| import OAI.AlgebraicGeometry.CharacterVarieties.Frames.PortValues |
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|
| namespace OAI |
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|
| noncomputable section |
| namespace IntegralCharacterVarieties.SurfacePresentation.Diagram |
| open scoped Classical Matrix |
| open OccurrenceIncidence MatrixExpression HomTransport |
|
|
| /-- The evaluated framed flag equation depends only on the ports and ranks, not on the |
| boundary enumeration. -/ |
| lemma valuesFromPorts_seamHolds_canonical |
| {F S V K R : Type} {arity : S → ℕ} [Field K] [CommRing R] [Finite S] |
| (D : Diagram F S V arity) (frames : D.PortFrames (R:=K)) |
| (side : D.SideValues (R:=K)) (handle : D.HandleValues (R:=K)) |
| (s : S) (φ : R →+* K) : |
| SameFramedFlag (D.seamGrade s) |
| (matrixUnitEquiv ((D.seamLeft s).eval φ (D.valuesFromPorts frames side handle))) |
| (matrixUnitEquiv ((D.seamRight s).eval φ (D.valuesFromPorts frames side handle))) ↔ |
| SameFramedFlag ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).seamGrade s) |
| (matrixUnitEquiv |
| (((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).seamLeft s).eval φ |
| ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).valuesFromPorts |
| frames side handle))) |
| (matrixUnitEquiv |
| (((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).seamRight s).eval φ |
| ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).valuesFromPorts |
| frames side handle))) := by |
| have hl : (D.seamLeft s).eval φ (D.valuesFromPorts frames side handle) = |
| ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).seamLeft s).eval φ |
| ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).valuesFromPorts |
| frames side handle) := by |
| simp only [seamLeft, Term.eval] |
| unfold parentWord |
| erw [Term.eval_cast_unit φ (D.valuesFromPorts frames side handle) (D.seamRank s), |
| Term.eval_cast_unit φ |
| ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).valuesFromPorts |
| frames side handle) (D.seamRank s)] |
| rfl |
| have hr : (D.seamRight s).eval φ (D.valuesFromPorts frames side handle) = |
| ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).seamRight s).eval φ |
| ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).valuesFromPorts |
| frames side handle) := rfl |
| rw [hl, hr] |
| rfl |
|
|
| end IntegralCharacterVarieties.SurfacePresentation.Diagram |
| end |
|
|
| end OAI |
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