openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /PointwisePolygonTables.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.TupleChartDomain | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PointwisePolygonTables | |
| variable {a m : ℕ} | |
| noncomputable def conditionalAlternatingHom (U : polygonAlgebra a) : | |
| alternatingGroup (Fin m) →* polygonAlternatingGroup a m := | |
| ((conditionalHom U).comp (alternatingGroup (Fin m)).subtype).codRestrict | |
| (polygonAlternatingGroup a m) (fun s => conditionalHom_alternating_mem U s.property) | |
| theorem conditionalHom_disjoint_commute (U V : polygonAlgebra a) | |
| (hUV : Disjoint U.val V.val) (σ ρ : Equiv.Perm (Fin m)) : | |
| Commute (conditionalHom U σ) (conditionalHom V ρ) := by | |
| change _ * _ = _ * _ | |
| apply Subtype.ext | |
| apply Equiv.ext | |
| intro p | |
| change conditionalPerm U σ (conditionalPerm V ρ p) = | |
| conditionalPerm V ρ (conditionalPerm U σ p) | |
| by_cases hu : p.2 ∈ U.val <;> by_cases hv : p.2 ∈ V.val | |
| · exact False.elim (Set.disjoint_left.mp hUV hu hv) | |
| all_goals simp [conditionalPerm, hu, hv] | |
| structure PolygonLabelling (a : ℕ) (Ω : Type*) where | |
| label : GenericSquare a → Ω | |
| fiber_mem : ∀ ω, {p | label p = ω} ∈ polygonAlgebra a | |
| noncomputable def PolygonLabelling.fiber {Ω : Type*} (v : PolygonLabelling a Ω) | |
| (ω : Ω) : polygonAlgebra a := ⟨{p | v.label p = ω},v.fiber_mem ω⟩ | |
| theorem PolygonLabelling.fiber_disjoint {Ω : Type*} (v : PolygonLabelling a Ω) | |
| {ω ν : Ω} (h : ω ≠ ν) : Disjoint (v.fiber ω).val (v.fiber ν).val := by | |
| apply Set.disjoint_left.mpr | |
| intro p hp hq | |
| exact h (hp.symm.trans hq) | |
| noncomputable def PolygonLabelling.permutationHom {Ω : Type*} (v : PolygonLabelling a Ω) : | |
| (Ω → alternatingGroup (Fin m)) →* Equiv.Perm (TrackPoint a m) where | |
| toFun σ := { | |
| toFun := fun p => ((σ (v.label p.2)).val p.1,p.2) | |
| invFun := fun p => ((σ (v.label p.2)).val.symm p.1,p.2) | |
| left_inv := by intro p; simp | |
| right_inv := by intro p; simp } | |
| map_one' := by apply Equiv.ext; intro p; rfl | |
| map_mul' := by intro σ ρ; apply Equiv.ext; intro p; rfl | |
| @[simp] theorem PolygonLabelling.permutationHom_apply {Ω : Type*} | |
| (v : PolygonLabelling a Ω) (σ : Ω → alternatingGroup (Fin m)) (p : TrackPoint a m) : | |
| v.permutationHom σ p = ((σ (v.label p.2)).val p.1,p.2) := rfl | |
| noncomputable def PolygonLabelling.tableHom {Ω : Type*} [Fintype Ω] | |
| (v : PolygonLabelling a Ω) : | |
| (Ω → alternatingGroup (Fin m)) →* polygonAlternatingGroup a m := | |
| MonoidHom.noncommPiCoprod (fun ω => conditionalAlternatingHom (v.fiber ω)) (by | |
| intro ω ν h s t | |
| change _ * _ = _ * _ | |
| apply Subtype.ext | |
| exact (conditionalHom_disjoint_commute (v.fiber ω) (v.fiber ν) | |
| (v.fiber_disjoint h) s.val t.val).eq) | |
| @[simp] theorem PolygonLabelling.tableHom_single {Ω : Type*} [Fintype Ω] [DecidableEq Ω] | |
| (v : PolygonLabelling a Ω) (ω : Ω) (s : alternatingGroup (Fin m)) : | |
| v.tableHom (Pi.mulSingle ω s) = conditionalAlternatingHom (v.fiber ω) s := | |
