| import OAI.Combinatorics.EditApproximation.Geometry.PrefixCosts |
| import OAI.Combinatorics.EditApproximation.Numerics.RationalRefinement |
| import OAI.Combinatorics.EditApproximation.Definitions |
|
|
|
|
| namespace OAI |
|
|
| /-! Rational execution and exact real casts of the finite band predictor. -/ |
|
|
| namespace EditApproximation |
|
|
| open Finset |
|
|
|
|
|
|
| theorem rationalConnectionBudget_cast {n : ℕ} (start finish : ℚ) (states : List (TargetInterval n)) : |
| (rationalConnectionBudget start states finish : ℝ) = realConnectionBudget start states finish := by |
| induction states generalizing start with |
| | nil => simp only [rationalConnectionBudget, realConnectionBudget, Rat.cast_abs, Rat.cast_sub] |
| | cons state rest ih => |
| simp only [rationalConnectionBudget, realConnectionBudget, Rat.cast_add, Rat.cast_abs, |
| Rat.cast_sub, Rat.cast_natCast, ih] |
|
|
|
|
|
|
| theorem rationalWidePrefixCost_cast {nx ny M : ℕ} (parent : TargetInterval nx) |
| (r : TargetInterval ny) (states : Fin M → TargetInterval ny) |
| (child : Fin M → TargetInterval ny → ℚ) (j : ℕ) (p : ℚ) : |
| (rationalWidePrefixCost parent r states child j p : ℝ) = |
| widePrefixCost parent r states (fun i s => (child i s : ℝ)) j p := by |
| simp only [rationalWidePrefixCost, widePrefixCost, Rat.cast_add, Rat.cast_sum, |
| rationalConnectionBudget_cast, Rat.cast_natCast] |
| congr 1 |
| apply sum_congr rfl |
| intro i hi |
| split_ifs <;> simp |
|
|
|
|
|
|
| theorem rationalWidePrefixMinimum_cast {nx ny M : ℕ} (parent : TargetInterval nx) |
| (r : TargetInterval ny) (actions : List (Fin M → TargetInterval ny)) (hnonempty : actions ≠ []) |
| (child : Fin M → TargetInterval ny → ℚ) (j : ℕ) (p : ℚ) : |
| (rationalWidePrefixMinimum parent r actions hnonempty child j p : ℝ) = |
| widePrefixMinimum parent r actions hnonempty (fun i s => (child i s : ℝ)) j p := by |
| rw [rationalWidePrefixMinimum, rationalMinimum_cast] |
| simp only [widePrefixMinimum, rationalWidePrefixCost_cast, List.map_map, Function.comp_def] |
|
|
| end EditApproximation |
|
|
| end OAI |
|
|
