext l | |
| simp only [l.property,Bool.false_eq_true,dite_false] | |
| def smallDisplay (small : C → Bool) (M : C → ℕ) (a : SmallProfiles small M) : SmallView small → ℕ := | |
| fun p => if p.2 then M p.1.val-(a p.1).val else (a p.1).val | |
| lemma partition_profile (small : C → Bool) (M : C → ℕ) | |
| (a : SmallProfiles small M×PhysicalAssignments small M) : | |
| profile M ((partition small M).symm a)= | |
| physicalProfile small M (smallDisplay small M a.1) a.2 := by | |
| funext p | |
| rcases p with ⟨c,b⟩ | |
| by_cases hs : small c=true | |
| · simp only [profile,partition,Equiv.coe_fn_symm_mk,physicalProfile,hs,dite_eq_left,smallDisplay] | |
| · simp only [profile,partition,Equiv.coe_fn_symm_mk,physicalProfile,hs,Bool.false_eq_true,dite_false] | |
| variable [Fintype C] | |
| /-- This exact finite product split removes all artificial labels and | |
| enumerations from the companion's terminal normalizer. -/ | |
| theorem sum_eq_small_large (small : C → Bool) (M : C → ℕ) | |
| (F : (C×Bool → ℕ) → ℝ) : | |
| (∑ a : Profiles M,F (profile M a))= | |
| ∑ s : SmallProfiles small M,∑ a : PhysicalAssignments small M, | |
| F (physicalProfile small M (smallDisplay small M s) a) := by | |
| have he := Fintype.sum_equiv (partition small M) | |
| (fun a => F (profile M a)) | |
| (fun a => F (physicalProfile small M (smallDisplay small M a.1) a.2)) | |
| (by | |
| intro a | |
| have h := partition_profile small M ((partition small M) a) | |
| simpa only [Equiv.symm_apply_apply] using congrArg F h) | |
| rw [he,Fintype.sum_prod_type] | |
| end ContingencyTables.BalancedCellFibres | |
| end OAI | |