File size: 3,104 Bytes
f0f05de | 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 | 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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