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Formal differential representations

Lax825748.DifferentialRepresentation · concepts/Lax825748/DifferentialRepresentation.lean · lax-825748

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    Natural Language Statement

    Definition

    A formal differential representation of dimension dd consists of one formal vector field for each letter and an arbitrary formal output series hh. The operators act on the right: the word xyxy applies the field for xx first and then the field for yy. Its generated coefficient at ww is the constant term of hμwh\mu w.

    Specifying the fields is equivalent to specifying the monoid homomorphism μ\mu. All state series and vector-field coefficients are formal.

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    Lean source view on GitHub

    1import Lax825748.FormalSeries
    2
    3/-!
    4---
    5title: Formal differential representations
    6type: definition
    7---
    8A formal differential representation of dimension dd consists of one
    9formal vector field for each letter and an arbitrary formal output series hh.
    10The operators act on the right: the word xyxy applies the field for xx
    11first and then the field for yy. Its generated coefficient at ww is
    12the constant term of hμwh\mu w.
    13
    14Specifying the fields is equivalent to specifying the monoid homomorphism
    15μ\mu. All state series and vector-field coefficients are formal.
    16-/
    17
    18noncomputable section
    19
    20namespace Lax825748.DifferentialRepresentation
    21
    22open Series FormalSeries
    23
    24structure Representation (X : Type) (d : ℕ) where
    25 fields : X → VectorField d
    26 output : StateSeries d
    27
    28noncomputable def act {X : Type} {d : ℕ} (R : Representation X d)
    29 (f : StateSeries d) (w : Word X) : StateSeries d :=
    30 w.toList.foldl (fun k x => applyVectorField (R.fields x) k) f
    31
    32noncomputable def generatingSeries {X : Type} {d : ℕ}
    33 (R : Representation X d) : NCSeries X :=
    34 fun w => MvPowerSeries.constantCoeff (act R R.output w)
    35
    36def Represents {X : Type} {d : ℕ} (R : Representation X d)
    37 (g : NCSeries X) : Prop := generatingSeries R = g
    38
    39def DifferentiallyProduced {X : Type} (g : NCSeries X) : Prop :=
    40 ∃ d, ∃ R : Representation X d, Represents R g
    41
    42end Lax825748.DifferentialRepresentation
    43
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