openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /BooleanIntegerLaw.lean
| import OAI.Combinatorics.ContingencyTables.Transport.IntegerEquivLaw | |
| import OAI.Combinatorics.ContingencyTables.Transport.RationalMapProducts | |
| namespace OAI | |
| /-! # Independent Boolean words combined with cached integer laws -/ | |
| namespace ContingencyTables.IntegerLaw | |
| open FirstSuccess FairBits ResidualMixture | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| def boolean (n : ℕ) : IntegerLaw (Fin n → Bool) (2^n) where | |
| count _ := 1 | |
| total := by simp | |
| lemma toRational_boolean (n : ℕ) : | |
| (boolean n).toRational (by positivity)=uniformLaw (α := Fin n → Bool) := by | |
| apply RationalLaw.ext | |
| funext x | |
| simp [toRational,boolean,uniformLaw,uniformMass] | |
| variable {B C : Type*} [Fintype B] [Fintype C] [DecidableEq C] | |
| /-- A short proposal word and an independently drawn cached completion | |
| reserve their full combined word length. -/ | |
| def booleanFinish {s f : ℕ} (p : IntegerLaw B (2^f)) | |
| (finish : (Fin s → Bool) → B → C) : IntegerLaw C (2^(s+f)) := | |
| (((boolean s).product p).push (fun z => finish z.1 z.2)).rebase (pow_add 2 s f).symm | |
| lemma toRational_booleanFinish {s f : ℕ} (p : IntegerLaw B (2^f)) | |
| (draw : (Fin f → Bool) → B) (finish : (Fin s → Bool) → B → C) | |
| (hp : p.toRational (by positivity)=mapLaw (uniformLaw (α := Fin f → Bool)) draw) : | |
| (booleanFinish p finish).toRational (by positivity)= | |
| mapLaw (uniformLaw (α := Fin (s+f) → Bool)) | |
| (fun bits => finish (splitWordEquiv s f bits).1 (draw (splitWordEquiv s f bits).2)) := by | |
| classical | |
| unfold booleanFinish | |
| rw [toRational_rebase,toRational_push,toRational_product,toRational_boolean,hp] | |
| have he := mapLaw_product (uniformLaw (α := Fin s → Bool)) | |
| (uniformLaw (α := Fin f → Bool)) id draw | |
| rw [mapLaw_id] at he | |
| rw [←he,mapLaw_comp] | |
| have hs := map_equiv (splitWordEquiv s f) | |
| (uniformLaw (α := Fin (s+f) → Bool)) | |
| (fun z => finish z.1 (draw z.2)) | |
| rw [uniform_equiv,←uniform_product] at hs | |
| exact hs | |
| all_goals positivity | |
| end | |
| end ContingencyTables.IntegerLaw | |
| end OAI | |