openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /ExposedWitnessProbability.lean
| 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 | |