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760f90e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | import Mathlib
namespace OAI
namespace Superstring
abbrev Symbol := List Bool
abbrev Word := List Symbol
abbrev Instance := List Word
/-- Prefix markers encode the list spine, including its final boundary. -/
def encodeList {α : Type} (enc : α → List Bool) : List α → List Bool
| [] => [true]
| a :: as => false :: (enc a ++ encodeList enc as)
def encodeSymbol : Symbol → List Bool := encodeList (fun b => [b])
def encodeWord : Word → List Bool := encodeList encodeSymbol
def encodeInstance : Instance → List Bool := encodeList encodeWord
/-- Ordinary contiguous substring containment, including the empty word. -/
def IsCommonSuperstring (S : Instance) (T : Word) : Prop :=
∀ s ∈ S, s <:+: T
/-- The optimum in symbols, not in bits. -/
noncomputable def opt (S : Instance) : ℕ :=
sInf {n : ℕ | ∃ T : Word, IsCommonSuperstring S T ∧ T.length = n}
/-- A deterministic finite-stack-machine implementation, with a polynomial bound
on bit-level running time in the explicitly encoded input length. -/
def HasPolynomialImplementation (f : Instance → Word) : Prop :=
∃ M : Turing.TM2ComputableInPolyTime encodeInstance encodeWord f,
∀ k : M.tm.K, Finite (M.tm.Γ k)
end Superstring
end OAI
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