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110f4f3 | 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open scoped NNReal
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology
open MeasureTheory
open scoped ContDiff ENNReal
open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory
open scoped Topology ContDiff Manifold Bundle ENNReal
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology Manifold
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Bundle
open scoped Topology Bundle
open scoped Topology
open Function Manifold Set
open Manifold Bundle
open scoped Manifold Bundle
open Set
open Set Filter
open scoped Topology ContDiff
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter
open scoped ContDiff Topology
namespace WeakMTWTransport
lemma contDiffAt_uncurry_iteratedDeriv {E F : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{f : E → ℝ → F} {x : E} {t : ℝ}
(hf : ContDiffAt ℝ ∞ (Function.uncurry f) (x,t)) (k : ℕ) :
ContDiffAt ℝ ∞ (fun q : E×ℝ => iteratedDeriv k (f q.1) q.2) (x,t) := by
induction k with
| zero =>
apply hf.congr_of_eventuallyEq
exact Filter.Eventually.of_forall (fun q => congrFun (iteratedDeriv_zero (f := f q.1)) q.2)
| succ k ih =>
have hcomp : ContDiffAt ℝ ∞
(Function.uncurry (fun q : E×ℝ => fun s : ℝ => iteratedDeriv k (f q.1) s)) ((x,t),t) :=
ih.comp ((x,t),t) ((contDiffAt_fst.fst).prodMk contDiffAt_snd)
have hd : ContDiffAt ℝ ∞
(fun q : E×ℝ => fderiv ℝ (iteratedDeriv k (f q.1)) q.2) (x,t) :=
hcomp.fderiv contDiffAt_snd (by simp)
simpa only [iteratedDeriv_succ,deriv] using hd.clm_apply (contDiffAt_const (c := (1:ℝ)))
lemma contDiffAt_iteratedDeriv_parameter {E F : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{f : E → ℝ → F} {x : E} {t : ℝ}
(hf : ContDiffAt ℝ ∞ (Function.uncurry f) (x,t)) (k : ℕ) :
ContDiffAt ℝ ∞ (fun q : E => iteratedDeriv k (f q) t) x :=
(contDiffAt_uncurry_iteratedDeriv hf k).comp x (contDiffAt_id.prodMk contDiffAt_const)
end WeakMTWTransport
end
end OAI
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