| import OAI.Geometry.HarmonicGrowth.ScalarODE |
| import OAI.Geometry.HarmonicGrowth.Taylor |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℝ R] |
|
|
| / |
| size (a-a_*) times the long pulse duration. -/ |
| lemma slow_residual_identity (A₀ p p₀ v θ τ : ℝ) (S Sdot Err : R) : |
| (A₀ • (1:R)+((p₀+2*θ)*τ) • S+Err)- |
| p • (v • (1:R)+τ • S)-(v • (1:R)+τ • S)*(v • (1:R)+τ • S)- |
| ((A₀-p*v-v^2) • (1:R)+(τ^2) • Sdot) = |
| Err+((p₀+2*θ-p-2*v)*τ) • S-(τ^2) • (S*S+Sdot) := by |
| simp only [add_mul,mul_add,smul_mul_assoc,mul_smul_comm,one_mul,mul_one,smul_smul,smul_add] |
| module |
|
|
| / |
| p are independent of the CURRENT control parameter. -/ |
| lemma slow_residual_jet_identity (p p₀ v θ τ : ℝ) (S Sh Shdot Errh : R) : |
| (((p₀+2*θ)*τ) • Sh+Errh)-p • (τ • Sh)- |
| (τ • Sh)*(v • (1:R)+τ • S)-(v • (1:R)+τ • S)*(τ • Sh)- |
| (τ^2) • Shdot = |
| Errh+((p₀+2*θ-p-2*v)*τ) • Sh-(τ^2) • (Sh*S+S*Sh+Shdot) := by |
| simp only [add_mul,mul_add,smul_mul_assoc,mul_smul_comm,one_mul,mul_one,smul_smul,smul_add] |
| module |
|
|
| / |
| lemma slow_residual_bound {Err S Sdot : R} {C δ τ ε : ℝ} |
| (hC : 0 ≤ C) (hδ : 0 ≤ δ) (hτ : 0 ≤ τ) |
| (he : ‖Err‖ ≤ C*(δ*τ+τ^2)) (hε : |ε| ≤ 3*C*δ) |
| (hS : ‖S‖ ≤ C) (hSd : ‖Sdot‖ ≤ C) : |
| ‖Err+(ε*τ) • S-(τ^2) • (S*S+Sdot)‖ ≤ |
| (3*C^2+2*C)*(δ*τ+τ^2) := by |
| have hn : ‖S*S+Sdot‖ ≤ C^2+C := by |
| calc |
| _ ≤ ‖S*S‖+‖Sdot‖ := norm_add_le _ _ |
| _ ≤ ‖S‖*‖S‖+C := add_le_add (norm_mul_le _ _) hSd |
| _ ≤ C*C+C := by nlinarith [mul_self_le_mul_self (norm_nonneg S) hS] |
| _ = _ := by ring |
| calc |
| _ ≤ ‖Err‖+‖(ε*τ) • S‖+‖(τ^2) • (S*S+Sdot)‖ := |
| (norm_sub_le _ _).trans (add_le_add_left (norm_add_le _ _) _) |
| _ = ‖Err‖+|ε| *τ*‖S‖+τ^2*‖S*S+Sdot‖ := by |
| simp only [norm_smul,Real.norm_eq_abs,abs_mul,abs_of_nonneg hτ,abs_of_nonneg (sq_nonneg τ),mul_assoc] |
| _ ≤ C*(δ*τ+τ^2)+(3*C*δ)*τ*C+τ^2*(C^2+C) := by |
| gcongr |
| _ ≤ _ := by nlinarith [mul_nonneg (sq_nonneg C) (sq_nonneg τ),mul_nonneg hC (mul_nonneg hδ hτ)] |
|
|
| lemma slow_residual_jet_bound {Errh S Sh Shdot : R} {C δ τ ε : ℝ} |
| (hC : 0 ≤ C) (hδ : 0 ≤ δ) (hτ : 0 ≤ τ) |
| (he : ‖Errh‖ ≤ C*(δ*τ+τ^2)) (hε : |ε| ≤ 3*C*δ) |
| (hS : ‖S‖ ≤ C) (hSh : ‖Sh‖ ≤ C) (hShd : ‖Shdot‖ ≤ C) : |
| ‖Errh+(ε*τ) • Sh-(τ^2) • (Sh*S+S*Sh+Shdot)‖ ≤ |
| (3*C^2+2*C)*(δ*τ+τ^2) := by |
| have hn : ‖Sh*S+S*Sh+Shdot‖ ≤ 2*C^2+C := by |
| calc |
| _ ≤ ‖Sh*S+S*Sh‖+‖Shdot‖ := norm_add_le _ _ |
| _ ≤ (‖Sh*S‖+‖S*Sh‖)+C := add_le_add (norm_add_le _ _) hShd |
| _ ≤ (‖Sh‖*‖S‖+‖S‖*‖Sh‖)+C := by gcongr <;> exact norm_mul_le _ _ |
| _ ≤ (C*C+C*C)+C := by gcongr |
| _ = _ := by ring |
| calc |
| _ ≤ ‖Errh‖+‖(ε*τ) • Sh‖+‖(τ^2) • (Sh*S+S*Sh+Shdot)‖ := |
| (norm_sub_le _ _).trans (add_le_add_left (norm_add_le _ _) _) |
| _ = ‖Errh‖+|ε| *τ*‖Sh‖+τ^2*‖Sh*S+S*Sh+Shdot‖ := by |
| simp only [norm_smul,Real.norm_eq_abs,abs_mul,abs_of_nonneg hτ,abs_of_nonneg (sq_nonneg τ),mul_assoc] |
| _ ≤ C*(δ*τ+τ^2)+(3*C*δ)*τ*C+τ^2*(2*C^2+C) := by gcongr |
| _ ≤ _ := by nlinarith [mul_nonneg (sq_nonneg C) (sq_nonneg τ),mul_nonneg hC (mul_nonneg hδ hτ)] |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| open HarmonicCounterexample.PulseTaylor |
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℝ R] |
|
|
| / |
| Riccati denominator. The angular infinitesimal is r B Id+D. -/ |
| def bergerSlowGenerator (r B β p₀ θ z : ℝ) (D : R) : R := |
| (β/(p₀+2*θ)*z) • (r • (B • (1:R))+D) |
|
|
| / |
| exact Berger comparison. The constant is chosen before the scalar history, |
| small deviation, duration, pulse position and parameters. All nonlinear |
| angular remainders are proved, not hypotheses. -/ |
| theorem berger_comparison_residuals (r B β p₀ θ : ℝ) (D : R) |
| {M Z K : ℝ} (hM : 0 ≤ M) (hZ : 0 ≤ Z) (hK : 0 ≤ K) |
| (hc : p₀+2*θ ≠ 0) : |
| ∃ C : ℝ,0 ≤ C ∧ ∀ α p v δ τ z zd zh zhd : ℝ, |
| 0 ≤ δ → 0 ≤ τ → |α| ≤ M → |α-β| ≤ δ → |p-p₀| ≤ δ → |v-θ| ≤ K*δ → |
| |z| ≤ Z → |zd| ≤ Z → |zh| ≤ Z → |zhd| ≤ Z → |
| 1+τ*z ∈ Set.Icc (1/2:ℝ) (3/2) → |
| let S := bergerSlowGenerator r B β p₀ θ z D |
| let Sd := bergerSlowGenerator r B β p₀ θ zd D |
| let Sh := bergerSlowGenerator r B β p₀ θ zh D |
| let Shd := bergerSlowGenerator r B β p₀ θ zhd D |
| let Q := v • (1:R)+τ • S |
| ‖α • angular r (B • (1:R)) D (1+τ*z)-p • Q-Q*Q- |
| ((α*B-p*v-v^2) • (1:R)+(τ^2) • Sd)‖ ≤ C*(δ*τ+τ^2) ∧ |
| ‖(τ*zh) • (α • angularD r (B • (1:R)) D (1+τ*z))- |
| p • (τ • Sh)-(τ • Sh)*Q-Q*(τ • Sh)-(τ^2) • Shd‖ ≤ C*(δ*τ+τ^2) := by |
| obtain ⟨C₀,hC₀,hrem⟩ := berger_slow_pulse_remainders r (B • (1:R)) D hM hZ |
| let L := r • (B • (1:R))+D |
| let A := |β/(p₀+2*θ)| *Z*‖L‖ |
| let C₁ := C₀+A+K+1 |
| have hA : 0 ≤ A := by dsimp [A]; positivity |
| have hC₁ : 0 ≤ C₁ := by dsimp [C₁];linarith |
| have h₀ : C₀ ≤ C₁ := by dsimp [C₁];linarith |
| have h₁ : A ≤ C₁ := by dsimp [C₁];linarith |
| have h₂ : K ≤ C₁ := by dsimp [C₁];linarith |
| have h₃ : 1 ≤ C₁ := by dsimp [C₁];linarith |
| refine ⟨3*C₁^2+2*C₁,by positivity,?_⟩ |
| intro α p v δ τ z zd zh zhd hδ hτ hα hscale hp hv hz hzd hzh hzhd hq |
| dsimp only |
| have hSn (x : ℝ) (hx : |x| ≤ Z) : ‖bergerSlowGenerator r B β p₀ θ x D‖ ≤ C₁ := by |
| calc |
| _ = |β/(p₀+2*θ)| *|x| *‖L‖ := by simp only [bergerSlowGenerator,norm_smul,Real.norm_eq_abs,abs_mul,L] |
| _ ≤ A := by dsimp [A];gcongr |
| _ ≤ C₁ := h₁ |
| have hε : |p₀+2*θ-p-2*v| ≤ 3*C₁*δ := by |
| have he : p₀+2*θ-p-2*v= -(p-p₀)-2*(v-θ) := by ring |
| rw [he] |
| calc |
| _ ≤ |-(p-p₀)|+|2*(v-θ)| := abs_sub _ _ |
