| import OAI.Combinatorics.EditApproximation.Geometry.ComparisonBandCoverage |
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| namespace OAI |
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|
| /-! |
| # Literal band endpoint geometry and rounding guards |
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| The full child endpoint diameter follows from the computed internal cuts |
| and zero connection, including the two outside endpoints. |
| -/ |
|
|
| namespace EditApproximation |
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|
| open Finset |
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|
| 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⟩ |
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|
| 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) |
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| end EditApproximation |
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| end OAI |
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