| import OAI.Probability.MatroidProphet.Reverse.Hazard |
| import OAI.Probability.MatroidProphet.Density.Birth |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Finset |
| variable {α : Type*} [Fintype α] [DecidableEq α] |
| attribute [local instance] Classical.propDecidable |
|
|
| structure ReverseTransition (α : Type*) where |
| time : ℤ |
| before : ℕ → Set α |
| after : ℕ → Set α |
|
|
| noncomputable def reverseTrace (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) : |
| ℕ → ℤ → (ℕ → Set α) → Finset α → List (ReverseTransition α) |
| | 0, _, _, _ => [] |
| | m+1, k, S, C => |
| ⟨k, S, (reverseStepTree M hE κ k D S G n).run C⟩ :: |
| reverseTrace M hE κ D G n m (k-1) ((reverseStepTree M hE κ k D S G n).run C) C |
|
|
| lemma reverseCost_le_of_trace (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (f g : ℤ → (ℕ → Set α) → (ℕ → Set α) → ℝ) |
| (m : ℕ) (k : ℤ) (S : ℕ → Set α) (C : Finset α) |
| (hfg : ∀ t ∈ reverseTrace M hE κ D G n m k S C, |
| f t.time t.before t.after ≤ g t.time t.before t.after) : |
| reverseCost M hE κ D G n f m k S C ≤ reverseCost M hE κ D G n g m k S C := by |
| induction m generalizing k S with |
| | zero => exact le_rfl |
| | succ m ih => |
| rw [reverseCost, reverseCost] |
| apply add_le_add (hfg ⟨k, S, (reverseStepTree M hE κ k D S G n).run C⟩ (by simp [reverseTrace])) |
| exact ih _ _ (fun t ht => hfg t (by simp only [reverseTrace, List.mem_cons]; exact Or.inr ht)) |
|
|
| noncomputable def traceHazards (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) (q : α → ℝ) (d : α) (h : ℕ) |
| (m : ℕ) (k : ℤ) (S : ℕ → Set α) : Finset ℝ := |
| univ.biUnion (fun C : Finset α => ((reverseTrace M hE κ D G n m k S C).map |
| (fun t => reverseHazard M hE κ D G q d h t.time t.before)).toFinset) |
|
|
| lemma mem_traceHazards (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) (q : α → ℝ) (d : α) (h : ℕ) |
| (m : ℕ) (k : ℤ) (S : ℕ → Set α) (C : Finset α) (t : ReverseTransition α) |
| (ht : t ∈ reverseTrace M hE κ D G n m k S C) : |
| reverseHazard M hE κ D G q d h t.time t.before ∈ traceHazards M hE κ D G n q d h m k S := by |
| exact mem_biUnion.mpr ⟨C, mem_univ C, List.mem_toFinset.mpr (List.mem_map.mpr ⟨t, ht, rfl⟩)⟩ |
|
|
| noncomputable def reverseFinal (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) : |
| ℕ → ℤ → (ℕ → Set α) → Finset α → (ℕ → Set α) |
| | 0, _, S, _ => S |
| | m+1, k, S, C => reverseFinal M hE κ D G n m (k-1) |
| ((reverseStepTree M hE κ k D S G n).run C) C |
|
|
| lemma reverseStepTree_path (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) (k : ℤ) (S : ℕ → Set α) (C : Finset α) |
| (hS : ∀ j ≤ n, S j = nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) j (k+1)) |
| {j : ℕ} (hj : j ≤ n) : |
| (reverseStepTree M hE κ k D S G n).run C j = |
| nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) j k := by |
| rw [reverseStepTree_run_coordinate M hE κ k D S G n hG C hj] |
| calc |
| _ = reverseNominal M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) |
| (fun i => nominalPath M hE κ D (fun l => (C : Set α) ∩ (G l : Set α)) i (k+1)) k j := by |
| apply reverseNominal_congr_before |
| · intro i hi; rfl |
| · intro i hi; exact hS i (by omega) |
