openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /FiveTrackGeneration.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.BooleanCoordinateGeneration | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.ConditionalBooleanGeneration | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section FiveTrackGeneration | |
| noncomputable def fiveTrackHom {m : ℕ} (ι : Fin 5 ↪ Fin m) : | |
| alternatingGroup (Fin 5) →* Equiv.Perm (Fin m) := | |
| (Equiv.Perm.viaEmbeddingHom ι).comp (alternatingGroup (Fin 5)).subtype | |
| instance fiveTrackPerfect : Group.IsPerfect (alternatingGroup (Fin 5)) := | |
| ⟨commutator_alternatingGroup_eq_top (by norm_num)⟩ | |
| theorem fiveTrack_sign {m : ℕ} (ι : Fin 5 ↪ Fin m) | |
| (g : alternatingGroup (Fin 5)) : fiveTrackHom ι g ∈ alternatingGroup (Fin m) := by | |
| classical | |
| change Equiv.Perm.sign (g.val.viaEmbedding ι) = 1 | |
| rw [Equiv.Perm.viaEmbedding,Equiv.Perm.sign_extendDomain] | |
| exact g.property | |
| theorem fiveTrack_support {m : ℕ} (ι : Fin 5 ↪ Fin m) | |
| (g : alternatingGroup (Fin 5)) : (fiveTrackHom ι g).support.card ≤ 5 := by | |
| classical | |
| have hsub : (fiveTrackHom ι g).support ⊆ Finset.univ.map ι := by | |
| intro i hi | |
| by_contra hn | |
| have hn' : i ∉ Set.range ι := by simpa using hn | |
| exact (Equiv.Perm.mem_support.mp hi) (Equiv.Perm.viaEmbedding_apply_of_notMem g.val ι i hn') | |
| simpa using Finset.card_le_card hsub | |
| theorem fiveTrack_initial_generated (a : ℕ) (r : CutRing) (m : ℕ) | |
| (ι : Fin 5 ↪ Fin (m+1)) (j : Fin 5) : | |
| ConditionalAvailable (fiveTrackHom ι) (sourceGenerated a r m) (initialTest a r j) := by | |
| intro g | |
| exact sourceGenerator_mem_generated a r m | |
| (Sum.inl (j,⟨fiveTrackHom ι g,fiveTrack_sign ι g,fiveTrack_support ι g⟩)) | |
| theorem fiveTrack_whole_generated (a : ℕ) (r : CutRing) (m : ℕ) | |
| (ι : Fin 5 ↪ Fin (m+1)) : | |
| ConditionalAvailable (fiveTrackHom ι) (sourceGenerated a r m) (wholePolygon a) := | |
| fiveTrack_initial_generated a r m ι 0 | |
| noncomputable def alphabetTranslation {m : ℕ} (ι : Fin 5 ↪ Fin m) | |
| (b : Fin m) (u : CutRing × CutRing) : Fin m → CutRing × CutRing := | |
| ∑ j : Fin 5, (Pi.single (ι j) u - Pi.single b u) | |
| theorem alphabetTranslation_generated (a : ℕ) (r : CutRing) (m : ℕ) | |
| (ι : Fin 5 ↪ Fin (m+1)) (b : Fin (m+1)) (u : CutRing × CutRing) : | |
| trackTranslation (a := a) (alphabetTranslation ι b u) ∈ sourceGenerated a r m := by | |
| classical | |
| unfold alphabetTranslation | |
| induction (Finset.univ : Finset (Fin 5)) using Finset.induction_on with | |
| | empty => simp | |
| | @insert i s hi ih => | |
| rw [Finset.sum_insert hi,trackTranslation_add] | |
| exact (sourceGenerated a r m).mul_mem (balanced_difference_generated a r m b (ι i) u) ih | |
| theorem alphabetTranslation_apply {m : ℕ} (ι : Fin 5 ↪ Fin m) | |
| (b : Fin m) (hb : b ∉ Set.range ι) (u : CutRing × CutRing) (j : Fin 5) : | |
| alphabetTranslation ι b u (ι j) = u := by | |
| classical | |
| have hb' : ∀ k, ι k ≠ b := fun k h => hb ⟨k,h⟩ | |
| simp [alphabetTranslation,Finset.sum_apply,Pi.single_apply,ι.injective.eq_iff,hb'] | |
| theorem conditional_translation_conjugate {a m : ℕ} (U : polygonAlgebra a) | |
| (σ : Equiv.Perm (Fin m)) (d : Fin m → CutRing × CutRing) (u : CutRing × CutRing) | |
| (hd : ∀ i, σ i ≠ i → d i = u) : | |
