File size: 3,737 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 | 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
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