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Counting all bounded-dimensional protected spaces

Lax342547.SmallSubspaceCount · concepts/Lax342547/SmallSubspaceCount.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    Padding an actual basis gives a generator encoding of every subspace of dimension at most r and the uniform count 2 to the ambient-dimension times r.

    Concept map
    3 concepts; 10 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 padded_generators proven

    2 small_subspace_card proven

    Lean source view on GitHub

    1import Lax342547.FiniteLinearLaw
    2import Mathlib.LinearAlgebra.Basis.VectorSpace
    3
    4/-!
    5---
    6title: Counting all bounded-dimensional protected spaces
    7type: lemma
    8---
    9Padding an actual basis gives a generator encoding of every subspace of dimension at most r and the uniform count 2 to the ambient-dimension times r.
    10-/
    11
    12namespace Lax342547.SmallSubspaceCount
    13
    14open Lax342547.MomentSpace
    15
    16axiom padded_generators {K V : Type} [Field K] [AddCommGroup V] [Module K V]
    17 [FiniteDimensional K V] (S : Submodule K V) (r : ℕ) (hr : Module.finrank K S ≤ r) :
    18 ∃ f : Fin r → V, Submodule.span K (Set.range f) = S
    19
    20axiom small_subspace_card {H : Type} [Fintype H] (r : ℕ) :
    21 Nat.card {S : Submodule Binary (H → Binary) // Module.finrank Binary S ≤ r} ≤
    22 2^(Fintype.card H*r)
    23
    24end Lax342547.SmallSubspaceCount
    25
    Show ProofShow Proof
    Builds on
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    From Mathlib

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