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@ -0,0 +1,414 @@
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/-
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This files defines `RegularSupportBasis`, which is a basis of the topological space α,
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made up of finite intersections of `RegularSupport α g` for `g : HomeoGroup α`.
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-/
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import Mathlib.Topology.Basic
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import Mathlib.Topology.Homeomorph
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import Rubin.LocallyDense
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import Rubin.Topology
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import Rubin.Support
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import Rubin.RegularSupport
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import Rubin.HomeoGroup
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namespace Rubin
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section RegularSupportBasis.Prelude
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variable {α : Type _}
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variable [TopologicalSpace α]
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variable [DecidableEq α]
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/--
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Maps a "seed" of homeorphisms in α to the intersection of their regular support in α.
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Note that the condition that the resulting set is non-empty is introduced later in `RegularSupportBasis₀`
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--/
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def RegularSupportInter₀ (S : Finset (HomeoGroup α)): Set α :=
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⋂ (g ∈ S), RegularSupport α g
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theorem RegularSupportInter₀.eq_sInter (S : Finset (HomeoGroup α)) :
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RegularSupportInter₀ S = ⋂₀ ((fun (g : HomeoGroup α) => RegularSupport α g) '' S) :=
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by
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rw [Set.sInter_image]
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rfl
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/--
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This is a predecessor type to `RegularSupportBasis`, where equality is defined on the `seed` used, rather than the `val`.
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--/
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structure RegularSupportBasis₀ (α : Type _) [TopologicalSpace α] where
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seed : Finset (HomeoGroup α)
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val_nonempty : Set.Nonempty (RegularSupportInter₀ seed)
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theorem RegularSupportBasis₀.eq_iff_seed_eq (S T : RegularSupportBasis₀ α) : S = T ↔ S.seed = T.seed := by
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-- Spooky :3c
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rw [mk.injEq]
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def RegularSupportBasis₀.val (S : RegularSupportBasis₀ α) : Set α := RegularSupportInter₀ S.seed
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theorem RegularSupportBasis₀.val_def (S : RegularSupportBasis₀ α) : S.val = RegularSupportInter₀ S.seed := rfl
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@[simp]
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theorem RegularSupportBasis₀.nonempty (S : RegularSupportBasis₀ α) : Set.Nonempty S.val := S.val_nonempty
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@[simp]
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theorem RegularSupportBasis₀.regular (S : RegularSupportBasis₀ α) : Regular S.val := by
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rw [S.val_def]
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rw [RegularSupportInter₀.eq_sInter]
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apply regular_sInter
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· have set_decidable : DecidableEq (Set α) := Classical.typeDecidableEq (Set α)
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let fin : Finset (Set α) := S.seed.image ((fun g => RegularSupport α g))
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apply Set.Finite.ofFinset fin
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simp
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· intro S S_in_set
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simp at S_in_set
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let ⟨g, ⟨_, Heq⟩⟩ := S_in_set
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rw [<-Heq]
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exact regularSupport_regular α g
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lemma RegularSupportInter₀.mul_seed (seed : Finset (HomeoGroup α)) [DecidableEq (HomeoGroup α)] (f : HomeoGroup α):
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RegularSupportInter₀ (Finset.image (fun g => f * g * f⁻¹) seed) = f •'' RegularSupportInter₀ seed :=
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by
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unfold RegularSupportInter₀
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simp
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conv => {
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rhs
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ext; lhs; ext x; ext; lhs
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ext
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rw [regularSupport_smulImage]
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}
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variable [DecidableEq (HomeoGroup α)]
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/--
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A `HomeoGroup α` group element `f` acts on an `RegularSupportBasis₀ α` set `S`,
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by mapping each element `g` of `S.seed` to `f * g * f⁻¹`
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--/
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instance homeoGroup_smul₂ : SMul (HomeoGroup α) (RegularSupportBasis₀ α) where
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smul := fun f S => ⟨
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(Finset.image (fun g => f * g * f⁻¹) S.seed),
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by
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rw [RegularSupportInter₀.mul_seed]
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simp
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exact S.val_nonempty
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⟩
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theorem RegularSupportBasis₀.smul_seed (f : HomeoGroup α) (S : RegularSupportBasis₀ α) :
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(f • S).seed = (Finset.image (fun g => f * g * f⁻¹) S.seed) := rfl
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theorem RegularSupportBasis₀.smul_val (f : HomeoGroup α) (S : RegularSupportBasis₀ α) :
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(f • S).val = RegularSupportInter₀ (Finset.image (fun g => f * g * f⁻¹) S.seed) := rfl
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theorem RegularSupportBasis₀.smul_val' (f : HomeoGroup α) (S : RegularSupportBasis₀ α) :
