File size: 2,330 Bytes
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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Computability.Superstring.WalkOutput
namespace OAI
namespace Superstring
open BitCode
theorem bitsList_eq_encodeList {α : Type} [Coded α] (e : α → List Bool)
(he : ∀ a, bits a = e a) (xs : List α) : bits xs = encodeList e xs := by
induction xs with
| nil => rfl
| cons a xs ih =>
change false :: (bits a ++ bits xs) = false :: (e a ++ encodeList e xs)
rw [he,ih]
theorem bitsSymbol_eq (s : Symbol) : bits s = encodeSymbol s :=
bitsList_eq_encodeList _ (fun _ => rfl) s
theorem bitsWord_eq (w : Word) : bits w = encodeWord w :=
bitsList_eq_encodeList _ bitsSymbol_eq w
theorem bitsInstance_eq (S : Instance) : bits S = encodeInstance S :=
bitsList_eq_encodeList _ bitsWord_eq S
def answer : Instance → Word := Executable.solve
theorem answer_poly : Poly answer := by
unfold answer
apply Executable.poly_solve
convert (Poly.ltList (α:=Bool) Poly.ltBool) using 1
funext q
exact decide_eq_decide.mpr Iff.rfl
theorem answer_implementation : HasPolynomialImplementation answer := by
have hc := RunsInPolyTime.certificate answer_poly
have hi : (bits : Instance → List Bool) = encodeInstance := funext bitsInstance_eq
have ho : (bits : Word → List Bool) = encodeWord := funext bitsWord_eq
change ∃ M : Turing.TM2ComputableInPolyTime bits bits answer, ∀ k : M.tm.K, Finite (M.tm.Γ k) at hc
rw [hi,ho] at hc
exact hc
theorem weight_le_opt_with (S : Instance) (d : DecidableEq Symbol) :
@Counts.Data.weight Symbol d inferInstance (@Input.data Symbol d S) ≤ opt S := by
have he : d = @instDecidableEqList Bool instDecidableEqBool := Subsingleton.elim _ _
rw [he]
exact weight_le_opt S
theorem answer_spec (S : Instance) : IsCommonSuperstring S (answer S) ∧
(answer S).length ≤ 2*opt S := by
obtain ⟨ht,hl⟩ := Executable.solve_spec S
refine ⟨ht, ?_⟩
apply hl.trans
exact Nat.mul_le_mul_left 2 (weight_le_opt_with S LinearOrder.toDecidableEq)
/-- The deterministic algorithm with its exact finite-alphabet bit-machine
certificate, contiguous coverage, and factor-two bound in symbols. -/
theorem main : ∃ f : Instance → Word,
HasPolynomialImplementation f ∧
∀ S, IsCommonSuperstring S (f S) ∧ (f S).length ≤ 2 * opt S :=
⟨answer, answer_implementation, answer_spec⟩
end Superstring
end OAI
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