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5d5048e | 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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import OAI.Combinatorics.LowSpaceSubsetSum.Counting.FiniteProductSamplerCoupling
import OAI.Combinatorics.LowSpaceSubsetSum.Counting.NestedPrimes
namespace OAI
/-! # Bounded fresh-prime calls for the nested three-factor sampler -/
noncomputable section
namespace LowSpaceSubsetSum
attribute [local instance] Classical.propDecidable
private instance factorNonempty (ell : ℕ) : Nonempty ↥(factorRange ell) :=
(factorRange_nonempty ell).to_subtype
def factorOneEquiv {ell : ℕ} (h : ell < 4) : Unit ≃ ↥(factorRange ell) where
toFun _ := ⟨1, by simp [factorRange, h]⟩
invFun _ := ()
left_inv _ := rfl
right_inv p := by
apply Subtype.ext
have hp : p.val = 1 := by simpa [factorRange, h] using p.property
exact hp.symm
def factorPrimeEquiv {ell : ℕ} (h : 4 ≤ ell) : ↥(primeRange ell) ≃ ↥(factorRange ell) where
toFun p := ⟨p.val, by simp [factorRange, show ¬ell < 4 by omega, p.property]⟩
invFun p := ⟨p.val, by simpa [factorRange, show ¬ell < 4 by omega] using p.property⟩
left_inv _ := rfl
right_inv _ := rfl
def factorCoupling (n ell : ℕ) : SamplerCoupling ↥(factorRange ell) :=
if h : ell < 4 then (SamplerCoupling.pure ()).map (factorOneEquiv h)
else (primeCoupling (by omega : 4 ≤ ell) n).map (factorPrimeEquiv (by omega))
theorem factorCoupling_uniform (n ell : ℕ) :
(factorCoupling n ell).idealLaw = FiniteLaw.uniform := by
by_cases h : ell < 4
· rw [factorCoupling, dite_eq_left h, SamplerCoupling.map_idealLaw,
SamplerCoupling.pure_uniform]
exact FiniteLaw.map_equiv_uniform (factorOneEquiv h)
· rw [factorCoupling, dite_eq_right h, SamplerCoupling.map_idealLaw,
primeCoupling_uniform]
let : Nonempty ↥(primeRange ell) := (primeRange_nonempty (by omega : 4 ≤ ell)).to_subtype
exact FiniteLaw.map_equiv_uniform (factorPrimeEquiv (by omega : 4 ≤ ell))
theorem factorCoupling_failure {n ell : ℕ} (hell : ell ≤ n) :
(factorCoupling n ell).failure ≤ Real.exp (-((n : ℝ) ^ 2 / 4)) := by
by_cases h : ell < 4
· rw [factorCoupling, dite_eq_left h, SamplerCoupling.map_failure,
SamplerCoupling.pure_failure]
positivity
· rw [factorCoupling, dite_eq_right h, SamplerCoupling.map_failure]
exact primeCoupling_failure (by omega) hell
def tripleFactorEquiv (a b c : ℕ) :
((↥(factorRange a) × ↥(factorRange b)) × ↥(factorRange c)) ≃
↥(tripleRange a b c) where
toFun z := ⟨((z.1.1.val, z.1.2.val), z.2.val),
Finset.mem_product.mpr ⟨Finset.mem_product.mpr ⟨z.1.1.property, z.1.2.property⟩,
z.2.property⟩⟩
invFun z := ((⟨z.val.1.1, (Finset.mem_product.mp (Finset.mem_product.mp z.property).1).1⟩,
⟨z.val.1.2, (Finset.mem_product.mp (Finset.mem_product.mp z.property).1).2⟩),
⟨z.val.2, (Finset.mem_product.mp z.property).2⟩)
left_inv _ := rfl
right_inv _ := rfl
def triplePrimeCoupling (n a b c : ℕ) : SamplerCoupling ↥(tripleRange a b c) :=
(((factorCoupling n a).prod (factorCoupling n b)).prod (factorCoupling n c)).map
(tripleFactorEquiv a b c)
theorem triplePrimeCoupling_uniform (n a b c : ℕ) :
(triplePrimeCoupling n a b c).idealLaw =
@FiniteLaw.uniform _ inferInstance (tripleRange_nonempty a b c).to_subtype := by
let : Nonempty ↥(tripleRange a b c) := (tripleRange_nonempty a b c).to_subtype
rw [triplePrimeCoupling, SamplerCoupling.map_idealLaw,
SamplerCoupling.prod_uniform _ _
(SamplerCoupling.prod_uniform _ _ (factorCoupling_uniform n a)
(factorCoupling_uniform n b)) (factorCoupling_uniform n c)]
exact FiniteLaw.map_equiv_uniform (tripleFactorEquiv a b c)
theorem triplePrimeCoupling_failure {n a b c : ℕ}
(ha : a ≤ n) (hb : b ≤ n) (hc : c ≤ n) :
(triplePrimeCoupling n a b c).failure ≤ 3 * Real.exp (-((n : ℝ) ^ 2 / 4)) := by
rw [triplePrimeCoupling, SamplerCoupling.map_failure]
have h1 := SamplerCoupling.prod_failure_le
((factorCoupling n a).prod (factorCoupling n b)) (factorCoupling n c)
have h2 := SamplerCoupling.prod_failure_le (factorCoupling n a) (factorCoupling n b)
have h3 := factorCoupling_failure ha
have h4 := factorCoupling_failure hb
have h5 := factorCoupling_failure hc
linarith
end LowSpaceSubsetSum
end
end OAI
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