| import OAI.Probability.MatroidProphet.Groups |
| import OAI.Probability.MatroidProphet.Residual.Statistics |
| import OAI.Probability.MatroidProphet.ProductSplit |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet.MainAlgorithm |
| open Finset |
| variable {n : ℕ} |
| attribute [local instance] Classical.propDecidable |
|
|
| noncomputable def trueGroup (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) : Finset (Fin n) := |
| univ.filter fun e => candidate M d s e (s e) ∧ s e = some i |
|
|
| @[simp] lemma mem_trueGroup (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (e : Fin n) : |
| e ∈ trueGroup M d s i ↔ candidate M d s e (s e) ∧ s e = some i := by |
| simp [trueGroup] |
|
|
| lemma trueGroup_disjoint (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) : Pairwise (fun i j => Disjoint (trueGroup M d s i) (trueGroup M d s j)) := by |
| intro i j hij |
| apply disjoint_left.mpr |
| intro e hei hej |
| exact hij (Option.some.inj (((mem_trueGroup M d s i e).mp hei).2.symm.trans |
| ((mem_trueGroup M d s j e).mp hej).2)) |
|
|
| lemma trueGroup_disjoint_H (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) : Disjoint (trueGroup M d s i) d.H := by |
| apply disjoint_left.mpr |
| intro e he hh |
| exact ((mem_trueGroup M d s i e).mp he).1.2.1 hh |
|
|
| lemma mem_listed_iff (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) : |
| i ∈ listed M d s ↔ ∃ e ∈ d.D, e ∈ trueGroup M d s i := by |
| simp only [listed, mem_biUnion, mem_trueGroup] |
| constructor |
| · rintro ⟨e, he, hi⟩ |
| by_cases hc : candidate M d s e (s e) |
| · simp only [ite_eq_left hc, Option.mem_toFinset] at hi |
| exact ⟨e, he, hc, hi⟩ |
| · simp only [ite_eq_right hc, Finset.notMem_empty] at hi |
| · rintro ⟨e, he, hc, hi⟩ |
| exact ⟨e, he, by rw [ite_eq_left hc]; simp [hi]⟩ |
|
|
| def withMasks (d : MainMasks n) (A B C : Finset (Fin n)) : MainMasks n := |
| { d with D := A, C := B, T := C } |
|
|
| @[simp] lemma candidate_withMasks (M : Matroid (Fin n)) (d : MainMasks n) |
| (A B C : Finset (Fin n)) (s : Fin n → Option ℤ) (e : Fin n) (v : Option ℤ) : |
| candidate M (withMasks d A B C) s e v ↔ candidate M d s e v := Iff.rfl |
|
|
| @[simp] lemma trueGroup_withMasks (M : Matroid (Fin n)) (d : MainMasks n) |
| (A B C : Finset (Fin n)) (s : Fin n → Option ℤ) (i : ℤ) : |
| trueGroup M (withMasks d A B C) s i = trueGroup M d s i := rfl |
|
|
| lemma groups_withMasks_guards (M : Matroid (Fin n)) (d : MainMasks n) |
| (A B C B' C' : Finset (Fin n)) (s : Fin n → Option ℤ) : |
| groups M (withMasks d A B C) s = groups M (withMasks d A B' C') s := rfl |
|
|
| def focalMasks (d : MainMasks n) (U A B C : Finset (Fin n)) : MainMasks n := |
| { d with D := (d.D \ U) ∪ A, C := (d.C \ U) ∪ B, T := (d.T \ U) ∪ C } |
|
|
| @[simp] lemma candidate_focalMasks (M : Matroid (Fin n)) (d : MainMasks n) |
| (U A B C : Finset (Fin n)) (s : Fin n → Option ℤ) (e : Fin n) (v : Option ℤ) : |
| candidate M (focalMasks d U A B C) s e v ↔ candidate M d s e v := Iff.rfl |
|
|
| @[simp] lemma trueGroup_focalMasks (M : Matroid (Fin n)) (d : MainMasks n) |
