File size: 1,239 Bytes
92416a2 | 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Computability.VertexCover.Machines.NatDiv
namespace OAI
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namespace VertexCover.Machine
def finCode {n : ℕ} (i : Fin n) : List Bool := natBits i.val
theorem finCode_injective (n : ℕ) : Function.Injective (finCode (n := n)) := by
intro a b h
apply Fin.ext
have := congrArg List.length h
simpa [finCode] using this
noncomputable def Poly.finMod (n : ℕ) (hn : 0 < n) :
Poly natBits (finCode (n := n)) (fun i => (⟨i%n,Nat.mod_lt i hn⟩ : Fin n)) :=
(((Poly.identity natBits).pair (Poly.const natBits natBits n)).comp Poly.natMod).encodeCongr
id (fun _ => rfl) (fun _ => rfl)
noncomputable def Poly.finVal (n : ℕ) : Poly (finCode (n := n)) natBits Fin.val :=
(Poly.identity natBits).encodeCongr Fin.val (fun _ => rfl) (fun _ => rfl)
end VertexCover.Machine
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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