| _ = |p-p₀|+2*|v-θ| := by rw [abs_neg,abs_mul,abs_of_nonneg (by norm_num : (0:ℝ) ≤ 2)] |
| _ ≤ δ+2*(K*δ) := by gcongr |
| _ ≤ 3*C₁*δ := by nlinarith [mul_le_mul_of_nonneg_right h₂ hδ,mul_le_mul_of_nonneg_right h₃ hδ] |
| obtain ⟨he,hej⟩ := hrem α β δ τ z zh hδ hτ hα hscale hz hzh hq |
| have hfac (x : ℝ) : ((p₀+2*θ)*τ) • bergerSlowGenerator r B β p₀ θ x D= |
| (β*τ*x) • L := by |
| rw [bergerSlowGenerator,smul_smul] |
| change (((p₀+2*θ)*τ)*(β/(p₀+2*θ)*x)) • L=(β*τ*x) • L |
| congr 1 |
| field_simp |
| let Err := α • angular r (B • (1:R)) D (1+τ*z)-α • (B • (1:R))-(β*τ*z) • L |
| let Errh := (τ*zh) • (α • angularD r (B • (1:R)) D (1+τ*z)-β • L) |
| have he' : ‖Err‖ ≤ C₁*(δ*τ+τ^2) := he.trans (mul_le_mul_of_nonneg_right h₀ (by positivity)) |
| have hej' : ‖Errh‖ ≤ C₁*(δ*τ+τ^2) := hej.trans (mul_le_mul_of_nonneg_right h₀ (by positivity)) |
| have hvId : (α*B) • (1:R)+((p₀+2*θ)*τ) • bergerSlowGenerator r B β p₀ θ z D+Err= |
| α • angular r (B • (1:R)) D (1+τ*z) := by |
| rw [hfac] |
| dsimp only [Err] |
| simp only [smul_smul] |
| abel |
| have hjId : ((p₀+2*θ)*τ) • bergerSlowGenerator r B β p₀ θ zh D+Errh= |
| (τ*zh) • (α • angularD r (B • (1:R)) D (1+τ*z)) := by |
| rw [hfac] |
| dsimp only [Errh] |
| simp only [smul_sub,smul_smul] |
| rw [show τ*zh*β=β*τ*zh by ring] |
| abel |
| constructor |
| · rw [← hvId,slow_residual_identity] |
| exact slow_residual_bound hC₁ hδ hτ he' hε (hSn z hz) (hSn zd hzd) |
| · rw [← hjId,slow_residual_jet_identity] |
| exact slow_residual_jet_bound hC₁ hδ hτ hej' hε (hSn z hz) (hSn zh hzh) (hSn zhd hzhd) |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| open Set |
|
|
| / |
| its limiting root by O(delta), uniformly in the pulse length. -/ |
| theorem berger_scalar_baseline_close {α p : ℝ → ℝ} |
| (hα : Continuous α) (hp : Continuous p) |
| {M Mp B β p₀ θ γ δ T : ℝ} |
| (hM : 0 ≤ M) (hMp : 0 ≤ Mp) (hθ : 0 ≤ θ) (hγ : 0 < γ) (hδ : 0 ≤ δ) |
| (hαn : ∀ t,|α t| ≤ M) (hpn : ∀ t,|p t| ≤ Mp) |
| (hAp : ∀ t,0 ≤ t → 0 ≤ α t*B) (hroot : β*B-p₀*θ-θ^2=0) |
| (hscale : ∀ t ∈ Icc 0 T,|α t-β| ≤ δ) |
| (hpt : ∀ t ∈ Icc 0 T,|p t-p₀| ≤ δ) |
| (hpl : ∀ t ∈ Icc 0 T,γ ≤ p t) : |
| ∀ t ∈ Icc 0 T,|scalarBaseline (fun s => α s*B) p θ t-θ| ≤ |
| ((|B|+|θ|)/γ)*δ := by |
| have hAB (t : ℝ) : |α t*B| ≤ M*|B| := by rw [abs_mul];gcongr;exact hαn t |
| have hF (t : ℝ) (ht : t ∈ Icc 0 T) : |
| |α t*B-p t*θ-θ^2| ≤ (|B|+|θ|)*δ := by |
| have he : α t*B-p t*θ-θ^2=(α t-β)*B-(p t-p₀)*θ := by nlinarith [hroot] |
| rw [he] |
| calc |
| _ ≤ |(α t-β)*B|+|(p t-p₀)*θ| := abs_sub _ _ |
| _ = |α t-β| *|B|+|p t-p₀| *|θ| := by rw [abs_mul,abs_mul] |
| _ ≤ δ*|B|+δ*|θ| := by gcongr;exact hscale t ht;exact hpt t ht |
| _ = _ := by ring |
| have hh := actual_scalar_baseline_close (hα.mul continuous_const) hp |
| (show 0 ≤ M*|B| by positivity) hMp hAB hpn hAp hθ hγ |
| (show 0 ≤ (|B|+|θ|)*δ by positivity) hpl hF |
| intro t ht |
| exact (hh t ht).trans_eq (by ring) |
|
|
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℝ R] |
|
|
| / |
| The resulting C is fixed BEFORE all histories, scalar coefficient functions, |
| period lengths, pulse positions and control parameters. This is the analytic |
| uniformity required for recursive attainment, not a compactness claim about |
| an unspecified incoming Riccati matrix. -/ |
| theorem actual_berger_comparison_residuals (r B β p₀ θ γ : ℝ) (D : R) |
| {M Z : ℝ} (hM : 0 ≤ M) (hZ : 0 ≤ Z) (hθ : 0 ≤ θ) (hγ : 0 < γ) |
| (hc : p₀+2*θ ≠ 0) (hroot : β*B-p₀*θ-θ^2=0) : |
| ∃ C : ℝ,0 ≤ C ∧ ∀ (α p : ℝ → ℝ) (Mp δ T τ : ℝ), |
| Continuous α → Continuous p → 0 ≤ Mp → 0 ≤ δ → 0 ≤ τ → |
| (∀ t,|α t| ≤ M) → (∀ t,|p t| ≤ Mp) → |
| (∀ t,0 ≤ t → 0 ≤ α t*B) → |
| (∀ t ∈ Icc 0 T,|α t-β| ≤ δ) → |
| (∀ t ∈ Icc 0 T,|p t-p₀| ≤ δ) → |
| (∀ t ∈ Icc 0 T,γ ≤ p t) → |
| ∀ t ∈ Icc 0 T,∀ z zd zh zhd : ℝ, |
| |z| ≤ Z → |zd| ≤ Z → |zh| ≤ Z → |zhd| ≤ Z → |
| 1+τ*z ∈ Icc (1/2:ℝ) (3/2) → |
| let v := scalarBaseline (fun s => α s*B) p θ t |
| let S := bergerSlowGenerator r B β p₀ θ z D |
| let Sd := bergerSlowGenerator r B β p₀ θ zd D |
| let Sh := bergerSlowGenerator r B β p₀ θ zh D |
| let Shd := bergerSlowGenerator r B β p₀ θ zhd D |
| let Q := v • (1:R)+τ • S |
| ‖α t • PulseTaylor.angular r (B • (1:R)) D (1+τ*z)-p t • Q-Q*Q- |
| ((α t*B-p t*v-v^2) • (1:R)+(τ^2) • Sd)‖ ≤ C*(δ*τ+τ^2) ∧ |
| ‖(τ*zh) • (α t • PulseTaylor.angularD r (B • (1:R)) D (1+τ*z))- |
| p t • (τ • Sh)-(τ • Sh)*Q-Q*(τ • Sh)-(τ^2) • Shd‖ ≤ C*(δ*τ+τ^2) := by |
| obtain ⟨C,hC,hres⟩ := berger_comparison_residuals r B β p₀ θ D hM hZ |
| (show 0 ≤ (|B|+|θ|)/γ by positivity) hc |
| refine ⟨C,hC,?_⟩ |
| intro α p Mp δ T τ hα hp hMp hδ hτ hαn hpn hAp hscale hpt hpl t ht z zd zh zhd hz hzd hzh hzhd hq |
| exact hres (α t) (p t) (scalarBaseline (fun s => α s*B) p θ t) δ τ z zd zh zhd |
| hδ hτ (hαn t) (hscale t ht) (hpt t ht) |
| (berger_scalar_baseline_close hα hp hM hMp hθ hγ hδ hαn hpn hAp hroot hscale hpt hpl t ht) |
| hz hzd hzh hzhd hq |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℝ R] |
|
|
| / |
| of the rescaled comparison, not a syntactically assigned jet. -/ |
| lemma bergerComparison_time_deriv (r B β p₀ θ τ : ℝ) (D : R) |
| {v z : ℝ → ℝ} {v' z' t : ℝ} |
| (hv : HasDerivAt v v' t) (hz : HasDerivAt z z' (τ*t)) : |
| HasDerivAt (fun s => v s • (1:R)+τ • bergerSlowGenerator r B β p₀ θ (z (τ*s)) D) |
| (v' • (1:R)+(τ^2) • bergerSlowGenerator r B β p₀ θ z' D) t := by |
