| import Mathlib |
| import OAI.MathematicalPhysics.PrefixFlows.Effective |
|
|
| namespace OAI |
|
|
| /-! Separation of the initial loading pulse from the repeated template. -/ |
|
|
| noncomputable section |
| open scoped ContDiff |
| namespace PrefixFlows |
| @[simp] theorem point_time (t : ℝ) (x : Space) : point t x 0 = t := rfl |
|
|
| @[simp] theorem position_point (t : ℝ) (x : Space) : position (point t x) = x := by |
| ext i |
| fin_cases i <;> rfl |
|
|
| theorem assemble_during_loading (template : Machine → Field) |
| (loading : Machine → List ℕ → Field) (M : Machine) (w : List ℕ) |
| (t : ℝ) (x : Space) (ht : t < 1) : |
| assemble template loading M w (point t x) = loading M w (point t x) := by |
| simp only [assemble, point_time] |
| exact ite_eq_left ht |
|
|
| theorem assemble_after_loading (template : Machine → Field) |
| (loading : Machine → List ℕ → Field) (M : Machine) (w : List ℕ) |
| (t : ℝ) (x : Space) (ht : 1 ≤ t) : |
| assemble template loading M w (point t x) = template M (point (Int.fract t) x) := by |
| simp only [assemble, point_time, ite_eq_right (not_lt_of_ge ht), position_point] |
|
|
| theorem assemble_period_one (template : Machine → Field) |
| (loading : Machine → List ℕ → Field) (M : Machine) (w : List ℕ) |
| (t : ℝ) (ht : 1 ≤ t) (x : Space) : |
| assemble template loading M w (point (t + 1) x) = |
| assemble template loading M w (point t x) := by |
| rw [assemble_after_loading template loading M w (t + 1) x (by linarith), |
| assemble_after_loading template loading M w t x ht, Int.fract_add_one] |
|
|
| theorem assemble_input_only_in_loading (template : Machine → Field) |
| (loading : Machine → List ℕ → Field) (M : Machine) (v w : List ℕ) |
| (t : ℝ) (ht : 1 ≤ t) (x : Space) : |
| assemble template loading M v (point t x) = assemble template loading M w (point t x) := by |
| rw [assemble_after_loading template loading M v t x ht, |
| assemble_after_loading template loading M w t x ht] |
| end PrefixFlows |
| end |
|
|
| end OAI |
|
|
