| import OAI.AlgebraicGeometry.CharacterVarieties.Frames.Endpoints |
| import OAI.AlgebraicGeometry.CharacterVarieties.Frames.CutFrames |
| import OAI.AlgebraicGeometry.CharacterVarieties.Cutting.GraftValues |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace IntegralCharacterVarieties.NamedBandGrades.IdentifiedBand |
| open scoped Classical |
| open TwoFlagBand OccurrenceIncidence VertexTable |
| variable {K : Type} [Field K] {n r : ℕ} {s : Fin n → ℕ} |
| {f h : (((i : Fin n) × Fin (s i)) → K) ≃ₗ[K] (Fin r → K)} |
| (w : IdentifiedBand s f h) |
|
|
| lemma mirrorColor_eq (T : MatrixIso K (Fin r) (Fin r)) |
| (c : Option (Secondary n n)) (hc : c≠none) : w.mirrorColorGauge T c=w.colorGauge c := by |
| cases c with |
| | none => exact (hc rfl).elim |
| | some c => rfl |
|
|
| lemma child_color_ne (v : Fin (w.shape.atomicBand.length+1)) |
| (p : (w.shape.atomicBand.kind v).table.Port) (a : (w.shape.atomicBand.kind v).table.Child p) : |
| (w.shape.atomicBand.decoration v).color ⟨p,some a⟩≠none := by |
| obtain ⟨d,hd⟩ := w.shape.atomicBand_fresh v |
| intro hh |
| have he : ((w.shape.atomicBand.kind v).freshDecoration d).color ⟨p,some a⟩=none := hd ▸ hh |
| rcases ((w.shape.atomicBand.kind v).fresh_color_none d _).mp he with hh|hh <;> cases hh |
| |
| lemma mirror_basis (T : MatrixIso K (Fin r) (Fin r)) |
| (v : Fin (w.shape.atomicBand.length+1)) (z) |
| (hz : (w.shape.atomicBand.decoration v).color z≠none) : |
| ((w.mirrorVertexBases T v).basis ((w.shape.atomicBand.kind v).mirrorEnd z).1 |
| ((w.shape.atomicBand.kind v).mirrorEnd z).2).reindex |
| (finCongr ((w.shape.vertexAtoms v).mirror.rank_of_atoms (w.shape.vertexAtoms v) |
| ((w.shape.vertexAtoms v).mirror_atoms z))).symm |
| (finCongr ((w.shape.vertexAtoms v).mirror.rank_of_atoms (w.shape.vertexAtoms v) |
| ((w.shape.vertexAtoms v).mirror_atoms z))).symm = |
| (w.vertexBases v).basis z.1 z.2 := by |
| let e := (w.shape.atomicBand.kind v).mirrorEnd |
| let c := (w.shape.atomicBand.decoration v).color z |
| let c' := (w.shape.atomicBand.decoration v).color (e.symm (e z)) |
| have he : c'=c := congrArg (w.shape.atomicBand.decoration v).color (e.symm_apply_apply z) |
| have hn : c'≠none := fun h => hz (he.symm.trans h) |
| have hm := w.mirrorColor_eq T c' hn |
| have hr := (w.shape.vertexAtoms v).mirror.rank_of_atoms (w.shape.vertexAtoms v) |
| ((w.shape.vertexAtoms v).mirror_atoms z) |
| have hh := MatrixIso.finite_basis_transport (freshRank r w.shape.secondaryRank) w.colorGauge |
| c c' ((w.shape.vertexAtoms v).rank z) ((w.shape.vertexAtoms v).mirror.rank (e z)) |
| (w.shape.vertexAtoms_rank v z) (w.shape.vertexAtoms_rank v (e.symm (e z))) he.symm hr.symm |
| exact (congrArg (fun g => (g.reindex |
| (finCongr (w.shape.vertexAtoms_rank v (e.symm (e z)))) |
| (finCongr (w.shape.vertexAtoms_rank v (e.symm (e z))))).reindex |
