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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Classification.AlgebraicDescent
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation
universe u v
lemma hom_finite_preimage_of_finite_kernel {G H : Type*} [Group G] [Group H]
(f : G →* H) (hk : (f.ker : Set G).Finite) (S : Set H) (hS : S.Finite) :
(f ⁻¹' S).Finite := by
let _ := hk.to_subtype
have hfib : ∀ a : H, (f ⁻¹' {a}).Finite := by
intro a
by_cases ha : (f ⁻¹' {a}).Nonempty
· obtain ⟨x,hx⟩ := ha
change f x=a at hx
rw [←hx]
let _ := Finite.of_equiv _ (f.fiberEquivKer x).symm
exact Set.toFinite _
· rw [Set.not_nonempty_iff_eq_empty.mp ha]
exact Set.finite_empty
apply (hS.biUnion (fun a _ => hfib a)).subset
exact fun x hx => Set.mem_iUnion₂.mpr ⟨f x,hx,rfl⟩
lemma hom_finite_composite_kernel {G H K : Type*} [Group G] [Group H] [Group K]
(f : G →* H) (g : H →* K)
(hf : (f.ker : Set G).Finite) (hg : (g.ker : Set H).Finite) :
((g.comp f).ker : Set G).Finite :=
hom_finite_preimage_of_finite_kernel f hf g.ker hg
theorem AlgebraicDescent.of_finite_kernel {N : Type u} [Group N]
(hN : AlgebraicDescent N) :
∀ {G : Type v} [Group G] (ρ : G →* N), Surjective ρ →
(ρ.ker : Set G).Finite → AlgebraicDescent G := by
induction hN with
| @finite N _ _ =>
intro G _ ρ _ hk
let _ := hk.to_subtype
let _ := ρ.finite_iff_finite_ker_range.mpr ⟨inferInstance,inferInstance⟩
exact AlgebraicDescent.finite G
| @cyclic N _ φ hφ hf =>
intro G _ ρ hρ hk
exact AlgebraicDescent.cyclic G (φ.comp ρ) (hφ.comp hρ)
(hom_finite_composite_kernel ρ φ hk hf)
| @plane N _ φ hφ lower ih =>
intro G _ ρ hρ hk
apply AlgebraicDescent.plane G (φ.comp ρ) (hφ.comp hρ)
intro J hJ
let f := ρ.domRestrict J
have hf : f.range≤φ.ker := by
rintro n ⟨j,rfl⟩
exact hJ j.property
apply ih f.range hf f.rangeRestrict f.rangeRestrict_surjective
apply (hk.preimage Subtype.val_injective.injOn).subset
intro j hj
exact congrArg Subtype.val hj
lemma AlgebraicDescent.of_equiv {G : Type u} {H : Type v} [Group G] [Group H]
(hH : AlgebraicDescent H) (e : G ≃* H) : AlgebraicDescent G := by
apply hH.of_finite_kernel e.toMonoidHom e.surjective
have he : (e.toMonoidHom.ker : Set G)={1} := by
ext g
change e g=1 ↔ g=1
rw [←e.map_one,e.injective.eq_iff]
rw [he]
exact Set.finite_singleton _
end CriticalPercolation
end
end
end
end OAI
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