Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Actual tensor cover decomposition and dimension

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    A checked quotient decomposition of every covered map gives the dimension bound for the whole actual cover space.

    Concept map
    8 concepts; 1 descendant hidden
    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 cover_space_dimension proven

    2 covered_decomposition proven

    Lean source view on GitHub

    1import Lax342547.CoverProjection
    2import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
    3
    4/-!
    5---
    6title: Actual tensor cover decomposition and dimension
    7type: lemma
    8---
    9A checked quotient decomposition of every covered map gives the dimension bound for the whole actual cover space.
    10-/
    11
    12namespace Lax342547.CoverSpace
    13
    14open Lax342547.CoverProjection
    15
    16noncomputable def coverAssembly {K V W : Type} [Field K]
    17 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    18 (S : Submodule K W) (T : Submodule K (Module.Dual K V)) :
    19 ((V →ₗ[K] S) × ((V ⧸ T.dualCoannihilator) →ₗ[K] W)) →ₗ[K] (V →ₗ[K] W) where
    20 toFun z := S.subtype.comp z.1+z.2.comp T.dualCoannihilator.mkQ
    21 map_add' _ _ := by simp [LinearMap.comp_add,LinearMap.add_comp]; abel
    22 map_smul' _ _ := by simp [LinearMap.comp_smul,LinearMap.smul_comp]
    23
    24noncomputable def coverOperator {K V W : Type} [Field K]
    25 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    26 (S : Submodule K W) (T : Submodule K (Module.Dual K V)) :
    27 (V →ₗ[K] W) →ₗ[K] (T.dualCoannihilator →ₗ[K] (W ⧸ S)) where
    28 toFun M := projection M S T
    29 map_add' _ _ := by simp [projection,LinearMap.comp_add,LinearMap.add_comp]
    30 map_smul' _ _ := by simp [projection,LinearMap.comp_smul,LinearMap.smul_comp]
    31
    32axiom covered_decomposition {K V W : Type} [Field K]
    33 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    34 (S : Submodule K W) (T : Submodule K (Module.Dual K V))
    35 (M : V →ₗ[K] W) (hM : projection M S T = 0) :
    36 M ∈ LinearMap.range (coverAssembly S T)
    37
    38axiom cover_space_dimension {K V W : Type} [Field K]
    39 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    40 [FiniteDimensional K V] [FiniteDimensional K W]
    41 (S : Submodule K W) (T : Submodule K (Module.Dual K V)) :
    42 Module.finrank K (LinearMap.ker (coverOperator S T)) ≤
    43 Module.finrank K V*Module.finrank K S+Module.finrank K T*Module.finrank K W
    44
    45end Lax342547.CoverSpace
    46
    Show ProofShow Proof
    Builds on
    Used by
    From Mathlib

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…