Documentation

Complexitylib.Circuits.DepthThree.LowerBound.Internal.TapeHash

Unrestricted depth-three lower bound: tape hash #

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

Control states of the hash routine.

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    def Complexity.DepthThreeLowerBound.TapeHash.program :
    TapeMultiProgram State

    Transition function of the hash routine.

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      def Complexity.DepthThreeLowerBound.TapeHash.start :
      State

      Initial control state of the hash routine.

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        def Complexity.DepthThreeLowerBound.TapeHash.done :
        State

        Terminal control state of the hash routine.

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          def Complexity.DepthThreeLowerBound.TapeHash.resultStore (σ : TapeStore) (r : ℕ) :
          TapeStore

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

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            theorem Complexity.DepthThreeLowerBound.TapeHash.cost_le (d r : ℕ) :
            10 * r + 8 + 1 + (r * (64 * (d + r + 1) ^ 2 + 4 * r + 12) + (4 * r + 8)) ≤ 200 * (d + r + 1) ^ 3
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            theorem Complexity.DepthThreeLowerBound.TapeHash.runs_hash (σ : TapeStore) (d r : ℕ) (hd : σ TapeRegister.dataLength = List.replicate d true) (hr : σ TapeRegister.ringDegree = List.replicate r true) (ha : σ TapeRegister.indexA = []) (hb : σ TapeRegister.indexB = []) (hc : σ TapeRegister.indexC = []) (hs : σ TapeRegister.scratchA = []) (hh : σ TapeRegister.hashBits = []) (hkey : (σ TapeRegister.keyBits).length = d + r - 1) (hdata : (σ TapeRegister.dataBits).length = d) :
            RunsIn program.step (TapeMultiProgram.cfg start (storeTapes σ)) (TapeMultiProgram.cfg done (storeTapes (resultStore σ r))) (200 * (d + r + 1) ^ 3)
            source
            theorem Complexity.DepthThreeLowerBound.TapeHash.runs_wordHash (σ : TapeStore) {d r : ℕ} (u : BitWord (d + r - 1)) (x : BitWord d) (hd : σ TapeRegister.dataLength = List.replicate d true) (hr : σ TapeRegister.ringDegree = List.replicate r true) (ha : σ TapeRegister.indexA = []) (hb : σ TapeRegister.indexB = []) (hc : σ TapeRegister.indexC = []) (hs : σ TapeRegister.scratchA = []) (hh : σ TapeRegister.hashBits = []) (hkey : σ TapeRegister.keyBits = List.ofFn u) (hdata : σ TapeRegister.dataBits = List.ofFn x) :
            RunsIn program.step (TapeMultiProgram.cfg start (storeTapes σ)) (TapeMultiProgram.cfg done (storeTapes (Function.update (Function.update σ TapeRegister.hashBits (List.ofFn (wordHash u x))) TapeRegister.indexC (List.replicate r true)))) (200 * (d + r + 1) ^ 3)
            source
            @[simp]
            theorem Complexity.DepthThreeLowerBound.TapeHash.program_done (h : TapeHeads) :
            program done h = none