File size: 4,856 Bytes
2f887db | 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 | import OAI.Computability.PerfectCompleteness.Decoding.LeftDecoder
import OAI.Computability.PerfectCompleteness.Foundations.TupleIndexEmitterLemmas
namespace OAI
namespace PerfectCompleteness.AffineDecoderMeeting
noncomputable section
open scoped BigOperators Classical
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Integration.BinaryLinear (F2)
section Product
variable {L R : Type*} [Fintype L] [Fintype R]
theorem product_probability_lower
(left : FiniteDistribution L) (right : FiniteDistribution R)
(event : L × R → Bool) (good : R → Bool) (p : ℝ)
(hpoint : ∀ y, good y = true →
p ≤ left.probability (fun x => event (x, y))) :
p * right.probability good ≤ (left.product right).probability event := by
rw [UsefulAdvice.product_probability_advice_first]
change p * (∑ y, if good y then right.weight y else 0) ≤
∑ y, right.weight y * left.probability (fun x => event (x, y))
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro y _
by_cases hy : good y = true
· rw [ite_eq_left hy, mul_comm p]
exact mul_le_mul_of_nonneg_left (hpoint y hy) (right.nonnegative y)
· rw [ite_eq_right hy, mul_zero]
exact mul_nonneg (right.nonnegative y) (left.probability_nonnegative _)
end Product
section Algebra
variable {H H' : Type*} [AddCommGroup H] [Module F2 H]
[AddCommGroup H'] [Module F2 H']
(i : H' →ₗ[F2] H) (B : Submodule F2 H)
(C : Submodule F2 (Module.Dual F2 (H ⧸ B)))
(center : Module.Dual F2 H)
def nativeQ : Submodule F2 (Module.Dual F2 H') :=
(PulledColumnSpace.pulled B C).map i.dualMap
def nativeCenter : Module.Dual F2 H' := i.dualMap center
theorem candidate_mem_nativeCoset (q : PulledColumnSpace.Candidates B C center) :
i.dualMap q.val - nativeCenter i center ∈ nativeQ i B C := by
change i.dualMap q.val - i.dualMap center ∈
(PulledColumnSpace.pulled B C).map i.dualMap
rw [← map_sub]
exact Submodule.mem_map_of_mem q.property
theorem exists_candidate_of_mem_nativeCoset (decoded : Module.Dual F2 H')
(hdecoded : decoded - nativeCenter i center ∈ nativeQ i B C) :
∃ q : PulledColumnSpace.Candidates B C center, i.dualMap q.val = decoded := by
obtain ⟨q, hq, heq⟩ := Submodule.mem_map.mp hdecoded
refine ⟨PulledColumnSpace.candidateEquiv B C center ⟨q, hq⟩, ?_⟩
change i.dualMap (center + q) = decoded
rw [map_add, heq]
change i.dualMap center + (decoded - i.dualMap center) = decoded
abel
theorem nativeCoset_iff_exists_candidate (decoded : Module.Dual F2 H') :
decoded - nativeCenter i center ∈ nativeQ i B C ↔
∃ q : PulledColumnSpace.Candidates B C center, i.dualMap q.val = decoded := by
constructor
· exact exists_candidate_of_mem_nativeCoset i B C center decoded
· rintro ⟨q, rfl⟩
exact candidate_mem_nativeCoset i B C center q
theorem finrank_nativeQ_le [FiniteDimensional F2 H] :
Module.finrank F2 (nativeQ i B C) ≤ Module.finrank F2 C := by
calc
Module.finrank F2 (nativeQ i B C) ≤
Module.finrank F2 (PulledColumnSpace.pulled B C) :=
Submodule.finrank_map_le i.dualMap (PulledColumnSpace.pulled B C)
_ = Module.finrank F2 C := PulledColumnSpace.finrank_pulled B C
end Algebra
variable {H H' : Type*} [AddCommGroup H] [Module F2 H]
[AddCommGroup H'] [Module F2 H']
[FiniteDimensional F2 H] [FiniteDimensional F2 H']
local instance leftDualFintype : Fintype (Module.Dual F2 H) :=
LeftDecoder.dualFintype (V := H)
local instance nativeDualFintype : Fintype (Module.Dual F2 H') :=
LeftDecoder.dualFintype (V := H')
variable (i : H' →ₗ[F2] H) (B : Submodule F2 H)
(C : Submodule F2 (Module.Dual F2 (H ⧸ B)))
(center : Module.Dual F2 H)
omit [FiniteDimensional F2 H'] in
theorem native_point_probability_lower {r : Nat}
(hC : Module.finrank F2 C ≤ r) (decoded : Module.Dual F2 H')
(hdecoded : decoded - nativeCenter i center ∈ nativeQ i B C) :
1 / (2 : ℝ) ^ r ≤ (LeftDecoder.affineLaw B C center).probability
(fun q => decide (i.dualMap q = decoded)) := by
apply LeftDecoder.affineLaw_success_lower B C center hC
obtain ⟨q, hq⟩ := exists_candidate_of_mem_nativeCoset i B C center decoded hdecoded
exact ⟨q, decide_eq_true hq⟩
theorem meeting_probability_lower {r : Nat}
(hC : Module.finrank F2 C ≤ r)
(right : FiniteDistribution (Module.Dual F2 H')) :
(1 / (2 : ℝ) ^ r) * right.probability
(fun decoded => decide (decoded - nativeCenter i center ∈ nativeQ i B C)) ≤
((LeftDecoder.affineLaw B C center).product right).probability
(fun pair => decide (i.dualMap pair.1 = pair.2)) := by
apply product_probability_lower
intro decoded hdecoded
exact native_point_probability_lower i B C center hC decoded (of_decide_eq_true hdecoded)
end
end PerfectCompleteness.AffineDecoderMeeting
end OAI
|