openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /ForceGerm.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.InitializedFluid | |
| namespace OAI | |
| /-! The flat-coordinate force depends only on a germ. No smoothness away | |
| from the point is needed for this locality statement. -/ | |
| noncomputable section | |
| open Filter | |
| open scoped Topology | |
| open ShearFlows | |
| namespace ForcedComputation | |
| theorem force_congr_germ {U V : Velocity} {t : ℝ} {x : Space} | |
| (he : U =ᶠ[𝓝 (t, x)] V) (ν : ℝ) : force ν U (t, x) = force ν V (t, x) := by | |
| have ht : (fun s => U (s, x)) =ᶠ[𝓝 t] fun s => V (s, x) := | |
| he.comp_tendsto (continuous_id.prodMk continuous_const).continuousAt | |
| have hx : (fun z => U (t, z)) =ᶠ[𝓝 x] fun z => V (t, z) := | |
| he.comp_tendsto (continuous_const.prodMk continuous_id).continuousAt | |
| have hd (j : Fin 3) : derivative (fun z => U (t, z)) j =ᶠ[𝓝 x] | |
| derivative (fun z => V (t, z)) j := | |
| (hx.fderiv (𝕜 := ℝ)).mono (fun z hz => congrArg (fun A => A (basis j)) hz) | |
| have hl : laplacian (fun z => U (t, z)) x = laplacian (fun z => V (t, z)) x := by | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| exact congrArg (fun A => A (basis j)) ((hd j).fderiv_eq (𝕜 := ℝ)) | |
| change deriv (fun s => U (s, x)) t - ν • laplacian (fun z => U (t, z)) x = | |
| deriv (fun s => V (s, x)) t - ν • laplacian (fun z => V (t, z)) x | |
| rw [ht.deriv_eq, hl] | |
| end ForcedComputation | |
| end | |
| end OAI | |