| import Mathlib |
| import OAI.Algebra.GroupRing.WordCursor |
| import OAI.Algebra.GroupRing.ConeBoundaryPaths |
| import OAI.Algebra.GroupRing.ConeChains |
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| namespace OAI |
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| section |
| namespace TorsionFreeZeroDivisors.ConeSigned |
| open scoped Classical |
| open PortSubdivision GraphPresentation GraphConePresentation |
| noncomputable section |
| variable {V D S : Type} (Γ : PortGraph V D) (R : Routes Γ) (label : D→S×Bool) |
| def pack : (S×Bool) ⊕ (ConePicture.Radial Γ R × Bool) ≃ (Generator (S:=S) Γ R × Bool) where |
| toFun | .inl (s,b)=>(.inl s,b) | .inr (v,b)=>(.inr v,b) |
| invFun | (.inl s,b)=>.inl (s,b) | (.inr v,b)=>.inr (v,b) |
| left_inv x := by rcases x with ⟨s,b⟩|⟨v,b⟩ <;> rfl |
| right_inv x := by rcases x with ⟨s|v,b⟩ <;> rfl |
| lemma pack_inverse (a : (S×Bool) ⊕ (ConePicture.Radial Γ R × Bool)) : |
| pack (S:=S) Γ R (ConePicture.inverse Γ R WordPairing.inverseLetter a)= |
| WordPairing.inverseLetter (pack Γ R a) := by |
| rcases a with ⟨s,b⟩|⟨v,b⟩ <;> rfl |
| def labels : ConePicture.Dart Γ R→Generator (S:=S) Γ R × Bool := pack Γ R ∘ ConePicture.labels Γ R label |
| lemma labels_reverse (hl : ∀d,label (Γ.reverse d)=WordPairing.inverseLetter (label d)) |
| (d : ConePicture.Dart Γ R) : |
| labels Γ R label ((ConePicture.graph Γ R).reverse d)=WordPairing.inverseLetter (labels Γ R label d) := by |
| dsimp only [labels,Function.comp_apply] |
| rw [ConePicture.label_reverse Γ R label WordPairing.inverseLetter hl,pack_inverse] |
| def unpack {O : Type}[Fintype O] |
| (a : SphericalPicture.Data (O:=O) (ConePicture.graph Γ R) (labels Γ R label) WordPairing.inverseLetter) : |
| SphericalPicture.Data (O:=O) (ConePicture.graph Γ R) (ConePicture.labels Γ R label) |
| (ConePicture.inverse Γ R WordPairing.inverseLetter) where |
| rotation := a.rotation |
| partner := a.partner |
| involutive := a.involutive |
| fixed_free := a.fixed_free |
| dart := a.dart |
| consistent := a.consistent |
| compatible i := by |
| apply (pack Γ R).injective |
| rw [pack_inverse] |
| exact a.compatible i |
| genusZero := a.genusZero |
| lemma reduces_to_empty (hl : ∀d,label (Γ.reverse d)=WordPairing.inverseLetter (label d)) |
| (hex : ∀{P : Type}[Fintype P][Nonempty P] |
| (a : SphericalPicture.Data (O:=P) Γ label WordPairing.inverseLetter),¬a.Reduced) |
| {O : Type}[Fintype O] |
| (a : SphericalPicture.Data (O:=O) (ConePicture.graph Γ R) (labels Γ R label) WordPairing.inverseLetter) : |
| SphericalPicture.ReducesToEmpty _ _ _ a := by |
| apply SphericalPicture.reduces_of_no_reduced WordPairing.inverseLetter_involutive (labels_reverse Γ R label hl) |
| intro P _ _ b hb |
| exact hex (ConePicture.oldData (unpack Γ R label b) hb) |
| (ConePicture.oldData_reduced _ hb) |
| def freeLetter {A : Type} (a : A×Bool) : FreeGroup A := FreeGroup.mk [a] |
| lemma freeLetter_inverse {A : Type} (a : A×Bool) : |
| freeLetter (WordPairing.inverseLetter a)=(freeLetter a)⁻¹ := by |
| change FreeGroup.mk [WordPairing.inverseLetter a]=(FreeGroup.mk [a])⁻¹ |
| rw [FreeGroup.inv_mk] |
| rfl |
| lemma freeLetter_map {A B : Type} (f : A→B) (a : A×Bool) : |
| FreeGroup.map f (freeLetter a)=freeLetter (f a.1,a.2) := by |
| simp only [freeLetter,FreeGroup.map.mk,List.map_singleton] |
| local notation "letters" => freeLetter ∘ label |
| local notation "G" => GraphConePresentation.Group Γ letters R |
| def eval (a : Generator (S:=S) Γ R × Bool) : G := PresentedGroup.mk _ (freeLetter a) |
| lemma eval_inverse (a : Generator (S:=S) Γ R × Bool) : |
| eval Γ R label (WordPairing.inverseLetter a)=(eval Γ R label a)⁻¹ := by |
| simp only [eval,freeLetter_inverse,map_inv] |
| lemma eval_pack (a : (S×Bool) ⊕ (ConePicture.Radial Γ R × Bool)) : |
| eval Γ R label (pack Γ R a)=ConeChains.eval Γ R label freeLetter a := by |
| rcases a with ⟨s,b⟩|⟨v,b⟩ |
| · change PresentedGroup.mk _ (freeLetter (Sum.inl s,b))= |
| oldMap Γ letters R (freeLetter (s,b)) |
| rw [oldMap,MonoidHom.comp_apply,old,freeLetter_map] |
| · cases b |
| · change PresentedGroup.mk _ (FreeGroup.of (Sum.inr v))⁻¹= |
| (radialValue Γ letters R v)⁻¹ |
| rw [map_inv,radialValue,radial,dite_eq_right v.property] |
| · change PresentedGroup.mk _ (FreeGroup.of (Sum.inr v))=radialValue Γ letters R v |
| rw [radialValue,radial,dite_eq_right v.property] |
| lemma wordLabel_eq (d : ConePicture.Dart Γ R) : |
| freeLetter (labels Γ R label d)=ConePicture.wordLabel Γ R letters d := by |
| rcases d with d|⟨v,b⟩ |
| · change freeLetter (Sum.inl (label d).1,(label d).2)=FreeGroup.map Sum.inl (freeLetter (label d)) |
| exact (freeLetter_map _ _).symm |
| · cases b <;> rfl |
| lemma value_mk {A H : Type} [Group H] (q : FreeGroup A→*H) (w : List (A×Bool)) : |
| WordCursor.value (fun a=>q (freeLetter a)) w=q (FreeGroup.mk w) := by |
| induction w with |
| | nil=>exact (map_one q).symm |
| | cons a w ih=>rw [WordCursor.value_cons,ih,show a::w=[a]++w from rfl,←FreeGroup.mul_mk,map_mul];rfl |
| lemma eval_value (w : List (Generator (S:=S) Γ R × Bool)) : |
| WordCursor.value (eval Γ R label) w=PresentedGroup.mk _ (FreeGroup.mk w) := |
| value_mk _ w |
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| end |
| end TorsionFreeZeroDivisors.ConeSigned |
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| end |
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| end OAI |
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