| have hτ : HasDerivAt (fun s : ℝ => τ*s) τ t := by simpa only [mul_one,id_eq] using (hasDerivAt_id t).const_mul τ |
| have hh := (hv.smul_const (1:R)).add |
| ((((hz.comp t hτ).const_mul (β/(p₀+2*θ))).smul_const (r • (B • (1:R))+D)).const_smul τ) |
| apply hh.congr_deriv |
| simp only [bergerSlowGenerator,smul_smul] |
| module |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℝ H] |
|
|
| / |
| the exact scalar baseline is independent of the current control parameter. -/ |
| lemma bergerComparison_parameter_deriv (r B β p₀ θ τ v : ℝ) (D : R) |
| {z : H → ℝ} {z' : H →L[ℝ] ℝ} {x : H} (hz : HasFDerivAt z z' x) : |
| HasFDerivAt (fun y => v • (1:R)+τ • bergerSlowGenerator r B β p₀ θ (z y) D) |
| (z'.smulRight (τ • bergerSlowGenerator r B β p₀ θ 1 D)) x := by |
| have h1 := (hz.const_mul (β/(p₀+2*θ))).smul_const (r • (B • (1:R))+D) |
| have hh := (h1.const_smul τ).const_add (v • (1:R)) |
| apply hh.congr_fderiv |
| ext h |
| simp only [bergerSlowGenerator,ContinuousLinearMap.smulRight_apply, |
| smul_apply,smul_smul,mul_one,smul_eq_mul] |
| module |
|
|
| / |
| including the current scale alpha and slow-pulse factor tau. -/ |
| lemma bergerAngular_parameter_deriv (r B α τ : ℝ) (D : R) |
| {z : H → ℝ} {z' : H →L[ℝ] ℝ} {x : H} (hz : HasFDerivAt z z' x) |
| (hq : 1+τ*z x ≠ 0) : |
| HasFDerivAt (fun y => α • PulseTaylor.angular r (B • (1:R)) D (1+τ*z y)) |
| (z'.smulRight ((α*τ) • PulseTaylor.angularD r (B • (1:R)) D (1+τ*z x))) x := by |
| have hh := (((PulseTaylor.angular_hasDerivAt r (B • (1:R)) D hq).hasFDerivAt).comp x |
| ((hz.const_mul τ).const_add 1)).const_smul α |
| apply hh.congr_fderiv |
| ext h |
| simp only [ContinuousLinearMap.smulRight_apply,smul_apply, |
| ContinuousLinearMap.comp_apply,ContinuousLinearMap.toSpanSingleton_apply, |
| smul_smul,smul_eq_mul] |
| module |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| open scoped InnerProductSpace |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| / |
| positive scalar baseline. Symmetry is not used or silently assumed here. -/ |
| lemma scalar_plus_small_positive {S : E →L[ℝ] E} {v τ C : ℝ} |
| (hτ : 0 ≤ τ) (hSn : ‖S‖ ≤ C) (hgap : τ*C ≤ v) : |
| ∀ x : E,0 ≤ inner ℝ x ((v • (1:E →L[ℝ] E)+τ • S) x) := by |
| intro x |
| have hn := S.le_opNorm x |
| have hb : ‖S x‖ ≤ C*‖x‖ := hn.trans (mul_le_mul_of_nonneg_right hSn (norm_nonneg x)) |
| have hh := neg_le_of_abs_le (abs_real_inner_le_norm x (S x)) |
| have he := mul_le_mul_of_nonneg_left hb (norm_nonneg x) |
| have hτh := mul_le_mul_of_nonneg_left hh hτ |
| have ht := mul_le_mul_of_nonneg_left he hτ |
| have hv := mul_le_mul_of_nonneg_right hgap (sq_nonneg ‖x‖) |
| simp only [add_apply,smul_apply, |
| one_apply_eq_self,inner_add_right,inner_smul_right, real_inner_self_eq_norm_sq] |
| nlinarith |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| open MeasureTheory Set |
|
|
| / |
| negative-time zeros. No global nonvanishing hypothesis is needed for FTC. -/ |
| lemma scalarBaseline_stronglyMeasurable {A p : ℝ → ℝ} |
| (hA : Continuous A) (hp : Continuous p) {MA Mp : ℝ} |
| (hMA : 0 ≤ MA) (hMp : 0 ≤ Mp) (hAn : ∀ t,|A t| ≤ MA) |
| (hpn : ∀ t,|p t| ≤ Mp) (θ : ℝ) : |
| StronglyMeasurable (scalarBaseline A p θ) := by |
| have hy : Continuous (fun t => (scalarPhase A p 1 θ t).1) := |
| continuous_iff_continuousAt.2 fun t => |
| (scalarPhase_value_deriv hA hp hMA hMp hAn hpn 1 θ t).continuousAt |
| have hv : Continuous (fun t => (scalarPhase A p 1 θ t).2) := |
| continuous_iff_continuousAt.2 fun t => |
| (scalarPhase_velocity_deriv hA hp hMA hMp hAn hpn 1 θ t).continuousAt |
| exact (hv.measurable.div hy.measurable).stronglyMeasurable |
|
|
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℝ R] |
|
|
| / |
| from PB flow, not chosen as an unproved solution of a Riccati equation. -/ |
| def actualBergerComparison (α p : ℝ → ℝ) (r B β p₀ θ τ : ℝ) (D : R) |
| (z : ℝ → ℝ) (t : ℝ) : R := |
| scalarBaseline (fun s => α s*B) p θ t • (1:R)+ |
| τ • bergerSlowGenerator r B β p₀ θ (z (τ*t)) D |
|
|
| lemma actualBergerComparison_deriv {α p : ℝ → ℝ} |
| (hα : Continuous α) (hp : Continuous p) {M Mp B : ℝ} |
| (hM : 0 ≤ M) (hMp : 0 ≤ Mp) (hαn : ∀ t,|α t| ≤ M) |
| (hpn : ∀ t,|p t| ≤ Mp) (hAp : ∀ t,0 ≤ t → 0 ≤ α t*B) |
| {θ : ℝ} (hθ : 0 ≤ θ) (r β p₀ τ : ℝ) (D : R) |
| {z : ℝ → ℝ} {zd t : ℝ} (ht : 0 ≤ t) (hz : HasDerivAt z zd (τ*t)) : |
| let v := scalarBaseline (fun s => α s*B) p θ t |
| HasDerivAt (actualBergerComparison α p r B β p₀ θ τ D z) |
| ((α t*B-p t*v-v^2) • (1:R)+(τ^2) • bergerSlowGenerator r B β p₀ θ zd D) t := by |
| have hn (s : ℝ) : |α s*B| ≤ M*|B| := by rw [abs_mul];gcongr;exact hαn s |
| exact bergerComparison_time_deriv r B β p₀ θ τ D |
| (actual_scalar_baseline_deriv (hα.mul continuous_const) hp (by positivity) hMp hn hpn hAp hθ ht) hz |
|
|
| lemma actualBergerComparison_initial (α p : ℝ → ℝ) (r B β p₀ θ τ : ℝ) (D : R) |
| {z : ℝ → ℝ} (hz : z 0=0) : |
| actualBergerComparison α p r B β p₀ θ τ D z 0=θ • (1:R) := by |
| simp only [actualBergerComparison,scalarBaseline_init,mul_zero,hz,bergerSlowGenerator, |
| zero_smul,smul_zero,add_zero] |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℝ H] |
|
|
| / |
| PB-based expression. The baseline does not depend on the current parameter. -/ |
| lemma actualBergerComparison_parameter_deriv (α p : ℝ → ℝ) (r B β p₀ θ τ : ℝ) (D : R) |
| {z : H → ℝ → ℝ} {z' : H →L[ℝ] ℝ} {x : H} (t : ℝ) |
| (hz : HasFDerivAt (fun y => z y (τ*t)) z' x) : |
| HasFDerivAt (fun y => actualBergerComparison α p r B β p₀ θ τ D (z y) t) |