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(f • S).val = f •'' S.val :=
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by
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unfold val
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rw [<-RegularSupportInter₀.mul_seed]
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rw [RegularSupportBasis₀.smul_seed]
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-- We define a "preliminary" group action from `HomeoGroup α` to `RegularSupportBasis₀`
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instance homeoGroup_mulAction₂ : MulAction (HomeoGroup α) (RegularSupportBasis₀ α) where
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one_smul := by
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intro S
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rw [RegularSupportBasis₀.eq_iff_seed_eq]
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rw [RegularSupportBasis₀.smul_seed]
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simp
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mul_smul := by
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intro f g S
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rw [RegularSupportBasis₀.eq_iff_seed_eq]
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repeat rw [RegularSupportBasis₀.smul_seed]
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rw [Finset.image_image]
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ext x
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simp
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group
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end RegularSupportBasis.Prelude
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/--
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A partially-ordered set, associated to Rubin's proof.
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Any element in that set is made up of a `seed`,
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such that `val = RegularSupportInter₀ seed` and `Set.Nonempty val`.
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Actions on this set are first defined in terms of `RegularSupportBasis₀`,
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as the proofs otherwise get hairy with a bunch of `Exists.choose` appearing.
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--/
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structure RegularSupportBasis (α : Type _) [TopologicalSpace α] where
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val : Set α
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val_has_seed : ∃ po_seed : RegularSupportBasis₀ α, po_seed.val = val
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namespace RegularSupportBasis
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variable {α : Type _}
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variable [TopologicalSpace α]
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variable [DecidableEq α]
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theorem eq_iff_val_eq (S T : RegularSupportBasis α) : S = T ↔ S.val = T.val := by
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rw [mk.injEq]
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def fromSeed (seed : RegularSupportBasis₀ α) : RegularSupportBasis α := ⟨
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seed.val,
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⟨seed, seed.val_def⟩
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⟩
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noncomputable def full_seed (S : RegularSupportBasis α) : RegularSupportBasis₀ α :=
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(Exists.choose S.val_has_seed)
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noncomputable def seed (S : RegularSupportBasis α) : Finset (HomeoGroup α) :=
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S.full_seed.seed
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@[simp]
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theorem full_seed_seed (S : RegularSupportBasis α) : S.full_seed.seed = S.seed := rfl
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@[simp]
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theorem fromSeed_val (seed : RegularSupportBasis₀ α) :
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(fromSeed seed).val = seed.val :=
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by
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unfold fromSeed
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simp
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@[simp]
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theorem val_from_seed (S : RegularSupportBasis α) : RegularSupportInter₀ S.seed = S.val := by
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unfold seed full_seed
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rw [<-RegularSupportBasis₀.val_def]
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rw [Exists.choose_spec S.val_has_seed]
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@[simp]
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theorem val_from_seed₂ (S : RegularSupportBasis α) : S.full_seed.val = S.val := by
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unfold full_seed
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rw [RegularSupportBasis₀.val_def]
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nth_rw 2 [<-val_from_seed]
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unfold seed full_seed
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rfl
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-- Allows one to prove properties of RegularSupportBasis without jumping through `Exists.choose`-shaped hoops
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theorem prop_from_val {p : Set α → Prop}
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(any_seed : ∀ po_seed : RegularSupportBasis₀ α, p po_seed.val) :
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∀ (S : RegularSupportBasis α), p S.val :=
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by
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intro S
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rw [<-val_from_seed]
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have res := any_seed S.full_seed
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rw [val_from_seed₂] at res
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rw [val_from_seed]
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exact res
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@[simp]
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theorem nonempty : ∀ (S : RegularSupportBasis α), Set.Nonempty S.val :=
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prop_from_val RegularSupportBasis₀.nonempty
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@[simp]
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theorem regular : ∀ (S : RegularSupportBasis α), Regular S.val :=
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prop_from_val RegularSupportBasis₀.regular
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variable [DecidableEq (HomeoGroup α)]
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instance homeoGroup_smul₃ : SMul (HomeoGroup α) (RegularSupportBasis α) where
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smul := fun f S => ⟨
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f •'' S.val,
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by
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use f • S.full_seed
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rw [RegularSupportBasis₀.smul_val']
