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Regular functions on types under string representation

Lax709149.RegularUnderRepresentation · concepts/Lax709149/RegularUnderRepresentation.lean · lax-709149

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    Natural Language Statement

    Definition

    A type-to-type function f:A→Bf : A \to B is regular under string representation (Definition C.5.2 of Transducers) if there is a regular string-to-string function f′f' over the eight-letter alphabet making the square commute: f′f' maps the representation of aa to the representation of f(a)f(a), for every a∈Aa \in A. Rational functions between types are defined in the same way. Nothing is required of f′f' on the strings that represent no element.

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    1import Lax132576.RationalFunctions
    2import Lax916827.RegularFunctions
    3import Lax709149.Types
    4
    5/-!
    6---
    7title: Regular functions on types under string representation
    8type: definition
    9---
    10A type-to-type function f:A→Bf : A \to B is *regular under string representation*
    11(Definition C.5.2 of *Transducers*) if there is a regular string-to-string
    12function f′f' over the eight-letter alphabet making the square commute: f′f'
    13maps the representation of aa to the representation of f(a)f(a), for every
    14a∈Aa \in A. Rational functions between types are defined in the same way. Nothing
    15is required of f′f' on the strings that represent no element.
    16
    17# Formalization notes
    18
    19`IsRegularUnderRepr f` asks for a regular `f' : List Sym8 → List Sym8` with
    20`f' (A.repr a) = B.repr (f a)` for all `a`; `IsRationalUnderRepr` for a rational
    21one.
    22-/
    23
    24namespace Lax709149.RegularUnderRepresentation
    25
    26open Lax132576.RationalFunctions Lax916827.RegularFunctions Lax709149.Types
    27
    28/-- A type-to-type function is regular under string representation if a regular
    29string-to-string function maps the representation of `a` to the representation
    30of `f a`. -/
    31def IsRegularUnderRepr {A B : Ty} (f : A.Elt → B.Elt) : Prop :=
    32 ∃ f' : List Sym8 → List Sym8, IsRegularFun f' ∧ ∀ a : A.Elt, f' (A.repr a) = B.repr (f a)
    33
    34/-- A type-to-type function is rational under string representation. -/
    35def IsRationalUnderRepr {A B : Ty} (f : A.Elt → B.Elt) : Prop :=
    36 ∃ f' : List Sym8 → List Sym8, IsRationalFun f' ∧ ∀ a : A.Elt, f' (A.repr a) = B.repr (f a)
    37
    38end Lax709149.RegularUnderRepresentation
    39
    Formalization notes

    IsRegularUnderReprfIsRegularUnderRepr f asks for a regular f′:ListSym8→ListSym8f' : List Sym8 → List Sym8 with f′(A.repra)=B.repr(fa)f' (A.repr a) = B.repr (f a) for all aa; IsRationalUnderReprIsRationalUnderRepr for a rational one.

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