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Tiny-cover pairing matrix rank

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

proven

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    Natural Language Statement

    Lemma

    The actual pairing matrix factors through the covered tensor space; its rank is bounded by the two mode dimensions times their ambient dimensions.

    Concept map
    9 concepts
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

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

    1 covered_pairing_rank proven

    2 pairing_matrix_rank_le proven

    Lean source view on GitHub

    1import Lax342547.CoverSpace
    2import Mathlib.LinearAlgebra.Matrix.Rank
    3import Mathlib.LinearAlgebra.Finsupp.LSum
    4
    5/-!
    6---
    7title: Tiny-cover pairing matrix rank
    8type: lemma
    9---
    10The actual pairing matrix factors through the covered tensor space; its rank is bounded by the two mode dimensions times their ambient dimensions.
    11-/
    12
    13namespace Lax342547.PairingRank
    14
    15open Lax342547.CoverProjection
    16open scoped BigOperators
    17
    18axiom pairing_matrix_rank_le {K V Ω : Type} [Field K] [Fintype Ω]
    19 [AddCommGroup V] [Module K V] [FiniteDimensional K V]
    20 (S : Submodule K V) (v : Ω → V) (hv : ∀ a, v a ∈ S) (φ : Ω → Module.Dual K V) :
    21 (Matrix.of (fun a b => φ b (v a))).rank ≤ Module.finrank K S
    22
    23axiom covered_pairing_rank {K V W Ω : Type} [Field K] [Fintype Ω]
    24 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    25 [FiniteDimensional K V] [FiniteDimensional K W]
    26 (S : Submodule K W) (T : Submodule K (Module.Dual K V))
    27 (Δ : Ω → V →ₗ[K] W) (φ : Ω → Module.Dual K (V →ₗ[K] W))
    28 (hΔ : ∀ a, projection (Δ a) S T = 0) :
    29 (Matrix.of (fun a b => φ b (Δ a))).rank ≤
    30 Module.finrank K V*Module.finrank K S+Module.finrank K T*Module.finrank K W
    31
    32end Lax342547.PairingRank
    33
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