| (U A B C : Finset (Fin n)) (s : Fin n → Option ℤ) (i : ℤ) : |
| trueGroup M (focalMasks d U A B C) s i = trueGroup M d s i := rfl |
|
|
| lemma listed_focalMasks (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) : |
| listed M (focalMasks d (trueGroup M d s i) A B C) s = |
| (listed M d s \ {i}) ∪ (if A.Nonempty then {i} else ∅) := by |
| ext j |
| rw [mem_listed_iff] |
| simp only [focalMasks, mem_union, mem_sdiff, |
| mem_singleton, mem_listed_iff] |
| by_cases hne : A.Nonempty |
| · simp only [ite_eq_left hne, mem_singleton] |
| constructor |
| · rintro ⟨e, he | he, hej⟩ |
| · refine Or.inl ⟨⟨e, he.1, hej⟩, ?_⟩ |
| intro hji |
| exact he.2 (hji ▸ hej) |
| · exact Or.inr (Option.some.inj |
| (((mem_trueGroup M d s j e).mp hej).2.symm.trans |
| ((mem_trueGroup M d s i e).mp (hA he)).2)) |
| · rintro (⟨⟨e, heD, hej⟩, hji⟩ | rfl) |
| · refine ⟨e, Or.inl ⟨heD, ?_⟩, hej⟩ |
| intro hei |
| exact hji (Option.some.inj |
| (((mem_trueGroup M d s j e).mp hej).2.symm.trans |
| ((mem_trueGroup M d s i e).mp hei).2)) |
| · obtain ⟨e, he⟩ := hne |
| exact ⟨e, Or.inr he, hA he⟩ |
| · simp only [ite_eq_right hne, Finset.notMem_empty, or_false] |
| constructor |
| · rintro ⟨e, he | he, hej⟩ |
| · refine ⟨⟨e, he.1, hej⟩, ?_⟩ |
| intro hji |
| exact he.2 (hji ▸ hej) |
| · exact False.elim (hne ⟨e, he⟩) |
| · rintro ⟨⟨e, heD, hej⟩, hji⟩ |
| refine ⟨e, Or.inl ⟨heD, ?_⟩, hej⟩ |
| intro hei |
| exact hji (Option.some.inj |
| (((mem_trueGroup M d s j e).mp hej).2.symm.trans |
| ((mem_trueGroup M d s i e).mp hei).2)) |
|
|
| noncomputable def focalReference (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) : MainMasks n := |
| focalMasks d (trueGroup M d s i) (trueGroup M d s i) ∅ ∅ |
|
|
| lemma groups_focalMasks (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hne : A.Nonempty) : |
| groups M (focalMasks d (trueGroup M d s i) A B C) s = |
| groups M (focalReference M d s i) s := by |
| have hU : (trueGroup M d s i).Nonempty := hne.mono hA |
| unfold groups focalReference |
| rw [listed_focalMasks M d s i A B C hA, |
| listed_focalMasks M d s i _ ∅ ∅ Subset.rfl, ite_eq_left hne, ite_eq_left hU] |
|
|
| lemma focal_mem_reference_groups (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (hU : (trueGroup M d s i).Nonempty) : |
| i ∈ groups M (focalReference M d s i) s := by |
| rw [mem_groups] |
| unfold focalReference |
| rw [listed_focalMasks M d s i _ ∅ ∅ Subset.rfl, ite_eq_left hU] |
| exact mem_union_right _ (mem_singleton_self i) |
|
|
| lemma groupMask_at_key (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (mask : Finset (Fin n)) (h : ℕ) (i : ℤ) |
| (hi : (groups M d s)[h]? = some i) : |
| groupMask M d s mask h = ((mask ∩ trueGroup M d s i : Finset (Fin n)) : Set (Fin n)) := by |
| ext e |
| simp only [groupMask, Set.mem_ofPred_eq, Finset.mem_coe, mem_inter, mem_trueGroup, hi] |
|
|
| lemma replaceMask_inter_self (X U A : Finset (Fin n)) (hA : A ⊆ U) : |
| ((X \ U) ∪ A) ∩ U = A := by |
| ext e |
| simp only [mem_inter, mem_union, mem_sdiff] |
| constructor |
| · rintro ⟨he | he, heU⟩ |
