Documentation

Complexitylib.Circuits.DepthThree.LowerBound.Internal.TapeHashInitialize

Unrestricted depth-three lower bound: tape hash initialize #

Adapted from OpenAI's formalization at revision adc7f1241b42e322a6451854ab7e4b4c146bf78a. Source: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/DepthThree/TapeHashInitialize.lean

Imports, namespaces, module visibility, documentation, and formatting are adapted for complexitylib. The source theorem statements and mathematical proofs are retained.

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@[reducible, inline]
abbrev Complexity.DepthThreeLowerBound.TapeHashInitialize.State :
Type

Control states of the hash initialize routine.

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    def Complexity.DepthThreeLowerBound.TapeHashInitialize.program :
    TapeMultiProgram State

    Transition function of the hash initialize routine.

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      def Complexity.DepthThreeLowerBound.TapeHashInitialize.start :
      State

      Initial control state of the hash initialize routine.

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        def Complexity.DepthThreeLowerBound.TapeHashInitialize.done :
        State

        Terminal control state of the hash initialize routine.

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          theorem Complexity.DepthThreeLowerBound.TapeHashInitialize.runs_initialize (σ : TapeStore) (r : ℕ) (hr : σ TapeRegister.ringDegree = List.replicate r true) (hs : σ TapeRegister.scratchA = []) (hh : σ TapeRegister.hashBits = []) :
          RunsIn program.step (TapeMultiProgram.cfg start (storeTapes σ)) (TapeMultiProgram.cfg done (storeTapes (Function.update σ TapeRegister.hashBits (List.replicate r false)))) (10 * r + 8)
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          @[simp]
          theorem Complexity.DepthThreeLowerBound.TapeHashInitialize.program_done (h : TapeHeads) :
          program done h = none