| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.AlignedEndingRegularity |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology Real |
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| lemma endingJoin_overlap {u : Coord3 → Fin 2 → ℝ} {lam k J L K : ℝ} |
| (hk : 0<k) (hJ : 0<J) |
| (hu : ∀ x : Coord3,4≤x 0 → u x=![x 0,exp (-lam*x 0)*cos (k*x 2)]) |
| {x : Coord3} (hx : x 0∈Icc (6:ℝ) 7) : |
| u=ᶠ[𝓝 x] axialPair (delayedEndingValue lam k J L K 7) := by |
| have h1 : ∀ᶠ y : Coord3 in 𝓝 x,4<y 0 := |
| (isOpen_lt continuous_const (continuous_apply 0)).mem_nhds (show 4<x 0 by linarith [hx.1]) |
| have h2 : ∀ᶠ y : Coord3 in 𝓝 x,k*(y 0-7)<1 := |
| (isOpen_lt (continuous_const.mul ((continuous_apply 0).sub continuous_const)) continuous_const).mem_nhds (show k*(x 0-7)<1 by |
| have hh := mul_nonpos_of_nonneg_of_nonpos hk.le (sub_nonpos.mpr hx.2) |
| linarith) |
| filter_upwards [h1,h2] with y hy1 hy2 |
| rw [hu y hy1.le] |
| simp only [axialPair,delayedEndingValue_initial_of_bound hk hJ y hy2.le] |
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| lemma matchedEndingPair_overlap {u : Coord3 → Fin 2 → ℝ} {lam k J L K : ℝ} {x : Coord3} |
| (he : u=ᶠ[𝓝 x] axialPair (delayedEndingValue lam k J L K 7)) : |
| matchedEndingPair u lam k J L K=ᶠ[𝓝 x] u := by |
| filter_upwards [he] with y hy |
| dsimp only [matchedEndingPair] |
| split_ifs |
| · rfl |
| · exact hy.symm |
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| lemma matchedEndingPair_left_deriv {u : Coord3 → Fin 2 → ℝ} {lam k J L K : ℝ} {x : Coord3} |
| (hx : x 0<13/2) : |
| fderiv ℝ (matchedEndingPair u lam k J L K) x=fderiv ℝ u x := |
| (axial_paste_eventually_left (f:=u) (g:=axialPair (delayedEndingValue lam k J L K 7)) |
| (continuous_apply 0) hx).fderiv_eq |
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| lemma matchedEndingPair_right_deriv {u : Coord3 → Fin 2 → ℝ} {lam k J L K : ℝ} {x : Coord3} |
| (hx : 13/2<x 0) : |
| fderiv ℝ (matchedEndingPair u lam k J L K) x=fderiv ℝ (axialPair (delayedEndingValue lam k J L K 7)) x := |
| (axial_paste_eventually_right (f:=u) (g:=axialPair (delayedEndingValue lam k J L K 7)) |
| (continuous_apply 0) hx).fderiv_eq |
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| end ScalarConductivity |
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| end |
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| end OAI |
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