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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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | 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
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