| import OAI.Combinatorics.BalancedRyser.Local.AlignedProjectedExposure |
| import OAI.Combinatorics.BalancedRyser.Local.InjectionExposureMixture |
|
|
| namespace OAI |
|
|
| namespace Balanced.Local.TokenExposure |
| open AlignedListProbability PartialAlignment InjectionExposure |
| noncomputable section |
| open scoped Classical |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
| variable {b κ : ℕ} {Z : Type*} [Fintype Z] |
| variable [Nonempty (DirectionToken b κ ↪ Direction F)] |
|
|
| theorem original_projected_left_probability_le_two_div |
| (P : EqualityPattern b κ) (bits : Bits b) (p : Point F) (i : Fin b) |
| (J : Finset (Fin 4)) (hJ : J.card ≤ 1) (j : Fin 4) (hj : j ∈ J) |
| (K : Finset (DirectionToken b κ)) |
| (htarget : Sum.inl i ∉ K) (hold : Sum.inr (P.classOf i j) ∈ K) |
| (noise : (DirectionToken b κ ↪ Direction F) → PMF Z) |
| (hnoise : DependsOnExposed K noise) |
| (a : Z → Fin 4 → Bool → F) |
| (x : Z → {j : Fin 4 // j ∈ freePositions J} → F) |
| (m : Z → Line F) (hK : K.card ≤ 5 * b) |
| (hb : 10 * b ≤ Fintype.card F) : |
| (((PMF.uniformOfFintype (DirectionToken b κ ↪ Direction F)).bind |
| (fun e => (noise e).map (fun z => (e,z)))).toOuterMeasure |
| {ez | (tupleDirections P ez.1 i j, a ez.2 j (bits i j)) ≠ m ez.2 ∧ |
| pointOutput P bits p i J hJ a x ez.1 ez.2 j ≠ p ∧ |
| OnLine (pointOutput P bits p i J hJ a x ez.1 ez.2 j) (m ez.2)}).toReal ≤ |
| 2 / (Fintype.card F : ℝ) := by |
| apply original_probability_le_of_auxiliary K noise |
| intro base |
| exact projected_left_probability_le_two_div P bits p i J hJ j hj K base |
| htarget hold noise hnoise a x m hK hb |
|
|
| theorem original_projected_right_probability_le_two_div |
| (P : EqualityPattern b κ) (bits : Bits b) (p : Point F) (i : Fin b) |
| (J : Finset (Fin 4)) (hJ : J.card ≤ 1) (j : Fin 4) (hj : j+1 ∈ J) |
| (K : Finset (DirectionToken b κ)) |
| (htarget : Sum.inl i ∉ K) (hold : Sum.inr (P.classOf i (j+1)) ∈ K) |
| (noise : (DirectionToken b κ ↪ Direction F) → PMF Z) |
| (hnoise : DependsOnExposed K noise) |
| (a : Z → Fin 4 → Bool → F) |
| (x : Z → {j : Fin 4 // j ∈ freePositions J} → F) |
| (m : Z → Line F) (hK : K.card ≤ 5 * b) |
| (hb : 10 * b ≤ Fintype.card F) : |
| (((PMF.uniformOfFintype (DirectionToken b κ ↪ Direction F)).bind |
| (fun e => (noise e).map (fun z => (e,z)))).toOuterMeasure |
| {ez | (tupleDirections P ez.1 i (j+1), a ez.2 (j+1) (!(bits i (j+1)))) ≠ m ez.2 ∧ |
| pointOutput P bits p i J hJ a x ez.1 ez.2 j ≠ p ∧ |
| OnLine (pointOutput P bits p i J hJ a x ez.1 ez.2 j) (m ez.2)}).toReal ≤ |
| 2 / (Fintype.card F : ℝ) := by |
| apply original_probability_le_of_auxiliary K noise |
| intro base |
| exact projected_right_probability_le_two_div P bits p i J hJ j hj K base |
| htarget hold noise hnoise a x m hK hb |
|
|
| end |
| end Balanced.Local.TokenExposure |
|
|
| end OAI |
|
|
