| import OAI.Geometry.IsometricImmersion.Taylor.TaylorCauchyData | |
| import OAI.Geometry.IsometricImmersion.Darboux.QSameStripMargins | |
| import OAI.Geometry.IsometricImmersion.Calculus.ShearHeightJets | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function | |
| open scoped ContDiff Topology Matrix | |
| namespace SmoothLocal.Taylor | |
| open SmoothLocal.Geometry SmoothLocal.HighEquation SmoothLocal.Pulse SmoothLocal.Flow | |
| def initialShearedStateBudget (Z q0 : ℝ) : ℝ := | |
| max 1 ((2*(1+|q0|))^2*Z) | |
| theorem initialShearedStateBudget_nonneg (Z q0 : ℝ) : 0 ≤ initialShearedStateBudget Z q0 := | |
| zero_le_one.trans (le_max_left _ _) | |
| theorem actual_initial_sheared_state_bound | |
| {z : Coord → ℝ} {U : Set Coord} (hz : ContDiffOn ℝ ∞ z U) | |
| (hU : IsOpen U) (hSU : modelSquare ⊆ U) {Z : ℝ} (hZ : 0 ≤ Z) | |
| (hjet : ∀ k ≤ 8, ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k z p‖ ≤ Z) | |
| (q0 left right a : ℝ) | |
| (hcut : ∀ x ∈ Ioo left right, inverseShearCoordinates q0 ![x,a] ∈ modelSquare) | |
| {x : ℝ} (hx : x ∈ Ioo left right) (hpB : ‖(![x,a] : Coord)‖ ≤ 1) : | |
| ‖qSolutionJet (linearCauchy a (heightCauchyValue (heightInShearCoordinates z q0) a) | |
| (heightCauchyVelocity (heightInShearCoordinates z q0) a)) ![x,a]‖ ≤ | |
| initialShearedStateBudget Z q0 := by | |
| have hz2 : CoordinateBound z modelSquare 2 Z := by | |
| apply coordinateBound_of_frechet_bounds hz hU hSU | |
| intro k hk p hp | |
| exact hjet k (by omega) p hp | |
| have hzs := heightInShearCoordinates_contDiffOn hz q0 | |
| have hV := hU.preimage (inverseShearCoordinates_contDiff q0).continuous | |
| rw [linearCauchy_height_initial_state hzs hV left right a | |
| (fun x hx => hSU (hcut x hx)) hx] | |
| have hb := SmoothLocal.Pulse.CoordinateBound.inverse_shear hz2 hz hU hSU hZ q0 | |
| apply qSolutionJet_norm_bound_of_words (by norm_num) hpB | |
| intro ds hds | |
| exact hb ds hds _ (hcut x hx) | |
| end SmoothLocal.Taylor | |
| end | |
| end OAI | |