openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Comparison /ComparisonUpperVelocity.lean
| 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 | |