| conditionalHom (⟨translate a (-u) ⁻¹' U.val, | |
| polygon_preimage_translate (-u) U.property⟩ : polygonAlgebra a) σ = | |
| trackTranslation d * conditionalHom U σ * (trackTranslation d)⁻¹ := by | |
| classical | |
| apply Subtype.ext | |
| apply Equiv.ext | |
| rintro ⟨i,p⟩ | |
| change conditionalPerm (⟨_,_⟩ : polygonAlgebra a) σ (i,p) = | |
| trackTranslationPerm d (conditionalPerm U σ ((trackTranslationPerm d).symm (i,p))) | |
| by_cases hi : σ i = i | |
| · simp [conditionalPerm,trackTranslationPerm,hi,← translate_add] | |
| · have hi' : σ (σ i) ≠ σ i := fun h => hi (σ.injective h) | |
| have hdi := hd i hi | |
| have hdsi := hd (σ i) hi' | |
| by_cases hp : translate a (-u) p ∈ U.val <;> | |
| simp_all [conditionalPerm,trackTranslationPerm,← translate_add] | |
| theorem fiveTrack_translate_generated (a : ℕ) (r : CutRing) (m : ℕ) | |
| (ι : Fin 5 ↪ Fin (m+1)) (b : Fin (m+1)) (hb : b ∉ Set.range ι) | |
| (U : polygonAlgebra a) | |
| (hU : ConditionalAvailable (fiveTrackHom ι) (sourceGenerated a r m) U) | |
| (u : CutRing × CutRing) : | |
| ConditionalAvailable (fiveTrackHom ι) (sourceGenerated a r m) | |
| ⟨translate a u ⁻¹' U.val,polygon_preimage_translate u U.property⟩ := by | |
| intro g | |
| let d := alphabetTranslation ι b (-u) | |
| have hd : ∀ i, fiveTrackHom ι g i ≠ i → d i = -u := by | |
| intro i hi | |
| have hi' : i ∈ Set.range ι := by | |
| by_contra hn | |
| exact hi (Equiv.Perm.viaEmbedding_apply_of_notMem g.val ι i hn) | |
| obtain ⟨j,rfl⟩ := hi' | |
| exact alphabetTranslation_apply ι b hb (-u) j | |
| have he := conditional_translation_conjugate U (fiveTrackHom ι g) d (-u) hd | |
| simp only [neg_neg] at he | |
| rw [he] | |
| have ht := alphabetTranslation_generated a r m ι b (-u) | |
| exact (sourceGenerated a r m).mul_mem ((sourceGenerated a r m).mul_mem ht (hU g)) | |
| ((sourceGenerated a r m).inv_mem ht) | |
| noncomputable def fiveTrackBoolean (a : ℕ) (r : CutRing) (m : ℕ) | |
| (ι : Fin 5 ↪ Fin (m+1)) : BooleanSubalgebra (Set (GenericSquare a)) := | |
| conditionalBoolean (fiveTrackHom ι) (sourceGenerated a r m) | |
| (fiveTrack_whole_generated a r m ι) | |
| theorem fiveTrackBoolean_translate (a : ℕ) (r : CutRing) (m : ℕ) | |
| (ι : Fin 5 ↪ Fin (m+1)) (b : Fin (m+1)) (hb : b ∉ Set.range ι) | |
| (u : CutRing × CutRing) {U : Set (GenericSquare a)} | |
| (hU : U ∈ fiveTrackBoolean a r m ι) : | |
| translate a u ⁻¹' U ∈ fiveTrackBoolean a r m ι := by | |
| obtain ⟨hU,hU'⟩ := hU | |
| exact ⟨polygon_preimage_translate u hU, | |
| fiveTrack_translate_generated a r m ι b hb ⟨U,hU⟩ hU' u⟩ | |
| theorem fiveTrackBoolean_initial (a : ℕ) (r : CutRing) (m : ℕ) | |
| (ι : Fin 5 ↪ Fin (m+1)) (j : Fin 5) : | |
| (initialTest a r j).val ∈ fiveTrackBoolean a r m ι := | |
| ⟨(initialTest a r j).property,fiveTrack_initial_generated a r m ι j⟩ | |
| theorem fiveTrackBoolean_coordinate (a : ℕ) (r : CutRing) (m : ℕ) | |
| (ι : Fin 5 ↪ Fin (m+1)) (b : Fin (m+1)) (hb : b ∉ Set.range ι) | |
| (j : Fin 2) (z : CutRing) : | |
| halfPlane a (Fin.castLE (by omega) j) z ∈ fiveTrackBoolean a r m ι := by | |
| apply coordinate_cuts_generated _ j | |
| (fun u _ hU => fiveTrackBoolean_translate a r m ι b hb u hU) | |
| fin_cases j | |
| · exact fiveTrackBoolean_initial a r m ι 1 | |
| · exact fiveTrackBoolean_initial a r m ι 2 | |
| end FiveTrackGeneration | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |