File size: 2,679 Bytes
9b6e014 | 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 | import OAI.Probability.ContinuumTransition.MaskedComparison
namespace OAI
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
def prepend (a : ℝ) (σ : ℕ → ℝ) : ℕ → ℝ
| 0 => a
| n+1 => σ n
@[simp] theorem prepend_zero (a : ℝ) (σ : ℕ → ℝ) : prepend a σ 0 = a := rfl
@[simp] theorem prepend_succ (a : ℝ) (σ : ℕ → ℝ) (n : ℕ) : prepend a σ (n+1) = σ n := rfl
theorem summable_prepend (a : ℝ) {σ : ℕ → ℝ} (hs : Summable σ) : Summable (prepend a σ) := by
exact (summable_nat_add_iff 1).mp hs
theorem tsum_prepend (a : ℝ) {σ : ℕ → ℝ} (hs : Summable σ) :
∑' n, prepend a σ n = a+∑' n, σ n := by
exact (summable_prepend a hs).tsum_eq_zero_add
theorem prepend_nonneg {a : ℝ} {σ : ℕ → ℝ} (ha : 0 ≤ a) (hσ : ∀ n, 0 ≤ σ n) :
∀ n, 0 ≤ prepend a σ n
| 0 => ha
| n+1 => hσ n
theorem prepend_ge_one {a : ℝ} {σ : ℕ → ℝ} (ha : 1 ≤ a) (hσ : ∀ n, 1 ≤ σ n) :
∀ n, 1 ≤ prepend a σ n
| 0 => ha
| n+1 => hσ n
theorem attraction_prepend (t : ℝ) {a S : ℝ} {σ R : ℕ → ℝ}
(hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) (r : ℝ) :
attraction t (prepend a σ) (prepend S R) r = t*a*scaledKernelProfile S r+attraction t σ R r := by
unfold attraction
have he : (fun n => prepend a σ n*scaledKernelProfile (prepend S R n) r) =
prepend (a*scaledKernelProfile S r) (fun n => σ n*scaledKernelProfile (R n) r) := by
funext n; cases n <;> rfl
rw [he, tsum_prepend _ (summable_attraction_terms hσ hR hs r)]
ring
theorem physicalProfile_prepend (t : ℝ) {a S : ℝ} {σ R : ℕ → ℝ}
(hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) (r : ℝ) :
physicalProfile t (prepend a σ) (prepend S R) r = physicalProfile t σ R r-t*a*scaledKernelProfile S r := by
unfold physicalProfile
rw [attraction_prepend t hσ hR hs]
ring
theorem attraction_zero (t : ℝ) (R : ℕ → ℝ) (r : ℝ) : attraction t (fun _ => 0) R r = 0 := by
simp [attraction]
theorem cutReference_prepend (t a S : ℝ) {σ R : ℕ → ℝ}
(hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) (q : ℕ) (x y : Space) :
cutReference t (prepend 0 σ) (prepend a (fun _ => 0)) (prepend S R) q x y =
blockCut q (fun x y => physicalProfile t σ R ‖x-y‖) x y-t*a*scaledKernelProfile S ‖x-y‖ := by
unfold cutReference
simp only [physicalProfile_prepend t hσ hR hs, mul_zero, zero_mul, sub_zero]
rw [attraction_prepend t (fun _ => le_refl 0) hR (summable_zero), attraction_zero, add_zero]
end ContinuumTemperature
end
end OAI
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