| import OAI.Dynamics.StandardMap.SegmentObservations |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open MeasureTheory Set Filter Metric |
| open scoped Topology ENNReal |
| namespace StandardMapEntropy |
| lemma wedge_quarterTurn_both (a b : ℂ) : wedge (quarterTurn a) (quarterTurn b)=wedge a b := by |
| simp [wedge,quarterTurn] |
| ring |
| lemma wedge_chain_bound (s : ℕ → ℂ) (e : ℕ → ℝ) (a b : ℕ) (hab : a ≤ b) |
| (hu : ∀ i, a ≤ i → i ≤ b → ‖s i‖=1) |
| (he : ∀ i, a ≤ i → i < b → |wedge (s i) (s (i+1))| ≤ e i) : |
| |wedge (s a) (s b)| ≤ ∑ i ∈ Finset.Ico a b, e i := by |
| induction b,hab using Nat.le_induction with |
| | base => simp [wedge_refl] |
| | succ b hab ih => |
| have hh:=ih (fun i hai hib => hu i hai (by omega)) (fun i hai hib => he i hai (by omega)) |
| have ht:=wedge_sine_triangle (s a) (s b) (s (b+1)) (hu a le_rfl (by omega)) |
| (hu b hab (by omega)) (hu (b+1) (by omega) le_rfl) |
| rw [Finset.sum_Ico_succ_top hab] |
| exact ht.trans (add_le_add hh (he b hab (by omega))) |
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| lemma stable_direction_step {A B : ℂ →L[ℝ] ℂ} (hA : PlaneAreaPreserving A) |
| (a b : ℂ) (ha : ‖a‖=1) (_hb : ‖b‖=1) |
| (hAa : ‖A a‖=‖A‖) (hBb : ‖B (quarterTurn b)‖=‖B‖⁻¹) |
| (M : ℝ) (hB : 0 < ‖B‖) (hstep : ∀ z, ‖A z‖ ≤ M*‖B z‖) : |
| |wedge (quarterTurn a) (quarterTurn b)| ≤ M/(‖A‖*‖B‖) := by |
| have hpos : 0 < ‖A‖ := lt_of_lt_of_le zero_lt_one hA.singular_pair.choose_spec.2.2.2.2 |
| have hp:=expanded_projection_bound hA a ha hAa (quarterTurn b) |
| rw [dot_quarterTurn_right] at hp |
| have hs:=hstep (quarterTurn b) |
| rw [hBb] at hs |
| rw [wedge_quarterTurn_both] |
| apply (le_div_iff₀ (mul_pos hpos hB)).mpr |
| rw [abs_neg] at hp |
| have hh := mul_le_mul_of_nonneg_right (hp.trans hs) hB.le |
| calc |
| _ = (‖A‖*|wedge a b|)*‖B‖ := by ring |
| _ ≤ (M*‖B‖⁻¹)*‖B‖ := hh |
| _ = M := by field_simp |
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| lemma stable_prefix_finite_bound (A : ℕ → ℂ →L[ℝ] ℂ) (a : ℕ → ℂ) |
| (D : ℕ → ℝ) (M : ℝ) (l B : ℕ) (hl : 1 ≤ l) (hlB : l ≤ B) |
| (harea : ∀ i, 1 ≤ i → i ≤ B → PlaneAreaPreserving (A i)) |
| (hunit : ∀ i, 1 ≤ i → i ≤ B → ‖a i‖=1) |
| (hnorm : ∀ i, 1 ≤ i → i ≤ B → ‖A i‖=D i) |
| (hmax : ∀ i, 1 ≤ i → i ≤ B → ‖A i (a i)‖=D i) |
| (hstable : ∀ i, 1 ≤ i → i ≤ B → ‖A i (quarterTurn (a i))‖=(D i)⁻¹) |
| (hstep : ∀ i, 1 ≤ i → i < B → ∀ z, ‖A i z‖ ≤ M*‖A (i+1) z‖) : |
| ‖A l (quarterTurn (a B))‖ ≤ (D l)⁻¹+D l*∑ i ∈ Finset.Ico l B, M/(D i*D (i+1)) := by |
| have hpos (i : ℕ) (hi : 1 ≤ i) (hiB : i ≤ B) : 0 < ‖A i‖ := |
| lt_of_lt_of_le zero_lt_one (harea i hi hiB).singular_pair.choose_spec.2.2.2.2 |
| have hchain:=wedge_chain_bound (fun i => quarterTurn (a i)) |
| (fun i => M/(D i*D (i+1))) l B hlB |
| (fun i hli hiB => by rw [norm_quarterTurn]; exact hunit i (by omega) hiB) |
| (by |
| intro i hli hiB |
| have hh:=stable_direction_step (harea i (by omega) (by omega)) (a i) (a (i+1)) |
| (hunit i (by omega) (by omega)) (hunit (i+1) (by omega) (by omega)) |
| (by rw [hmax i (by omega) (by omega),hnorm i (by omega) (by omega)]) |
| (by rw [hstable (i+1) (by omega) (by omega),hnorm (i+1) (by omega) (by omega)]) |
| M (hpos (i+1) (by omega) (by omega)) (hstep i (by omega) hiB) |
| simpa only [hnorm i (by omega) (by omega),hnorm (i+1) (by omega) (by omega)] using hh) |
| rw [wedge_quarterTurn_both] at hchain |
| have hup:=two_singular_upper (harea l hl hlB) (a l) (hunit l hl hlB) |
| (by rw [hmax l hl hlB,hnorm l hl hlB]) (quarterTurn (a B)) |
| rw [dot_quarterTurn_right,abs_neg,hnorm l hl hlB] at hup |
| have hw:=abs_wedge_le_norm_mul (a l) (quarterTurn (a B)) |
| rw [norm_quarterTurn,hunit l hl hlB,hunit B (by omega) le_rfl,mul_one] at hw |
| have hD : 0 ≤ D l := by rw [← hnorm l hl hlB]; exact norm_nonneg _ |
| calc |
| _ ≤ D l*|wedge (a l) (a B)|+(D l)⁻¹*|wedge (a l) (quarterTurn (a B))| := hup |
| _ ≤ D l*(∑ i ∈ Finset.Ico l B, M/(D i*D (i+1)))+(D l)⁻¹*1 := |
| add_le_add (mul_le_mul_of_nonneg_left hchain hD) (mul_le_mul_of_nonneg_left hw (inv_nonneg.mpr hD)) |
| _ = _ := by ring |
| end StandardMapEntropy |
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| end OAI |
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