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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 99 100 101 102 103 104 105 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphActionCoordinates
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E Λ Q : Type*} (G : BondGraph V E)
[Group Λ] [Group Q] [MulAction Λ V] [MulAction Λ E]
lemma finite_orbit_quotient_of_representatives (S : Set V) (hS : S.Finite)
(hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) :
Finite (MulAction.orbitRel.Quotient Λ V) := by
let : Setoid V := MulAction.orbitRel Λ V
let _ := hS.to_subtype
apply Finite.of_surjective (fun s : S => Quotient.mk' s.val)
intro q
induction q using Quotient.inductionOn with
| h v =>
obtain ⟨s,hs,hvs⟩ := hcover v
exact ⟨⟨s,hs⟩,Quotient.sound ((MulAction.orbitRel Λ V).symm (MulAction.orbitRel_apply.mpr hvs))⟩
lemma finite_representatives_of_orbit_quotient
[Finite (MulAction.orbitRel.Quotient Λ V)] :
∃ S : Set V, S.Finite ∧ (∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) := by
let : Setoid V := MulAction.orbitRel Λ V
let S := Set.range (Quotient.out : MulAction.orbitRel.Quotient Λ V → V)
refine ⟨S,Set.finite_range _,?_⟩
intro v
refine ⟨Quotient.out (Quotient.mk' v),⟨Quotient.mk' v,rfl⟩,?_⟩
apply MulAction.orbitRel_apply.mp
exact Quotient.exact' (Quotient.out_eq' (Quotient.mk' v)).symm
lemma finiteIndex_action_representatives (S : Set V) (hS : S.Finite)
(hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s)
(L : Subgroup Λ) (hL : L.index≠0) :
∃ R : Set V, R.Finite ∧ (∀ v, ∃ r∈R, v∈MulAction.orbit L r) := by
let _ := finite_orbit_quotient_of_representatives S hS hcover
let _ := L.finite_quotient_of_finite_quotient_of_index_ne_zero (X := V) hL
exact finite_representatives_of_orbit_quotient
def quotientSubgroupMap (ρ : Λ →* Q) (N : Subgroup Q) : N.comap ρ →* N where
toFun g := ⟨ρ g.val,g.property⟩
map_one' := Subtype.ext (ρ.map_one)
map_mul' g h := Subtype.ext (ρ.map_mul g.val h.val)
lemma quotientSubgroupMap_surjective (ρ : Λ →* Q) (hρ : Surjective ρ)
(N : Subgroup Q) : Surjective (quotientSubgroupMap ρ N) := by
intro n
obtain ⟨g,hg⟩ := hρ n.val
exact ⟨⟨g,by change ρ g∈N; rw [hg]; exact n.property⟩,Subtype.ext hg⟩
lemma quotientSubgroupMap_kernel_orbits (ρ : Λ →* Q) (N : Subgroup Q)
(hf : ∀ v : V, (MulAction.orbit ρ.ker v).Finite) :
∀ v : V, (MulAction.orbit (quotientSubgroupMap ρ N).ker v).Finite := by
intro v
apply (hf v).subset
rintro w ⟨g,hg⟩
exact ⟨⟨g.val.val,congrArg Subtype.val g.property⟩,hg⟩
lemma quotientSubgroupMap_stabilizer_images (ρ : Λ →* Q) (N : Subgroup Q)
(hf : ∀ v : V, ((MulAction.stabilizer Λ v).map ρ : Set Q).Finite) :
∀ v : V, ((MulAction.stabilizer (N.comap ρ) v).map
(quotientSubgroupMap ρ N) : Set N).Finite := by
intro v
apply (Set.Finite.preimage Subtype.val_injective.injOn (hf v)).subset
rintro n ⟨g,hg,rfl⟩
exact ⟨g.val,hg,rfl⟩
lemma subgroup_preservesEnds (ha : G.PreservesEnds (Λ := Λ)) (L : Subgroup Λ) :
G.PreservesEnds (Λ := L) := fun g e => ha g.val e
lemma finiteIndex_quotient_transfer
(ha : G.PreservesEnds (Λ := Λ))
(S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s)
(ρ : Λ →* Q) (hρ : Surjective ρ)
(hf : ∀ v : V, (MulAction.orbit ρ.ker v).Finite)
(hs : ∀ v : V, ((MulAction.stabilizer Λ v).map ρ : Set Q).Finite)
(N : Subgroup Q) (hN : N.index≠0) :
G.PreservesEnds (Λ := N.comap ρ) ∧
(∃ R : Set V, R.Finite ∧ ∀ v, ∃ r∈R, v∈MulAction.orbit (N.comap ρ) r) ∧
Surjective (quotientSubgroupMap ρ N) ∧
(∀ v : V, (MulAction.orbit (quotientSubgroupMap ρ N).ker v).Finite) ∧
(∀ v : V, ((MulAction.stabilizer (N.comap ρ) v).map
(quotientSubgroupMap ρ N) : Set N).Finite) := by
refine ⟨G.subgroup_preservesEnds ha (N.comap ρ),?_,
quotientSubgroupMap_surjective ρ hρ N,
quotientSubgroupMap_kernel_orbits ρ N hf,
quotientSubgroupMap_stabilizer_images ρ N hs⟩
exact finiteIndex_action_representatives S hS hcover (N.comap ρ)
(by rwa [Subgroup.index_comap_of_surjective _ hρ])
end CriticalPercolation.BondGraph
end
end
end
end OAI
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