| import OAI.Combinatorics.BalancedRyser.Local.AlignedUsedObservable |
| import OAI.Combinatorics.BalancedRyser.Local.LocalUsedEventUnion |
| |
| namespace OAI |
|
|
| namespace Balanced.Local.UsedEventUnion |
| |
| open AlignedUnused AlignedGlobal RootChildPartition |
| noncomputable section |
| attribute [local instance] Classical.propDecidable |
|
|
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
| variable {b κ : ℕ} |
|
|
| def usedGeometricGuards (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (M : MergedPairs F) : Prop := |
| Aligned P a bits p M ∧ Function.Injective (prescribedPoint P a bits M) ∧ |
| (∀ ij, prescribedPoint P a bits M ij ≠ p) ∧ |
| (∀ i c s, lineThrough (a.1 i) p ≠ constituent M (a.2 c) s) |
|
|
| theorem admissible_usedGeometricGuards (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (M : MergedPairs F) (h : Admissible P a bits p M) : |
| usedGeometricGuards P a bits p M := by |
| refine ⟨h.1, h.2.1, h.2.2.1, ?_⟩ |
| intro i c s heq |
| apply h.2.2.2.1 i |
| rw [heq] |
| exact Finset.mem_image.mpr ⟨(a.2 c,s), Finset.mem_univ _, rfl⟩ |
|
|
| omit [Fintype F] in |
| theorem usedGeometricGuards_agree (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (M N : MergedPairs F) |
| (hMN : ∀ d ∈ usedDirections P a, M d = N d) : |
| usedGeometricGuards P a bits p M ↔ usedGeometricGuards P a bits p N := by |
| have hp : prescribedPoint P a bits M = prescribedPoint P a bits N := by |
| funext ij |
| exact UnusedOffsetProbability.prescribedPoint_eq_of_used_agreement (P := P) (bits := bits) M N hMN ij |
| have hc (c : Fin (4*b-κ)) : M (a.2 c) = N (a.2 c) := |
| hMN _ (Finset.mem_image.mpr ⟨c, Finset.mem_univ _, rfl⟩) |
| simp only [usedGeometricGuards, Aligned, hp, constituent, hc] |
|
|
| def geometricUsedTest (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (c : Fin (4*b-κ)) (ij : Fin b × Fin 4) (s : Bool) |
| (M : MergedPairs F) : Prop := |
| usedGeometricGuards P a bits p M ∧ (∀ j, P.classOf ij.1 j ≠ c) ∧ |
| OnLine (prescribedPoint P a bits M ij) (constituent M (a.2 c) s) |
|
|
| def geometricCollinearTest (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (i j k : Fin b × Fin 4) (M : MergedPairs F) : Prop := |
| usedGeometricGuards P a bits p M ∧ |
| i.1 ≠ j.1 ∧ i.1 ≠ k.1 ∧ j.1 ≠ k.1 ∧ |
| prescribedPoint P a bits M i ≠ prescribedPoint P a bits M j ∧ |
| prescribedPoint P a bits M i ≠ prescribedPoint P a bits M k ∧ |
| prescribedPoint P a bits M j ≠ prescribedPoint P a bits M k ∧ |
| OnLine (prescribedPoint P a bits M k) |
| (joiningLine (prescribedPoint P a bits M i) (prescribedPoint P a bits M j)) ∧ |
| (∀ t, joiningLine (prescribedPoint P a bits M i) (prescribedPoint P a bits M j) ≠ |
| lineThrough (a.1 t) p) ∧ |
| (∀ c s, joiningLine (prescribedPoint P a bits M i) (prescribedPoint P a bits M j) ≠ |
| constituent M (a.2 c) s) |
|
|
| theorem usedConstituentTest_implies_geometric (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (c : Fin (4*b-κ)) (ij : Fin b × Fin 4) (s : Bool) (M : MergedPairs F) |
| (h : usedConstituentTest P a bits p c ij s M) : |
| geometricUsedTest P a bits p c ij s M := |
| ⟨admissible_usedGeometricGuards P a bits p M h.1, h.2⟩ |
|
|
