File size: 2,864 Bytes
a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Probability.RadialTransition.IdealAttainment
namespace OAI
universe uAlpha
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
def prepend {α : Type uAlpha} (v : α) (f : ℕ → α) : ℕ → α := fun n => Nat.casesOn n v f
@[simp] lemma prepend_zero {α : Type uAlpha} (v : α) (f : ℕ → α) : prepend v f 0 = v := rfl
@[simp] lemma prepend_succ {α : Type uAlpha} (v : α) (f : ℕ → α) (n : ℕ) : prepend v f (n+1) = f n := rfl
lemma summable_prepend (v : ℝ) {f : ℕ → ℝ} (hf : Summable f) : Summable (prepend v f) := by
apply (summable_nat_add_iff 1).mp
simpa only [prepend_succ] using hf
lemma tsum_prepend (v : ℝ) {f : ℕ → ℝ} (hf : Summable f) :
(∑' n, prepend v f n) = v+∑' n, f n := by
simpa only [prepend_zero, prepend_succ] using (summable_prepend v hf).tsum_eq_zero_add
lemma prepend_nonneg {a : ℝ} {b : ℕ → ℝ} (ha : 0 ≤ a) (hb : ∀ n, 0 ≤ b n) :
∀ n, 0 ≤ prepend a b n := by
intro n; cases n with
| zero => exact ha
| succ n => exact hb n
lemma prepend_range {R : ℝ} {S : ℕ → ℝ} (hR : 1 ≤ R) (hS : ∀ n, 1 ≤ S n) :
∀ n, 1 ≤ prepend R S n := by
intro n; cases n with
| zero => exact hR
| succ n => exact hS n
lemma attractionProfile_prepend {a R : ℝ} {b S : ℕ → ℝ}
(hb : ∀ n, 0 ≤ b n) (hS : ∀ n, 1 ≤ S n) (hs : Summable b) (r : ℝ) :
attractionProfile (prepend a b) (prepend R S) r = a*scaledBumpProfile R r+attractionProfile b S r := by
unfold attractionProfile
have he : (fun n => prepend a b n*scaledBumpProfile (prepend R S n) r) =
prepend (a*scaledBumpProfile R r) (fun n => b n*scaledBumpProfile (S n) r) := by
funext n; cases n <;> rfl
rw [he, tsum_prepend _ (summable_attractionProfile hb hS hs r)]
lemma familyPotential_prepend {a R t : ℝ} {b S : ℕ → ℝ}
(hb : ∀ n, 0 ≤ b n) (hS : ∀ n, 1 ≤ S n) (hs : Summable b) :
familyPotential t (prepend a b) (prepend R S) = familyPotential t b S+singleAttraction R (t*a) := by
funext r
simp only [familyPotential, Pi.add_apply, attractionProfile_prepend hb hS hs, singleAttraction]
ring
lemma familyCellMajorant_prepend {a R t : ℝ} {b S : ℕ → ℝ}
(hb : ∀ n, 0 ≤ b n) (hS : ∀ n, 1 ≤ S n) (hs : Summable b) (i j : Cell) :
familyCellMajorant t (prepend a b) (prepend R S) i j =
familyCellMajorant t b S i j+t*a*cellMajorant R bumpIntegral⁻¹ i j := by
unfold familyCellMajorant
have he : (fun n => prepend a b n * cellMajorant (prepend R S n) bumpIntegral⁻¹ i j) =
prepend (a*cellMajorant R bumpIntegral⁻¹ i j) (fun n => b n*cellMajorant (S n) bumpIntegral⁻¹ i j) := by
funext n; cases n <;> rfl
rw [he, tsum_prepend _ (summable_cellMajorant hb hS hs i j)]
ring
end RadialTransition
end
end OAI
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