openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Local /AlignedGlobalGeneratorTest.lean
| import OAI.Combinatorics.BalancedRyser.Incidence.AlignedGeneratorIncidence | |
| import OAI.Combinatorics.BalancedRyser.Local.AlignedHistoryKernel | |
| import OAI.Combinatorics.BalancedRyser.Local.InjectionExposureMixture | |
| import OAI.Combinatorics.BalancedRyser.Local.LocalInjectionFreeLaw | |
| import OAI.Combinatorics.BalancedRyser.Local.LocalPrescribedGeometry | |
| namespace OAI | |
| namespace Balanced.Local.AlignedGlobal | |
| open RootChildPartition InjectionExposure TokenExposure UsedEventUnion | |
| noncomputable section | |
| open scoped Classical | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| variable {b κ : ℕ} | |
| omit [Fintype F] [DecidableEq F] in | |
| theorem constituent_eq_purposeful_line (P : EqualityPattern b κ) | |
| (a : DirectionAssignment F b κ) (ha : Eligible P a) | |
| (bits : Bits b) (M : MergedPairs F) (i : Fin b) (j : Fin 4) (s : Bool) : | |
| constituent M (a.2 (P.classOf i j)) s = | |
| lineThrough (a.2 (P.classOf i j)) | |
| (prescribedPoint P a bits M (i,purposefulIndex (bits i) j s)) := by | |
| apply line_eq_lineThrough | |
| unfold purposefulIndex | |
| split_ifs with hs | |
| · simpa only [hs] using PrescribedGeometry.prescribedPoint_on_left (M := M) ha i j | |
| · have hs' : s = !(bits i j) := by | |
| cases s <;> cases h : bits i j <;> simp_all | |
| have hj : j - 1 + 1 = j := by fin_cases j <;> decide | |
| simpa only [hj, hs'] using | |
| PrescribedGeometry.prescribedPoint_on_right (M := M) ha i (j-1) | |
| variable [Nonempty (DirectionToken b κ ↪ Direction F)] | |
| theorem global_later_generator_probability_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 : Fin b) | |
| (hphase : i ∈ R ∨ R ⊆ S) | |
| (K : Finset (DirectionToken b κ)) | |
| (hKt : ∀ r ∈ S, Sum.inl r ∈ K) | |
| (hKg : ∀ c ∈ S.biUnion P.classSet, Sum.inr c ∈ K) | |
| (hi : Sum.inl i ∈ K) (j : Fin 4) | |
| (hnew : Sum.inr (P.classOf i j) ∉ K) (s : Bool) | |
| (r : {r : Fin b // r ∈ S}) (t : Fin 4) | |
| (hK : K.card ≤ 5*b) (hb : 10*b ≤ Fintype.card F) : | |
| injectionFreeProbability P R (fun e x => | |
| let a := (tokenInjectionEquiv F b κ e).1 | |
| let M := injectedGenerated P hκ R hD bits p e x | |
| prescribedPoint P a bits M (i,purposefulIndex (bits i) j s) ≠ | |
| prescribedPoint P a bits M (r.1,t) ∧ | |
| OnLine (prescribedPoint P a bits M (r.1,t)) | |
| (constituent M (a.2 (P.classOf i j)) s)) ≤ | |
| 2 / (Fintype.card F : ℝ) := by | |
| let noise := currentNoise P hκ R S hD bits p i | |
| let endpoint := fun z : (FreeIndex P R → F) × | |
| (PrefixData F P S × (Fin 4 → Point F)) => | |
| z.2.2 (purposefulIndex (bits i) j s) | |
| let previous := fun z : (FreeIndex P R → F) × | |
| (PrefixData F P S × (Fin 4 → Point F)) => z.2.1.1 r t | |
| let E : Set ((DirectionToken b κ ↪ Direction F) × | |
| ((FreeIndex P R → F) × (PrefixData F P S × (Fin 4 → Point F)))) := | |
| {ez | endpoint ez.2 ≠ previous ez.2 ∧ | |
| OnLine (previous ez.2) (lineThrough (ez.1 (Sum.inr (P.classOf i j))) (endpoint ez.2))} | |
