Documentation

Complexitylib.Circuits.DepthThree.LowerBound.Internal.TapeHashStep

Unrestricted depth-three lower bound: tape hash step #

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

Control states of the hash step routine.

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    def Complexity.DepthThreeLowerBound.TapeHashStep.program :
    TapeMultiProgram State

    Transition function of the hash step routine.

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      def Complexity.DepthThreeLowerBound.TapeHashStep.start :
      State

      Initial control state of the hash step routine.

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        def Complexity.DepthThreeLowerBound.TapeHashStep.done :
        State

        Terminal control state of the hash step routine.

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          def Complexity.DepthThreeLowerBound.TapeHashStep.resultStore (σ : TapeStore) (i j : ℕ) :
          TapeStore

          Tape-store contents at the result stage of the hash step routine.

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            theorem Complexity.DepthThreeLowerBound.TapeHashStep.runs_step (σ : TapeStore) (i j : ℕ) (ha : σ TapeRegister.indexA = List.replicate (i + j) true) (hb : σ TapeRegister.indexB = List.replicate j true) (hc : σ TapeRegister.indexC = List.replicate i true) (hkey : i + j < (σ TapeRegister.keyBits).length) (hdata : j < (σ TapeRegister.dataBits).length) (hout : i < (σ TapeRegister.hashBits).length) :
            RunsIn program.step (TapeMultiProgram.cfg start (storeTapes σ)) (TapeMultiProgram.cfg done (storeTapes (resultStore σ i j))) (4 * i + 4 * j + 17)
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            @[simp]
            theorem Complexity.DepthThreeLowerBound.TapeHashStep.program_done (h : TapeHeads) :
            program done h = none