| import OAI.NumberTheory.TwoPoint.Bounds.ActualSlotClasses | |
| import OAI.NumberTheory.TwoPoint.Bounds.PositiveWordProbability | |
| namespace OAI | |
| /-! Actual code assignments preserve prime pools and the distinct-prime weight. -/ | |
| namespace TwoPointCorrelations | |
| open Finset | |
| open scoped Classical | |
| variable {R : ℕ} | |
| lemma actualSlotPrime_mem_tuplePool (w : Fin R → SignedStep) (P : Finset ℕ) | |
| (hP : ∀ i, (w i).tuple.primeFactors ⊆ P) (s : ActualPrimeSlot w) | |
| (hs : actualSlotKind w s = false) : actualSlotPrime w s ∈ P := by | |
| rcases s with ⟨i, p | p⟩ | |
| · exact hP i p.property | |
| · cases hs | |
| lemma actualSlotPrime_mem_paddingPool (w : Fin R → SignedStep) (Q : Finset ℕ) | |
| (hQ : ∀ i, (w i).padding.primeFactors ⊆ Q) (s : ActualPrimeSlot w) | |
| (hs : actualSlotKind w s = true) : actualSlotPrime w s ∈ Q := by | |
| rcases s with ⟨i, p | p⟩ | |
| · cases hs | |
| · exact hQ i p.property | |
| lemma actualPrimeSlotData_prime_image (w : Fin R → SignedStep) : | |
| univ.image (actualPrimeSlotData w).prime = univ.image (actualSlotPrime w) := by | |
| ext p | |
| constructor | |
| · rintro hp | |
| obtain ⟨i, _, rfl⟩ := mem_image.mp hp | |
| exact mem_image.mpr ⟨(actualPrimeSlotEquiv w).symm i, mem_univ _, rfl⟩ | |
| · rintro hp | |
| obtain ⟨s, _, rfl⟩ := mem_image.mp hp | |
| refine mem_image.mpr ⟨actualPrimeSlotEquiv w s, mem_univ _, ?_⟩ | |
| change actualSlotPrime w ((actualPrimeSlotEquiv w).symm (actualPrimeSlotEquiv w s)) = _ | |
| rw [Equiv.symm_apply_apply] | |
| lemma actualPrimeSlot_support (w : Fin R → SignedStep) | |
| (ht : ∀ i, (w i).tuple ≠ 0) (hq : ∀ i, (w i).padding ≠ 0) : | |
| wordDivisorPrimeSupport (List.ofFn w) = univ.image (actualSlotPrime w) := by | |
| ext p | |
| simp only [wordDivisorPrimeSupport, mem_biUnion, List.mem_toFinset, List.mem_ofFn] | |
| constructor | |
| · rintro ⟨a, ⟨i, rfl⟩, hp⟩ | |
| rw [Nat.primeFactors_mul (hq i) (ht i), mem_union] at hp | |
| rcases hp with hp | hp | |
| · exact mem_image.mpr ⟨⟨i, Sum.inr ⟨p, hp⟩⟩, mem_univ _, rfl⟩ | |
| · exact mem_image.mpr ⟨⟨i, Sum.inl ⟨p, hp⟩⟩, mem_univ _, rfl⟩ | |
| · rintro hp | |
| obtain ⟨⟨i, p | p⟩, _, rfl⟩ := mem_image.mp hp | |
| · refine ⟨w i, ⟨i, rfl⟩, ?_⟩ | |
| rw [Nat.primeFactors_mul (hq i) (ht i), mem_union] | |
| exact Or.inr p.property | |
| · refine ⟨w i, ⟨i, rfl⟩, ?_⟩ | |
| rw [Nat.primeFactors_mul (hq i) (ht i), mem_union] | |
| exact Or.inl p.property | |
| /-- The encoded assignment pays exactly one reciprocal per distinct prime. -/ | |
| theorem actual_code_reciprocal_weight (w : Fin R → SignedStep) | |
| (ht : ∀ i, (w i).tuple ≠ 0) (hq : ∀ i, (w i).padding ≠ 0) : | |
| (∏ z : (actualPrimeSlotData w).code.usedClasses, | |
| ((actualPrimeSlotData w).prime z.val : ℝ)⁻¹) = | |
| ∏ p ∈ wordDivisorPrimeSupport (List.ofFn w), (p : ℝ)⁻¹ := by | |
| rw [actualPrimeSlot_support w ht hq, ← actualPrimeSlotData_prime_image] | |
| rw [← PrimeSlotData.code_value_image, prod_image] | |
| exact fun z _ t _ h => (actualPrimeSlotData w).code_value_injective h | |
| /-- Actual tuple and padding classes inhabit their separate numerical pools. -/ | |
| theorem actual_code_pool_assignments (w : Fin R → SignedStep) (P Q : Finset ℕ) | |
| (hP : ∀ i, (w i).tuple.primeFactors ⊆ P) | |
| (hQ : ∀ i, (w i).padding.primeFactors ⊆ Q) : | |
| ∃ a : (actualPrimeSlotData w).code.tupleClasses → P, | |
| ∃ b : {z : (actualPrimeSlotData w).code.usedClasses // | |
| z ∉ (actualPrimeSlotData w).code.tupleClasses} → Q, | |
| joinCoordinates (actualPrimeSlotData w).code.tupleClasses | |
| (fun z => (a z).val) (fun z => (b z).val) = | |
| fun z => (actualPrimeSlotData w).prime z.val := by | |
| let c := (actualPrimeSlotData w).code | |
| have htuple (z : c.tupleClasses) : (actualPrimeSlotData w).prime z.val.val ∈ P := by | |
| apply actualSlotPrime_mem_tuplePool w P hP | |
| ((actualPrimeSlotEquiv w).symm z.val.val) | |
| exact (mem_filter.mp z.property).2 | |
| have hpadding (z : {z : c.usedClasses // z ∉ c.tupleClasses}) : | |
| (actualPrimeSlotData w).prime z.val.val ∈ Q := by | |
| apply actualSlotPrime_mem_paddingPool w Q hQ | |
| ((actualPrimeSlotEquiv w).symm z.val.val) | |
| have hn : ¬c.2.2.2 z.val.val = false := by | |
| intro hz | |
| exact z.property (mem_filter.mpr ⟨mem_univ _, hz⟩) | |
| change c.2.2.2 z.val.val = true | |
| cases he : c.2.2.2 z.val.val <;> simp_all | |
| refine ⟨(fun z => ⟨_, htuple z⟩), (fun z => ⟨_, hpadding z⟩), ?_⟩ | |
| funext z | |
| dsimp only [joinCoordinates] | |
| split_ifs <;> rfl | |
| end TwoPointCorrelations | |
| end OAI | |