| have hbound : (((PMF.uniformOfFintype (DirectionToken b κ ↪ Direction F)).bind | |
| (fun e => (noise e).map (fun z => (e,z)))).toOuterMeasure E).toReal ≤ | |
| 2 / (Fintype.card F : ℝ) := by | |
| apply original_probability_le_of_auxiliary K noise E | |
| intro base | |
| exact (auxiliary_guarded_joining_probability_le K base | |
| (Sum.inr (P.classOf i j)) hnew noise | |
| (currentNoise_dependsOnExposed P hκ R S hD hS bits p i hphase K hKt hKg hi) | |
| endpoint previous).trans | |
| (inverse_remaining_le_two_div (Fintype.card F) K.card b Fintype.card_pos hK hb) | |
| have hevent : (fun e x => | |
| let a := (tokenInjectionEquiv F b κ e).1 | |
| let M := injectedGenerated P hκ R hD bits p e x | |
| prescribedPoint P a bits M (i,purposefulIndex (bits i) j s) ≠ | |
| prescribedPoint P a bits M (r.1,t) ∧ | |
| OnLine (prescribedPoint P a bits M (r.1,t)) | |
| (constituent M (a.2 (P.classOf i j)) s)) = | |
| (fun e x => | |
| let a := (tokenInjectionEquiv F b κ e).1 | |
| let M := injectedGenerated P hκ R hD bits p e x | |
| prescribedPoint P a bits M (i,purposefulIndex (bits i) j s) ≠ | |
| prescribedPoint P a bits M (r.1,t) ∧ | |
| OnLine (prescribedPoint P a bits M (r.1,t)) | |
| (lineThrough (e (Sum.inr (P.classOf i j))) | |
| (prescribedPoint P a bits M (i,purposefulIndex (bits i) j s)))) := by | |
| funext e x | |
| dsimp only | |
| rw [constituent_eq_purposeful_line P _ | |
| (allTokensDistinct_eligible P _ (tokenInjectionEquiv F b κ e).2)] | |
| rfl | |
| rw [hevent] | |
| unfold injectionFreeProbability | |
| rw [UniformConditionalProduct.uniform_product_eq_bind] | |
| simpa only [noise, currentNoise, E, endpoint, previous, PMF.map_comp, | |
| PMF.toOuterMeasure_bind_apply, PMF.toOuterMeasure_map_apply, | |
| Set.preimage_ofPred_eq, prefixObservation, injectedGenerated, Function.comp_apply] using hbound | |
| theorem geometricUsedTest_later_generator_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 : Fin b) | |
| (hphase : i ∈ R ∨ R ⊆ S) | |
| (K : Finset (DirectionToken b κ)) | |
| (hKt : ∀ r ∈ S, Sum.inl r ∈ K) | |
| (hKg : ∀ c ∈ S.biUnion P.classSet, Sum.inr c ∈ K) | |
| (hi : Sum.inl i ∈ K) (j : Fin 4) | |
| (hnew : Sum.inr (P.classOf i j) ∉ K) (s : Bool) | |
| (r : {r : Fin b // r ∈ S}) (t : Fin 4) | |
| (hK : K.card ≤ 5*b) (hb : 10*b ≤ Fintype.card F) : | |
| injectionFreeProbability P R (fun e x => | |
| geometricUsedTest P (tokenInjectionEquiv F b κ e).1 bits p | |
| (P.classOf i j) (r.1,t) s (injectedGenerated P hκ R hD bits p e x)) ≤ | |
| 2 / (Fintype.card F : ℝ) := by | |
| apply le_trans _ (global_later_generator_probability_le P hκ R S hD hS bits p i | |
| hphase K hKt hKg hi j hnew s r t hK hb) | |
| apply ENNReal.toReal_mono | |
| · rw [PMF.toOuterMeasure_apply] | |
| exact PMF.tsum_coe_indicator_ne_top _ _ | |
| · apply PMF.toOuterMeasure_mono | |
| intro ex hx | |
| refine ⟨?_, hx.1.2.2⟩ | |
| intro heq | |
| have hip := hx.1.1.2.1 heq | |
| have hir : i = r.1 := congrArg Prod.fst hip | |
| exact hx.1.2.1 j (by simp only [hir]) | |
| end | |
| end Balanced.Local.AlignedGlobal | |
| end OAI | |