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2fca478 | 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 | import Mathlib
import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
namespace OAI
/-! Bootstrap Calculus. -/
section
noncomputable section
namespace FixedBootstrapCalculus
variable {E F G H : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
[NormedAddCommGroup H] [NormedSpace ℝ H]
lemma derivative_compatibility (B : E → E →L[ℝ] F) (C : G → G →L[ℝ] H)
(L : E →L[ℝ] G) (M : F →L[ℝ] H)
(hB : Differentiable ℝ B) (hC : Differentiable ℝ C)
(he : ∀ y v, C (L y) (L v) = M (B y v)) (u w v : E) :
fderiv ℝ C (L u) (L w) (L v) = M (fderiv ℝ B u w v) := by
have hL := ((hC (L u)).hasFDerivAt.comp u L.hasFDerivAt).clm_apply
(hasFDerivAt_const (L v) u)
have hR := M.hasFDerivAt.comp u
((hB u).hasFDerivAt.clm_apply (hasFDerivAt_const v u))
have he' : (fun y => C (L y) (L v)) = (fun y => M (B y v)) := funext (fun y => he y v)
simp only [Function.comp_def] at hL hR
rw [he'] at hL
have hh := congrArg (fun T : E →L[ℝ] H => T w) (hL.unique hR)
simpa only [add_apply, ContinuousLinearMap.comp_apply,
zero_apply, map_zero, zero_add, ContinuousLinearMap.flip_apply] using hh
lemma differentiated_commutator (B : E → E →L[ℝ] F)
(L P : G →L[ℝ] E) (Q M : H →L[ℝ] F) (V N : G → H)
(hB : Differentiable ℝ B) (hV : Differentiable ℝ V) (hN : Differentiable ℝ N)
(he : ∀ y, B (L y) (P y) = Q (V y) + M (N y)) (u w : G) :
B (L u) (P w) + fderiv ℝ B (L u) (L w) (P u) =
Q (fderiv ℝ V u w) + M (fderiv ℝ N u w) := by
have hL := ((hB (L u)).hasFDerivAt.comp u L.hasFDerivAt).clm_apply P.hasFDerivAt
have hR := (Q.hasFDerivAt.comp u (hV u).hasFDerivAt).add
(M.hasFDerivAt.comp u (hN u).hasFDerivAt)
have he' : (fun y => B (L y) (P y)) = (fun y => Q (V y) + M (N y)) := funext he
simp only [Function.comp_def] at hL hR
rw [he'] at hL
have hh := congrArg (fun T : G →L[ℝ] F => T w) (hL.unique hR)
simpa only [add_apply, ContinuousLinearMap.comp_apply,
ContinuousLinearMap.flip_apply] using hh
lemma lowered_remainder (M : H →L[ℝ] F) (a b c : F) (r e : H)
(h : a + b = c + M r) (hb : b = M e) : a = c + M (r-e) := by
rw [hb] at h
rw [map_sub]
exact (eq_sub_iff_add_eq.mpr h).trans (by abel)
end FixedBootstrapCalculus
end
end
end OAI
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