| MonoidHom.noncommPiCoprod_mulSingle _ _ _ | |
| theorem PolygonLabelling.tableHom_permutation {Ω : Type*} [Fintype Ω] | |
| (v : PolygonLabelling a Ω) : | |
| (polygonFullGroup a m).subtype.comp | |
| ((polygonAlternatingGroup a m).subtype.comp v.tableHom) = v.permutationHom := by | |
| classical | |
| apply MonoidHom.pi_ext | |
| intro ω s | |
| simp only [MonoidHom.comp_apply, v.tableHom_single] | |
| apply Equiv.ext | |
| intro p | |
| change conditionalPerm (v.fiber ω) s.val p = | |
| (((Pi.mulSingle ω s : Ω → alternatingGroup (Fin m)) (v.label p.2)).val p.1,p.2) | |
| by_cases hp : v.label p.2 = ω | |
| · simp [conditionalPerm, PolygonLabelling.fiber, hp] | |
| · simp [conditionalPerm, PolygonLabelling.fiber, hp] | |
| @[simp] theorem PolygonLabelling.tableHom_apply {Ω : Type*} [Fintype Ω] | |
| (v : PolygonLabelling a Ω) (σ : Ω → alternatingGroup (Fin m)) (p : TrackPoint a m) : | |
| (v.tableHom σ).val.val p = ((σ (v.label p.2)).val p.1,p.2) := by | |
| have he := DFunLike.congr_fun v.tableHom_permutation σ | |
| exact DFunLike.congr_fun he p | |
| theorem PolygonLabelling.tableHom_injective {Ω : Type*} [Fintype Ω] | |
| (v : PolygonLabelling a Ω) (hv : Function.Surjective v.label) : | |
| Function.Injective (v.tableHom (m := m)) := by | |
| intro s t h | |
| funext ω | |
| apply Subtype.ext | |
| apply Equiv.ext | |
| intro i | |
| obtain ⟨p,hp⟩ := hv ω | |
| have he := congrArg (fun g : polygonAlternatingGroup a m => g.val.val (i,p)) h | |
| simpa only [v.tableHom_apply, hp] using congrArg Prod.fst he | |
| noncomputable def polygonAssignment {ι : Type*} (U : ι → polygonAlgebra a) | |
| (p : GenericSquare a) : ι → Bool := by | |
| classical | |
| exact fun i => decide (p ∈ (U i).val) | |
| theorem polygonAssignment_fiber_mem {ι : Type*} [Finite ι] | |
| (U : ι → polygonAlgebra a) (ω : ι → Bool) : | |
| {p | polygonAssignment U p = ω} ∈ polygonAlgebra a := by | |
| classical | |
| have he : {p | polygonAssignment U p = ω} = | |
| ⋂ i, if ω i then (U i).val else (U i).valᶜ := by | |
| ext p | |
| simp only [Set.mem_ofPred_eq, Set.mem_iInter, funext_iff, polygonAssignment] | |
| apply forall_congr' | |
| intro i | |
| cases ω i <;> simp | |
| rw [he] | |
| apply BooleanSubalgebra.iInf_mem | |
| intro i | |
| cases ω i | |
| · exact BooleanSubalgebra.compl_mem (U i).property | |
| · exact (U i).property | |
| noncomputable def actualPolygonLabelling {ι : Type*} [Finite ι] | |
| (U : ι → polygonAlgebra a) : PolygonLabelling a (Set.range (polygonAssignment U)) where | |
| label p := ⟨polygonAssignment U p,⟨p,rfl⟩⟩ | |
| fiber_mem ω := by | |
| have he : {p | (⟨polygonAssignment U p,⟨p,rfl⟩⟩ : Set.range (polygonAssignment U)) = ω} = | |
| {p | polygonAssignment U p = ω.val} := by | |
| ext p | |
| exact Subtype.ext_iff | |
| rw [he] | |
| exact polygonAssignment_fiber_mem U ω.val | |
| theorem actualPolygonLabelling_surjective {ι : Type*} [Finite ι] | |
| (U : ι → polygonAlgebra a) : Function.Surjective (actualPolygonLabelling U).label := by | |
| rintro ⟨ω,p,rfl⟩ | |
| exact ⟨p,rfl⟩ | |
| end PointwisePolygonTables | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |