openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /RestrictedQuotient.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ComponentAction | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V Λ N T : Type*} [Group Λ] [Group N] [Group T] [MulAction Λ V] | |
| lemma rangeRestrict_kernel_orbits (f : Λ →* N) | |
| (hf : ∀ v : V, (MulAction.orbit f.ker v).Finite) : | |
| ∀ v : V, (MulAction.orbit f.rangeRestrict.ker v).Finite := by | |
| intro v | |
| apply (hf v).subset | |
| rintro w ⟨g,hg⟩ | |
| have hgm : f g.val=1 := congrArg Subtype.val g.property | |
| exact ⟨⟨g.val,hgm⟩,hg⟩ | |
| lemma rangeRestrict_stabilizer_images (f : Λ →* N) | |
| (hf : ∀ v : V, ((MulAction.stabilizer Λ v).map f : Set N).Finite) : | |
| ∀ v : V, ((MulAction.stabilizer Λ v).map f.rangeRestrict : Set f.range).Finite := by | |
| intro v | |
| apply (Set.Finite.preimage Subtype.val_injective.injOn (hf v)).subset | |
| rintro n ⟨g,hg,rfl⟩ | |
| exact ⟨g,hg,rfl⟩ | |
| lemma component_quotient_le_kernel (ρ : Λ →* N) (φ : N →* T) (C : Set V) : | |
| (ρ.comp (componentKernel (φ.comp ρ) C).subtype).range≤φ.ker := by | |
| rintro n ⟨g,rfl⟩ | |
| exact g.property.1 | |
| lemma infinite_index_of_le_coordinate_kernel [Infinite T] | |
| (φ : N →* T) (hφ : Surjective φ) (J : Subgroup N) (hJ : J≤φ.ker) : | |
| J.index=0 := by | |
| have hr : Infinite φ.range := Infinite.of_surjective φ.range.subtype | |
| (by intro t; obtain ⟨n,rfl⟩ := hφ t; exact ⟨⟨φ n,⟨n,rfl⟩⟩,rfl⟩) | |
| have hk : φ.ker.index=0 := by rw [Subgroup.index_ker,Nat.card_eq_zero_of_infinite] | |
| exact Nat.eq_zero_of_zero_dvd (hk ▸ Subgroup.index_dvd_of_le hJ) | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |