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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 | import OAI.Combinatorics.BalancedRyser.Local.GlobalPartialAlignment
import OAI.Combinatorics.BalancedRyser.Local.ReuseTargetSum
namespace OAI
namespace Balanced.PartialAlignment
open TupleAlignment ConditionalRestriction
noncomputable section
attribute [local instance] Classical.propDecidable
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
def canonicalExternalAlignmentProbability (T : FourTuple F) (p : Point F)
(g : Direction F) (D : Finset (Direction F)) (a : MergedPairs F) : ℝ :=
(((offsetLaw F).filter {M | ∀ d, d ∈ D → M d = a d}
(canonical_exposure_supported D a)).toOuterMeasure
{M | (∀ i, OnLine (tuplePoint M T i) (lineThrough g p)) ∧
∀ i, tuplePoint M T i ≠ p}).toReal
theorem canonicalExternalAlignmentProbability_eq (T : FourTuple F)
(hT : Function.Injective T.generator) (p : Point F) (g : Direction F)
(D : Finset (Direction F)) (a : MergedPairs F) :
canonicalExternalAlignmentProbability T p g D a =
externalAlignmentProbability T.generator (adjacent_ne_of_injective T hT)
T.side p g (exposedTupleIndices T D) (fun i => a (T.generator i)) := by
let hd := adjacent_ne_of_injective T hT
let E : Set (Fin 4 → Bool → F) :=
{o | (∀ i, OnLine (tuplePositionEquiv T.generator hd T.side o i)
(lineThrough g p)) ∧ ∀ i, tuplePositionEquiv T.generator hd T.side o i ≠ p}
have h := congrArg (fun law => (law.toOuterMeasure E).toReal)
(global_exposed_tuple_law T hT D a)
rw [PMF.toOuterMeasure_map_apply, partialOffsets_conditional_law,
PMF.toOuterMeasure_map_apply] at h
simpa only [canonicalExternalAlignmentProbability, offsetLaw_eq_uniform,
externalAlignmentProbability, externalAlignmentEvent, E, Set.preimage_ofPred_eq,
tuplePositionEquiv_apply, tuplePoint, constituent] using h
theorem sum_canonicalExternalAlignmentProbability_le (T : FourTuple F)
(hT : Function.Injective T.generator) (p : Point F)
(D : Finset (Direction F)) (a : MergedPairs F) :
(∑ g ∈ (Finset.univ.filter (fun g : Direction F => ∀ i, T.generator i ≠ g)),
canonicalExternalAlignmentProbability T p g D a) ≤
(if (fixedPositions (exposedTupleIndices T D)).card = 0 then
(Fintype.card F : ℝ) + 1 else 1) *
((Fintype.card F : ℝ)⁻¹ ^ (4 - (fixedPositions (exposedTupleIndices T D)).card)) := by
simp_rw [canonicalExternalAlignmentProbability_eq T hT]
exact sum_externalAlignmentProbability_le T.generator
(adjacent_ne_of_injective T hT) T.side p (exposedTupleIndices T D)
(fun i => a (T.generator i))
end
end Balanced.PartialAlignment
end OAI
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