import Mathlib
import OAI.Computability.VertexCover.Machines.Primitives

namespace OAI

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namespace VertexCover.Machine

theorem prodBits_injective {α β : Type} {ea : α → List Bool} {eb : β → List Bool}
    (ha : Function.Injective ea) (hb : Function.Injective eb) :
    Function.Injective (prodBits ea eb) := by
  intro a b h
  apply Prod.ext
  · apply ha
    have hh := congrArg (Stream.output (Project.trans true false) (fun _ => []) 0) h
    simpa only [prodBits,Project.framed,Bool.false_eq_true,↓reduceIte,List.append_nil] using hh
  · apply hb
    have hh := congrArg (Stream.output (Project.trans false true) (fun _ => []) 0) h
    simpa only [prodBits,Project.framed,Bool.false_eq_true,↓reduceIte,List.nil_append] using hh

theorem boolBits_injective : Function.Injective boolBits := by
  intro a b h
  exact List.singleton_injective h

end VertexCover.Machine
end


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end OAI