| · exact False.elim (he.2 heU) |
| · exact he |
| · intro he |
| exact ⟨Or.inr he, hA he⟩ |
|
|
| lemma replaceMask_inter_other (X U A V : Finset (Fin n)) |
| (hA : A ⊆ U) (hUV : Disjoint U V) : |
| ((X \ U) ∪ A) ∩ V = X ∩ V := by |
| ext e |
| simp only [mem_inter, mem_union, mem_sdiff] |
| constructor |
| · rintro ⟨he | he, heV⟩ |
| · exact ⟨he.1, heV⟩ |
| · exact False.elim (disjoint_left.mp hUV (hA he) heV) |
| · rintro ⟨heX, heV⟩ |
| exact ⟨Or.inl ⟨heX, fun heU => disjoint_left.mp hUV heU heV⟩, heV⟩ |
|
|
| noncomputable def focalIndex (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) : ℕ := |
| (groups M (focalReference M d s i) s).idxOf i |
|
|
| lemma focal_group_key (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hne : A.Nonempty) : |
| (groups M (focalMasks d (trueGroup M d s i) A B C) s)[focalIndex M d s i]? = some i := by |
| rw [groups_focalMasks M d s i A B C hA hne] |
| exact groups_get?_idxOf M _ s i (focal_mem_reference_groups M d s i (hne.mono hA)) |
|
|
| lemma focal_group_mask (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C X Y : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hne : A.Nonempty) |
| (hY : Y ⊆ trueGroup M d s i) : |
| groupMask M (focalMasks d (trueGroup M d s i) A B C) s |
| ((X \ trueGroup M d s i) ∪ Y) (focalIndex M d s i) = (Y : Set (Fin n)) := by |
| rw [groupMask_at_key M _ s _ _ i (focal_group_key M d s i A B C hA hne), |
| trueGroup_focalMasks, replaceMask_inter_self X _ Y hY] |
|
|
| lemma lower_group_mask (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C X Y Z : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hne : A.Nonempty) |
| (hY : Y ⊆ trueGroup M d s i) (hZ : Z ⊆ trueGroup M d s i) |
| (j : ℕ) (hj : j < focalIndex M d s i) : |
| groupMask M (focalMasks d (trueGroup M d s i) A B C) s |
| ((X \ trueGroup M d s i) ∪ Y) j = |
| groupMask M (focalReference M d s i) s |
| ((X \ trueGroup M d s i) ∪ Z) j := by |
| have hi := groups_get?_idxOf M (focalReference M d s i) s i |
| (focal_mem_reference_groups M d s i (hne.mono hA)) |
| obtain ⟨k, hk, hki⟩ := groups_earlier_key M (focalReference M d s i) s hi hj |
| have hk' : (groups M (focalMasks d (trueGroup M d s i) A B C) s)[j]? = some k := by |
| rwa [groups_focalMasks M d s i A B C hA hne] |
| have hdisj := trueGroup_disjoint M d s (ne_of_gt hki) |
| rw [groupMask_at_key M _ s _ j k hk', groupMask_at_key M _ s _ j k hk] |
| simp only [focalReference, trueGroup_focalMasks] |
| rw [replaceMask_inter_other X _ Y _ hY hdisj, |
| replaceMask_inter_other X _ Z _ hZ hdisj] |
|
|
| end MatroidProphet.MainAlgorithm |
|
|
| namespace MatroidProphet |
| open Set |
| variable {α : Type*} [Fintype α] |
|
|
| lemma guardedPath_congr_prefix (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C D' C' : ℕ → Set α) (h : ℕ) |
| (hD : ∀ j < h, D j = D' j) (hC : ∀ j < h, C j = C' j) (k : ℤ) : |
| guardedPath M hE κ D C h k = guardedPath M hE κ D' C' h k := by |
| induction h generalizing k with |
| | zero => rfl |
| | succ h ih => |
| simp only [guardedPath, hD h (Nat.lt_succ_self h), hC h (Nat.lt_succ_self h), |
