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Rank of an actual linear map sum

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

proven

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    Natural Language Statement

    Lemma

    Stacking maps identifies the joint row span, and addition loses at most the actual mode-space deficit. This gives the rank-deficit inequality for the actual sum.

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

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

    1 stacked_rank proven

    2 sum_rank_deficit proven

    Lean source view on GitHub

    1import Lax342547.SpanDeficits
    2import Lax342547.RestrictionRank
    3import Lax342547.BlockDeficits
    4
    5/-!
    6---
    7title: Rank of an actual linear map sum
    8type: lemma
    9---
    10Stacking maps identifies the joint row span, and addition loses at most the actual mode-space deficit. This gives the rank-deficit inequality for the actual sum.
    11-/
    12
    13namespace Lax342547.SumRank
    14
    15open Lax342547.SpanDeficits
    16open scoped BigOperators
    17
    18axiom stacked_rank {K V W ι : Type} [Field K] [Fintype ι]
    19 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    20 [FiniteDimensional K V] [FiniteDimensional K W]
    21 (M : ι → V →ₗ[K] W) (S : ι → Submodule K W) (hS : ∀ i, LinearMap.range (M i) ≤ S i) :
    22 Module.finrank K (LinearMap.range (LinearMap.pi (fun i => (M i).codRestrict (S i)
    23 (fun v => hS i (LinearMap.mem_range_self _ v))))) =
    24 Module.finrank K (⨆ i, LinearMap.range (M i).dualMap : Submodule K (Module.Dual K V))
    25
    26axiom sum_rank_deficit {K V W ι : Type} [Field K] [Fintype ι]
    27 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    28 [FiniteDimensional K V] [FiniteDimensional K W]
    29 (M : ι → V →ₗ[K] W) (S : ι → Submodule K W) (hS : ∀ i, LinearMap.range (M i) ≤ S i) :
    30 (∑ i, Module.finrank K (LinearMap.range (M i))) ≤
    31 Module.finrank K (LinearMap.range (∑ i, M i))+
    32 ((∑ i, Module.finrank K (S i))-Module.finrank K (⨆ i, S i : Submodule K W))+
    33 ((∑ i, Module.finrank K (LinearMap.range (M i).dualMap))-
    34 Module.finrank K (⨆ i, LinearMap.range (M i).dualMap : Submodule K (Module.Dual K V)))
    35
    36end Lax342547.SumRank
    37
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