File size: 4,988 Bytes
e63eb7f | 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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.Combinatorics.EditApproximation.Numerics.RationalPrefix
import OAI.Combinatorics.EditApproximation.Geometry.Bands
import OAI.Combinatorics.EditApproximation.Geometry.WideActions
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-!
# The literal finite predicted band
The band is a rationally decidable filter of the enumerated wide actions.
Its internal-cut diameter follows from the actual finite prefix predictor,
without assuming a predictor Lipschitz contract.
-/
namespace EditApproximation
theorem computedBandActions_mem {nx ny M : ℕ} (parent : TargetInterval nx)
(r : TargetInterval ny) (b P : ℕ) (tau theta velocity : ℚ)
(child : Fin M → TargetInterval ny → ℚ) (states : Fin M → TargetInterval ny)
(hstates : states ∈ computedBandActions parent r b P tau theta velocity child) :
∃ hnonempty : wideActionList M parent r b P ≠ [],
states ∈ wideActionList M parent r b P ∧ |velocity| ≤ 1 - tau ∧
bandActionPredicate parent r b tau theta velocity (wideActionList M parent r b P) hnonempty child states := by
by_cases hempty : wideActionList M parent r b P = []
· simp only [computedBandActions, dite_eq_left hempty, List.not_mem_nil] at hstates
· by_cases hvelocity : |velocity| ≤ 1 - tau
· simp only [computedBandActions, dite_eq_right hempty, ite_eq_left hvelocity] at hstates
obtain ⟨hmem, hpredicate⟩ := List.mem_filter.mp hstates
exact ⟨hempty, hmem, hvelocity, of_decide_eq_true hpredicate⟩
· simp only [computedBandActions, dite_eq_right hempty, ite_eq_right hvelocity,
List.not_mem_nil] at hstates
theorem computedBandActions_complete {nx ny M : ℕ} (parent : TargetInterval nx)
(r : TargetInterval ny) (b P : ℕ) (tau theta velocity : ℚ)
(child : Fin M → TargetInterval ny → ℚ) (states : Fin M → TargetInterval ny)
(hnonempty : wideActionList M parent r b P ≠ [])
(hstates : states ∈ wideActionList M parent r b P) (hvelocity : |velocity| ≤ 1 - tau)
(hpredicate : bandActionPredicate parent r b tau theta velocity
(wideActionList M parent r b P) hnonempty child states) :
states ∈ computedBandActions parent r b P tau theta velocity child := by
simp only [computedBandActions, dite_eq_right hnonempty, ite_eq_left hvelocity]
exact List.mem_filter.mpr ⟨hstates, decide_eq_true hpredicate⟩
theorem computedBand_cut_diameter {nx ny M : ℕ} (parent : TargetInterval nx)
(r : TargetInterval ny) (b P : ℕ) (tau theta velocity : ℚ) (htau : 0 < tau)
(child : Fin M → TargetInterval ny → ℚ) (states states' : Fin M → TargetInterval ny)
(hstates : states ∈ computedBandActions parent r b P tau theta velocity child)
(hstates' : states' ∈ computedBandActions parent r b P tau theta velocity child)
(j : Fin M) (hj : 0 < j.val) :
(Nat.dist (states j).lo (states' j).lo : ℝ) ≤ 2 * (theta : ℝ) * b / M := by
obtain ⟨hnonempty, hmem, hvelocity, hpredicate⟩ :=
computedBandActions_mem parent r b P tau theta velocity child states hstates
obtain ⟨hnonempty', hmem', hvelocity', hpredicate'⟩ :=
computedBandActions_mem parent r b P tau theta velocity child states' hstates'
let predictor := fun p : ℝ => widePrefixMinimum parent r (wideActionList M parent r b P)
hnonempty (fun i s => (child i s : ℝ)) j.val p
have hvelocityR : |(velocity : ℝ)| ≤ 1 - (tau : ℝ) := by exact_mod_cast hvelocity
have htauR : (0 : ℝ) < tau := by exact_mod_cast htau
have hband (s : Fin M → TargetInterval ny)
(hs : bandActionPredicate parent r b tau theta velocity (wideActionList M parent r b P) hnonempty child s) :
inPredictedBand predictor ((r.lo : ℝ) - parent.lo) velocity tau
((theta : ℝ) * b / M) (bandCutShift parent s j) := by
have h := hs.2.2 j hj
have hR : ((|bandCutShift parent s j - ((r.lo : ℚ) - parent.lo) - velocity *
rationalWidePrefixMinimum parent r (wideActionList M parent r b P) hnonempty child j.val
(bandCutShift parent s j)| : ℚ) : ℝ) ≤ ((tau * theta * b / M : ℚ) : ℝ) :=
Rat.cast_le.mpr h
simp only [Rat.cast_abs, Rat.cast_sub, Rat.cast_natCast, Rat.cast_mul, Rat.cast_div,
rationalWidePrefixMinimum_cast] at hR
simpa only [inPredictedBand, predictor, mul_assoc, mul_div_assoc] using hR
have hdiam := predicted_band_diameter htauR hvelocityR
(widePrefixMinimum_lipschitz parent r (wideActionList M parent r b P) hnonempty
(fun i s => (child i s : ℝ)) j.val _ _)
(hband states hpredicate) (hband states' hpredicate')
have hpoint : ((bandCutShift parent states j : ℚ) : ℝ) -
((bandCutShift parent states' j : ℚ) : ℝ) = (states j).lo - ((states' j).lo : ℝ) := by
simp only [bandCutShift, Rat.cast_sub, Rat.cast_natCast]
ring
rw [hpoint, natural_distance_cast] at hdiam
simpa only [mul_div_assoc, mul_assoc] using hdiam
end EditApproximation
end OAI
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