| 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 | |