| ih (fun j hj => hD j (Nat.lt_succ_of_lt hj)) (fun j hj => hC j (Nat.lt_succ_of_lt hj))] |
|
|
| lemma nominalPath_congr_prefix (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C D' C' : ℕ → Set α) (h : ℕ) |
| (hD : ∀ j ≤ h, D j = D' j) (hC : ∀ j < h, C j = C' j) (k : ℤ) : |
| nominalPath M hE κ D C h k = nominalPath M hE κ D' C' h k := by |
| rw [nominalPath, nominalPath, hD h le_rfl, |
| guardedPath_congr_prefix M hE κ D C D' C' h (fun j hj => hD j hj.le) hC k] |
|
|
| lemma lowerCompetition_congr_prefix (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C T D' C' T' : ℕ → Set α) (h : ℕ) |
| (hD : ∀ j < h, D j = D' j) (hC : ∀ j < h, C j = C' j) |
| (hT : ∀ j < h, T j = T' j) (k : ℤ) : |
| lowerCompetition M hE κ D C T k h = lowerCompetition M hE κ D' C' T' k h := by |
| induction h with |
| | zero => rfl |
| | succ h ih => |
| rw [lowerCompetition, lowerCompetition, hT h (Nat.lt_succ_self h), |
| ih (fun j hj => hD j (Nat.lt_succ_of_lt hj)) (fun j hj => hC j (Nat.lt_succ_of_lt hj)) |
| (fun j hj => hT j (Nat.lt_succ_of_lt hj)), |
| nominalPath_congr_prefix M hE κ D C D' C' h |
| (fun j hj => hD j (Nat.lt_succ_of_le hj)) (fun j hj => hC j (Nat.lt_succ_of_lt hj)), |
| nominalPath_congr_prefix M hE κ D C D' C' h |
| (fun j hj => hD j (Nat.lt_succ_of_le hj)) (fun j hj => hC j (Nat.lt_succ_of_lt hj))] |
|
|
| namespace MainAlgorithm |
| open Finset |
| variable {n : ℕ} |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma focal_prefix_masks (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hB : B ⊆ trueGroup M d s i) |
| (hC : C ⊆ trueGroup M d s i) (hne : A.Nonempty) |
| (j : ℕ) (hj : j < focalIndex M d s i) : |
| let f := focalMasks d (trueGroup M d s i) A B C |
| let r := focalReference M d s i |
| groupMask M f s f.D j = groupMask M r s r.D j ∧ |
| groupMask M f s f.C j = groupMask M r s r.C j ∧ |
| groupMask M f s f.T j = groupMask M r s r.T j := by |
| exact ⟨lower_group_mask M d s i A B C d.D A _ hA hne hA Finset.Subset.rfl j hj, |
| lower_group_mask M d s i A B C d.C B ∅ hA hne hB (empty_subset _) j hj, |
| lower_group_mask M d s i A B C d.T C ∅ hA hne hC (empty_subset _) j hj⟩ |
|
|
| lemma focal_incoming_path (M : Matroid (Fin n)) (hE : M.E = Set.univ) (κ : ℕ) |
| (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hB : B ⊆ trueGroup M d s i) |
| (hC : C ⊆ trueGroup M d s i) (hne : A.Nonempty) (k : ℤ) : |
| let f := focalMasks d (trueGroup M d s i) A B C |
| let r := focalReference M d s i |
| guardedPath M hE κ (groupMask M f s f.D) (groupMask M f s f.C) (focalIndex M d s i) k = |
| guardedPath M hE κ (groupMask M r s r.D) (groupMask M r s r.C) (focalIndex M d s i) k := by |
| apply guardedPath_congr_prefix |
| · exact fun j hj => (focal_prefix_masks M d s i A B C hA hB hC hne j hj).1 |
| · exact fun j hj => (focal_prefix_masks M d s i A B C hA hB hC hne j hj).2.1 |
|
|
| lemma focal_lowerCompetition (M : Matroid (Fin n)) (hE : M.E = Set.univ) (κ : ℕ) |
| (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hB : B ⊆ trueGroup M d s i) |
| (hC : C ⊆ trueGroup M d s i) (hne : A.Nonempty) (k : ℤ) : |
