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Complexitylib.Circuits.DepthThree.LowerBound.Internal.TapeHornerMultiply

Unrestricted depth-three lower bound: tape horner multiply #

Adapted from OpenAI's formalization at revision adc7f1241b42e322a6451854ab7e4b4c146bf78a. Source: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/DepthThree/TapeHornerMultiply.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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structure Complexity.DepthThreeLowerBound.TapeHornerMultiply.Workspace (σ : TapeStore) (r : ℕ) :
Prop

Register allocation for the horner multiply routine.

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    @[reducible, inline]
    abbrev Complexity.DepthThreeLowerBound.TapeHornerMultiply.WorkState :
    Type

    Control states for the work stage of the horner multiply routine.

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

      Control states of the horner multiply routine.

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        def Complexity.DepthThreeLowerBound.TapeHornerMultiply.work :
        TapeMultiProgram WorkState

        Compose the main computational stages of the routine.

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          def Complexity.DepthThreeLowerBound.TapeHornerMultiply.program :
          TapeMultiProgram State

          Transition function of the horner multiply routine.

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            def Complexity.DepthThreeLowerBound.TapeHornerMultiply.start :
            State

            Initial control state of the horner multiply routine.

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              def Complexity.DepthThreeLowerBound.TapeHornerMultiply.done :
              State

              Terminal control state of the horner multiply routine.

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                def Complexity.DepthThreeLowerBound.TapeHornerMultiply.cost (r : ℕ) :
                ℕ

                Transition budget for the horner multiply routine.

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                  @[simp]
                  theorem Complexity.DepthThreeLowerBound.TapeHornerMultiply.program_done (h : TapeHeads) :
                  program done h = none
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                  theorem Complexity.DepthThreeLowerBound.TapeHornerMultiply.runs_multiply (σ : TapeStore) (p h a : List Bool) (r : ℕ) (hW : Workspace σ r) (hP : σ TapeRegister.polyBits = p) (hH : σ TapeRegister.hashBits = h) (hA : σ TapeRegister.accumBits = a) (hp : p.length = r) (hh : h.length = r) (ha : a.length = r) :
                  RunsIn program.step (TapeMultiProgram.cfg start (storeTapes σ)) (TapeMultiProgram.cfg done (storeTapes (Function.update σ TapeRegister.accumBits (multiplyBitLists r p h a)))) (cost r)