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The map lifting of a continuous function is continuous

Lax916827.MapLiftingContinuity · concepts/Lax916827/MapLiftingContinuity.lean · lax-916827

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    Natural Language Statement

    Theorem

    If a string-to-string function is continuous, then so is its map lifting (Lemma C.1.3 of Transducers). An automaton for the inverse image of a regular language under the map lifting guesses, for every block, the states of the automaton for the language at its two ends, and checks the transitions across the blocks by continuity of the lifted function and across the separators directly.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    In the paper

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    1import Lax765601.Continuity
    2import Lax765601.MapLifting
    3
    4/-!
    5---
    6title: The map lifting of a continuous function is continuous
    7type: theorem
    8---
    9If a string-to-string function is continuous, then so is its map lifting
    10(Lemma C.1.3 of *Transducers*). An automaton for the inverse image of a regular
    11language under the map lifting guesses, for every block, the states of the
    12automaton for the language at its two ends, and checks the transitions across
    13the blocks by continuity of the lifted function and across the separators
    14directly.
    15
    16# Formalization notes
    17
    18Both alphabets are assumed finite, as in the book.
    19-/
    20
    21namespace Lax916827.MapLiftingContinuity
    22
    23open Lax765601.Continuity Lax765601.MapLifting
    24
    25/-- The map lifting of a continuous function is continuous. -/
    26axiom continuous_mapLift {A B : Type} [Finite A] [Finite B] {f : List A → List B}
    27 (hf : Continuous f) : Continuous (mapLift f)
    28
    29end Lax916827.MapLiftingContinuity
    30
    Show Proof
    Formalization notes

    Both alphabets are assumed finite, as in the book.

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