File size: 2,039 Bytes
d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | 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
|