| theorem collinearTest_implies_geometric (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (i j k : Fin b × Fin 4) (M : MergedPairs F) |
| (h : collinearTest P a bits p i j k M) : |
| geometricCollinearTest P a bits p i j k M := |
| ⟨admissible_usedGeometricGuards P a bits p M h.1, h.2⟩ |
|
|
| omit [Fintype F] in |
| theorem geometricUsedTest_agree (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (c : Fin (4*b-κ)) (ij : Fin b × Fin 4) (s : Bool) |
| (M N : MergedPairs F) (hMN : ∀ d ∈ usedDirections P a, M d = N d) : |
| geometricUsedTest P a bits p c ij s M ↔ geometricUsedTest P a bits p c ij s N := by |
| have hp := UnusedOffsetProbability.prescribedPoint_eq_of_used_agreement (P := P) (bits := bits) M N hMN ij |
| have hc := hMN _ (Finset.mem_image.mpr ⟨c, Finset.mem_univ _, rfl⟩) |
| simp only [geometricUsedTest, usedGeometricGuards_agree P a bits p M N hMN, |
| hp, constituent, hc] |
|
|
| omit [Fintype F] in |
| theorem geometricCollinearTest_agree (P : EqualityPattern b κ) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) |
| (i j k : Fin b × Fin 4) (M N : MergedPairs F) |
| (hMN : ∀ d ∈ usedDirections P a, M d = N d) : |
| geometricCollinearTest P a bits p i j k M ↔ |
| geometricCollinearTest P a bits p i j k N := by |
| have hp := UnusedOffsetProbability.prescribedPoint_eq_of_used_agreement (P := P) (bits := bits) M N hMN |
| have hc (c : Fin (4*b-κ)) : M (a.2 c) = N (a.2 c) := |
| hMN _ (Finset.mem_image.mpr ⟨c, Finset.mem_univ _, rfl⟩) |
| simp only [geometricCollinearTest, usedGeometricGuards_agree P a bits p M N hMN, |
| hp, constituent, hc] |
|
|
| theorem conditioned_usedTest_le_uniform_free (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) |
| (a : DirectionAssignment F b κ) (ha : Eligible P a) |
| (bits : Bits b) (p : Point F) (c : Fin (4*b-κ)) (ij : Fin b × Fin 4) (s : Bool) : |
| conditionedOffsetProbability (Aligned P a bits p) |
| (usedConstituentTest P a bits p c ij s) ≤ |
| ((PMF.uniformOfFintype (FreeIndex P R → F)).toOuterMeasure |
| {x | geometricUsedTest P a bits p c ij s |
| (generatedZero P hκ R hD a ha bits p x)}).toReal := by |
| exact (UnusedOffsetProbability.conditionedOffsetProbability_mono (Aligned P a bits p) |
| (usedConstituentTest_implies_geometric P a bits p c ij s)).trans_eq |
| (conditioned_probability_eq_uniform_free P hκ R hD a ha bits p _ |
| (geometricUsedTest_agree P a bits p c ij s)) |
| |
| theorem conditioned_collinearTest_le_uniform_free (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) |
| (a : DirectionAssignment F b κ) (ha : Eligible P a) |
| (bits : Bits b) (p : Point F) (i j k : Fin b × Fin 4) : |
| conditionedOffsetProbability (Aligned P a bits p) |
| (collinearTest P a bits p i j k) ≤ |
| ((PMF.uniformOfFintype (FreeIndex P R → F)).toOuterMeasure |
| {x | geometricCollinearTest P a bits p i j k |
| (generatedZero P hκ R hD a ha bits p x)}).toReal := by |
| exact (UnusedOffsetProbability.conditionedOffsetProbability_mono (Aligned P a bits p) |
| (collinearTest_implies_geometric P a bits p i j k)).trans_eq |
| (conditioned_probability_eq_uniform_free P hκ R hD a ha bits p _ |
| (geometricCollinearTest_agree P a bits p i j k)) |
|
|
| end |
| end Balanced.Local.UsedEventUnion |
| |
| end OAI |
|
|
