openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphTopology /LocalProjectiveTopology.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Filter MeasureTheory Module Metric TopologicalSpace | |
| open scoped Classical Topology | |
| namespace CriticalPercolation.LocalProjective | |
| variable (K V : Type*) [NontriviallyNormedField K] [NormedAddCommGroup V] [NormedSpace K V] | |
| abbrev NZ := {v : V // v≠0} | |
| instance nzMulAction : MulAction Kˣ (NZ V) where | |
| smul a v := ⟨(a:K) • (v:V),smul_ne_zero a.ne_zero v.property⟩ | |
| one_smul v := Subtype.ext (one_smul K (v:V)) | |
| mul_smul a b v := Subtype.ext (mul_smul (a:K) (b:K) (v:V)) | |
| instance nzContinuousConstSMul : ContinuousConstSMul Kˣ (NZ V) := | |
| ⟨fun _ => (continuous_const.smul continuous_subtype_val).subtype_mk _⟩ | |
| abbrev Space := Quotient (MulAction.orbitRel Kˣ (NZ V)) | |
| def point (v : NZ V) : Space K V := Quotient.mk _ v | |
| lemma point_surjective : Surjective (point K V) := Quotient.mk_surjective | |
| lemma continuous_point : Continuous (point K V) := continuous_quot_mk | |
| lemma openQuotient_point : IsOpenQuotientMap (point K V) := | |
| MulAction.isOpenQuotientMap_quotientMk | |
| variable {K V} | |
| lemma point_eq_iff (v w : NZ V) : point K V v=point K V w ↔ | |
| ∃ a : K, a • (w:V)=(v:V) := by | |
| constructor | |
| · intro he | |
| obtain ⟨a,ha⟩ := Quotient.exact he | |
| exact ⟨a,congrArg Subtype.val ha⟩ | |
| · rintro ⟨a,ha⟩ | |
| have hane : a≠0 := by intro he; rw [he,zero_smul] at ha; exact v.property ha.symm | |
| apply Quotient.sound | |
| exact ⟨Units.mk0 a hane,Subtype.ext ha⟩ | |
| lemma point_eq_iff_dependent (v w : NZ V) : point K V v=point K V w ↔ | |
| ¬LinearIndependent K ![(v:V),(w:V)] := by | |
| rw [point_eq_iff,linearIndependent_fin2] | |
| change (∃ a : K, a • (w:V)=(v:V)) ↔ ¬((w:V)≠0 ∧ ∀ a : K, a • (w:V)≠(v:V)) | |
| constructor | |
| · rintro ⟨a,ha⟩ h | |
| exact h.2 a ha | |
| · intro h | |
| by_contra hn | |
| exact h ⟨w.property,fun a ha => hn ⟨a,ha⟩⟩ | |
| instance spaceT2 [CompleteSpace K] : T2Space (Space K V) := by | |
| apply (t2Space_iff_of_isOpenQuotientMap (openQuotient_point K V)).mpr | |
| have hc : Continuous (fun z : NZ V × NZ V => ![(z.1:V),(z.2:V)]) := by | |
| apply continuous_pi | |
| intro i | |
| fin_cases i <;> fun_prop | |
| convert (isOpen_setOfPred_linearIndependent (𝕜:=K) (E:=V) (ι:=Fin 2)).isClosed_compl.preimage hc using 1 | |
| ext z | |
| exact point_eq_iff_dependent z.1 z.2 | |
| variable (K V) | |
| instance spaceCompact [ProperSpace K] [FiniteDimensional K V] : CompactSpace (Space K V) := by | |
| let : ProperSpace V := FiniteDimensional.proper K V | |
| obtain ⟨c,hc⟩ := NormedField.exists_one_lt_norm K | |
| let r : ℝ := 1/‖c‖ | |
| have hr : 0<r := div_pos zero_lt_one (lt_trans zero_lt_one hc) | |
| let C : Set V := {v | r≤‖v‖ ∧ ‖v‖≤1} | |
| have hC : IsCompact C := by | |
| have he : C=Metric.closedBall (0:V) 1 ∩ {v | r≤‖v‖} := by | |
| ext v | |
| simp only [C,mem_ofPred_eq,mem_inter_iff,mem_closedBall,dist_zero_right,and_comm] | |
| rw [he] | |
| exact (isCompact_closedBall (0:V) 1).inter_right (isClosed_le continuous_const continuous_norm) | |
| let : CompactSpace C := isCompact_iff_compactSpace.mp hC | |
| let f : C → NZ V := fun v => ⟨v,fun he => by | |
| have hh := v.property.1 | |
| rw [he,norm_zero] at hh | |
| exact hr.not_ge hh⟩ | |
| have hf : Continuous f := continuous_subtype_val.subtype_mk _ | |
| apply Function.Surjective.compactSpace ((continuous_point K V).comp hf) | |
| intro q | |
| obtain ⟨v,rfl⟩ := point_surjective K V q | |
| obtain ⟨a,ha,hlt,hle,_⟩ := rescale_to_shell hc zero_lt_one (x:=(v:V)) v.property | |
| let w : C := ⟨a • (v:V),hle,hlt.le⟩ | |
| refine ⟨w,(point_eq_iff (f w) v).mpr ⟨a,rfl⟩⟩ | |
| instance spaceSecondCountable [ProperSpace K] [FiniteDimensional K V] : | |
| SecondCountableTopology (Space K V) := by | |
| let : ProperSpace V := FiniteDimensional.proper K V | |
| exact ContinuousConstSMul.secondCountableTopology | |
| instance spaceMetrizable [ProperSpace K] [FiniteDimensional K V] : | |
| MetrizableSpace (Space K V) := inferInstance | |
| end CriticalPercolation.LocalProjective | |
| end | |
| end | |
| end OAI | |