| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
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| open Set Filter |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
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| lemma fderiv_fst_eq_of_line_stationary {B : E×F → ℝ} {v a : E} {q : F} |
| (hB : DifferentiableAt ℝ B (v,q)) |
| (hline : HasDerivAt (fun s : ℝ => B (v+s • a,q)) 0 0) : |
| fderiv ℝ B (v,q) (a,0) = 0 := by |
| have hp : HasDerivAt (fun s : ℝ => (v+s • a,q)) (a,0) 0 := by |
| convert! (((hasDerivAt_const (0:ℝ) v).add ((hasDerivAt_id (0:ℝ)).smul_const a)).prodMk |
| (hasDerivAt_const (0:ℝ) q)) using 1; simp |
| have hb : HasFDerivAt B (fderiv ℝ B (v,q)) (v+(0:ℝ) • a,q) := by simpa using hB.hasFDerivAt |
| have h := hb.comp_hasDerivAt 0 hp |
| exact h.unique hline |
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| lemma fderiv_snd_eq_of_partial {B : E×F → ℝ} {v : E} {q : F} {D : F →L[ℝ] ℝ} |
| (hB : DifferentiableAt ℝ B (v,q)) |
| (hpart : HasFDerivAt (fun y : F => B (v,y)) D q) (u : F) : |
| fderiv ℝ B (v,q) (0,u) = D u := by |
| have hp := (hasFDerivAt_const v q).prodMk (hasFDerivAt_id (𝕜 := ℝ) q) |
| have h := hB.hasFDerivAt.comp q hp |
| have heq := h.unique hpart |
| exact congrArg (fun A : F →L[ℝ] ℝ => A u) heq |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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