totalDeficit_flat

formal source

 154
 155/-- If every hinge satisfies the flat-sum condition, the total deficit
 156    vanishes. -/
 157theorem totalDeficit_flat {nH : ℕ}
 158    (hinges : Fin nH → SimplicialHingeData)
 159    (hFlat : ∀ h : Fin nH,
 160      DihedralAngle.FlatSumCondition (hinges h).dihedrals) :
 161    totalDeficit hinges = 0 := by
 162  unfold totalDeficit
 163  apply Finset.sum_eq_zero
 164  intro h _
 165  have : (hinges h).deficit = 0 := by
 166    unfold SimplicialHingeData.deficit
 167    exact DihedralAngle.deficit_eq_zero_of_flat _ (hFlat h)
 168  rw [this]; ring
 169
 170/-! ## §6. Certificate -/
 171
 172/-- Phase C3 certificate. -/
 173structure SchlaefliCert where
 174  flat_total_zero : ∀ {nH : ℕ} (hinges : Fin nH → SimplicialHingeData),
 175    (∀ h, DihedralAngle.FlatSumCondition (hinges h).dihedrals) →
 176    totalDeficit hinges = 0
 177  schlaefli_kills_sum : ∀ {nH nE : ℕ}
 178    (hinges : Fin nH → SimplicialHingeData)
 179    (M : DeficitDerivativeMatrix nH nE),
 180    SchlaefliIdentity hinges M →
 181    ∀ e : Fin nE,
 182      (∑ h : Fin nH, (hinges h).area * M.dThetadL h e) = 0
 183
 184theorem schlaefliCert : SchlaefliCert where
 185  flat_total_zero := fun hinges hFlat => totalDeficit_flat hinges hFlat
 186  schlaefli_kills_sum := fun hinges M hS e => schlaefli_kills_dtheta hinges M hS e
 187