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simp
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⟩
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theorem smul_val (f : HomeoGroup α) (S : RegularSupportBasis α) :
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(f • S).val = f •'' S.val := rfl
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theorem smul_seed' (f : HomeoGroup α) (S : RegularSupportBasis α) (seed : Finset (HomeoGroup α)) :
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S.val = RegularSupportInter₀ seed →
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(f • S).val = RegularSupportInter₀ (Finset.image (fun g => f * g * f⁻¹) seed) :=
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by
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intro S_val_eq
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rw [smul_val]
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rw [RegularSupportInter₀.mul_seed]
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rw [S_val_eq]
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theorem smul_seed (f : HomeoGroup α) (S : RegularSupportBasis α) :
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(f • S).val = RegularSupportInter₀ (Finset.image (fun g => f * g * f⁻¹) S.seed) :=
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by
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apply smul_seed'
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symm
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exact val_from_seed S
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-- Note: we could potentially implement MulActionHom
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instance homeoGroup_mulAction₃ : MulAction (HomeoGroup α) (RegularSupportBasis α) where
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one_smul := by
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intro S
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rw [eq_iff_val_eq]
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repeat rw [smul_val]
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rw [one_smulImage]
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mul_smul := by
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intro S f g
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rw [eq_iff_val_eq]
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repeat rw [smul_val]
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rw [smulImage_mul]
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instance associatedPoset_le : LE (RegularSupportBasis α) where
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le := fun S T => S.val ⊆ T.val
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theorem le_def (S T : RegularSupportBasis α) : S ≤ T ↔ S.val ⊆ T.val := by
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rw [iff_eq_eq]
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rfl
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instance associatedPoset_partialOrder : PartialOrder (RegularSupportBasis α) where
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le_refl := fun S => (le_def S S).mpr (le_refl S.val)
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le_trans := fun S T U S_le_T S_le_U => (le_def S U).mpr (le_trans
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((le_def _ _).mp S_le_T)
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((le_def _ _).mp S_le_U)
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)
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le_antisymm := by
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intro S T S_le_T T_le_S
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rw [le_def] at S_le_T
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rw [le_def] at T_le_S
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rw [eq_iff_val_eq]
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apply le_antisymm <;> assumption
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theorem smul_mono {S T : RegularSupportBasis α} (f : HomeoGroup α) (S_le_T : S ≤ T) :
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f • S ≤ f • T :=
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by
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rw [le_def]
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repeat rw [smul_val]
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apply smulImage_mono
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assumption
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instance associatedPoset_coeSet : Coe (RegularSupportBasis α) (Set α) where
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coe := RegularSupportBasis.val
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def asSet (α : Type _) [TopologicalSpace α]: Set (Set α) :=
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{ S : Set α | ∃ T : RegularSupportBasis α, T.val = S }
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theorem asSet_def :
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RegularSupportBasis.asSet α = { S : Set α | ∃ T : RegularSupportBasis α, T.val = S } := rfl
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theorem mem_asSet (S : Set α) :
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S ∈ RegularSupportBasis.asSet α ↔ ∃ T : RegularSupportBasis α, T.val = S :=
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by
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rw [asSet_def]
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simp
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theorem mem_asSet' (S : Set α) :
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S ∈ RegularSupportBasis.asSet α ↔ Set.Nonempty S ∧ ∃ seed : Finset (HomeoGroup α), S = RegularSupportInter₀ seed :=
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by
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rw [asSet_def]
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simp
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constructor
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· intro ⟨T, T_eq⟩
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rw [<-T_eq]
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constructor
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simp
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let ⟨⟨seed, _⟩, eq⟩ := T.val_has_seed
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rw [RegularSupportBasis₀.val_def] at eq
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simp at eq
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use seed
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exact eq.symm
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· intro ⟨S_nonempty, ⟨seed, val_from_seed⟩⟩
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rw [val_from_seed] at S_nonempty
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use fromSeed ⟨seed, S_nonempty⟩
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rw [val_from_seed]
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simp
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rw [RegularSupportBasis₀.val_def]
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instance membership : Membership α (RegularSupportBasis α) where
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mem := fun x S => x ∈ S.val
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theorem mem_iff (x : α) (S : RegularSupportBasis α) : x ∈ S ↔ x ∈ (S : Set α) := iff_eq_eq ▸ rfl
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section Basis
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open Topology
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-- TODO: clean this lemma to not mention W anymore?
