File size: 2,516 Bytes
97c3405 | 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 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Combinatorics.ContingencyTables.Transport.RationalMapProducts
namespace OAI
/-! # Reserving a common word length for adaptive bounded draws -/
namespace ContingencyTables.FirstSuccess
open scoped BigOperators Classical
open ResidualMixture FairBits
noncomputable section
variable {A B : Type*} [Fintype A] [Fintype B] [DecidableEq A]
lemma mapLaw_product_fst (p : RationalLaw A) (q : RationalLaw B) :
mapLaw (productLaw p q) Prod.fst=p := by
apply RationalLaw.ext
funext x
simp only [mapLaw, productLaw, Fintype.sum_prod_type]
change (∑ a,∑ b,if a=x then p.mass a*q.mass b else 0)=p.mass x
have hs (a : A) : (∑ b,if a=x then p.mass a*q.mass b else 0)=if a=x then p.mass a else 0 := by
by_cases ha : a=x
· simp only [ha,ite_true,←Finset.mul_sum,q.total,mul_one]
· simp only [ha,ite_false,Finset.sum_const_zero]
simp_rw [hs]
simp
def prefixWord {k N : ℕ} (hk : k ≤ N) (bits : Fin N → Bool) : Fin k → Bool :=
fun i => bits (Fin.castLE hk i)
def paddingWordEquiv {k N : ℕ} (hk : k ≤ N) :
(Fin N → Bool) ≃ ((Fin k → Bool) × (Fin (N-k) → Bool)) :=
(Equiv.arrowCongr (finCongr (Nat.add_sub_of_le hk)).symm (Equiv.refl Bool)).trans
(splitWordEquiv k (N-k))
lemma paddingWordEquiv_fst {k N : ℕ} (hk : k ≤ N) (bits : Fin N → Bool) :
(paddingWordEquiv hk bits).1=prefixWord hk bits := by
funext i
change bits _=bits _
apply congrArg bits
apply Fin.ext
rfl
/-- Taking the needed prefix of a larger uniform word preserves the exact
law, so unused bits can be reserved even after the current state is known. -/
theorem prefixWord_uniform {k N : ℕ} (hk : k ≤ N) :
mapLaw (uniformLaw (α := Fin N → Bool)) (prefixWord hk)=uniformLaw (α := Fin k → Bool) := by
have hf : prefixWord hk=fun bits => (paddingWordEquiv hk bits).1 := by
funext bits
exact (paddingWordEquiv_fst hk bits).symm
rw [hf,←map_equiv (paddingWordEquiv hk),uniform_equiv,←uniform_product,mapLaw_product_fst]
variable {Y : Type*} [Fintype Y] [DecidableEq Y]
/-- An arbitrary finite draw keeps its law when padded to the common
bound; this applies separately at every state of an adaptive walk. -/
theorem padded_draw_law {k N : ℕ} (hk : k ≤ N) (draw : (Fin k → Bool) → Y) :
mapLaw (uniformLaw (α := Fin N → Bool)) (fun bits => draw (prefixWord hk bits))=
mapLaw (uniformLaw (α := Fin k → Bool)) draw := by
rw [←mapLaw_comp,prefixWord_uniform]
end
end ContingencyTables.FirstSuccess
end OAI
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