| (z'.smulRight (τ • bergerSlowGenerator r B β p₀ θ 1 D)) x := |
| bergerComparison_parameter_deriv r B β p₀ θ τ _ D hz |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| open scoped InnerProductSpace |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| lemma scalar_plus_small_coercive {S : E →L[ℝ] E} {v τ C : ℝ} |
| (hτ : 0 ≤ τ) (hSn : ‖S‖ ≤ C) : |
| ∀ x : E,(v-τ*C)*‖x‖^2 ≤ inner ℝ x ((v • (1:E →L[ℝ] E)+τ • S) x) := by |
| intro x |
| have hb : ‖S x‖ ≤ C*‖x‖ := (S.le_opNorm x).trans |
| (mul_le_mul_of_nonneg_right hSn (norm_nonneg x)) |
| have hh := neg_le_of_abs_le (abs_real_inner_le_norm x (S x)) |
| have he := mul_le_mul_of_nonneg_left hb (norm_nonneg x) |
| have hτh := mul_le_mul_of_nonneg_left hh hτ |
| have ht := mul_le_mul_of_nonneg_left he hτ |
| simp only [add_apply,smul_apply,one_apply_eq_self,inner_add_right,inner_smul_right, |
| real_inner_self_eq_norm_sq] |
| nlinarith |
|
|
| / |
| baseline estimate and smallness conditions, not assumed for the Riccati bound. -/ |
| lemma actualBergerComparison_coercive {α p : ℝ → ℝ} |
| (hα : Continuous α) (hp : Continuous p) {M Mp B β p₀ θ γ δ T τ r Z : ℝ} |
| (hM : 0 ≤ M) (hMp : 0 ≤ Mp) (hθ : 0 ≤ θ) (hγ : 0 < γ) (hδ : 0 ≤ δ) |
| (hτ : 0 ≤ τ) (hαn : ∀ t,|α t| ≤ M) (hpn : ∀ t,|p t| ≤ Mp) |
| (hAp : ∀ t,0 ≤ t → 0 ≤ α t*B) (hroot : β*B-p₀*θ-θ^2=0) |
| (hscale : ∀ t ∈ Set.Icc 0 T,|α t-β| ≤ δ) |
| (hpt : ∀ t ∈ Set.Icc 0 T,|p t-p₀| ≤ δ) |
| (hpl : ∀ t ∈ Set.Icc 0 T,γ ≤ p t) |
| (D : E →L[ℝ] E) {z : ℝ → ℝ} |
| (hz : ∀ t ∈ Set.Icc 0 T,|z (τ*t)| ≤ Z) |
| (hsmall : ((|B|+|θ|)/γ)*δ+ |
| τ*(|β/(p₀+2*θ)| * Z * ‖r • (B • (1:E →L[ℝ] E))+D‖) ≤ θ/2) : |
| ∀ t ∈ Set.Icc 0 T,∀ x : E, |
| (θ/2)*‖x‖^2 ≤ inner ℝ x (actualBergerComparison α p r B β p₀ θ τ D z t x) := by |
| intro t ht x |
| have hv := neg_le_of_abs_le |
| (berger_scalar_baseline_close hα hp hM hMp hθ hγ hδ hαn hpn hAp hroot hscale hpt hpl t ht) |
| have hn : ‖bergerSlowGenerator r B β p₀ θ (z (τ*t)) D‖ ≤ |
| |β/(p₀+2*θ)| * Z * ‖r • (B • (1:E →L[ℝ] E))+D‖ := by |
| simp only [bergerSlowGenerator,norm_smul,Real.norm_eq_abs,abs_mul] |
| gcongr |
| exact hz t ht |
| have hh := scalar_plus_small_coercive |
| (v := scalarBaseline (fun s => α s*B) p θ t) hτ hn x |
| have hnum : θ/2 ≤ scalarBaseline (fun s => α s*B) p θ t- |
| τ*(|β/(p₀+2*θ)| * Z * ‖r • (B • (1:E →L[ℝ] E))+D‖) := by linarith |
| exact (mul_le_mul_of_nonneg_right hnum (sq_nonneg ‖x‖)).trans hh |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| open Set Matrix |
| open scoped BigOperators InnerProductSpace Matrix.Norms.Frobenius |
| variable {E H ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [NormedAddCommGroup H] [NormedSpace ℝ H] |
| [Fintype ι] [DecidableEq ι] |
| local instance : NormedAddCommGroup (E →L[ℝ] E) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance : NormedSpace ℝ (E →L[ℝ] E) := ContinuousLinearMap.toNormedSpace |
| local instance : NormedAddCommGroup ((E →L[ℝ] E)×(E →L[ℝ] E)) := inferInstance |
| local instance : NormedSpace ℝ ((E →L[ℝ] E)×(E →L[ℝ] E)) := inferInstance |
| local instance : NormedAddCommGroup (((E →L[ℝ] E)×(E →L[ℝ] E)) →L[ℝ] ((E →L[ℝ] E)×(E →L[ℝ] E))) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance : NormedSpace ℝ (((E →L[ℝ] E)×(E →L[ℝ] E)) →L[ℝ] ((E →L[ℝ] E)×(E →L[ℝ] E))) := ContinuousLinearMap.toNormedSpace |
|
|
| local instance : NormedAddCommGroup ((E →L[ℝ] E) →L[ℝ] Matrix ι ι ℝ) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance : NormedSpace ℝ ((E →L[ℝ] E) →L[ℝ] Matrix ι ι ℝ) := ContinuousLinearMap.toNormedSpace |
|
|
| / |
| scalar Peano |
| Riccati and variational equations. The constant is chosen before all histories, |
| start times, durations, scalar backgrounds, and pulse shapes. -/ |
| theorem actual_berger_pulse_integrated (basis : OrthonormalBasis ι ℝ E) |
| (r B β p₀ θ γ : ℝ) (D : E →L[ℝ] E) {M Z : ℝ} |
| (hM : 0 ≤ M) (hZ : 0 ≤ Z) (hθ : 0 ≤ θ) (hγ : 0 < γ) |
| (hc : p₀+2*θ ≠ 0) (hroot : β*B-p₀*θ-θ^2=0) : |
| ∃ C : ℝ,1 ≤ C ∧ ∀ (A : H → ℝ → E →L[ℝ] E) (b : ℝ → ℝ) |
| (B' : H → ℝ → H →L[ℝ] OperatorPhase E →L[ℝ] OperatorPhase E) |
| (A' : H → ℝ → H →L[ℝ] E →L[ℝ] E) (x h : H) |
| (α p z zd zh zhd : ℝ → ℝ) (Mp MB MA Mb l B₀ a T δ ε : ℝ), |
| Continuous (A x) → Continuous b → Continuous (B' x) → |
| Continuous (fun t => A' x t h) → |
| (∀ t v w,B' x t v w=(0,A' x t v*w.1)) → |
| 0 ≤ MB → 0 ≤ MA → 0 ≤ Mb → |
| (∀ t,‖block (leftAction (A x)) b t‖ ≤ MB) → (∀ t,‖B' x t‖ ≤ MB) → |
| (∀ t,‖A x t‖ ≤ MA) → (∀ t,|b t| ≤ Mb) → |
| (∀ t v,0 ≤ inner ℝ v (A x t v)) → 0 < l → |
| (∀ t ≤ 0,∀ v,A x t v=(l*(l+B₀)) • v) → (∀ t ≤ 0,b t=B₀) → |
| Continuous α → Continuous p → 0 ≤ Mp → |
| (∀ t,|α t| ≤ M) → (∀ t,|p t| ≤ Mp) → (∀ t,0 ≤ t → 0 ≤ α t*B) → |
| 0 ≤ a → 1 ≤ T → 0 ≤ δ → 0 ≤ ε → |
| (∀ t ∈ Icc 0 T,|α t-β| ≤ δ) → (∀ t ∈ Icc 0 T,|p t-p₀| ≤ δ) → |
| (∀ t ∈ Icc 0 T,γ ≤ p t) → |
| (∀ u,HasDerivAt z (zd u) u) → (∀ u,HasDerivAt zh (zhd u) u) → |
| Continuous zd → Continuous zhd → z 0=0 → zh 0=0 → |
| (∀ t ∈ Icc 0 T,|z (T⁻¹*t)| ≤ Z ∧ |zd (T⁻¹*t)| ≤ Z ∧ |
| |zh (T⁻¹*t)| ≤ Z ∧ |zhd (T⁻¹*t)| ≤ Z) → |
| (∀ t ∈ Icc 0 T,1+T⁻¹*z (T⁻¹*t) ∈ Icc (1/2:ℝ) (3/2)) → |
| ((|B|+|θ|)/γ)*δ+T⁻¹*(|β/(p₀+2*θ)| *Z*‖r • (B • (1:E →L[ℝ] E))+D‖) ≤ θ/2 → |
| (∀ t ∈ Icc a (a+T),b t=p (t-a)) → |
| (∀ t ∈ Icc a (a+T),A x t= |
| α (t-a) • PulseTaylor.angular r (B • (1:E →L[ℝ] E)) D (1+T⁻¹*z (T⁻¹*(t-a)))) → |
| (∀ t ∈ Icc a (a+T),A' x t h= |
| (T⁻¹*zh (T⁻¹*(t-a))) • (α (t-a) • |
