| import OAI.Combinatorics.BalancedRyser.Incidence.AffineInjectionGuards |
| import OAI.Combinatorics.BalancedRyser.Local.AlignedTokenPointLaw |
|
|
| namespace OAI |
|
|
| namespace Balanced.InjectionExposure |
|
|
| noncomputable section |
| open scoped Classical |
|
|
| variable {I J Z : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] |
|
|
| theorem auxiliaryLaw_fst_mem_completions (K : Finset I) (base : I ↪ J) |
| (noise : (I ↪ J) → PMF Z) (x : (I ↪ J) × Z) |
| (hx : x ∈ (auxiliaryLaw K base noise).support) : x.1 ∈ completions K base := by |
| change x ∈ ((PMF.uniformOfFinset (completions K base) |
| (completions_nonempty K base)).bind |
| (fun e => (noise e).map (fun z => (e,z)))).support at hx |
| rw [PMF.mem_support_bind_iff] at hx |
| obtain ⟨e,he,hx⟩ := hx |
| rw [PMF.mem_support_map_iff] at hx |
| obtain ⟨z,hz,hx⟩ := hx |
| have he' : e ∈ completions K base := by simpa using he |
| have hf := congrArg Prod.fst hx |
| simpa only [← hf] using he' |
|
|
| end |
| end Balanced.InjectionExposure |
|
|
| 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] |
|
|
| def pointOutput (P : EqualityPattern b κ) (bits : Bits b) (p : Point F) |
| (i : Fin b) (J : Finset (Fin 4)) (hJ : J.card ≤ 1) |
| (a : Z → Fin 4 → Bool → F) |
| (x : Z → {j : Fin 4 // j ∈ freePositions J} → F) |
| (e : DirectionToken b κ ↪ Direction F) (z : Z) (j : Fin 4) : Point F := |
| generatedPositions (tupleDirections P e i) (tupleDirections_adjacent P e i) |
| (bits i) (e (Sum.inl i)) (tupleDirections_ne_target P e i) |
| (lineEval (e (Sum.inl i)) p) J hJ (a z) (x z) j |
|
|
| theorem projected_left_probability_le |
| (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)) (base : DirectionToken b κ ↪ Direction F) |
| (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) : |
| ((auxiliaryLaw K base noise).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 ≤ |
| 1 / ((Fintype.card F + 1 - K.card : ℕ) : ℝ) := by |
| let l : Z → Line F := fun z => (base (Sum.inr (P.classOf i j)), a z j (bits i j)) |
| have h := auxiliary_guarded_projection_probability_le K base (Sum.inl i) htarget |
| noise hnoise (fun _ => p) l m |
| apply le_trans _ h |
| apply ENNReal.toReal_mono |
| · rw [PMF.toOuterMeasure_apply] |
| exact PMF.tsum_coe_indicator_ne_top _ _ |
| · apply PMF.toOuterMeasure_mono |
| intro ez hez |
| have he := (mem_completions K base ez.1).mp |
| (auxiliaryLaw_fst_mem_completions K base noise ez hez.2) |
| have hd : tupleDirections P ez.1 i j = base (Sum.inr (P.classOf i j)) := |
| he _ hold |
| have hp : pointOutput P bits p i J hJ a x ez.1 ez.2 j = |
| lineIntersection (l ez.2) (lineThrough (ez.1 (Sum.inl i)) p) := by |
| change generatedPositions _ _ _ _ _ _ _ _ _ _ j = _ |
| rw [generatedPositions_old_left _ _ _ _ _ _ J hJ _ _ j hj] |
| rw [hd] |
| rfl |
| have hne : ez.1 (Sum.inl i) ≠ (l ez.2).1 := by |
| change ez.1 (Sum.inl i) ≠ base (Sum.inr (P.classOf i j)) |
| rw [← hd] |
| exact (tupleDirections_ne_target P ez.1 i j).symm |
| exact ⟨by simpa only [l, hd] using hez.1.1, hne, |
| by simpa only [hp] using hez.1.2.1, |
| by simpa only [hp] using hez.1.2.2⟩ |
|
|
| theorem projected_right_probability_le |
| (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)) (base : DirectionToken b κ ↪ Direction F) |
| (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) : |
| ((auxiliaryLaw K base noise).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 ≤ |
| 1 / ((Fintype.card F + 1 - K.card : ℕ) : ℝ) := by |
| let l : Z → Line F := fun z => |
| (base (Sum.inr (P.classOf i (j+1))), a z (j+1) (!(bits i (j+1)))) |
| have h := auxiliary_guarded_projection_probability_le K base (Sum.inl i) htarget |
| noise hnoise (fun _ => p) l m |
| apply le_trans _ h |
| apply ENNReal.toReal_mono |
| · rw [PMF.toOuterMeasure_apply] |
| exact PMF.tsum_coe_indicator_ne_top _ _ |
| · apply PMF.toOuterMeasure_mono |
| intro ez hez |
| have he := (mem_completions K base ez.1).mp |
| (auxiliaryLaw_fst_mem_completions K base noise ez hez.2) |
| have hd : tupleDirections P ez.1 i (j+1) = base (Sum.inr (P.classOf i (j+1))) := |
| he _ hold |
| have hp : pointOutput P bits p i J hJ a x ez.1 ez.2 j = |
| lineIntersection (l ez.2) (lineThrough (ez.1 (Sum.inl i)) p) := by |
| change generatedPositions _ _ _ _ _ _ _ _ _ _ j = _ |
| rw [generatedPositions_old_right _ _ _ _ _ _ J hJ _ _ j hj] |
| rw [hd] |
| rfl |
| have hne : ez.1 (Sum.inl i) ≠ (l ez.2).1 := by |
| change ez.1 (Sum.inl i) ≠ base (Sum.inr (P.classOf i (j+1))) |
| rw [← hd] |
| exact (tupleDirections_ne_target P ez.1 i (j+1)).symm |
| exact ⟨by simpa only [l, hd] using hez.1.1, hne, |
| by simpa only [hp] using hez.1.2.1, |
| by simpa only [hp] using hez.1.2.2⟩ |
|
|
| theorem 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)) (base : DirectionToken b κ ↪ Direction F) |
| (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) : |
| ((auxiliaryLaw K base noise).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 |
| exact (projected_left_probability_le P bits p i J hJ j hj K base |
| htarget hold noise hnoise a x m).trans |
| (inverse_remaining_le_two_div (Fintype.card F) K.card b |
| Fintype.card_pos hK hb) |
|
|
| theorem 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)) (base : DirectionToken b κ ↪ Direction F) |
| (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) : |
| ((auxiliaryLaw K base noise).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 |
| exact (projected_right_probability_le P bits p i J hJ j hj K base |
| htarget hold noise hnoise a x m).trans |
| (inverse_remaining_le_two_div (Fintype.card F) K.card b |
| Fintype.card_pos hK hb) |
|
|
| end |
| end Balanced.Local.TokenExposure |
|
|
| end OAI |
|
|
