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caec230 | 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 | 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
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