| PulseTaylor.angularD r (B • (1:E →L[ℝ] E)) D (1+T⁻¹*z (T⁻¹*(t-a))))) → |
| ‖orthogonalMatrix basis (slope (A x) b l a)-θ • (1:Matrix ι ι ℝ)‖ ≤ ε → |
| orthogonalMatrix basis (slopeJet A b B' l x a h)=0 → |
| let Q := fun t => orthogonalMatrix basis |
| (actualBergerComparison α p r B β p₀ θ T⁻¹ D z (t-a)) |
| let Qh := fun t => orthogonalMatrix basis |
| (T⁻¹ • bergerSlowGenerator r B β p₀ θ (zh (T⁻¹*(t-a))) D) |
| (∫ t in a..a+T,‖orthogonalMatrix basis (slope (A x) b l t)-Q t‖) ≤ |
| 2*C*(δ+T⁻¹+ε)/γ ∧ |
| (∫ t in a..a+T,‖orthogonalMatrix basis (slopeJet A b B' l x t h)-Qh t‖) ≤ |
| C*(δ+T⁻¹+ε)/γ+4*C^2*(δ+T⁻¹+ε)/γ^2 := by |
| obtain ⟨R,hR,hres⟩ := actual_berger_comparison_residuals r B β p₀ θ γ D hM hZ hθ hγ hc hroot |
| let G := |β/(p₀+2*θ)| *Z*‖r • (B • (1:E →L[ℝ] E))+D‖ |
| let L := orthogonalMatrix basis |
| let C := 1+‖L‖*(R+G) |
| have hG : 0 ≤ G := by dsimp [G];positivity |
| have hC₁ : 1 ≤ C := by dsimp [C];exact le_add_of_nonneg_right (mul_nonneg (norm_nonneg L) (add_nonneg hR hG)) |
| have hC : 0 ≤ C := zero_le_one.trans hC₁ |
| have hCR : ‖L‖*R ≤ C := by dsimp [C];nlinarith [norm_nonneg L,mul_nonneg (norm_nonneg L) hG] |
| have hCG : ‖L‖*G ≤ C := by dsimp [C];nlinarith [norm_nonneg L,mul_nonneg (norm_nonneg L) hR] |
| refine ⟨C,hC₁,?_⟩ |
| intro A b B' A' x h α p z zd zh zhd Mp MB MA Mb l B₀ a T δ ε |
| hA hb hB' hA' hBeval hMB hMA hMb hnB hnB' hnA hnb hAp hl hAc hbc |
| hα hp hMp hαn hpn hαp ha hT hδ hε hscale hpt hpl hzd hzhd hczd hczhd hz₀ hzh₀ |
| hzn hq hsmall hbmodel hAmodel hA'model hinit hjet |
| let τ := T⁻¹ |
| have hT₀ : 0 < T := lt_of_lt_of_le zero_lt_one hT |
| have hτ : 0 ≤ τ := inv_nonneg.mpr hT₀.le |
| let v := scalarBaseline (fun s => α s*B) p θ |
| let S := fun u => bergerSlowGenerator r B β p₀ θ u D |
| let q := actualBergerComparison α p r B β p₀ θ τ D z |
| let qd := fun t => (α t*B-p t*v t-(v t)^2) • (1:E →L[ℝ] E)+(τ^2) • S (zd (τ*t)) |
| let qh := fun t => τ • S (zh (τ*t)) |
| let qhd := fun t => (τ^2) • S (zhd (τ*t)) |
| let Q := fun t => L (q (t-a)) |
| let Qd := fun t => L (qd (t-a)) |
| let Qh := fun t => L (qh (t-a)) |
| let Qhd := fun t => L (qhd (t-a)) |
| have hshift (t : ℝ) (ht : t ∈ Icc a (a+T)) : t-a ∈ Icc 0 T := by |
| constructor <;> linarith [ht.1,ht.2] |
| have hdq (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivAt q (qd t) t := |
| actualBergerComparison_deriv hα hp hM hMp hαn hpn hαp hθ r β p₀ τ D ht.1 (hzd _) |
| have hdqh (t : ℝ) : HasDerivAt qh (qhd t) t := by |
| have htau : HasDerivAt (fun s : ℝ => τ*s) τ t := by |
| simpa only [mul_one,id_eq] using (hasDerivAt_id t).const_mul τ |
| have hh := ((((hzhd (τ*t)).comp t htau).const_mul (β/(p₀+2*θ))).smul_const |
| (r • (B • (1:E →L[ℝ] E))+D)).const_smul τ |
| apply hh.congr_deriv |
| dsimp [qh,qhd,S,bergerSlowGenerator] |
| simp only [smul_smul] |
| module |
| have hdQ (t : ℝ) (ht : t ∈ Icc a (a+T)) : HasDerivAt Q (Qd t) t := by |
| simpa only [Q,Qd,L,Function.comp_def,id_eq,one_smul] using orthogonalMatrix_hasDerivAt basis |
| ((hdq (t-a) (hshift t ht)).scomp t ((hasDerivAt_id t).sub_const a)) |
| have hdQh (t : ℝ) (_ht : t ∈ Icc a (a+T)) : HasDerivAt Qh (Qhd t) t := by |
| simpa only [Qh,Qhd,L,Function.comp_def,id_eq,one_smul] using orthogonalMatrix_hasDerivAt basis |
| ((hdqh (t-a)).scomp t ((hasDerivAt_id t).sub_const a)) |
| have hvc : ContinuousOn v (Icc 0 T) := fun t ht => by |
| have hn (s : ℝ) : |α s*B| ≤ M*|B| := by rw [abs_mul];gcongr;exact hαn s |
| exact (actual_scalar_baseline_deriv (hα.mul continuous_const) hp |
| (by positivity) hMp hn hpn hαp hθ ht.1).continuousAt.continuousWithinAt |
| have hSc : Continuous S := by unfold S bergerSlowGenerator;fun_prop |
| have hqdc : ContinuousOn qd (Icc 0 T) := by |
| exact (((hα.continuousOn.mul_const B).sub (hp.continuousOn.mul hvc)).sub |
| (hvc.pow 2)).smul continuousOn_const |>.add |
| ((hSc.comp (hczd.comp ((continuous_const : Continuous (fun _ : ℝ => τ)).mul continuous_id))).continuousOn.const_smul (τ^2)) |
| have hqhdc : Continuous qhd := |
| (hSc.comp (hczhd.comp ((continuous_const : Continuous (fun _ : ℝ => τ)).mul continuous_id))).const_smul (τ^2) |
| have hQdc : ContinuousOn Qd (Icc a (a+T)) := |
| L.continuous.comp_continuousOn (hqdc.comp (continuous_id.sub continuous_const).continuousOn hshift) |
| have hQhdc : ContinuousOn Qhd (Icc a (a+T)) := |
| (L.continuous.comp (hqhdc.comp (continuous_id.sub continuous_const))).continuousOn |
| have hQpos (t : ℝ) (ht : t ∈ Icc a (a+T)) (w : ι → ℝ) : |
| 0 ≤ ∑ i,∑ j,w i*Q t i j*w j := by |
| have hh := actualBergerComparison_coercive hα hp hM hMp hθ hγ hδ hτ hαn hpn hαp |
| hroot hscale hpt hpl D (fun s hs => (hzn s hs).1) hsmall (t-a) (hshift t ht) |
| have hm := orthogonalMatrix_coercive basis (q (t-a)) hh w |
| exact (mul_nonneg (by positivity) (Finset.sum_nonneg fun i _ => sq_nonneg (w i))).trans hm |
| have hSn (u : ℝ) (hu : |u| ≤ Z) : ‖S u‖ ≤ G := by |
| dsimp [S,bergerSlowGenerator,G] |
| rw [norm_smul,Real.norm_eq_abs,abs_mul] |
| gcongr |
| have hη : 0 ≤ δ+τ+ε := by positivity |
| have hbound : δ*τ+τ^2 ≤ (δ+τ+ε)/T := by |
| rw [div_eq_mul_inv] |
| change δ*τ+τ^2 ≤ (δ+τ+ε)*τ |
| nlinarith [mul_nonneg hε hτ] |
| have hρt (t : ℝ) (ht : t ∈ Icc a (a+T)) := |
| hres α p Mp δ T τ hα hp hMp hδ hτ hαn hpn hαp hscale hpt hpl |
| (t-a) (hshift t ht) (z (τ*(t-a))) (zd (τ*(t-a))) (zh (τ*(t-a))) (zhd (τ*(t-a))) |
| (hzn (t-a) (hshift t ht)).1 (hzn (t-a) (hshift t ht)).2.1 |
