| import OAI.AlgebraicGeometry.CharacterVarieties.Frames.FramedFlags |
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|
| namespace OAI |
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|
| /-! |
| |
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| The column and parent coordinate identifications transport local frame matrices |
| between the two ends of a surface seam. |
| -/ |
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|
| noncomputable section |
| namespace IntegralCharacterVarieties.SurfacePresentation.Diagram |
| open scoped Classical Matrix |
| open OccurrenceIncidence VertexTable MatrixExpression |
| variable {F S V R : Type} {arity : S → ℕ} [CommRing R] |
| (D : Diagram F S V arity) |
| lemma portParentMatch_index (p u : LocalPort V D.ports.kind) |
| (h : (D.ports.attach p).1=(D.ports.attach u).1) : |
| (D.portParentMatch p u h).trans (D.portRowIndex u)= |
| (D.portRowIndex p).trans (finCongr (congrArg D.seamDim h)) := by |
| apply Equiv.ext |
| intro i |
| apply Fin.ext |
| rfl |
| lemma localFrameUnit_match (f : D.PortFrames (R:=R)) (p u : LocalPort V D.ports.kind) |
| (h : (D.ports.attach p).1=(D.ports.attach u).1) |
| (hs : (f u).reindex (D.portColumnMatch p u h) (D.portParentMatch p u h)=f p) : |
| (MatrixIso.unit (D.localFrameUnit f u)).reindex |
| (finCongr (congrArg D.seamDim h)) (finCongr (congrArg D.seamDim h))= |
| MatrixIso.unit (D.localFrameUnit f p) := by |
| have hh := congrArg (fun z => z.reindex (D.portColumnIndex p).symm (D.portRowIndex p).symm) hs |
| simp only [MatrixIso.reindex_reindex_eq,D.portColumnMatch_from_index,D.portParentMatch_from_index] at hh |
| simpa only [localFrameUnit,MatrixIso.unit_toUnit,MatrixIso.reindex_reindex_eq] using hh |
| lemma localFrameUnit_rebase_match (f : D.PortFrames (R:=R)) (p u : LocalPort V D.ports.kind) |
| (h : (D.ports.attach p).1=(D.ports.attach u).1) |
| (hs : (f u).reindex (D.portColumnMatch p u h) (D.portParentMatch p u h)=f p) : |
| rebaseUnit (congrArg D.seamDim h).symm (D.localFrameUnit f u)=D.localFrameUnit f p := by |
| have hh := D.localFrameUnit_match f p u h hs |
| have he : finCongr (congrArg D.seamDim h)=finCongr (congrArg D.seamDim h).symm.symm := rfl |
| rw [he,←rebaseUnit_iso] at hh |
| exact congrArg MatrixIso.toUnit hh |
| lemma frameValues_match (f : D.PortFrames (R:=R)) (p u : LocalPort V D.ports.kind) |
| (h : (D.ports.attach p).1=(D.ports.attach u).1) |
| (hs : (f u).reindex (D.portColumnMatch p u h) (D.portParentMatch p u h)=f p) : |
| rebaseUnit (congrArg D.seamDim h).symm |
| (D.frameValues f (D.ports.attach u).1 (D.ports.attach u).2)= |
| D.frameValues f (D.ports.attach p).1 (D.ports.attach p).2 := by |
| rw [D.frameValues_attach,D.frameValues_attach] |
| exact D.localFrameUnit_rebase_match f p u h hs |
| lemma frameValues_seam_of_ports (f : D.PortFrames (R:=R)) |
| (p u : LocalPort V D.ports.kind) (s : S) |
| (hp : D.ports.attach p=(s,false)) (hu : D.ports.attach u=(s,true)) |
| (h : (D.ports.attach p).1=(D.ports.attach u).1) |
| (hs : (f u).reindex (D.portColumnMatch p u h) (D.portParentMatch p u h)=f p) : |
| D.frameValues f s false=D.frameValues f s true := by |
| have hh := D.frameValues_match f p u h hs |
| have aux {a b : S × Bool} (ha : a=(s,false)) (hb : b=(s,true)) |
| (he : a.1=b.1) |
| (ht : rebaseUnit (congrArg D.seamDim he).symm |
| (D.frameValues f b.1 b.2)=D.frameValues f a.1 a.2) : |
| D.frameValues f s false=D.frameValues f s true := by |
| subst a |
| subst b |
| simpa only [rebaseUnit_refl] using ht.symm |
| exact aux hp hu h hh |
| lemma attachedEnds_same (s : S) : |
| (D.ports.attach (D.ports.attach.symm (s,false))).1= |
| (D.ports.attach (D.ports.attach.symm (s,true))).1 := by |
| simp only [Equiv.apply_symm_apply] |
| lemma frameValues_seam_match (f : D.PortFrames (R:=R)) (s : S) |
| (hs : (f (D.ports.attach.symm (s,true))).reindex |
| (D.portColumnMatch (D.ports.attach.symm (s,false)) (D.ports.attach.symm (s,true)) |
| (D.attachedEnds_same s)) |
| (D.portParentMatch (D.ports.attach.symm (s,false)) (D.ports.attach.symm (s,true)) |
| (D.attachedEnds_same s))=f (D.ports.attach.symm (s,false))) : |
| D.frameValues f s false=D.frameValues f s true := by |
| have hh := D.frameValues_match f (D.ports.attach.symm (s,false)) |
| (D.ports.attach.symm (s,true)) (D.attachedEnds_same s) hs |
| have aux {a b : S × Bool} (ha : a=(s,false)) (hb : b=(s,true)) |
| (he : a.1=b.1) |
| (ht : rebaseUnit (congrArg D.seamDim he).symm |
| (D.frameValues f b.1 b.2)=D.frameValues f a.1 a.2) : |
| D.frameValues f s false=D.frameValues f s true := by |
| subst a |
| subst b |
| simpa only [rebaseUnit_refl] using ht.symm |
| exact aux (D.ports.attach.apply_symm_apply (s,false)) |
| (D.ports.attach.apply_symm_apply (s,true)) (D.attachedEnds_same s) hh |
| end IntegralCharacterVarieties.SurfacePresentation.Diagram |
| end |
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| end OAI |
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