| let f := focalMasks d (trueGroup M d s i) A B C |
| let r := focalReference M d s i |
| lowerCompetition M hE κ (groupMask M f s f.D) (groupMask M f s f.C) |
| (groupMask M f s f.T) k (focalIndex M d s i) = |
| lowerCompetition M hE κ (groupMask M r s r.D) (groupMask M r s r.C) |
| (groupMask M r s r.T) k (focalIndex M d s i) := by |
| apply lowerCompetition_congr_prefix |
| · exact fun j hj => (focal_prefix_masks M d s i A B C hA hB hC hne j hj).1 |
| · exact fun j hj => (focal_prefix_masks M d s i A B C hA hB hC hne j hj).2.1 |
| · exact fun j hj => (focal_prefix_masks M d s i A B C hA hB hC hne j hj).2.2 |
|
|
| lemma focal_nominal_path (M : Matroid (Fin n)) (hE : M.E = Set.univ) (κ : ℕ) |
| (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hB : B ⊆ trueGroup M d s i) |
| (hC : C ⊆ trueGroup M d s i) (hne : A.Nonempty) (k : ℤ) : |
| let f := focalMasks d (trueGroup M d s i) A B C |
| let r := focalReference M d s i |
| nominalPath M hE κ (groupMask M f s f.D) (groupMask M f s f.C) (focalIndex M d s i) k = |
| densityEnabled M hE κ (A : Set (Fin n)) (activation (focalIndex M d s i)) k |
| (guardedPath M hE κ (groupMask M r s r.D) (groupMask M r s r.C) (focalIndex M d s i) k) := by |
| dsimp only |
| rw [nominalPath, focal_incoming_path M hE κ d s i A B C hA hB hC hne k] |
| congr 1 |
| exact focal_group_mask M d s i A B C d.D A hA hne hA |
|
|
| lemma focal_observed_inter (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) (hB : B ⊆ trueGroup M d s i) : |
| let f := focalMasks d (trueGroup M d s i) A B C |
| (f.H ∪ f.D ∪ f.C) ∩ trueGroup M d s i = A ∪ B := by |
| dsimp only [focalMasks] |
| rw [Finset.union_inter_distrib_right, Finset.union_inter_distrib_right, |
| disjoint_iff_inter_eq_empty.mp (trueGroup_disjoint_H M d s i).symm, |
| replaceMask_inter_self _ _ _ hA, replaceMask_inter_self _ _ _ hB, Finset.empty_union] |
|
|
| lemma focal_listed_iff_nonempty (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (hA : A ⊆ trueGroup M d s i) : |
| i ∈ groups M (focalMasks d (trueGroup M d s i) A B C) s ↔ A.Nonempty := by |
| rw [mem_groups, listed_focalMasks M d s i A B C hA] |
| by_cases hne : A.Nonempty <;> simp [hne] |
|
|
| noncomputable def listedSafeStatistic (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (κ : ℕ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) |
| (Y : Set (Fin n)) (ε : Fin 2) : ℕ := |
| if i ∈ groups M d s then |
| safeLayerStatistic M hE κ (groupMask M d s d.D) (groupMask M d s d.C) |
| (groupMask M d s d.T) ((groups M d s).idxOf i) (trueGroup M d s i : Set (Fin n)) |
| ((d.D : Set (Fin n)) ∩ Y) ε |
| else 0 |
|
|
| noncomputable def listedRankStatistic (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (κ : ℕ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) (ε : Fin 2) : ℕ := |
| if i ∈ groups M d s then |
| nominalRankStatistic M hE κ (groupMask M d s d.D) (groupMask M d s d.C) |
| (groupMask M d s d.T) ((groups M d s).idxOf i) (trueGroup M d s i : Set (Fin n)) |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) ε |
| else 0 |
|
|
