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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 | import OAI.AlgebraicGeometry.CharacterVarieties.Frames.PortValues
namespace OAI
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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