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lemma proposition_3_2_subset (G : Type _) {α : Type _} [Group G] [TopologicalSpace α] [MulAction G α]
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[T2Space α] [LocallyCompactSpace α] [h_ld : LocallyDense G α] [HasNoIsolatedPoints α]
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[ContinuousMulAction G α]
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{U : Set α} (U_open : IsOpen U) {p : α} (p_in_U : p ∈ U) :
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∃ (W : Set α), W ∈ 𝓝 p ∧ closure W ⊆ U ∧
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∃ (g : G), g ∈ RigidStabilizer G W ∧ p ∈ RegularSupport α g ∧ RegularSupport α g ⊆ closure W :=
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by
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have U_in_nhds : U ∈ 𝓝 p := by
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rw [mem_nhds_iff]
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use U
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let ⟨W', W'_in_nhds, W'_ss_U, W'_compact⟩ := local_compact_nhds U_in_nhds
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-- This feels like black magic, but okay
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let ⟨W, _W_compact, W_closed, W'_ss_int_W, W_ss_U⟩ := exists_compact_closed_between W'_compact U_open W'_ss_U
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have W_cl_eq_W : closure W = W := IsClosed.closure_eq W_closed
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have W_in_nhds : W ∈ 𝓝 p := by
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rw [mem_nhds_iff]
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use interior W
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repeat' apply And.intro
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· exact interior_subset
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· simp
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· exact W'_ss_int_W (mem_of_mem_nhds W'_in_nhds)
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use W
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repeat' apply And.intro
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exact W_in_nhds
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{
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rw [W_cl_eq_W]
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exact W_ss_U
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}
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have p_in_int_W : p ∈ interior W := W'_ss_int_W (mem_of_mem_nhds W'_in_nhds)
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let ⟨g, g_in_rist, g_moves_p⟩ := get_moving_elem_in_rigidStabilizer G isOpen_interior p_in_int_W
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use g
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repeat' apply And.intro
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· apply rigidStabilizer_mono interior_subset
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simp
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exact g_in_rist
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· rw [<-mem_support] at g_moves_p
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apply support_subset_regularSupport
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exact g_moves_p
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· rw [rigidStabilizer_support] at g_in_rist
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apply subset_trans
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exact regularSupport_subset_closure_support
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apply closure_mono
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apply subset_trans
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exact g_in_rist
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exact interior_subset
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/--
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## Proposition 3.2 : RegularSupportBasis is a topological basis of `α`
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-/
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theorem isBasis (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α]
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[T2Space α] [LocallyCompactSpace α] [h_ld : LocallyDense G α] [HasNoIsolatedPoints α]
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[hc : ContinuousMulAction G α] :
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TopologicalSpace.IsTopologicalBasis (RegularSupportBasis.asSet α) :=
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by
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apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
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{
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intro U U_in_poset
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rw [RegularSupportBasis.mem_asSet] at U_in_poset
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let ⟨T, T_val⟩ := U_in_poset
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rw [<-T_val]
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exact T.regular.isOpen
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}
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intro p U p_in_U U_open
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let ⟨W, _, clW_ss_U, ⟨g, _, p_in_rsupp, rsupp_ss_clW⟩⟩ := proposition_3_2_subset G U_open p_in_U
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use RegularSupport α g
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repeat' apply And.intro
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· rw [RegularSupportBasis.mem_asSet']
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constructor
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exact ⟨p, p_in_rsupp⟩
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use {(ContinuousMulAction.toHomeomorph α g : HomeoGroup α)}
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unfold RegularSupportInter₀
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simp
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unfold RegularSupport
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rw [<-homeoGroup_support_eq_support_toHomeomorph g]
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· exact p_in_rsupp
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· apply subset_trans
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exact rsupp_ss_clW
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exact clW_ss_U
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end Basis
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end RegularSupportBasis
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end Rubin
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