openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Counting /PrimeSamplerCoupling.lean
| 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 | |