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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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.Combinatorics.EditApproximation.Geometry.ComparisonBandCoverage
namespace OAI
/-!
# Literal band endpoint geometry and rounding guards
The full child endpoint diameter follows from the computed internal cuts
and zero connection, including the two outside endpoints.
-/
namespace EditApproximation
open Finset
theorem connectionBudget_zero_head {n : ℕ} (states : List (TargetInterval n))
(start finish : ℕ) (hzero : connectionBudget start states finish = 0) :
(states.head?.map TargetInterval.lo).getD start = start := by
cases states with
| nil => rfl
| cons state rest =>
have hdistance : Nat.dist start state.lo = 0 := by
change Nat.dist start state.lo + connectionBudget state.hi rest finish = 0 at hzero
omega
exact (Nat.eq_of_dist_eq_zero hdistance).symm
/-- Each actual right endpoint is the next actual left endpoint, or the
parent's end when this is the last child. -/
theorem connectionBudget_zero_right {n : ℕ} (states : List (TargetInterval n))
(start finish : ℕ) (hzero : connectionBudget start states finish = 0)
(i : ℕ) (hi : i < states.length) :
states[i].hi = connectionPrefixBoundary states finish (i + 1) := by
induction states generalizing start i with
| nil => simp only [List.length_nil] at hi; omega
| cons state rest ih =>
have htail : connectionBudget state.hi rest finish = 0 := by
change Nat.dist start state.lo + connectionBudget state.hi rest finish = 0 at hzero
omega
cases i with
| zero =>
cases rest with
| nil =>
exact Nat.eq_of_dist_eq_zero htail
| cons next rest =>
have hdistance : Nat.dist state.hi next.lo = 0 := by
change Nat.dist state.hi next.lo + connectionBudget next.hi rest finish = 0 at htail
omega
exact Nat.eq_of_dist_eq_zero hdistance
| succ i =>
have hprev := ih state.hi htail i (by simp only [List.length_cons] at hi; omega)
simpa only [List.getElem_cons_succ, connectionPrefixBoundary, List.drop_succ_cons] using hprev
theorem zero_connection_left_endpoint {n M : ℕ} (r : TargetInterval n)
(states : Fin M → TargetInterval n) (hzero : endpointActionConnection r states = 0)
(i : Fin M) (hi : i.val = 0) : (states i).lo = r.lo := by
have hhead := connectionBudget_zero_head (List.ofFn states) r.lo r.hi hzero
have hM : 0 < M := by have := i.isLt; omega
simp only [List.head?_eq_getElem?, List.getElem?_ofFn, dite_eq_left hM,
Option.map_some, Option.getD_some] at hhead
have hequal : (⟨0, hM⟩ : Fin M) = i := Fin.ext hi.symm
rw [hequal] at hhead
exact hhead
theorem zero_connection_right_endpoint {n M : ℕ} (r : TargetInterval n)
(states : Fin M → TargetInterval n) (hzero : endpointActionConnection r states = 0)
(i : Fin M) :
(states i).hi = connectionPrefixBoundary (List.ofFn states) r.hi (i.val + 1) := by
have h := connectionBudget_zero_right (List.ofFn states) r.lo r.hi hzero i.val
(by simp only [List.length_ofFn]; exact i.isLt)
rw [List.getElem_ofFn] at h
exact h
/-- Full endpoint distance, including first/last endpoints, for every two
members of the literal computed band. -/
theorem computedBand_child_endpoint_diameter {nx ny M : ℕ} (parent : TargetInterval nx)
(r : TargetInterval ny) (b P : ℕ) (tau theta velocity : ℚ) (htau : 0 < tau) (htheta : 0 ≤ theta)
(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)
(i : Fin M) :
(targetEndpointDistance (states i) (states' i) : ℝ) ≤ 4 * (theta : ℝ) * b / M := by
have hthetaR : (0 : ℝ) ≤ theta := by exact_mod_cast htheta
have hthetaBound : 0 ≤ 2 * (theta : ℝ) * b / M := by positivity
obtain ⟨hnonempty, hwide, hv, hpredicate⟩ := computedBandActions_mem parent r b P tau theta velocity child states hstates
obtain ⟨hnonempty', hwide', hv', hpredicate'⟩ := computedBandActions_mem parent r b P tau theta velocity child states' hstates'
have hlo : (Nat.dist (states i).lo (states' i).lo : ℝ) ≤ 2 * (theta : ℝ) * b / M := by
by_cases hi : 0 < i.val
· exact computedBand_cut_diameter parent r b P tau theta velocity htau child states states' hstates hstates' i hi
· have hi0 : i.val = 0 := by omega
rw [zero_connection_left_endpoint r states hpredicate.1 i hi0,
zero_connection_left_endpoint r states' hpredicate'.1 i hi0, Nat.dist_self, Nat.cast_zero]
exact hthetaBound
have hhi : (Nat.dist (states i).hi (states' i).hi : ℝ) ≤ 2 * (theta : ℝ) * b / M := by
rw [zero_connection_right_endpoint r states hpredicate.1 i,
zero_connection_right_endpoint r states' hpredicate'.1 i]
by_cases hnext : i.val + 1 < M
· rw [connectionPrefixBoundary_ofFn states r.hi ⟨i.val + 1, hnext⟩,
connectionPrefixBoundary_ofFn states' r.hi ⟨i.val + 1, hnext⟩]
exact computedBand_cut_diameter parent r b P tau theta velocity htau child states states' hstates hstates'
⟨i.val + 1, hnext⟩ (by change 0 < i.val + 1; omega)
· have hdrop : M ≤ i.val + 1 := by omega
simp only [connectionPrefixBoundary, List.drop_eq_nil_iff.mpr
(show (List.ofFn states).length ≤ i.val + 1 by simpa only [List.length_ofFn] using hdrop),
List.drop_eq_nil_iff.mpr
(show (List.ofFn states').length ≤ i.val + 1 by simpa only [List.length_ofFn] using hdrop),
List.head?_nil, Option.map_none, Option.getD_none, Nat.dist_self, Nat.cast_zero]
exact hthetaBound
unfold targetEndpointDistance
push_cast
convert add_le_add hlo hhi using 1
ring
/-- Grid soundness of the literal local box enumeration. -/
theorem gridStateBox_on_grid {n g radius lo hi : ℕ} (hg : 0 < g)
(state : TargetInterval n) (hstate : state ∈ gridStateBox n g radius lo hi) :
g ∣ state.lo ∧ g ∣ state.hi := by
obtain ⟨s, hs, htail⟩ := List.mem_flatMap.mp hstate
obtain ⟨t, htpoint, hmade⟩ := List.mem_filterMap.mp htail
unfold makeTargetState at hmade
split_ifs at hmade with hvalid
· cases Option.some.inj hmade
exact ⟨(gridPointsBetween_sound hg hs).2.2, (gridPointsBetween_sound hg htpoint).2.2⟩
theorem wideActionList_on_grid {nx ny M b P : ℕ} (parent : TargetInterval nx)
(r : TargetInterval ny) (action : Fin M → TargetInterval ny)
(haction : action ∈ wideActionList M parent r b P) :
∀ i, seedGridSpacing b P ∣ (action i).lo ∧ seedGridSpacing b P ∣ (action i).hi := by
intro i
exact gridStateBox_on_grid (by unfold seedGridSpacing; omega) (action i)
(finiteAssignments_sound M _ action haction i)
end EditApproximation
end OAI
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