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Dimension and component-rank bounds for effective test spaces

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

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    Natural Language Statement

    Theorem

    The tensor space generated by two projected pin spaces has dimension at most the product of their dimensions. Its matrix ranks are bounded by either factor. Summing these estimates gives the effective-space bounds dim C <= K^2 and total component rank <= K from the actual pin budget.

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

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 effective_bounds proven

    2 tensor_dimension proven

    3 tensor_rank proven

    Lean source view on GitHub

    1import Lax342547.ProjectedPins
    2import Mathlib.LinearAlgebra.TensorProduct.Basic
    3import Mathlib.LinearAlgebra.Dimension.Constructions
    4
    5/-!
    6---
    7title: Dimension and component-rank bounds for effective test spaces
    8type: theorem
    9---
    10The tensor space generated by two projected pin spaces has dimension at
    11most the product of their dimensions. Its matrix ranks are bounded by
    12either factor. Summing these estimates gives the effective-space bounds
    13dim C <= K^2 and total component rank <= K from the actual pin budget.
    14-/
    15
    16namespace Lax342547.EffectiveBounds
    17
    18open Lax342547.MomentSpace Lax342547.ExactPins Lax342547.ProjectedPins
    19
    20axiom tensor_dimension {B : Type} [Fintype B] (S T : Submodule Binary (B → Binary)) :
    21 Module.finrank Binary (tensorSpace S T) ≤ Module.finrank Binary S * Module.finrank Binary T
    22
    23axiom tensor_rank {B : Type} [Fintype B] (S T : Submodule Binary (B → Binary))
    24 (M : Matrix B B Binary) (hM : M ∈ tensorSpace S T) :
    25 M.rank ≤ min (Module.finrank Binary S) (Module.finrank Binary T)
    26
    27axiom effective_bounds {Comp B H N : Type} [Fintype Comp] [Fintype B] [Fintype H]
    28 (X : Submodule Binary (Comp → Matrix B B Binary))
    29 (P : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N) (i : Fin 2) (K : ℕ) (hK : P.rank ≤ K) :
    30 Module.finrank Binary (effective X P i) ≤ K ^ 2 ∧
    31 ∀ x ∈ effective X P i, (∑ e, (x.val e).rank) ≤ K
    32
    33end Lax342547.EffectiveBounds
    34
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