| (finCongr hr.symm) (finCongr hr.symm)) hm).trans hh |
|
|
| lemma transverse_parent_color (v : Fin (w.shape.atomicBand.length+1)) |
| (p : (w.shape.atomicBand.kind v).table.Port) |
| (hin : p≠(w.shape.atomicBand.kind v).input) (hout : p≠(w.shape.atomicBand.kind v).output) : |
| (w.shape.atomicBand.decoration v).color ⟨p,none⟩≠none := by |
| obtain ⟨d,hd⟩ := w.shape.atomicBand_fresh v |
| intro hh |
| have he : ((w.shape.atomicBand.kind v).freshDecoration d).color ⟨p,none⟩=none := hd ▸ hh |
| rcases ((w.shape.atomicBand.kind v).fresh_color_none d _).mp he with hh|hh |
| · exact hin (congrArg Sigma.fst hh) |
| · exact hout (congrArg Sigma.fst hh) |
|
|
| /-- The transverse branch has smaller parent rank, so its parent and child transports are identities |
| under the mirror principal gauge. -/ |
| lemma transverse_frame_seam (T : MatrixIso K (Fin r) (Fin r)) |
| (v : Fin (w.shape.atomicBand.length+1)) (p : (w.shape.atomicBand.kind v).table.Port) |
| (hin : p≠(w.shape.atomicBand.kind v).input) (hout : p≠(w.shape.atomicBand.kind v).output) : |
| (w.mirrorVertexFrames T v ((w.shape.atomicBand.kind v).mirrorPort p)).reindex |
| ((w.shape.vertexAtoms v).seamColumns (w.shape.vertexAtoms v).mirror |
| p ((w.shape.atomicBand.kind v).mirrorPort p) ((w.shape.atomicBand.kind v).mirrorChild p) |
| (fun a => ((w.shape.vertexAtoms v).mirror_atoms ⟨p,some a⟩).symm)) |
| (finCongr ((w.shape.vertexAtoms v).rank_of_atoms (w.shape.vertexAtoms v).mirror |
| ((w.shape.vertexAtoms v).mirror_atoms ⟨p,none⟩).symm))=w.vertexFrames v p := by |
| apply (w.shape.vertexAtoms v).reframe_seam (w.shape.vertexAtoms v).mirror |
| (w.vertexBases v) (w.mirrorVertexBases T v) p ((w.shape.atomicBand.kind v).mirrorPort p) |
| ((w.shape.atomicBand.kind v).mirrorChild p) |
| ((w.shape.vertexAtoms v).mirror_atoms ⟨p,none⟩).symm |
| (fun a => ((w.shape.vertexAtoms v).mirror_atoms ⟨p,some a⟩).symm) |
| · exact w.mirror_basis T v ⟨p,none⟩ (w.transverse_parent_color v p hin hout) |
| · intro a |
| exact w.mirror_basis T v ⟨p,some a⟩ (w.child_color_ne v p a) |
|
|
| end IntegralCharacterVarieties.NamedBandGrades.IdentifiedBand |
| end |
|
|
| noncomputable section |
| namespace IntegralCharacterVarieties.SurfacePresentation.Diagram |
| open scoped Classical |
| open OccurrenceIncidence VertexTable TwoFlagBand NamedBandGrades |
| variable {F S V K : Type} {arity : S → ℕ} [Field K] |
| (D : Diagram F S V arity) (q : S) [Finite V] |
| {f h : (((i : Fin (arity q)) × Fin (D.childDim q i)) → K) ≃ₗ[K] |
| (Fin (D.rank (D.ports.facet ⟨q,none⟩)) → K)} |
| (w : IdentifiedBand (D.childDim q) f h) |
| local notation "C" => D.refinedCutDiagram q w.shape rfl w.rowRanks w.colRanks |
| lemma namedCut_orig_localRanks (v : Fin (w.shape.atomicBand.length+1)) : |
| (w.shape.vertexAtoms v).localRanks=(C).vertexRanks (.inr (.inl v)) := by |
| apply LocalRanks.ext_rank |
| funext p c |