| (hzn (t-a) (hshift t ht)).2.2.1 (hzn (t-a) (hshift t ht)).2.2.2 |
| (hq (t-a) (hshift t ht)) |
| have hρ (t : ℝ) (ht : t ∈ Icc a (a+T)) : |
| ‖L (A x t)-b t • Q t-Q t*Q t-Qd t‖ ≤ C*(δ+τ+ε)/T := by |
| have hh := (L.le_opNorm |
| (A x t-b t • q (t-a)-q (t-a)*q (t-a)-qd (t-a))) |
| have he : ‖A x t-b t • q (t-a)-q (t-a)*q (t-a)-qd (t-a)‖ ≤ R*(δ*τ+τ^2) := by |
| rw [hAmodel t ht,hbmodel t ht] |
| exact (hρt t ht).1 |
| have hh' := hh.trans (mul_le_mul_of_nonneg_left he (norm_nonneg L)) |
| change ‖orthogonalMatrix basis _‖ ≤ ‖L‖*(R*(δ*τ+τ^2)) at hh' |
| simp only [map_sub,map_smul,orthogonalMatrix_mul] at hh' |
| calc |
| _ ≤ ‖L‖*(R*(δ*τ+τ^2)) := hh' |
| _ = (‖L‖*R)*(δ*τ+τ^2) := by ring |
| _ ≤ C*((δ+τ+ε)/T) := mul_le_mul hCR hbound (by positivity) hC |
| _ = _ := by ring |
| have hρh (t : ℝ) (ht : t ∈ Icc a (a+T)) : |
| ‖L (A' x t h)-b t • Qh t-Qh t*Q t-Q t*Qh t-Qhd t‖ ≤ C*(δ+τ+ε)/T := by |
| have hh := L.le_opNorm |
| (A' x t h-b t • qh (t-a)-qh (t-a)*q (t-a)-q (t-a)*qh (t-a)-qhd (t-a)) |
| have he : ‖A' x t h-b t • qh (t-a)-qh (t-a)*q (t-a)-q (t-a)*qh (t-a)-qhd (t-a)‖ ≤ R*(δ*τ+τ^2) := by |
| rw [hA'model t ht,hbmodel t ht] |
| exact (hρt t ht).2 |
| have hh' := hh.trans (mul_le_mul_of_nonneg_left he (norm_nonneg L)) |
| change ‖orthogonalMatrix basis _‖ ≤ ‖L‖*(R*(δ*τ+τ^2)) at hh' |
| simp only [map_sub,map_smul,orthogonalMatrix_mul] at hh' |
| calc |
| _ ≤ ‖L‖*(R*(δ*τ+τ^2)) := hh' |
| _ = (‖L‖*R)*(δ*τ+τ^2) := by ring |
| _ ≤ C*((δ+τ+ε)/T) := mul_le_mul hCR hbound (by positivity) hC |
| _ = _ := by ring |
| have hQhn (t : ℝ) (ht : t ∈ Icc a (a+T)) : ‖Qh t‖ ≤ C/T := by |
| calc |
| _ ≤ ‖L‖*‖qh (t-a)‖ := L.le_opNorm _ |
| _ = ‖L‖*(τ*‖S (zh (τ*(t-a)))‖) := by rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg hτ] |
| _ ≤ ‖L‖*(τ*G) := by gcongr;exact hSn _ (hzn (t-a) (hshift t ht)).2.2.1 |
| _ = (‖L‖*G)/T := by dsimp [τ];ring |
| _ ≤ C/T := div_le_div_of_nonneg_right hCG hT₀.le |
| have hQi : Q a=θ • (1:Matrix ι ι ℝ) := by |
| dsimp only [Q,q,L] |
| rw [sub_self,actualBergerComparison_initial α p r B β p₀ θ τ D hz₀,map_smul,orthogonalMatrix_one] |
| have hQhi : Qh a=0 := by |
| dsimp [Qh,qh,S,τ,bergerSlowGenerator] |
| simp only [sub_self,mul_zero,hzh₀,zero_smul,smul_zero,map_zero] |
| have hini : ‖L (slope (A x) b l a)-Q a‖ ≤ C*(δ+τ+ε) := by |
| rw [hQi] |
| apply hinit.trans |
| have he : ε ≤ δ+τ+ε := by linarith |
| exact he.trans (le_mul_of_one_le_left hη hC₁) |
| exact actual_slow_pulse_integrated_on basis A b B' A' x h hA hb hB' hA' hBeval |
| hMB hMA hMb hnB hnB' hnA hnb hAp hl hAc hbc Q Qd Qh Qhd |
| hdQ hdQh hQdc hQhdc ha hT hγ hC hη |
| (fun t ht => by rw [hbmodel t ht];exact hpl (t-a) (hshift t ht)) |
| hQpos hρ hρh hQhn hini (by rw [hjet,hQhi]) |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped BigOperators Topology ENNReal ContDiff |
| open scoped Topology |
| open scoped Topology |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open Filter MeasureTheory Set |
| open Set |
|
|
| namespace HarmonicCounterexample.LinearODE |
| open Set Matrix |
| open scoped BigOperators InnerProductSpace Matrix.Norms.Frobenius |
| variable {E H ι κ : Type*} [Fintype κ] [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [NormedAddCommGroup H] [NormedSpace ℝ H] |
| [Fintype ι] [DecidableEq ι] |
| local instance : NormedAddCommGroup (E →L[ℝ] E) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance : NormedSpace ℝ (E →L[ℝ] E) := ContinuousLinearMap.toNormedSpace |
| local instance : NormedAddCommGroup ((E →L[ℝ] E)×(E →L[ℝ] E)) := inferInstance |
| local instance : NormedSpace ℝ ((E →L[ℝ] E)×(E →L[ℝ] E)) := inferInstance |
| local instance : NormedAddCommGroup (((E →L[ℝ] E)×(E →L[ℝ] E)) →L[ℝ] ((E →L[ℝ] E)×(E →L[ℝ] E))) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance : NormedSpace ℝ (((E →L[ℝ] E)×(E →L[ℝ] E)) →L[ℝ] ((E →L[ℝ] E)×(E →L[ℝ] E))) := ContinuousLinearMap.toNormedSpace |
|
|
| local instance : NormedAddCommGroup ((E →L[ℝ] E) →L[ℝ] Matrix ι ι ℝ) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance : NormedSpace ℝ ((E →L[ℝ] E) →L[ℝ] Matrix ι ι ℝ) := ContinuousLinearMap.toNormedSpace |
|
|
| / |
| round gaps. The smooth first-order coefficient and its derivative are supplied |
| by the explicit finite pulse family, not by differentiating a discontinuous |
| selector. Nonlinear residuals are produced uniformly over the FINITE library. |
| The one constant is chosen before every history, selector, shape and duration. -/ |
| theorem actual_switched_berger_estimates (basis : OrthonormalBasis ι ℝ E) |
| (r B β p₀ θ γ : ℝ) (D : κ → E →L[ℝ] E) {M Z : ℝ} |
| (hM : 0 ≤ M) (hZ : 0 ≤ Z) (hθ : 0 ≤ θ) (hγ : 0 < γ) |
| (hc : p₀+2*θ ≠ 0) (hroot : β*B-p₀*θ-θ^2=0) : |
| ∃ C : ℝ,1 ≤ C ∧ ∀ (A : H → ℝ → E →L[ℝ] E) (b : ℝ → ℝ) |
| (B' : H → ℝ → H →L[ℝ] OperatorPhase E →L[ℝ] OperatorPhase E) |
| (A' : H → ℝ → H →L[ℝ] E →L[ℝ] E) (x h : H) |
| (α p z zd zh zhd : ℝ → ℝ) (d : ℝ → κ) |
| (S Sd Sh Shd : ℝ → E →L[ℝ] E) (Mp MB MA Mb l B₀ a T δ ε : ℝ), |
| Continuous (A x) → Continuous b → Continuous (B' x) → |
| Continuous (fun t => A' x t h) → |
| (∀ t v w,B' x t v w=(0,A' x t v*w.1)) → |
| 0 ≤ MB → 0 ≤ MA → 0 ≤ Mb → |
| (∀ t,‖block (leftAction (A x)) b t‖ ≤ MB) → (∀ t,‖B' x t‖ ≤ MB) → |
| (∀ t,‖A x t‖ ≤ MA) → (∀ t,|b t| ≤ Mb) → |
| (∀ t v,0 ≤ inner ℝ v (A x t v)) → 0 < l → |
| (∀ t ≤ 0,∀ v,A x t v=(l*(l+B₀)) • v) → (∀ t ≤ 0,b t=B₀) → |
| Continuous α → Continuous p → 0 ≤ Mp → |
