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Lax194892.PrenexNormalForm

Every for-transducer has a prenex form

concepts/Lax194892/PrenexNormalForm.lean · lax-194892

proven

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    In the paper

    Theorem

    Every for-transducer is equivalent to one in prenex form (Lemma D.1.3 of Transducers): a block of nested loops over positions, in increasing or decreasing order, whose body is loop-free and outputs at most one letter per iteration, followed by a loop-free epilogue. Loops are pulled outwards one at a time, Boolean variables recording which iterations have already been executed.

    Lean source view on GitHub

    1import Lax194892.ForTransducers
    2
    3/-!
    4---
    5title: Every for-transducer has a prenex form
    6type: theorem
    7---
    8Every for-transducer is equivalent to one in prenex form (Lemma D.1.3 of
    9*Transducers*): a block of nested loops over positions, in increasing or
    10decreasing order, whose body is loop-free and outputs at most one letter per
    11iteration, followed by a loop-free epilogue. Loops are pulled outwards one at a
    12time, Boolean variables recording which iterations have already been executed.
    13
    14# Formalization notes
    15
    16Stated over any alphabets, for every program `P`; `PrenexForm` is that of
    17`ForTransducers`.
    18-/
    19
    20namespace Lax194892.PrenexNormalForm
    21
    22open Lax194892.ForTransducers
    23
    24/-- Every for-transducer program is equivalent to one in prenex form. -/
    25axiom exists_prenexForm {A B : Type} (P : ForProg A B) :
    26 ∃ P' : ForProg A B, P'.PrenexForm ∧ ∀ w, P'.eval w = P.eval w
    27
    28end Lax194892.PrenexNormalForm
    29
    Show Proof

    Formalization notes

    Stated over any alphabets, for every program PP; PrenexFormPrenexForm is that of ForTransducersForTransducers.

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