| import Mathlib | |
| namespace OAI | |
| open Set MeasureTheory | |
| open scoped Interval | |
| namespace RadialAlignment | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] | |
| theorem alignment_integral | |
| {α β c t : ℝ} {a b : E} {w : ℝ → E} {lam : ℝ → ℝ} | |
| (hlam : ContinuousOn lam (Icc α β)) | |
| (hw : ContinuousOn w (Icc α β)) | |
| (hder : ∀ x ∈ Ioo α β, HasDerivAt w (lam x • (a + x • b)) x) | |
| (hc : c ∈ Icc α β) (ht : t ∈ Icc α β) : | |
| w t - w c = (∫ x in c..t, lam x) • a + | |
| (∫ x in c..t, lam x * x) • b := by | |
| have hsub : uIcc c t ⊆ Icc α β := uIcc_subset_Icc hc ht | |
| have hl : ContinuousOn lam (uIcc c t) := hlam.mono hsub | |
| have hlin : ContinuousOn (fun x : ℝ => a + x • b) (uIcc c t) := | |
| continuousOn_const.add (continuousOn_id.smul continuousOn_const) | |
| have hFTC : (∫ x in c..t, lam x • (a + x • b)) = w t - w c := by | |
| apply intervalIntegral.integral_eq_sub_of_hasDeriv_right (hw.mono hsub) | |
| · intro x hx | |
| have hinner : x ∈ Ioo α β := by | |
| constructor | |
| · exact lt_of_le_of_lt (le_min hc.1 ht.1) hx.1 | |
| · exact lt_of_lt_of_le hx.2 (max_le hc.2 ht.2) | |
| exact (hder x hinner).hasDerivWithinAt | |
| · exact (hl.smul hlin).intervalIntegrable | |
| rw [← hFTC] | |
| have ha : IntervalIntegrable (fun x => lam x • a) volume c t := | |
| (hl.smul continuousOn_const).intervalIntegrable | |
| have hb : IntervalIntegrable (fun x => (lam x * x) • b) volume c t := | |
| ((hl.mul continuousOn_id).smul continuousOn_const).intervalIntegrable | |
| simp_rw [smul_add, smul_smul] | |
| rw [intervalIntegral.integral_add ha hb] | |
| simp only [intervalIntegral.integral_smul_const] | |
| theorem centered_integral | |
| {α β c t : ℝ} {lam : ℝ → ℝ} | |
| (hlam : ContinuousOn lam (Icc α β)) | |
| (hc : c ∈ Icc α β) (ht : t ∈ Icc α β) : | |
| (∫ x in c..t, lam x * (x - t)) = | |
| (∫ x in c..t, lam x * x) - t * (∫ x in c..t, lam x) := by | |
| have hl : ContinuousOn lam (uIcc c t) := hlam.mono (uIcc_subset_Icc hc ht) | |
| have h₁ : IntervalIntegrable (fun x => lam x * x) volume c t := | |
| (hl.mul continuousOn_id).intervalIntegrable | |
| have h₂ : IntervalIntegrable (fun x => t * lam x) volume c t := | |
| (continuousOn_const.mul hl).intervalIntegrable | |
| simp_rw [mul_sub, mul_comm (lam _) t] | |
| rw [intervalIntegral.integral_sub h₁ h₂, intervalIntegral.integral_const_mul] | |
| theorem frozen_intercept | |
| {α β c t ρ s : ℝ} {p₀ e a b : E} {w : ℝ → E} {lam : ℝ → ℝ} | |
| (hlam : ContinuousOn lam (Icc α β)) | |
| (hw : ContinuousOn w (Icc α β)) | |
| (hder : ∀ x ∈ Ioo α β, HasDerivAt w (lam x • (a + x • b)) x) | |
| (hc : c ∈ Icc α β) (ht : t ∈ Icc α β) : | |
| let u := ρ + s * (∫ x in c..t, lam x) | |
| let v := t * u + s * (∫ x in c..t, lam x * (x - t)) | |
| p₀ + ρ • (a + t • b) + s • (e + w t) - s • (e + w c) = | |
| p₀ + u • a + v • b := by | |
| dsimp only | |
| have hvec := alignment_integral hlam hw hder hc ht | |
| have hcenter := centered_integral hlam hc ht | |
| rw [hcenter] | |
| have hwt : w t = w c + | |
| ((∫ x in c..t, lam x) • a + (∫ x in c..t, lam x * x) • b) := by | |
| rw [← hvec] | |
| abel | |
| rw [hwt] | |
| module | |
| end RadialAlignment | |
| end OAI | |