| lemma listedSafeStatistic_focal_empty (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (κ : ℕ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) |
| (B C : Finset (Fin n)) (Y : Set (Fin n)) (ε : Fin 2) : |
| listedSafeStatistic M hE κ (focalMasks d (trueGroup M d s i) ∅ B C) s i Y ε = 0 := by |
| rw [listedSafeStatistic, ite_eq_right] |
| simp only [focal_listed_iff_nonempty M d s i ∅ B C (Finset.empty_subset _), |
| Finset.not_nonempty_empty, not_false_eq_true] |
|
|
| lemma listedRankStatistic_focal_empty (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (κ : ℕ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) |
| (B C : Finset (Fin n)) (ε : Fin 2) : |
| listedRankStatistic M hE κ (focalMasks d (trueGroup M d s i) ∅ B C) s i ε = 0 := by |
| rw [listedRankStatistic, ite_eq_right] |
| simp only [focal_listed_iff_nonempty M d s i ∅ B C (Finset.empty_subset _), |
| Finset.not_nonempty_empty, not_false_eq_true] |
|
|
| lemma listedSafeStatistic_focal_nonempty (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (κ : ℕ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) |
| (A B C : Finset (Fin n)) (Y : Set (Fin n)) (ε : Fin 2) |
| (hA : A ⊆ trueGroup M d s i) (hne : A.Nonempty) : |
| let f := focalMasks d (trueGroup M d s i) A B C |
| listedSafeStatistic M hE κ f s i Y ε = |
| safeLayerStatistic M hE κ (groupMask M f s f.D) (groupMask M f s f.C) |
| (groupMask M f s f.T) (focalIndex M d s i) (trueGroup M d s i : Set (Fin n)) |
| ((f.D : Set (Fin n)) ∩ Y) ε := by |
| dsimp only |
| rw [listedSafeStatistic, ite_eq_left ((focal_listed_iff_nonempty M d s i A B C hA).mpr hne), |
| groups_focalMasks M d s i A B C hA hne, trueGroup_focalMasks] |
| rfl |
|
|
| lemma listedRankStatistic_focal_nonempty (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (κ : ℕ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) |
| (A B C : Finset (Fin n)) (ε : Fin 2) |
| (hA : A ⊆ trueGroup M d s i) (hne : A.Nonempty) : |
| let f := focalMasks d (trueGroup M d s i) A B C |
| listedRankStatistic M hE κ f s i ε = |
| nominalRankStatistic M hE κ (groupMask M f s f.D) (groupMask M f s f.C) |
| (groupMask M f s f.T) (focalIndex M d s i) (trueGroup M d s i : Set (Fin n)) |
| ((f.H ∪ f.D ∪ f.C : Finset (Fin n)) : Set (Fin n)) ε := by |
| dsimp only |
| rw [listedRankStatistic, ite_eq_left ((focal_listed_iff_nonempty M d s i A B C hA).mpr hne), |
| groups_focalMasks M d s i A B C hA hne, trueGroup_focalMasks] |
| rfl |
|
|
| lemma focal_density_inter (M : Matroid (Fin n)) (d : MainMasks n) |
| (s : Fin n → Option ℤ) (i : ℤ) (A B C : Finset (Fin n)) |
| (Y : Set (Fin n)) (hY : Y ⊆ (trueGroup M d s i : Set (Fin n))) : |
| ((focalMasks d (trueGroup M d s i) A B C).D : Set (Fin n)) ∩ Y = |
| (A : Set (Fin n)) ∩ Y := by |
| ext e |
| simp only [Set.mem_inter_iff, Finset.mem_coe, focalMasks, Finset.mem_union, |
| Finset.mem_sdiff] |
| constructor |
| · rintro ⟨he | he, heY⟩ |
| · exact False.elim (he.2 (hY heY)) |
| · exact ⟨he, heY⟩ |
| · rintro ⟨heA, heY⟩ |
| exact ⟨Or.inr heA, heY⟩ |
|
|
| end MainAlgorithm |
| end MatroidProphet |
|
|
| end OAI |
|
|