| (∀ t,|α t| ≤ M) → (∀ t,|p t| ≤ Mp) → (∀ t,0 ≤ t → 0 ≤ α t*B) → |
| 0 ≤ a → 1 ≤ T → 0 ≤ δ → 0 ≤ ε → |
| (∀ t ∈ Icc 0 T,|α t-β| ≤ δ) → (∀ t ∈ Icc 0 T,|p t-p₀| ≤ δ) → |
| (∀ t ∈ Icc 0 T,γ ≤ p t) → |
| (∀ u,HasDerivAt S (Sd u) u) → (∀ u,HasDerivAt Sh (Shd u) u) → |
| Continuous Sd → Continuous Shd → S 0=0 → Sh 0=0 → |
| (∀ u,S u=bergerSlowGenerator r B β p₀ θ (z u) (D (d u)) ∧ |
| Sd u=bergerSlowGenerator r B β p₀ θ (zd u) (D (d u)) ∧ |
| Sh u=bergerSlowGenerator r B β p₀ θ (zh u) (D (d u)) ∧ |
| Shd u=bergerSlowGenerator r B β p₀ θ (zhd u) (D (d u))) → |
| (∀ t ∈ Icc 0 T,|z (T⁻¹*t)| ≤ Z ∧ |zd (T⁻¹*t)| ≤ Z ∧ |
| |zh (T⁻¹*t)| ≤ Z ∧ |zhd (T⁻¹*t)| ≤ Z) → |
| (∀ t ∈ Icc 0 T,1+T⁻¹*z (T⁻¹*t) ∈ Icc (1/2:ℝ) (3/2)) → |
| ((|B|+|θ|)/γ)*δ+T⁻¹*(∑ j : κ,|β/(p₀+2*θ)| *Z*‖r • (B • (1:E →L[ℝ] E))+D j‖) ≤ θ/2 → |
| (∀ t ∈ Icc a (a+T),b t=p (t-a)) → |
| (∀ t ∈ Icc a (a+T),A x t= |
| α (t-a) • PulseTaylor.angular r (B • (1:E →L[ℝ] E)) (D (d (T⁻¹*(t-a)))) (1+T⁻¹*z (T⁻¹*(t-a)))) → |
| (∀ t ∈ Icc a (a+T),A' x t h= |
| (T⁻¹*zh (T⁻¹*(t-a))) • (α (t-a) • |
| PulseTaylor.angularD r (B • (1:E →L[ℝ] E)) (D (d (T⁻¹*(t-a)))) (1+T⁻¹*z (T⁻¹*(t-a))))) → |
| ‖orthogonalMatrix basis (slope (A x) b l a)-θ • (1:Matrix ι ι ℝ)‖ ≤ ε → |
| orthogonalMatrix basis (slopeJet A b B' l x a h)=0 → |
| let Q := fun t => orthogonalMatrix basis |
| (scalarBaseline (fun s => α s*B) p θ (t-a) • (1:E →L[ℝ] E)+T⁻¹ • S (T⁻¹*(t-a))) |
| let Qh := fun t => orthogonalMatrix basis |
| (T⁻¹ • Sh (T⁻¹*(t-a))) |
| (∀ t ∈ Icc a (a+T),‖orthogonalMatrix basis (slope (A x) b l t)-Q t‖ ≤ |
| (C+C/γ)*(δ+T⁻¹+ε)) ∧ |
| (∫ t in a..a+T,‖orthogonalMatrix basis (slope (A x) b l t)-Q t‖) ≤ |
| 2*C*(δ+T⁻¹+ε)/γ ∧ |
| (∀ t ∈ Icc a (a+T),‖orthogonalMatrix basis (slopeJet A b B' l x t h)-Qh t‖ ≤ |
| (C+2*C*(C+C/γ))*(δ+T⁻¹+ε)/(γ*T)) ∧ |
| (∫ t in a..a+T,‖orthogonalMatrix basis (slopeJet A b B' l x t h)-Qh t‖) ≤ |
| C*(δ+T⁻¹+ε)/γ+4*C^2*(δ+T⁻¹+ε)/γ^2 := by |
| choose Rs hRs hres using fun j : κ => |
| actual_berger_comparison_residuals r B β p₀ θ γ (D j) hM hZ hθ hγ hc hroot |
| let R := ∑ j : κ,Rs j |
| let Gs := fun j : κ => |β/(p₀+2*θ)| *Z*‖r • (B • (1:E →L[ℝ] E))+D j‖ |
| let G := ∑ j : κ,Gs j |
| have hR : 0 ≤ R := Finset.sum_nonneg fun j _ => hRs j |
| have hRj (j : κ) : Rs j ≤ R := Finset.single_le_sum (fun i _ => hRs i) (Finset.mem_univ j) |
| have hGs (j : κ) : 0 ≤ Gs j := by dsimp [Gs];positivity |
| have hGj (j : κ) : Gs j ≤ G := Finset.single_le_sum (fun i _ => hGs i) (Finset.mem_univ j) |
| let L := orthogonalMatrix basis |
| let C := 1+‖L‖*(R+G) |
| have hG : 0 ≤ G := Finset.sum_nonneg fun j _ => hGs j |
| have hC₁ : 1 ≤ C := by dsimp [C];exact le_add_of_nonneg_right (mul_nonneg (norm_nonneg L) (add_nonneg hR hG)) |
| have hC : 0 ≤ C := zero_le_one.trans hC₁ |
| have hCR : ‖L‖*R ≤ C := by dsimp [C];nlinarith [norm_nonneg L,mul_nonneg (norm_nonneg L) hG] |
| have hCG : ‖L‖*G ≤ C := by dsimp [C];nlinarith [norm_nonneg L,mul_nonneg (norm_nonneg L) hR] |
| refine ⟨C,hC₁,?_⟩ |
| intro A b B' A' x h α p z zd zh zhd d S Sd Sh Shd Mp MB MA Mb l B₀ a T δ ε |
| hA hb hB' hA' hBeval hMB hMA hMb hnB hnB' hnA hnb hAp hl hAc hbc |
| hα hp hMp hαn hpn hαp ha hT hδ hε hscale hpt hpl hSd hShd hcSd hcShd hS₀ hSh₀ hshape |
| hzn hq hsmall hbmodel hAmodel hA'model hinit hjet |
| let τ := T⁻¹ |
| have hT₀ : 0 < T := lt_of_lt_of_le zero_lt_one hT |
| have hτ : 0 ≤ τ := inv_nonneg.mpr hT₀.le |
| let v := scalarBaseline (fun s => α s*B) p θ |
| let q := fun t => v t • (1:E →L[ℝ] E)+τ • S (τ*t) |
| let qd := fun t => (α t*B-p t*v t-(v t)^2) • (1:E →L[ℝ] E)+(τ^2) • Sd (τ*t) |
| let qh := fun t => τ • Sh (τ*t) |
| let qhd := fun t => (τ^2) • Shd (τ*t) |
| let Q := fun t => L (q (t-a)) |
| let Qd := fun t => L (qd (t-a)) |
| let Qh := fun t => L (qh (t-a)) |
| let Qhd := fun t => L (qhd (t-a)) |
| have hshift (t : ℝ) (ht : t ∈ Icc a (a+T)) : t-a ∈ Icc 0 T := by |
| constructor <;> linarith [ht.1,ht.2] |
| have hαBn (s : ℝ) : |α s*B| ≤ M*|B| := by rw [abs_mul];gcongr;exact hαn s |
| have hdv (t : ℝ) (ht : 0 ≤ t) : HasDerivAt v (α t*B-p t*v t-(v t)^2) t := |
| actual_scalar_baseline_deriv (hα.mul continuous_const) hp |
| (by positivity) hMp hαBn hpn hαp hθ ht |
| have hτd (t : ℝ) : HasDerivAt (fun s : ℝ => τ*s) τ t := by |
| simpa only [mul_one,id_eq] using (hasDerivAt_id t).const_mul τ |
| have hdq (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivAt q (qd t) t := by |
| have hh := ((hdv t ht.1).smul_const (1:E →L[ℝ] E)).add |
| (((hSd (τ*t)).scomp t (hτd t)).const_smul τ) |
| apply hh.congr_deriv |
| dsimp [qd] |
| simp only [smul_smul,pow_two] |
| have hdqh (t : ℝ) : HasDerivAt qh (qhd t) t := by |
| have hh := ((hShd (τ*t)).scomp t (hτd t)).const_smul τ |
| apply hh.congr_deriv |
| dsimp [qh,qhd] |
| simp only [smul_smul,pow_two] |
| have hdQ (t : ℝ) (ht : t ∈ Icc a (a+T)) : HasDerivAt Q (Qd t) t := by |
| simpa only [Q,Qd,L,Function.comp_def,id_eq,one_smul] using orthogonalMatrix_hasDerivAt basis |
| ((hdq (t-a) (hshift t ht)).scomp t ((hasDerivAt_id t).sub_const a)) |
| have hdQh (t : ℝ) (_ht : t ∈ Icc a (a+T)) : HasDerivAt Qh (Qhd t) t := by |
| simpa only [Qh,Qhd,L,Function.comp_def,id_eq,one_smul] using orthogonalMatrix_hasDerivAt basis |
| ((hdqh (t-a)).scomp t ((hasDerivAt_id t).sub_const a)) |
| have hvc : ContinuousOn v (Icc 0 T) := fun t ht => by |
| have hn (s : ℝ) : |α s*B| ≤ M*|B| := by rw [abs_mul];gcongr;exact hαn s |
| exact (actual_scalar_baseline_deriv (hα.mul continuous_const) hp |