| exact D.refinedCut_orig_ranks q w.shape rfl w.rowRanks w.colRanks v p c |
| lemma namedCut_mirror_localRanks (v : Fin (w.shape.atomicBand.length+1)) : |
| (w.shape.vertexAtoms v).mirror.localRanks=(C).vertexRanks (.inr (.inr v)) := by |
| apply LocalRanks.ext_rank |
| funext p c |
| exact D.refinedCut_mirror_ranks q w.shape rfl w.rowRanks w.colRanks v p c |
|
|
| def namedCutOrigFrames (v : Fin (w.shape.atomicBand.length+1)) := |
| LocalRanks.transportFrames (D.namedCut_orig_localRanks q w v) (w.vertexFrames v) |
| def namedCutMirrorFrames (T : MatrixIso K (Fin (D.rank (D.ports.facet ⟨q,none⟩))) |
| (Fin (D.rank (D.ports.facet ⟨q,none⟩)))) (v : Fin (w.shape.atomicBand.length+1)) := |
| LocalRanks.transportFrames (D.namedCut_mirror_localRanks q w v) (w.mirrorVertexFrames T v) |
|
|
| def namedCutVertexFrames (old : D.PortFrames (R:=K)) |
| (T : MatrixIso K (Fin (D.rank (D.ports.facet ⟨q,none⟩))) |
| (Fin (D.rank (D.ports.facet ⟨q,none⟩)))) |
| (v : V ⊕ (Fin (w.shape.atomicBand.length+1) ⊕ Fin (w.shape.atomicBand.length+1))) : |
| (p : ((C).ports.kind v).table.Port) → |
| MatrixIso K (((C).vertexRanks v).Columns p) (((C).vertexRanks v).Parent p) := by |
| rcases v with v|v|v |
| · exact D.refinedOldFrames q w.shape rfl w.rowRanks w.colRanks old v |
| · exact D.namedCutOrigFrames q w v |
| · exact D.namedCutMirrorFrames q w T v |
|
|
| def namedCutPortFrames (old : D.PortFrames (R:=K)) |
| (T : MatrixIso K (Fin (D.rank (D.ports.facet ⟨q,none⟩))) |
| (Fin (D.rank (D.ports.facet ⟨q,none⟩)))) : (C).PortFrames (R:=K) := |
| fun p => D.namedCutVertexFrames q w old T p.1 p.2 |
|
|
| lemma namedCutPortFrames_holds (old : D.PortFrames (R:=K)) |
| (T : MatrixIso K (Fin (D.rank (D.ports.facet ⟨q,none⟩))) |
| (Fin (D.rank (D.ports.facet ⟨q,none⟩)))) |
| (hold : ∀ v,(LocalRanks.comparison (D.ports.kind v) (D.vertexRanks v) |
| (fun p => old ⟨v,p⟩)).Holds) |
| (v) : (LocalRanks.comparison ((C).ports.kind v) ((C).vertexRanks v) |
| (fun p => D.namedCutPortFrames q w old T ⟨v,p⟩)).Holds := by |
| rcases v with v|v|v |
| · exact LocalRanks.transportFrames_holds _ _ (hold v) |
| · exact LocalRanks.transportFrames_holds _ _ (w.vertexFrames_holds v) |
| · exact LocalRanks.transportFrames_holds _ _ (w.mirrorVertexFrames_holds T v) |
| end IntegralCharacterVarieties.SurfacePresentation.Diagram |
| end |
|
|
| noncomputable section |
| namespace IntegralCharacterVarieties.SurfacePresentation.Diagram |
| open scoped Classical Matrix |
| open OccurrenceIncidence VertexTable TwoFlagBand NamedBandGrades |
| variable {F S V R : Type} {arity : S → ℕ} [CommRing R] |
| (D : Diagram F S V arity) (q : S) |
| {r : ℕ} (d : RankShape (arity q) (arity q) r) |
| (hp : D.rank (D.ports.facet ⟨q,none⟩)=r) |
| (hc : ∀ i,D.rank (D.ports.facet ⟨q,some i⟩)=d.secondaryRank (.row i)) |
| (hc' : ∀ i,D.rank (D.ports.facet ⟨q,some i⟩)=d.secondaryRank (.col i)) |
