| import OAI.Geometry.Arnold.FixedCount |
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| namespace OAI |
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| universe uE |
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| noncomputable section |
| open scoped BigOperators ContDiff |
| open scoped Topology NNReal |
| open Set Filter |
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| namespace ArnoldCounterexample.Midpoint |
| open scoped ContDiff Topology NNReal |
| variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
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| lemma resolvent_equilibrium (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖*C < 1) (t : ℝ) (z : E) (hz : X z = 0) : |
| resolvent X hX τ hτ (t,z) = z := by |
| symm |
| apply resolvent_unique |
| simp [hz] |
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| lemma step_equilibrium (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖*C < 1) (t : ℝ) (z : E) (hz : X z = 0) : |
| step X hX τ hτ (t,z) = z := by |
| rw [step_center, resolvent_equilibrium _ hX τ hτ t z hz, hz, smul_zero, add_zero] |
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| lemma resolvent_right_inverse (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖*C < 1) (t : ℝ) (z : E) : |
| resolvent X hX τ hτ (t, z - τ t • X z) = z := by |
| symm |
| apply resolvent_unique |
| simp |
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| lemma resolvent_kernel (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (hXs : ContDiff ℝ ∞ X) (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖*C < 1) |
| (hτs : ContDiff ℝ ∞ τ) (t : ℝ) (z v : E) (hz : X z = 0) |
| (hv : fderiv ℝ X z v = 0) : |
| fderiv ℝ (fun w ↦ resolvent X hX τ hτ (t,w)) z v = v := by |
| let R : E → E := fun w ↦ resolvent X hX τ hτ (t,w) |
| have hR : ContDiff ℝ ∞ R := (resolvent_smooth _ hX hXs _ hτ hτs).comp (contDiff_const.prodMk contDiff_id) |
| have hU := (hasFDerivAt_id (𝕜 := ℝ) z).sub ((hXs.differentiable (by simp) z).hasFDerivAt.const_smul (τ t)) |
| have he : (fun w ↦ R (w - τ t • X w)) = id := funext (resolvent_right_inverse _ hX τ hτ t) |
| have hUZ : z - τ t • X z = z := by simp [hz] |
| have hd := (hR.differentiable (by simp) (z - τ t • X z)).hasFDerivAt.comp z hU |
| change HasFDerivAt (fun w ↦ R (w - τ t • X w)) _ z at hd |
| rw [he,hUZ] at hd |
| have h := congrArg (fun L : E →L[ℝ] E ↦ L v) (hd.unique (hasFDerivAt_id z)) |
| simpa only [ContinuousLinearMap.comp_apply, sub_apply, |
| ContinuousLinearMap.id_apply, smul_apply, hv, smul_zero, sub_zero] using h |
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| lemma step_kernel (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (hXs : ContDiff ℝ ∞ X) (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖*C < 1) |
| (hτs : ContDiff ℝ ∞ τ) (t : ℝ) (z v : E) (hz : X z = 0) |
| (hv : fderiv ℝ X z v = 0) : |
| fderiv ℝ (fun w ↦ step X hX τ hτ (t,w)) z v = v := by |
| have hR : ContDiff ℝ ∞ (fun w ↦ resolvent X hX τ hτ (t,w)) := |
| (resolvent_smooth _ hX hXs _ hτ hτs).comp (contDiff_const.prodMk contDiff_id) |
| have hd := ((hR.differentiable (by simp) z).hasFDerivAt.const_smul (2:ℝ)).sub (hasFDerivAt_id z) |
| change HasFDerivAt (fun w ↦ step X hX τ hτ (t,w)) _ z at hd |
| rw [hd.fderiv] |
| change (2:ℝ) • fderiv ℝ (fun w ↦ resolvent X hX τ hτ (t,w)) z v - v = v |
| rw [resolvent_kernel _ hX hXs τ hτ hτs t z v hz hv] |
| module |
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| end ArnoldCounterexample.Midpoint |
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| end |
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| end OAI |
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