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7f5fab8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import OAI.AlgebraicGeometry.CharacterVarieties.Frames.FramedFlags
namespace OAI
/-!
# Transport of local frames along seams
The column and parent coordinate identifications transport local frame matrices
between the two ends of a surface seam.
-/
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
end OAI
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