| local notation "B" => D.ports.refinedBandForSeam q d |
| local notation "A" => D.ports.mapFacet (Sum.inl : F → D.ports.RefinedBandFacet q d) |
|
|
| /-- The selected original parent side changes from P to J; children retain their transports over the |
| coefficient ring. -/ |
| def cutOldSideValues (old : D.SideValues (R:=R)) (J : (Matrix (Fin (D.rank (D.ports.facet ⟨q,none⟩))) (Fin (D.rank (D.ports.facet ⟨q,none⟩))) R)ˣ) : D.SideValues (R:=R) |
| | ⟨s,none⟩ => if h : s=q then h.symm ▸ J else old ⟨s,none⟩ |
| | ⟨s,some i⟩ => old ⟨s,some i⟩ |
|
|
| lemma cutOldSideValues_selected (old : D.SideValues (R:=R)) (J : (Matrix (Fin (D.rank (D.ports.facet ⟨q,none⟩))) (Fin (D.rank (D.ports.facet ⟨q,none⟩))) R)ˣ) : |
| D.cutOldSideValues q old J ⟨q,none⟩=J := by simp only [cutOldSideValues,dite_true] |
| lemma cutOldSideValues_child (old : D.SideValues (R:=R)) (J : (Matrix (Fin (D.rank (D.ports.facet ⟨q,none⟩))) (Fin (D.rank (D.ports.facet ⟨q,none⟩))) R)ˣ) (s) (i) : |
| D.cutOldSideValues q old J ⟨s,some i⟩=old ⟨s,some i⟩ := rfl |
|
|
| /-- New strip sides have identity transport except for the mirror principal short seam, which |
| carries Y = T⁻¹ J⁻¹ P T. -/ |
| def cutShortSideValues (Y : (Matrix (Fin (D.rank (D.ports.facet ⟨q,none⟩))) (Fin (D.rank (D.ports.facet ⟨q,none⟩))) R)ˣ) (b : Bool) (i : Option (Fin (arity q))) : |
| FacetUnit (R:=R) D.rank (D.ports.facet ⟨q,i⟩) := by |
| cases i with |
| | none => exact if b then Y else 1 |
| | some _ => exact 1 |
|
|
| lemma refinedGenus_inl_le (f : F) : D.refinedGenus q d (.inl f)≤D.genus f := by |
| change (if f=D.ports.facet ⟨q,none⟩ then D.genus f-1 else D.genus f)≤D.genus f |
| split_ifs <;> omega |
| |
| variable [Finite V] |
| local notation "C" => D.refinedCutDiagram q d hp hc hc' |
| |
| def refinedCutSideValues (old : D.SideValues (R:=R)) (J Y : (Matrix (Fin (D.rank (D.ports.facet ⟨q,none⟩))) (Fin (D.rank (D.ports.facet ⟨q,none⟩))) R)ˣ) : (C).SideValues (R:=R) := |
| graftSideValues (A) (B).doublePatch (doubleDecoration (B).decoration) q |
| ((B).patch.double_signatureMatch (B).decoration (B).patch_signatureMatch (B).shortFirst (B).shortLast) |
| (fun i => by rw [(B).double_signature_plus]; cases i <;> rfl) |
| (fun i => by rw [(B).double_signature_minus]; cases i <;> rfl) |
| (D.ports.refinedRank q d D.rank) (D.cutOldSideValues q old J) (D.cutShortSideValues q Y) |
| |
| /-- The original handles are retained except for the last handle of the selected parent. New strip |
| facets are disks. -/ |
| def refinedCutHandleValues (old : D.HandleValues (R:=R)) : (C).HandleValues (R:=R) := by |
| intro f i b |
| cases f with |
| | inl f => exact old f ⟨i.val,i.isLt.trans_le (D.refinedGenus_inl_le q d f)⟩ b |
| | inr f => exact Fin.elim0 i |
| end IntegralCharacterVarieties.SurfacePresentation.Diagram |
| end |
|
|
| end OAI |
|
|
