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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 := ⟨∅, ∅⟩
/-- A leaf checks only forced-one indices in its own top block. Forced zeros
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⟩
/-- A compatible full mask yields a compatible leaf restriction because the
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
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