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Rank preservation under mode-cover avoidance

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

proven

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    Natural Language Statement

    Lemma

    Actual quotient and annihilator restriction preserve rank when the original column and row spaces avoid the cover.

    Concept map
    8 concepts; 2 descendants hidden
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 finrank_map_of_disjoint_kernel proven

    2 projection_rank_of_avoidance proven

    3 quotient_rank_of_avoidance proven

    4 restriction_rank_of_avoidance proven

    Lean source view on GitHub

    1import Lax342547.CoverProjection
    2
    3/-!
    4---
    5title: Rank preservation under mode-cover avoidance
    6type: lemma
    7---
    8Actual quotient and annihilator restriction preserve rank when the original column and row spaces avoid the cover.
    9-/
    10
    11namespace Lax342547.CoverAvoidance
    12
    13open Lax342547.CoverProjection
    14
    15axiom finrank_map_of_disjoint_kernel {K V W : Type} [Field K]
    16 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    17 (f : V →ₗ[K] W) (S : Submodule K V) (h : Disjoint S (LinearMap.ker f)) :
    18 Module.finrank K (S.map f) = Module.finrank K S
    19
    20axiom quotient_rank_of_avoidance {K V W : Type} [Field K]
    21 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    22 (M : V →ₗ[K] W) (S : Submodule K W) (h : Disjoint (LinearMap.range M) S) :
    23 Module.finrank K (LinearMap.range (S.mkQ.comp M)) = Module.finrank K (LinearMap.range M)
    24
    25axiom restriction_rank_of_avoidance {K V W : Type} [Field K]
    26 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    27 [FiniteDimensional K V] [FiniteDimensional K W]
    28 (M : V →ₗ[K] W) (T : Submodule K (Module.Dual K V))
    29 (h : Disjoint (LinearMap.range M.dualMap) T) :
    30 Module.finrank K (LinearMap.range (M.comp T.dualCoannihilator.subtype)) =
    31 Module.finrank K (LinearMap.range M)
    32
    33axiom projection_rank_of_avoidance {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 : V →ₗ[K] W) (S : Submodule K W) (T : Submodule K (Module.Dual K V))
    37 (hS : Disjoint (LinearMap.range M) S) (hT : Disjoint (LinearMap.range M.dualMap) T) :
    38 Module.finrank K (LinearMap.range (projection M S T)) = Module.finrank K (LinearMap.range M)
    39
    40end Lax342547.CoverAvoidance
    41
    Show ProofShow ProofShow ProofShow Proof
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