| (by positivity) hMp hn hpn hαp hθ ht.1).continuousAt.continuousWithinAt |
| have hqdc : ContinuousOn qd (Icc 0 T) := by |
| exact (((hα.continuousOn.mul_const B).sub (hp.continuousOn.mul hvc)).sub |
| (hvc.pow 2)).smul continuousOn_const |>.add |
| ((hcSd.comp (continuous_const.mul continuous_id)).continuousOn.const_smul (τ^2)) |
| have hqhdc : Continuous qhd := |
| (hcShd.comp (continuous_const.mul continuous_id)).const_smul (τ^2) |
| have hQdc : ContinuousOn Qd (Icc a (a+T)) := |
| L.continuous.comp_continuousOn (hqdc.comp (continuous_id.sub continuous_const).continuousOn hshift) |
| have hQhdc : ContinuousOn Qhd (Icc a (a+T)) := |
| (L.continuous.comp (hqhdc.comp (continuous_id.sub continuous_const))).continuousOn |
| have hgen (j : κ) (u : ℝ) (hu : |u| ≤ Z) : |
| ‖bergerSlowGenerator r B β p₀ θ u (D j)‖ ≤ G := by |
| apply le_trans _ (hGj j) |
| dsimp [bergerSlowGenerator,Gs] |
| rw [norm_smul,Real.norm_eq_abs,abs_mul] |
| gcongr |
| have hSn (t : ℝ) (ht : t ∈ Icc 0 T) : ‖S (τ*t)‖ ≤ G := by |
| rw [(hshape (τ*t)).1];exact hgen _ _ (hzn t ht).1 |
| have hShn (t : ℝ) (ht : t ∈ Icc 0 T) : ‖Sh (τ*t)‖ ≤ G := by |
| rw [(hshape (τ*t)).2.2.1];exact hgen _ _ (hzn t ht).2.2.1 |
| have hQpos (t : ℝ) (ht : t ∈ Icc a (a+T)) (w : ι → ℝ) : |
| 0 ≤ ∑ i,∑ j,w i*Q t i j*w j := by |
| have hh (y : E) : (θ/2)*‖y‖^2 ≤ inner ℝ y (q (t-a) y) := by |
| have hv := neg_le_of_abs_le |
| (berger_scalar_baseline_close hα hp hM hMp hθ hγ hδ hαn hpn hαp hroot hscale hpt hpl |
| (t-a) (hshift t ht)) |
| have hco := scalar_plus_small_coercive (v := v (t-a)) hτ (hSn (t-a) (hshift t ht)) y |
| have hnum : θ/2 ≤ v (t-a)-τ*G := by dsimp only [G,Gs,τ] at *;linarith |
| exact (mul_le_mul_of_nonneg_right hnum (sq_nonneg ‖y‖)).trans hco |
| have hm := orthogonalMatrix_coercive basis (q (t-a)) hh w |
| exact (mul_nonneg (by positivity) (Finset.sum_nonneg fun i _ => sq_nonneg (w i))).trans hm |
| have hη : 0 ≤ δ+τ+ε := by positivity |
| have hbound : δ*τ+τ^2 ≤ (δ+τ+ε)/T := by |
| rw [div_eq_mul_inv] |
| change δ*τ+τ^2 ≤ (δ+τ+ε)*τ |
| nlinarith [mul_nonneg hε hτ] |
| have hρt (t : ℝ) (ht : t ∈ Icc a (a+T)) := |
| hres (d (τ*(t-a))) α p Mp δ T τ hα hp hMp hδ hτ hαn hpn hαp hscale hpt hpl |
| (t-a) (hshift t ht) (z (τ*(t-a))) (zd (τ*(t-a))) (zh (τ*(t-a))) (zhd (τ*(t-a))) |
| (hzn (t-a) (hshift t ht)).1 (hzn (t-a) (hshift t ht)).2.1 |
| (hzn (t-a) (hshift t ht)).2.2.1 (hzn (t-a) (hshift t ht)).2.2.2 |
| (hq (t-a) (hshift t ht)) |
| have hρ (t : ℝ) (ht : t ∈ Icc a (a+T)) : |
| ‖L (A x t)-b t • Q t-Q t*Q t-Qd t‖ ≤ C*(δ+τ+ε)/T := by |
| have hh := (L.le_opNorm |
| (A x t-b t • q (t-a)-q (t-a)*q (t-a)-qd (t-a))) |
| have he : ‖A x t-b t • q (t-a)-q (t-a)*q (t-a)-qd (t-a)‖ ≤ R*(δ*τ+τ^2) := by |
| rw [hAmodel t ht,hbmodel t ht] |
| dsimp only [q,qd] |
| rw [(hshape (τ*(t-a))).1,(hshape (τ*(t-a))).2.1] |
| exact (hρt t ht).1.trans (mul_le_mul_of_nonneg_right (hRj _) (by positivity)) |
| have hh' := hh.trans (mul_le_mul_of_nonneg_left he (norm_nonneg L)) |
| change ‖orthogonalMatrix basis _‖ ≤ ‖L‖*(R*(δ*τ+τ^2)) at hh' |
| simp only [map_sub,map_smul,orthogonalMatrix_mul] at hh' |
| calc |
| _ ≤ ‖L‖*(R*(δ*τ+τ^2)) := hh' |
| _ = (‖L‖*R)*(δ*τ+τ^2) := by ring |
| _ ≤ C*((δ+τ+ε)/T) := mul_le_mul hCR hbound (by positivity) hC |
| _ = _ := by ring |
| have hρh (t : ℝ) (ht : t ∈ Icc a (a+T)) : |
| ‖L (A' x t h)-b t • Qh t-Qh t*Q t-Q t*Qh t-Qhd t‖ ≤ C*(δ+τ+ε)/T := by |
| have hh := L.le_opNorm |
| (A' x t h-b t • qh (t-a)-qh (t-a)*q (t-a)-q (t-a)*qh (t-a)-qhd (t-a)) |
| have he : ‖A' x t h-b t • qh (t-a)-qh (t-a)*q (t-a)-q (t-a)*qh (t-a)-qhd (t-a)‖ ≤ R*(δ*τ+τ^2) := by |
| rw [hA'model t ht,hbmodel t ht] |
| dsimp only [q,qh,qhd] |
| rw [(hshape (τ*(t-a))).1,(hshape (τ*(t-a))).2.2.1,(hshape (τ*(t-a))).2.2.2] |
| exact (hρt t ht).2.trans (mul_le_mul_of_nonneg_right (hRj _) (by positivity)) |
| have hh' := hh.trans (mul_le_mul_of_nonneg_left he (norm_nonneg L)) |
| change ‖orthogonalMatrix basis _‖ ≤ ‖L‖*(R*(δ*τ+τ^2)) at hh' |
| simp only [map_sub,map_smul,orthogonalMatrix_mul] at hh' |
| calc |
| _ ≤ ‖L‖*(R*(δ*τ+τ^2)) := hh' |
| _ = (‖L‖*R)*(δ*τ+τ^2) := by ring |
| _ ≤ C*((δ+τ+ε)/T) := mul_le_mul hCR hbound (by positivity) hC |
| _ = _ := by ring |
| have hQhn (t : ℝ) (ht : t ∈ Icc a (a+T)) : ‖Qh t‖ ≤ C/T := by |
| calc |
| _ ≤ ‖L‖*‖qh (t-a)‖ := L.le_opNorm _ |
| _ = ‖L‖*(τ*‖Sh (τ*(t-a))‖) := by rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg hτ] |
| _ ≤ ‖L‖*(τ*G) := by gcongr;exact hShn (t-a) (hshift t ht) |
| _ = (‖L‖*G)/T := by dsimp [τ];ring |
| _ ≤ C/T := div_le_div_of_nonneg_right hCG hT₀.le |
| have hQi : Q a=θ • (1:Matrix ι ι ℝ) := by |
| dsimp only [Q,q,L,v] |
| simp only [sub_self,scalarBaseline_init,mul_zero,hS₀,smul_zero,add_zero,map_smul,orthogonalMatrix_one] |
| have hQhi : Qh a=0 := by |
| dsimp only [Qh,qh,L] |
| simp only [sub_self,mul_zero,hSh₀,smul_zero,map_zero] |
| have hini : ‖L (slope (A x) b l a)-Q a‖ ≤ C*(δ+τ+ε) := by |
| rw [hQi] |
| apply hinit.trans |
| have he : ε ≤ δ+τ+ε := by linarith |
| exact he.trans (le_mul_of_one_le_left hη hC₁) |
| exact actual_slow_pulse_estimates_on basis A b B' A' x h hA hb hB' hA' hBeval |
| hMB hMA hMb hnB hnB' hnA hnb hAp hl hAc hbc Q Qd Qh Qhd |
| hdQ hdQh hQdc hQhdc ha hT hγ hC hη |
| (fun t ht => by rw [hbmodel t ht];exact hpl (t-a) (hshift t ht)) |
| hQpos hρ hρh hQhn hini (by rw [hjet,hQhi]) |
|
|
| end HarmonicCounterexample.LinearODE |
|
|
| end |
|
|
| end OAI |
|
|
