| import OAI.Combinatorics.LowSpaceSubsetSum.Counting.Model |
|
|
| namespace OAI |
|
|
| /-! # Testing a partial indexed mask before generating raw streams -/ |
|
|
| namespace LowSpaceSubsetSum |
|
|
| structure MaskRestriction (n : ℕ) where |
| ones : Finset (Fin n) |
| zeros : Finset (Fin n) |
|
|
| namespace MaskRestriction |
|
|
| variable {n : ℕ} (P : MaskRestriction n) |
|
|
| def empty : MaskRestriction n := ⟨∅, ∅⟩ |
|
|
| / |
| can be tested against its selected top mask directly. -/ |
| def Compatible (domain selected : Finset (Fin n)) : Prop := |
| P.ones ∩ domain ⊆ selected ∧ Disjoint P.zeros selected |
| |
| instance (domain selected : Finset (Fin n)) : Decidable (P.Compatible domain selected) := |
| inferInstanceAs (Decidable (_ ∧ _)) |
| |
| def accept (domain selected : Finset (Fin n)) : Bool := decide (P.Compatible domain selected) |
| |
| @[simp] theorem accept_eq_true (domain selected : Finset (Fin n)) : |
| P.accept domain selected = true ↔ P.Compatible domain selected := by simp [accept] |
| |
| @[simp] theorem empty_compatible (domain selected : Finset (Fin n)) : |
| (empty (n := n)).Compatible domain selected := by simp [empty, Compatible] |
| |
| theorem union_compatible {A B S T : Finset (Fin n)} |
| (hS : P.Compatible A S) (hT : P.Compatible B T) : |
| P.Compatible (A ∪ B) (S ∪ T) := by |
| constructor |
| · intro i hi |
| obtain ⟨hiP, hi⟩ := Finset.mem_inter.mp hi |
| rcases Finset.mem_union.mp hi with hi | hi |
| · exact Finset.mem_union_left _ (hS.1 (Finset.mem_inter.mpr ⟨hiP, hi⟩)) |
| · exact Finset.mem_union_right _ (hT.1 (Finset.mem_inter.mpr ⟨hiP, hi⟩)) |
| · exact Finset.disjoint_union_right.mpr ⟨hS.2, hT.2⟩ |
| |
| / |
| other leaf cannot contain an index in this leaf's disjoint top domain. -/ |
| theorem left_compatible {A B S T : Finset (Fin n)} |
| (hd : Disjoint A B) (hT : T ⊆ B) (h : P.Compatible (A ∪ B) (S ∪ T)) : |
| P.Compatible A S := by |
| constructor |
| · intro i hi |
| obtain ⟨hiP, hiA⟩ := Finset.mem_inter.mp hi |
| have hiST := h.1 (Finset.mem_inter.mpr ⟨hiP, Finset.mem_union_left _ hiA⟩) |
| rcases Finset.mem_union.mp hiST with hiS | hiT |
| · exact hiS |
| · exact (Finset.disjoint_left.mp hd hiA (hT hiT)).elim |
| · exact h.2.mono_right Finset.subset_union_left |
| |
| theorem right_compatible {A B S T : Finset (Fin n)} |
| (hd : Disjoint A B) (hS : S ⊆ A) (h : P.Compatible (A ∪ B) (S ∪ T)) : |
| P.Compatible B T := by |
| apply P.left_compatible hd.symm hS |
| simpa only [Finset.union_comm] using h |
| |
| end MaskRestriction |
| end LowSpaceSubsetSum |
| |
| end OAI |
| |
