Lax Lean Archive

Lie rank is the least formal realization dimension

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    Every formal differential representation of dimension ee generates a series of finite Lie rank at most ee. Lie rank dd is equivalent to dd being the least dimension of a formal differential representation. This includes constant series, whose minimal dimension is zero.

    Concept map
    11 concepts; 4 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 least_dimension proven

    2 rank_bound proven

    Lean source view on GitHub

    1import Lax825748.FormalRealization
    2
    3/-!
    4---
    5title: Lie rank is the least formal realization dimension
    6type: theorem
    7---
    8Every formal differential representation of dimension ee generates a series
    9of finite Lie rank at most ee. Lie rank dd is equivalent to dd being
    10the least dimension of a formal differential
    11representation. This includes constant series, whose minimal dimension
    12is zero.
    13-/
    14
    15noncomputable section
    16
    17namespace Lax825748.FormalMinimality
    18
    19open Series LieRank DifferentialRepresentation
    20
    21def IsLeastDimension {X : Type} (g : NCSeries X) (d : ℕ) : Prop :=
    22 (∃ R : Representation X d, Represents R g) ∧
    23 ∀ e, (∃ R : Representation X e, Represents R g) → d ≤ e
    24
    25axiom rank_bound {X : Type} {e : ℕ} (R : Representation X e) :
    26 ∃ d, HasLieRank (generatingSeries R) d ∧ d ≤ e
    27
    28axiom least_dimension {X : Type} [Finite X] (g : NCSeries X) (d : ℕ) :
    29 HasLieRank g d ↔ IsLeastDimension g d
    30
    31end Lax825748.FormalMinimality
    32
    Show ProofShow Proof
    Builds on
    Used by
    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…