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Offsets constructed from the actual common projection

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

proven

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    Natural Language Statement

    Lemma

    The actual shared projected endpoint tensor supplies its own minimal factorization; its covered offsets have total dimension at most twice the projected rank.

    Concept map
    10 concepts
    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 actual_common_projection_offsets proven

    2 nonzero_difference_has_offset proven

    Lean source view on GitHub

    1import Lax342547.QuotientFactorLifts
    2
    3/-!
    4---
    5title: Offsets constructed from the actual common projection
    6type: lemma
    7---
    8The actual shared projected endpoint tensor supplies its own minimal factorization; its covered offsets have total dimension at most twice the projected rank.
    9-/
    10
    11namespace Lax342547.ProjectedOffsets
    12
    13
    14
    15axiom actual_common_projection_offsets {K V W : Type} [Field K]
    16 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    17 [FiniteDimensional K V] [FiniteDimensional K W]
    18 (S : Submodule K W) (T : Submodule K (Module.Dual K V)) (M₁ M₂ : V →ₗ[K] W)
    19 (hS₁ : Disjoint (LinearMap.range M₁) S) (hT₁ : Disjoint (LinearMap.range M₁.dualMap) T)
    20 (hS₂ : Disjoint (LinearMap.range M₂) S) (hT₂ : Disjoint (LinearMap.range M₂.dualMap) T)
    21 (hcommon : Lax342547.CoverProjection.projection M₁ S T = Lax342547.CoverProjection.projection M₂ S T) :
    22 ∃ P₁ P₂ : LinearMap.range (Lax342547.CoverProjection.projection M₁ S T) →ₗ[K] W,
    23 ∃ Q₁ Q₂ : V →ₗ[K] LinearMap.range (Lax342547.CoverProjection.projection M₁ S T),
    24 P₁.comp Q₁ = M₁ ∧ P₂.comp Q₂ = M₂ ∧
    25 LinearMap.range (P₁-P₂) ≤ S ∧ LinearMap.range (Q₁-Q₂).dualMap ≤ T ∧
    26 Disjoint (LinearMap.range P₁) S ∧ Disjoint (LinearMap.range Q₂.dualMap) T ∧
    27 M₁-M₂ = (P₁-P₂).comp Q₂+P₁.comp (Q₁-Q₂) ∧
    28 Module.finrank K (LinearMap.range (P₁-P₂))+
    29 Module.finrank K (LinearMap.range (Q₁-Q₂).dualMap) ≤
    30 2*Module.finrank K (LinearMap.range (Lax342547.CoverProjection.projection M₁ S T))
    31
    32axiom nonzero_difference_has_offset {K U V W : Type} [Field K]
    33 [AddCommGroup U] [Module K U] [AddCommGroup V] [Module K V]
    34 [AddCommGroup W] [Module K W]
    35 (P₁ P₂ : U →ₗ[K] W) (Q₁ Q₂ : V →ₗ[K] U)
    36 (hne : P₁.comp Q₁ ≠ P₂.comp Q₂) : P₁-P₂ ≠ 0 ∨ Q₁-Q₂ ≠ 0
    37
    38end Lax342547.ProjectedOffsets
    39
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