| import OAI.Combinatorics.BalancedRyser.Local.PartialAlignment |
| import OAI.Combinatorics.BalancedRyser.Probability.ConditionalRestriction |
| import OAI.Combinatorics.BalancedRyser.Probability.GlobalTupleLaw |
|
|
| 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 exposedTupleIndices (T : FourTuple F) (D : Finset (Direction F)) : Finset (Fin 4) := |
| Finset.univ.filter (fun i => T.generator i ∈ D) |
|
|
| theorem offsetLaw_eq_uniform : offsetLaw F = PMF.uniformOfFintype (MergedPairs F) := by |
| ext M |
| simp only [offsetLaw, PMF.uniformOfFintype_apply, Fintype.card_fun, |
| Fintype.card_option, Fintype.card_bool] |
|
|
| theorem global_exposed_tuple_law (T : FourTuple F) |
| (hT : Function.Injective T.generator) (D : Finset (Direction F)) (a : MergedPairs F) : |
| ((PMF.uniformOfFintype (MergedPairs F)).filter |
| {M | ∀ d, d ∈ D → M d = a d} (fixed_supported (fun d => d ∈ D) a)).map |
| (fun M i => M (T.generator i)) = |
| (PMF.uniformOfFintype (Fin 4 → Bool → F)).filter |
| {o | ∀ i ∈ exposedTupleIndices T D, o i = a (T.generator i)} |
| (partialOffsets_supported (exposedTupleIndices T D) (fun i => a (T.generator i))) := by |
| simpa only [exposedTupleIndices, Finset.mem_filter, Finset.mem_univ, true_and, |
| Function.Embedding.coeFn_mk] using |
| uniform_conditional_restrict (A := Bool → F) ⟨T.generator, hT⟩ |
| (fun d => d ∈ D) a |
|
|
| theorem global_conditional_alignment_probability_le (T : FourTuple F) |
| (hT : Function.Injective T.generator) (g : Direction F) |
| (hg : ∀ i, T.generator i ≠ g) (c : F) |
| (D : Finset (Direction F)) (a : MergedPairs F) : |
| (((PMF.uniformOfFintype (MergedPairs F)).filter |
| {M | ∀ d, d ∈ D → M d = a d} (fixed_supported (fun d => d ∈ D) a)).toOuterMeasure |
| {M | ∀ i, OnLine (tuplePoint M T i) (g,c)}).toReal ≤ |
| (Fintype.card F : ℝ)⁻¹ ^ (4 - (fixedPositions (exposedTupleIndices T D)).card) := 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) (g,c)} |
| have h := congrArg (fun law => (law.toOuterMeasure E).toReal) |
| (global_exposed_tuple_law T hT D a) |
| rw [PMF.toOuterMeasure_map_apply] at h |
| calc |
| _ = (((PMF.uniformOfFintype (Fin 4 → Bool → F)).filter |
| {o | ∀ i ∈ exposedTupleIndices T D, o i = a (T.generator i)} |
| (partialOffsets_supported (exposedTupleIndices T D) |
| (fun i => a (T.generator i)))).toOuterMeasure E).toReal := h |
| _ ≤ _ := original_conditional_alignment_probability_le |
| T.generator hd T.side g hg c (exposedTupleIndices T D) (fun i => a (T.generator i)) |
|
|
| theorem canonical_exposure_supported (D : Finset (Direction F)) (a : MergedPairs F) : |
| ∃ M ∈ {M : MergedPairs F | ∀ d, d ∈ D → M d = a d}, |
| M ∈ (offsetLaw F).support := by |
| rw [offsetLaw_eq_uniform] |
| exact fixed_supported (fun d => d ∈ D) a |
|
|
| theorem canonical_conditional_alignment_probability_le (T : FourTuple F) |
| (hT : Function.Injective T.generator) (g : Direction F) |
| (hg : ∀ i, T.generator i ≠ g) (c : F) |
| (D : Finset (Direction F)) (a : MergedPairs F) : |
| (((offsetLaw F).filter {M | ∀ d, d ∈ D → M d = a d} |
| (canonical_exposure_supported D a)).toOuterMeasure |
| {M | ∀ i, OnLine (tuplePoint M T i) (g,c)}).toReal ≤ |
| (Fintype.card F : ℝ)⁻¹ ^ (4 - (fixedPositions (exposedTupleIndices T D)).card) := by |
| simpa only [offsetLaw_eq_uniform] using |
| global_conditional_alignment_probability_le T hT g hg c D a |
|
|
| end |
| end Balanced.PartialAlignment |
|
|
| end OAI |
|
|
