| import OAI.Probability.MatroidProphet.PathRankCost |
| import OAI.Probability.MatroidProphet.Residual.Timeline |
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| namespace OAI |
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| namespace MatroidProphet |
| open Set Finset |
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| lemma sum_spaced_potential_drop (p : ℤ → ℕ) (hp : Antitone p) |
| (L : Finset ℤ) (lo hi : ℤ) (hlohi : lo ≤ hi) |
| (hlo : ∀ b ∈ L, lo + 2 ≤ b) (hhi : ∀ b ∈ L, b ≤ hi) |
| (hgap : ∀ b ∈ L, ∀ c ∈ L, b < c → b + 2 ≤ c) : |
| (∑ b ∈ L, (p (b-2) - p b)) + p hi ≤ p lo := by |
| induction L using Finset.induction_on_max generalizing lo hi with |
| | empty => simpa using hp hlohi |
| | @insert a L hmax ih => |
| have ha : a ∉ L := fun hm => (lt_irrefl a) (hmax a hm) |
| have hla : lo ≤ a-2 := by have := hlo a (mem_insert_self a L); omega |
| have harest : ∀ b ∈ L, b ≤ a-2 := by |
| intro b hb |
| have hb' := hmax b hb |
| have hg := hgap b (mem_insert_of_mem hb) a (mem_insert_self a L) hb' |
| omega |
| have hrec := ih lo (a-2) hla |
| (fun b hb => hlo b (mem_insert_of_mem hb)) harest |
| (fun b hb c hc hbc => hgap b (mem_insert_of_mem hb) c (mem_insert_of_mem hc) hbc) |
| have hahi := hhi a (mem_insert_self a L) |
| have hphi := hp hahi |
| have hpaa := hp (show a-2 ≤ a by omega) |
| rw [sum_insert ha] |
| omega |
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| variable {α : Type*} [Fintype α] |
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| theorem rankCost_spaced (M : Matroid α) (J : Set α) (L : Finset ℤ) |
| (lo hi : ℤ) (hlohi : lo ≤ hi) |
| (hlo : ∀ b ∈ L, lo+2 ≤ b) (hhi : ∀ b ∈ L, b ≤ hi) |
| (hgap : ∀ b ∈ L, ∀ c ∈ L, b < c → b+2 ≤ c) |
| (X A B S : ℤ → Set α) (hX : Monotone X) |
| (hXA : ∀ b ∈ L, X (b-2) ⊆ A b) |
| (hAB : ∀ b ∈ L, A b ⊆ B b) |
| (hB : ∀ b ∈ L, B b ⊆ M.closure (A b ∪ J)) |
| (hY : ∀ b ∈ L, A b ∪ S b ⊆ X b) : |
| (∑ b ∈ L, conditionalRank M (S b) (A b)) ≤ |
| (∑ b ∈ L, conditionalRank M (S b) (B b)) + natRank M J := by |
| have hp : Antitone (fun b => conditionalRank M J (X b)) := |
| fun _ _ hab => conditionalRank_antitone M J (hX hab) |
| have hd := sum_spaced_potential_drop _ hp L lo hi hlohi hlo hhi hgap |
| have hs : (∑ b ∈ L, conditionalRank M (S b) (A b)) ≤ |
| ∑ b ∈ L, (conditionalRank M (S b) (B b) + |
| (conditionalRank M J (X (b-2)) - conditionalRank M J (X b))) := by |
| apply sum_le_sum |
| intro b hb |
| have hc := rankCost_step M J (S b) (A b) (B b) (X (b-2)) (X b) |
| (hXA b hb) (hAB b hb) (hB b hb) (hY b hb) |
| have hm := conditionalRank_antitone M (S b) (hAB b hb) |
| omega |
| rw [sum_add_distrib] at hs |
| have hb := conditionalRank_le_rank M J (X lo) |
| omega |
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| theorem actual_path_rank_cost_finset (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (hκ : 0 < κ) (D C T : ℕ → Set α) (h later : ℕ) |
| (L : Finset ℤ) (lo hi : ℤ) (hlohi : lo ≤ hi) |
| (hlo : ∀ b ∈ L, lo+2 ≤ b) (hhi : ∀ b ∈ L, b ≤ hi) |
| (hgap : ∀ b ∈ L, ∀ c ∈ L, b < c → b+2 ≤ c) |
| (S : ℤ → Set α) (hS : ∀ b, S b ⊆ nominalPath M hE κ D C h b) : |
| κ * (∑ b ∈ L, conditionalRank M (S b) |
| (nominalPath M hE κ D C h (b-2) ∪ lowerCompetition M hE κ D C T b h)) ≤ |
| κ * (∑ b ∈ L, conditionalRank M (S b) |
| (guardedPath M hE κ D C (h+1+later) (b-2) ∪ lowerCompetition M hE κ D C T b h)) + |
| κ * (C h).ncard + |
| ∑ j ∈ range later, (κ * (C (h+1+j)).ncard + (D (h+1+j)).ncard) := by |
| obtain ⟨J, hJcost, hJ⟩ := exists_commonPathBudget M hE κ hκ D C h later |
| have hXfinal (b : ℤ) : nominalPath M hE κ D C h b ⊆ |
| guardedPath M hE κ D C (h+1+later) b := by |
| apply (nominal_subset_guarded M hE κ D C h b).trans |
| have hm : Monotone (fun j => guardedPath M hE κ D C j b) := |
| monotone_nat_of_le_succ (fun j => (guarded_subset_nominal M hE κ D C j b).trans |
| (nominal_subset_guarded M hE κ D C j b)) |
| exact hm (by omega) |
| have hr := rankCost_spaced M J L lo hi hlohi hlo hhi hgap |
| (nominalPath M hE κ D C h) |
| (fun b => nominalPath M hE κ D C h (b-2) ∪ lowerCompetition M hE κ D C T b h) |
| (fun b => guardedPath M hE κ D C (h+1+later) (b-2) ∪ lowerCompetition M hE κ D C T b h) |
| S (nominalPath_mono M hE κ D C h) |
| (fun _ _ => Set.subset_union_left) |
| (fun b _ => Set.union_subset_union_left _ (hXfinal _)) |
| (by |
| intro b hb |
| apply Set.union_subset |
| · exact (hJ _).trans (M.closure_subset_closure (by |
| intro e he |
| rcases he with he | he |
| · exact Or.inl (Or.inl he) |
| · exact Or.inr he)) |
| · intro e he |
| exact M.mem_closure_of_mem (Or.inl (Or.inr he)) (by simp [hE])) |
| (by |
| intro b hb |
| apply Set.union_subset |
| · apply Set.union_subset |
| · exact nominalPath_mono M hE κ D C h (by omega) |
| · exact (lowerCompetition_subset_guarded M hE κ D C T _ h).trans |
| (guarded_subset_nominal M hE κ D C h _) |
| · exact hS _) |
| have hmul := Nat.mul_le_mul_left κ hr |
| rw [Nat.mul_add] at hmul |
| have hcard := natRank_le_ncard M J |
| have hbound := (Nat.mul_le_mul_left κ hcard).trans hJcost |
| omega |
|
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| noncomputable def parityPoints (a : ℤ) (ε : Fin 2) : Finset ℤ := |
| Finset.univ.image (fun b : Pivots.ParityWindow a ε => b.val.val) |
|
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| lemma mem_parityPoints (a : ℤ) (ε : Fin 2) (b : ℤ) : |
| b ∈ parityPoints a ε ↔ a-2 ≤ b ∧ b ≤ 1 ∧ b % 2 = (ε.val : ℤ) := by |
| classical |
| constructor |
| · intro hb |
| obtain ⟨c, _, rfl⟩ := Finset.mem_image.mp hb |
| exact ⟨c.val.property.1, c.val.property.2, c.property⟩ |
| · rintro ⟨hl, hr, hm⟩ |
| exact Finset.mem_image.mpr ⟨⟨⟨b, ⟨hl, hr⟩⟩, hm⟩, Finset.mem_univ _, rfl⟩ |
|
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| lemma sum_parityPoints {β : Type*} [AddCommMonoid β] (a : ℤ) (ε : Fin 2) (f : ℤ → β) : |
| (∑ b ∈ parityPoints a ε, f b) = ∑ b : Pivots.ParityWindow a ε, f b.val.val := by |
| classical |
| apply Finset.sum_image |
| intro x _ y _ hxy |
| exact Subtype.ext (Subtype.ext hxy) |
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| end MatroidProphet |
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| end OAI |
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