openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Local /AlignedProjectedCollinearTest.lean
| import OAI.Combinatorics.BalancedRyser.Incidence.AlignedGlobalProjectedIncidence | |
| import OAI.Combinatorics.BalancedRyser.Local.LocalInjectionFreeLaw | |
| namespace OAI | |
| namespace Balanced.Local.AlignedGlobal | |
| open RootChildPartition UsedEventUnion | |
| noncomputable section | |
| open scoped Classical | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| variable {b κ : ℕ} | |
| theorem dependent_currentOld_cases (P : EqualityPattern b κ) | |
| (R : Finset (Fin b)) (k : Fin b × Fin 4) | |
| (hk : k ∈ dependentOccurrences P R) : | |
| k.2 ∈ currentOldIndices P R k.1 ∨ | |
| k.2 + 1 ∈ currentOldIndices P R k.1 := by | |
| rcases (mem_dependentOccurrences P R k).mp hk with ⟨hn, hl | hr⟩ | |
| · exact Or.inl (by simpa [currentOldIndices, hn, rootGeneratorIndices] using hl) | |
| · exact Or.inr (by simpa [currentOldIndices, hn, rootGeneratorIndices] using hr) | |
| theorem geometricCollinearTest_projected_le | |
| (P : EqualityPattern b κ) (hκ : κ ≤ 2) | |
| (R S : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) | |
| (hS : S ⊆ R ∨ R ⊆ S) (bits : Bits b) (p : Point F) | |
| (i j k : Fin b × Fin 4) (hi : i.1 ∈ S) (hj : j.1 ∈ S) | |
| (hdep : k ∈ dependentOccurrences P R) | |
| (hJ : (currentOldIndices P R k.1).card ≤ 1) | |
| (hOld : ∀ t ∈ currentOldIndices P R k.1, | |
| P.classOf k.1 t ∈ S.biUnion P.classSet) | |
| (K : Finset (DirectionToken b κ)) | |
| (hKt : ∀ r ∈ S, Sum.inl r ∈ K) | |
| (hKg : ∀ c ∈ S.biUnion P.classSet, Sum.inr c ∈ K) | |
| (htarget : Sum.inl k.1 ∉ K) | |
| (hK : K.card ≤ 5*b) (hb : 10*b ≤ Fintype.card F) | |
| [Nonempty (DirectionToken b κ ↪ Direction F)] : | |
| injectionFreeProbability P R (fun e x => | |
| geometricCollinearTest P (tokenInjectionEquiv F b κ e).1 bits p i j k | |
| (injectedGenerated P hκ R hD bits p e x)) ≤ | |
| 2 / (Fintype.card F : ℝ) := by | |
| let m : ProjectionData (F := F) P R S → Line F := fun z => | |
| joiningLine (z.2.1 ⟨i.1,hi⟩ i.2) (z.2.1 ⟨j.1,hj⟩ j.2) | |
| have hmono (c : Fin (4*b-κ)) (s : Bool) : | |
| injectionFreeProbability P R (fun e x => | |
| geometricCollinearTest P (tokenInjectionEquiv F b κ e).1 bits p i j k | |
| (injectedGenerated P hκ R hD bits p e x)) ≤ | |
| ((PMF.uniformOfFintype ((DirectionToken b κ ↪ Direction F) × | |
| (FreeIndex P R → F))).toOuterMeasure | |
| {ex | let a := tokenInjectionEquiv F b κ ex.1 | |
| let M := generatedZero P hκ R hD a.1 | |
| (allTokensDistinct_eligible P a.1 a.2) bits p ex.2 | |
| let z := projectionObservation P hκ R S hD bits p ex.1 ex.2 | |
| constituent M (ex.1 (Sum.inr c)) s ≠ m z ∧ | |
| prescribedPoint P a.1 bits M k ≠ p ∧ | |
| OnLine (prescribedPoint P a.1 bits M k) (m z)}).toReal := by | |
| unfold injectionFreeProbability | |
| apply ENNReal.toReal_mono | |
| · rw [PMF.toOuterMeasure_apply] | |
| exact PMF.tsum_coe_indicator_ne_top _ _ | |
| · apply (PMF.uniformOfFintype _).toOuterMeasure.mono | |
| intro ex hex | |
| rcases hex with ⟨hg, hij, hik, hjk, hxy, hxz, hyz, hinc, ht, hu⟩ | |
| refine ⟨?_, hg.2.2.1 k, ?_⟩ | |
| · exact (hu c s).symm | |
| · exact hinc | |
| rcases dependent_currentOld_cases P R k hdep with hl | hr | |
| · exact (hmono (P.classOf k.1 k.2) (bits k.1 k.2)).trans | |
| (global_projected_left_probability_le_two_div P hκ R S hD hS bits p | |
| k.1 hJ hOld k.2 hl K hKt hKg htarget m hK hb) | |
| · exact (hmono (P.classOf k.1 (k.2+1)) (!(bits k.1 (k.2+1)))).trans | |
| (global_projected_right_probability_le_two_div P hκ R S hD hS bits p | |
| k.1 hJ hOld k.2 hr K hKt hKg htarget m hK hb) | |
| theorem collinearTest_injectionAverage_projected_le | |
| (P : EqualityPattern b κ) (hκ : κ ≤ 2) | |
| (R S : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) | |
| (hS : S ⊆ R ∨ R ⊆ S) (bits : Bits b) (p : Point F) | |
| (i j k : Fin b × Fin 4) (hi : i.1 ∈ S) (hj : j.1 ∈ S) | |
| (hdep : k ∈ dependentOccurrences P R) | |
| (hJ : (currentOldIndices P R k.1).card ≤ 1) | |
| (hOld : ∀ t ∈ currentOldIndices P R k.1, | |
| P.classOf k.1 t ∈ S.biUnion P.classSet) | |
| (K : Finset (DirectionToken b κ)) | |
| (hKt : ∀ r ∈ S, Sum.inl r ∈ K) | |
| (hKg : ∀ c ∈ S.biUnion P.classSet, Sum.inr c ∈ K) | |
| (htarget : Sum.inl k.1 ∉ K) | |
| (hK : K.card ≤ 5*b) (hb : 10*b ≤ Fintype.card F) | |
| [Nonempty (DirectionToken b κ ↪ Direction F)] : | |
| injectionAverage P bits p (fun a => collinearTest P a bits p i j k) ≤ | |
| 2 / (Fintype.card F : ℝ) := | |
| (collinearTest_injectionAverage_le_free P hκ R hD bits p i j k).trans | |
| (geometricCollinearTest_projected_le P hκ R S hD hS bits p i j k hi hj | |
| hdep hJ hOld K hKt hKg htarget hK hb) | |
| end | |
| end Balanced.Local.AlignedGlobal | |
| end OAI | |