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Rank of a lifted tensor sum

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

proven

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    Natural Language Statement

    Lemma

    A lifted block sum has rank at least its number of nonzero blocks minus the actual column and row span deficits.

    Concept map
    6 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 lifted_sum_rank_deficit proven

    2 nonzero_block_count_deficit proven

    3 nonzero_map_rank proven

    Lean source view on GitHub

    1import Lax342547.RankLoss
    2import Lax342547.DirectRanks
    3import Lax342547.SpanDeficits
    4
    5/-!
    6---
    7title: Rank of a lifted tensor sum
    8type: lemma
    9---
    10A lifted block sum has rank at least its number of nonzero blocks minus the actual column and row span deficits.
    11-/
    12
    13namespace Lax342547.BlockDeficits
    14
    15open Lax342547.SpanDeficits
    16open scoped BigOperators
    17
    18noncomputable def liftedSum {K V W ι : Type} [Field K] [Fintype ι]
    19 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    20 (S : ι → Submodule K W) (T : ι → Submodule K V)
    21 (M : ∀ i, Module.Dual K (T i) →ₗ[K] S i) : Module.Dual K V →ₗ[K] W := by
    22 classical
    23 let B := (LinearMap.lsum K (fun i => T i) K).symm.toLinearMap.comp (addition T).dualMap
    24 exact (addition S).comp ((LinearMap.piMap M).comp B)
    25
    26axiom lifted_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 (S : ι → Submodule K W) (T : ι → Submodule K V)
    30 (M : ∀ i, Module.Dual K (T i) →ₗ[K] S i) :
    31 (∑ i, Module.finrank K (LinearMap.range (M i))) ≤
    32 Module.finrank K (LinearMap.range (liftedSum S T M)) +
    33 ((∑ i, Module.finrank K (S i))-Module.finrank K (⨆ i, S i : Submodule K W)) +
    34 ((∑ i, Module.finrank K (T i))-Module.finrank K (⨆ i, T i : Submodule K V))
    35
    36axiom nonzero_map_rank {K V W : Type} [Field K]
    37 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    38 [FiniteDimensional K W] (M : V →ₗ[K] W) (hM : M ≠ 0) :
    39 1 ≤ Module.finrank K (LinearMap.range M)
    40
    41axiom nonzero_block_count_deficit {K V W ι : Type} [Field K] [Fintype ι]
    42 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    43 [FiniteDimensional K V] [FiniteDimensional K W]
    44 (S : ι → Submodule K W) (T : ι → Submodule K V)
    45 (M : ∀ i, Module.Dual K (T i) →ₗ[K] S i) : by
    46 classical
    47 exact (Finset.univ.filter (fun i => M i ≠ 0)).card ≤
    48 Module.finrank K (LinearMap.range (liftedSum S T M)) +
    49 ((∑ i, Module.finrank K (S i))-Module.finrank K (⨆ i, S i : Submodule K W)) +
    50 ((∑ i, Module.finrank K (T i))-Module.finrank K (⨆ i, T i : Submodule K V))
    51
    52end Lax342547.BlockDeficits
    53
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