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59d1c80 | 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 | import OAI.Geometry.IsometricImmersion.Taylor.TaylorUpperCauchyBounds
import OAI.Geometry.IsometricImmersion.Comparison.ActualComparisonEquation
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Taylor
theorem comparison_upper_velocity_jet_bound {P z : Coord → ℝ} {U : Set Coord} {I : Set ℝ}
(hU : IsOpen U) (hI : IsOpen I) (hP : ContDiffOn ℝ ∞ P U) (hz : ContDiffOn ℝ ∞ z U)
(b : ℝ) (hupper : ∀ x ∈ I, (![x,b] : Coord) ∈ U) {x : ℝ} (hx : x ∈ I)
(N : ℕ) {BZ BP : ℝ}
(hZB : ‖iteratedFDeriv ℝ N (heightCauchyVelocity z b) x‖ ≤ BZ)
(hPB : ‖iteratedFDeriv ℝ N (heightCauchyVelocity P b) x‖ ≤ BP) :
‖iteratedFDeriv ℝ N (heightCauchyVelocity (comparisonDifference P z) b) x‖ ≤ BZ+BP := by
have hzv := heightCauchyVelocity_contDiffOn hz hU b hupper
have hPv := heightCauchyVelocity_contDiffOn hP hU b hupper
have heq : heightCauchyVelocity (comparisonDifference P z) b =ᶠ[𝓝 x]
(heightCauchyVelocity z b-heightCauchyVelocity P b) := by
filter_upwards [hI.mem_nhds hx] with y hy
exact CoordinateBound.iterated_sub hz hP hU [1] (hupper y hy)
rw [(heq.iteratedFDeriv ℝ N).self_of_nhds,iteratedFDeriv_sub_apply
((hzv.contDiffAt (hI.mem_nhds hx)).of_le (WithTop.coe_le_coe.mpr le_top))
((hPv.contDiffAt (hI.mem_nhds hx)).of_le (WithTop.coe_le_coe.mpr le_top))]
exact (norm_sub_le _ _).trans (add_le_add hZB hPB)
end SmoothLocal.HighEquation
end
end OAI
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