| _ = _ := reverseNominal_agrees M hE κ D _ k j |
|
|
| lemma reverseFinal_path (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) |
| (m : ℕ) (k : ℤ) (S : ℕ → Set α) (C : Finset α) |
| (hS : ∀ j ≤ n, S j = nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) j (k+1)) |
| {j : ℕ} (hj : j ≤ n) : |
| reverseFinal M hE κ D G n m k S C j = |
| nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) j (k+1-m) := by |
| induction m generalizing k S with |
| | zero => simpa only [reverseFinal, Nat.cast_zero, sub_zero] using hS j hj |
| | succ m ih => |
| rw [reverseFinal] |
| have hs' : ∀ i ≤ n, (reverseStepTree M hE κ k D S G n).run C i = |
| nominalPath M hE κ D (fun l => (C : Set α) ∩ (G l : Set α)) i (k-1+1) := by |
| intro i hi |
| simpa only [sub_add_cancel] using reverseStepTree_path M hE κ D G n hG k S C hS hi |
| rw [ih (k-1) _ hs'] |
| congr 1 |
| omega |
|
|
| lemma reverseFinal_from_univ (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) |
| (h : ℕ) (hh : h ≤ n) (C : Finset α) : |
| reverseFinal M hE κ D G n (pathHorizon h + 1) (-1) (fun _ => Set.univ) C h = M.closure ∅ := by |
| rw [reverseFinal_path M hE κ D G n hG (pathHorizon h+1) (-1) (fun _ => Set.univ) C |
| (fun j hj => by simpa only [neg_add_cancel] using |
| (nominalPath_nonnegative M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) j (by omega : (0:ℤ) ≤ -1+1)).symm) hh] |
| apply nominalPath_early |
| unfold pathHorizon activation |
| omega |
|
|
| noncomputable def exitCost (d : α) (h : ℕ) (_ : ℤ) (S S' : ℕ → Set α) : ℝ := |
| if d ∈ S h ∧ d ∉ S' h then 1 else 0 |
|
|
| lemma reverseCost_exit_telescope (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) (d : α) (h : ℕ) |
| (m : ℕ) (k : ℤ) (S : ℕ → Set α) (closed : ReverseClosed M hE κ D k S) (C : Finset α) : |
| reverseCost M hE κ D G n (exitCost d h) m k S C = |
| (if d ∈ S h then 1 else 0) - (if d ∈ reverseFinal M hE κ D G n m k S C h then 1 else 0) := by |
| induction m generalizing k S with |
| | zero => by_cases hs : d ∈ S h <;> simp [reverseCost, reverseFinal, hs] |
| | succ m ih => |
| rw [reverseCost, reverseFinal, ih (k-1) _ (reverseStepTree_closed M hE κ k D S G n C)] |
| have hc := reverseStepTree_contracts M hE κ k D S G n closed C h |
| by_cases hs : d ∈ S h <;> by_cases hn : d ∈ (reverseStepTree M hE κ k D S G n).run C h |
| · simp [exitCost, hs, hn] |
| · simp [exitCost, hs, hn]; ring |
| · exact (hs (hc hn)).elim |
| · simp [exitCost, hs, hn] |
|
|
| lemma reverse_exit_total_one (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) (q : α → ℝ) |
| (d : α) (hd : d ∉ M.closure ∅) (h : ℕ) (hh : h ≤ n) : |
| reverseMeanCost M hE κ D G n q (exitCost d h) (pathHorizon h+1) (-1) (fun _ => Set.univ) = 1 := by |
| unfold reverseMeanCost |
| have hp : ∀ C : Finset α, reverseCost M hE κ D G n (exitCost d h) |
| (pathHorizon h+1) (-1) (fun _ => Set.univ) C = 1 := by |
| intro C |
| rw [reverseCost_exit_telescope M hE κ D G n d h (pathHorizon h+1) (-1) _ |
| (reverseClosed_univ M hE κ D (-1)) C, reverseFinal_from_univ M hE κ D G n hG h hh C] |
| simp [hd] |
| exact (bitsExpectation_congr q univ (fun C _ => hp C)).trans (bitsExpectation_const q univ 1) |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
