| import OAI.Combinatorics.BalancedRyser.Local.AlignedListProbability |
| import OAI.Combinatorics.BalancedRyser.Probability.EqualityPatternOrdering |
|
|
| namespace OAI |
|
|
| namespace Balanced.Local |
|
|
| open AlignedListProbability |
| open scoped BigOperators |
|
|
| noncomputable section |
| attribute [local instance] Classical.propDecidable |
|
|
| universe u |
| variable {F : Type u} [Field F] [Fintype F] [DecidableEq F] |
|
|
| omit [Field F] [Fintype F] in |
|
|
| theorem smallReuseOrder_prescribed {b κ : ℕ} (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (ha : Function.Injective a.2) |
| (bits : Bits b) (l : List (Fin b)) (h : SmallReuseOrder P.classSet l) : |
| SmallReuseOrder (fun i => generatorSet (prescribedTuple P a bits i)) l := by |
| induction l with |
| | nil => trivial |
| | cons i l ih => |
| refine ⟨ih h.1, ?_⟩ |
| rw [P.prescribed_generator_overlap_card a ha bits i l.toFinset] |
| exact h.2 |
|
|
| theorem alignment_probability {b κ : ℕ} (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (a : DirectionAssignment F b κ) (ha : Eligible P a) |
| (bits : Bits b) (p : Point F) : |
| offsetProbability (Aligned P a bits p) = |
| 1 / (Fintype.card F : ℝ) ^ (4 * b) := by |
| obtain ⟨l, hnd, hfull, horder⟩ := P.exists_smallReuseOrder hκ |
| have hlen : l.length = b := by |
| rw [← List.toFinset_card_of_nodup hnd, hfull] |
| simp |
| have hall (i : Fin b) : i ∈ l := by |
| apply List.mem_toFinset.mp |
| rw [hfull] |
| exact Finset.mem_univ _ |
| have hgenerators (i : Fin b) : Function.Injective (prescribedTuple P a bits i).generator := |
| ha.2.1.comp (P.withinTuple i) |
| have htargets (i : Fin b) (j : Fin 4) : |
| (prescribedTuple P a bits i).generator j ≠ (lineThrough (a.1 i) p).1 := |
| ha.2.2 i j |
| have hprob := list_alignment_probability (prescribedTuple P a bits) |
| (fun i => lineThrough (a.1 i) p) l (fun i _ => hgenerators i) |
| (fun i _ => htargets i) (smallReuseOrder_prescribed P a ha.2.1 bits l horder) |
| have hevent : |
| {M | ListAligned (prescribedTuple P a bits) (fun i => lineThrough (a.1 i) p) l M} = |
| {M | Aligned P a bits p M} := by |
| ext M |
| constructor |
| · intro h ij |
| exact h ij.1 (hall ij.1) ij.2 |
| · intro h i _ j |
| exact h (i,j) |
| rw [hevent, hlen] at hprob |
| calc |
| offsetProbability (Aligned P a bits p) = |
| ((PMF.uniformOfFintype (MergedPairs F)).toOuterMeasure |
| {M | Aligned P a bits p M}).toReal := |
| (FiniteUniformFibers.probability_eq_sum (Aligned P a bits p)).symm |
| _ = ((Fintype.card F : ℝ)⁻¹ ^ 4) ^ b := hprob |
| _ = 1 / (Fintype.card F : ℝ) ^ (4 * b) := by |
| rw [← pow_mul, inv_pow, one_div] |
|
|
| theorem alignmentNormalization_proof : AlignmentNormalizationResult.{u} := by |
| intro F _ _ _ b κ _ _ hκ P a ha bits p |
| exact alignment_probability P hκ a ha bits p |
|
|
| theorem alignment_probability_pos {b κ : ℕ} (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (a : DirectionAssignment F b κ) (ha : Eligible P a) |
| (bits : Bits b) (p : Point F) : |
| 0 < offsetProbability (Aligned P a bits p) := by |
| rw [alignment_probability P hκ a ha bits p] |
| positivity |
|
|
| end |
| end Balanced.Local |
|
|
| end OAI |
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|
