openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /ConditionalTranslationStabilizers.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.FiveTrackGeneration | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section ConditionalTranslationStabilizers | |
| variable {a m : ℕ} | |
| theorem trackTranslation_conditional_commute (U : polygonAlgebra a) | |
| (σ : Equiv.Perm (Fin m)) (d : Fin m → CutRing × CutRing) | |
| (hd : ∀ i, σ i ≠ i → d i = 0) : | |
| Commute (trackTranslation d) (conditionalHom U σ) := by | |
| have he := conditional_translation_conjugate U σ d 0 hd | |
| have hU : (⟨translate a (-0) ⁻¹' U.val,polygon_preimage_translate (-0) U.property⟩ : | |
| polygonAlgebra a) = U := by | |
| apply Subtype.ext | |
| ext p | |
| simp only [Set.mem_preimage,neg_zero,translate_zero] | |
| rw [hU] at he | |
| have hx := congrArg (fun z => z * trackTranslation d) he | |
| change _*_ = _*_ | |
| simpa only [mul_assoc,inv_mul_cancel,mul_one] using hx.symm | |
| theorem conditional_conjugation_support (U : polygonAlgebra a) | |
| (σ : Equiv.Perm (Fin m)) (d e : Fin m → CutRing × CutRing) | |
| (hde : ∀ i, σ i ≠ i → d i = e i) : | |
| trackTranslation d * conditionalHom U σ * (trackTranslation d)⁻¹ = | |
| trackTranslation e * conditionalHom U σ * (trackTranslation e)⁻¹ := by | |
| have hc := trackTranslation_conditional_commute U σ (d-e) (by | |
| intro i hi | |
| change d i - e i = 0 | |
| rw [hde i hi,sub_self]) | |
| rw [trackTranslation_sub] at hc | |
| have hcomm : Commute (trackTranslation (a := a) d) (trackTranslation e) := by | |
| change _*_ = _*_ | |
| rw [← trackTranslation_add,← trackTranslation_add,add_comm d e] | |
| have he := congrArg (fun z => trackTranslation e * z * (trackTranslation d)⁻¹) hc.eq | |
| rw [hcomm.inv_right.eq] at he | |
| have hh : trackTranslation e * ((trackTranslation e)⁻¹ * trackTranslation d * conditionalHom U σ) * | |
| (trackTranslation d)⁻¹ = trackTranslation d * conditionalHom U σ * (trackTranslation d)⁻¹ := by group | |
| rw [hh] at he | |
| convert he using 1; group | |
| end ConditionalTranslationStabilizers | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |