| import OAI.Probability.MatroidProphet.Pivots.Phases |
| import OAI.Probability.MatroidProphet.Residual.Generators |
| import Mathlib.Data.Int.Interval |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| namespace Pivots |
|
|
| open Set |
|
|
| abbrev IntWindow (a : ℤ) := {k : ℤ // k ∈ Set.Icc (a - 2) 1} |
|
|
| noncomputable instance (a : ℤ) : Fintype (IntWindow a) := Fintype.ofFinite _ |
|
|
| abbrev ParityWindow (a : ℤ) (ε : Fin 2) := {k : IntWindow a // k.val % 2 = (ε.val : ℤ)} |
|
|
| noncomputable instance (a : ℤ) (ε : Fin 2) : Fintype (ParityWindow a ε) := Fintype.ofFinite _ |
|
|
| def roundParity (ε : Fin 2) (k : ℤ) : ℤ := if k % 2 = (ε.val : ℤ) then k else k + 1 |
|
|
| lemma roundParity_bounds (ε : Fin 2) (k : ℤ) : k ≤ roundParity ε k ∧ roundParity ε k ≤ k + 1 := by |
| unfold roundParity |
| split <;> omega |
|
|
| lemma roundParity_mod (ε : Fin 2) (k : ℤ) : roundParity ε k % 2 = (ε.val : ℤ) := by |
| have hε := ε.isLt |
| unfold roundParity |
| split <;> omega |
|
|
| lemma roundParity_le_iff (ε : Fin 2) (k b : ℤ) (hb : b % 2 = (ε.val : ℤ)) : |
| roundParity ε k ≤ b ↔ k ≤ b := by |
| have hε := ε.isLt |
| unfold roundParity |
| split <;> omega |
|
|
| lemma roundParity_lt_iff (ε : Fin 2) (k b : ℤ) (hb : b % 2 = (ε.val : ℤ)) : |
| roundParity ε k < b ↔ k ≤ b - 2 := by |
| have hε := ε.isLt |
| unfold roundParity |
| split <;> omega |
|
|
| variable {α G : Type*} [Fintype α] [LinearOrder α] [Fintype G] |
|
|
| def windowPlacement (a : ℤ) (ν : G → ℤ) (hν : ∀ g, a ≤ ν g ∧ ν g ≤ 0) : G → IntWindow a := |
| fun g => ⟨ν g, by have := hν g; constructor <;> omega⟩ |
|
|
| def parityPlacement (a : ℤ) (ε : Fin 2) (ν : G → ℤ) |
| (hν : ∀ g, a ≤ ν g ∧ ν g ≤ 0) : G → ParityWindow a ε := |
| fun g => ⟨⟨roundParity ε (ν g), by |
| have := hν g |
| have := roundParity_bounds ε (ν g) |
| constructor <;> omega⟩, roundParity_mod ε (ν g)⟩ |
|
|
| def generatedPath (M : Matroid α) (F : ℤ → Set α) (g : G → α) (ν : G → ℤ) (k : ℤ) : Set α := |
| M.closure (F k ∪ g '' {i | ν i ≤ k}) |
|
|
| lemma window_prefix (M : Matroid α) (F : ℤ → Set α) (hF : Monotone F) |
| (a : ℤ) (g : G → α) (ν : G → ℤ) (hν : ∀ i, a ≤ ν i ∧ ν i ≤ 0) (k : IntWindow a) : |
| M.closure (blockPrefix (fun t : IntWindow a => F t.val) g (windowPlacement a ν hν) k) = |
| generatedPath M F g ν k.val := by |
| rw [blockPrefix_eq (fun t : IntWindow a => F t.val) (fun _ _ h => hF h)] |
| rfl |
|
|
| lemma parity_upperPath |
| {α : Type u_1} {G : Type u_2} [Fintype α] [LinearOrder α] [Fintype G] |
| (M : Matroid α) (F : ℤ → Set α) (a : ℤ) (ε : Fin 2) |
| (g : G → α) (ν : G → ℤ) (hν : ∀ i, a ≤ ν i ∧ ν i ≤ 0) (b : ParityWindow a ε) : |
| upperPath M (fun t : ParityWindow a ε => F t.val.val) g (parityPlacement a ε ν hν) b = |
| generatedPath M F g ν b.val.val := by |
| have hs : {i | parityPlacement a ε ν hν i ≤ b} = {i | ν i ≤ b.val.val} := by |
| ext i |
| exact roundParity_le_iff ε (ν i) b.val.val b.property |
| unfold upperPath generatedPath |
| rw [hs] |
|
|
| lemma parity_lowerPath |
| {α : Type u_1} {G : Type u_2} [Fintype α] [LinearOrder α] [Fintype G] |
| (M : Matroid α) (hE : M.E = univ) |
| (F : ℤ → Set α) (hF : Monotone F) (a : ℤ) |
| (hFe : ∀ k < a, F k = M.closure ∅) (ε : Fin 2) |
| (g : G → α) (ν : G → ℤ) (hν : ∀ i, a ≤ ν i ∧ ν i ≤ 0) (b : ParityWindow a ε) : |
| lowerPath M (fun t : ParityWindow a ε => F t.val.val) g (parityPlacement a ε ν hν) b = |
| generatedPath M F g ν (b.val.val - 2) := by |
| have hgen : g '' {i | parityPlacement a ε ν hν i < b} = g '' {i | ν i ≤ b.val.val - 2} := by |
| congr 1 |
| ext i |
| exact roundParity_lt_iff ε (ν i) b.val.val b.property |
| unfold lowerPath generatedPath |
| rw [hgen] |
| apply Subset.antisymm |
| · apply M.closure_subset_closure |
| apply union_subset_union _ Subset.rfl |
| intro e he |
| obtain ⟨c, hc, he⟩ := mem_iUnion.mp he |>.imp fun _ h => mem_iUnion.mp h |
| have hcb : c.val.val ≤ b.val.val - 2 := by |
| have hb := b.property |
| have hcmod := c.property |
| change c.val.val < b.val.val at hc |
| omega |
| exact hF hcb he |
| · apply (M.closure_subset_closure ?_).trans_eq (M.closure_closure _) |
| apply union_subset |
| · intro e he |
| by_cases hlo : a - 2 ≤ b.val.val - 2 |
| · let c : ParityWindow a ε := ⟨⟨b.val.val - 2, ⟨hlo, by have := b.val.property.2; omega⟩⟩, |
| by change (b.val.val - 2) % 2 = (ε.val : ℤ); have := b.property; omega⟩ |
| apply M.subset_closure _ (by simp [hE]) |
| left |
| refine mem_iUnion.mpr ⟨c, mem_iUnion.mpr ⟨?_, he⟩⟩ |
| change b.val.val - 2 < b.val.val |
| omega |
| · have hearly : b.val.val - 2 < a := by omega |
| rw [hFe _ hearly] at he |
| exact M.closure_subset_closure (empty_subset _) he |
| · exact (subset_union_right.trans (M.subset_closure _ (by simp [hE]))) |
|
|
| lemma window_before (M : Matroid α) (hE : M.E = univ) |
| (F : ℤ → Set α) (hF : Monotone F) (a : ℤ) |
| (hFe : ∀ k < a, F k = M.closure ∅) |
| (g : G → α) (ν : G → ℤ) (hν : ∀ i, a ≤ ν i ∧ ν i ≤ 0) (k : IntWindow a) : |
| M.closure (blockBefore (fun t : IntWindow a => F t.val) g (windowPlacement a ν hν) k) = |
| generatedPath M F g ν (k.val - 1) := by |
| rw [blockBefore_eq_iUnion] |
| have hgen : g '' {i | windowPlacement a ν hν i < k} = g '' {i | ν i ≤ k.val - 1} := by |
| congr 1 |
| ext i |
| change ν i < k.val ↔ ν i ≤ k.val - 1 |
| omega |
| rw [hgen] |
| unfold generatedPath |
| apply Subset.antisymm |
| · apply M.closure_subset_closure |
| apply union_subset_union _ Subset.rfl |
| intro e he |
| obtain ⟨c, hc, he⟩ := mem_iUnion.mp he |>.imp fun _ h => mem_iUnion.mp h |
| apply hF (show c.val ≤ k.val - 1 from by change c.val < k.val at hc; omega) he |
| · apply (M.closure_subset_closure ?_).trans_eq (M.closure_closure _) |
| apply union_subset |
| · intro e he |
| by_cases hlo : a - 2 ≤ k.val - 1 |
| · let c : IntWindow a := ⟨k.val - 1, ⟨hlo, by have := k.property.2; omega⟩⟩ |
| apply M.subset_closure _ (by simp [hE]) |
| left |
| refine mem_iUnion.mpr ⟨c, mem_iUnion.mpr ⟨?_, he⟩⟩ |
| change k.val - 1 < k.val |
| omega |
| · have hearly : k.val - 1 < a := by omega |
| rw [hFe _ hearly] at he |
| exact M.closure_subset_closure (empty_subset _) he |
| · exact (subset_union_right.trans (M.subset_closure _ (by simp [hE]))) |
|
|
| end Pivots |
| end MatroidProphet |
|
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| end OAI |
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