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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 | import OAI.Combinatorics.BalancedRyser.Candidates.CandidateExposedWitness
import OAI.Combinatorics.BalancedRyser.Probability.FreshTupleConstraints
namespace OAI
namespace Balanced.ExposedWitnessProbability
open CandidateValidityWitness CandidateExposedWitness FreshWitnessProbability
open FreshTupleConstraints
open scoped BigOperators
noncomputable section
attribute [local instance] Classical.propDecidable
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
omit [DecidableEq F] in
theorem exposedLaw_support_fixed (D : Finset (Direction F)) (a M : MergedPairs F)
(hM : M ∈ (exposedLaw D a).support) (d : Direction F) (hd : d ∈ D) :
M d = a d := by
exact ((PMF.mem_support_filter_iff _).mp hM).1 d hd
theorem real_probability_mono_support {Ω : Type*} (law : PMF Ω) {E G : Set Ω}
(h : E ∩ law.support ⊆ G) :
(law.toOuterMeasure E).toReal ≤ (law.toOuterMeasure G).toReal := by
apply ENNReal.toReal_mono
· rw [PMF.toOuterMeasure_apply]
exact law.tsum_coe_indicator_ne_top _
· exact law.toOuterMeasure_mono h
theorem successful_opposite_probability_le
(D : Finset (Direction F)) (a : MergedPairs F) (T : FourTuple F)
(hT : Function.Injective T.generator) (hfresh : ∀ i, T.generator i ∉ D)
(S : Line F) (d : Direction F) (hdD : d ∈ D) (hd : d ≠ S.1)
(i : Fin 4) :
((exposedLaw D a).toOuterMeasure {M | SuccessfulOn M T S ∧
PairWitness M d (tuplePoint M T i) (tuplePoint M T (i + 2))}).toReal ≤
2 * (Fintype.card F : ℝ)⁻¹ ^ 6 := by
let E (s : Bool) : Set (MergedPairs F) := {M |
(∀ k, OnLine (tuplePoint M T k) S) ∧
tuplePoint M T i = forcedPoint a S d s ∧
tuplePoint M T (i + 2) = forcedPoint a S d (!s)}
have hij : i ≠ i + 2 := by fin_cases i <;> decide
calc
_ ≤ ((exposedLaw D a).toOuterMeasure {M | ∃ s, M ∈ E s}).toReal := by
apply real_probability_mono_support
rintro M ⟨⟨hS, hw⟩, hM⟩
obtain ⟨s, hs, ht⟩ := (successful_opposite_iff_fixed_positions hS hd
(exposedLaw_support_fixed D a M hM d hdD) i).mp hw
exact ⟨s, hS.2.2.2, hs, ht⟩
_ ≤ ∑ s : Bool, ((exposedLaw D a).toOuterMeasure (E s)).toReal :=
probability_exists_le_sum _ _
_ = _ := by
have hE (s : Bool) : ((exposedLaw D a).toOuterMeasure (E s)).toReal =
(Fintype.card F : ℝ)⁻¹ ^ 6 :=
two_pinned_positions_probability D a T hT hfresh S i (i + 2) hij
_ _ (forcedPoint_on_target a S d hd s)
(forcedPoint_on_target a S d hd (!s))
simp only [hE, Finset.sum_const, Finset.card_univ, Fintype.card_bool,
nsmul_eq_mul, Nat.cast_ofNat]
theorem successful_external_probability_le
(D : Finset (Direction F)) (a : MergedPairs F) (T : FourTuple F)
(hT : Function.Injective T.generator) (hfresh : ∀ i, T.generator i ∉ D)
(S : Line F) (d : Direction F) (hdD : d ∈ D) (hd : d ≠ S.1)
(i : Fin 4) (R : MergedPairs F → Point F → Prop) :
((exposedLaw D a).toOuterMeasure {M | SuccessfulOn M T S ∧
∃ x, OnLine x S ∧ PairWitness M d x (tuplePoint M T i) ∧ R M x}).toReal ≤
2 * (Fintype.card F : ℝ)⁻¹ ^ 5 := by
let E (s : Bool) : Set (MergedPairs F) := {M |
(∀ k, OnLine (tuplePoint M T k) S) ∧
tuplePoint M T i = forcedPoint a S d (!s)}
calc
_ ≤ ((exposedLaw D a).toOuterMeasure {M | ∃ s, M ∈ E s}).toReal := by
apply real_probability_mono_support
rintro M ⟨⟨hS, hw⟩, hM⟩
obtain ⟨s, hs, _⟩ := (exists_external_iff_forcedPoint hd
(exposedLaw_support_fixed D a M hM d hdD) (hS.2.2.2 i) (R M)).mp hw
exact ⟨s, hS.2.2.2, hs⟩
_ ≤ ∑ s : Bool, ((exposedLaw D a).toOuterMeasure (E s)).toReal :=
probability_exists_le_sum _ _
_ = _ := by
have hE (s : Bool) : ((exposedLaw D a).toOuterMeasure (E s)).toReal =
(Fintype.card F : ℝ)⁻¹ ^ 5 :=
one_pinned_position_probability D a T hT hfresh S i _
(forcedPoint_on_target a S d hd (!s))
simp only [hE, Finset.sum_const, Finset.card_univ, Fintype.card_bool,
nsmul_eq_mul, Nat.cast_ofNat]
end
end Balanced.ExposedWitnessProbability
end OAI
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