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Rank loss under two restrictions

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

proven

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    Natural Language Statement

    Lemma

    A composition loses at most the output kernel dimension and the input range codimension, including the two quotient modes of a tensor.

    Concept map
    3 concepts
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

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

    1 composition_rank_loss proven

    2 matrix_sandwich_rank_loss proven

    3 quotient_tensor_rank_loss proven

    Lean source view on GitHub

    1import Lax342547.RestrictionRank
    2import Lax342547.QuotientRank
    3
    4/-!
    5---
    6title: Rank loss under two restrictions
    7type: lemma
    8---
    9A composition loses at most the output kernel dimension and the input range codimension, including the two quotient modes of a tensor.
    10-/
    11
    12namespace Lax342547.RankLoss
    13
    14
    15
    16axiom composition_rank_loss {K U V W X : Type} [Field K]
    17 [AddCommGroup U] [Module K U] [AddCommGroup V] [Module K V]
    18 [AddCommGroup W] [Module K W] [AddCommGroup X] [Module K X]
    19 [FiniteDimensional K V] [FiniteDimensional K W]
    20 (A : W →ₗ[K] X) (M : V →ₗ[K] W) (B : U →ₗ[K] V) :
    21 Module.finrank K (LinearMap.range M) ≤
    22 Module.finrank K (LinearMap.range (A.comp (M.comp B))) +
    23 Module.finrank K (LinearMap.ker A) +
    24 (Module.finrank K V-Module.finrank K (LinearMap.range B))
    25
    26axiom matrix_sandwich_rank_loss {K I J P Q : Type} [Field K]
    27 [Fintype I] [Fintype J] [Fintype P] [Fintype Q]
    28 (A : Matrix P I K) (M : Matrix I J K) (B : Matrix Q J K) :
    29 M.rank ≤ (A*M*B.transpose).rank +
    30 Module.finrank K (LinearMap.ker A.mulVecLin) +
    31 Module.finrank K (LinearMap.ker B.mulVecLin)
    32
    33axiom quotient_tensor_rank_loss {K V W : Type} [Field K]
    34 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    35 [FiniteDimensional K V] [FiniteDimensional K W]
    36 (M : Module.Dual K V →ₗ[K] W) (S : Submodule K W) (T : Submodule K V) :
    37 Module.finrank K (LinearMap.range M) ≤
    38 Module.finrank K (LinearMap.range (S.mkQ.comp (M.comp T.mkQ.dualMap))) +
    39 Module.finrank K S+Module.finrank K T
    40
    41end Lax342547.RankLoss
    42
    Show ProofShow ProofShow Proof
    Builds on
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    From Mathlib

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