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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | 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
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