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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
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