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Combined column and row mode exposure

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

proven

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    Natural Language Statement

    Lemma

    Products of the two mode spaces have additive dimensions, gains, and deficits, allowing one greedy exposed set to control both modes.

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

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 deficit_product proven

    2 gain_product proven

    3 product_space_dimension proven

    4 span_after_product proven

    5 sum_gain_product proven

    Lean source view on GitHub

    1import Lax342547.ActualDeficits
    2import Mathlib.LinearAlgebra.Prod
    3
    4/-!
    5---
    6title: Combined column and row mode exposure
    7type: lemma
    8---
    9Products of the two mode spaces have additive dimensions, gains, and deficits, allowing one greedy exposed set to control both modes.
    10-/
    11
    12namespace Lax342547.CombinedSpans
    13
    14open Lax342547.GreedySpans Lax342547.GreedyTails
    15open scoped BigOperators
    16
    17noncomputable def productSpaceEquiv {K V W : Type} [Field K] [AddCommGroup V] [Module K V]
    18 [AddCommGroup W] [Module K W] (C : Submodule K V) (R : Submodule K W) :
    19 (C.prod R) ≃ₗ[K] C × R :=
    20 { toFun := fun v => (⟨v.val.1,v.property.1⟩,⟨v.val.2,v.property.2⟩)
    21 invFun := fun v => ⟨(v.1.val,v.2.val),v.1.property,v.2.property⟩
    22 left_inv := fun _ => rfl
    23 right_inv := fun _ => rfl
    24 map_add' := fun _ _ => rfl
    25 map_smul' := fun _ _ => rfl }
    26
    27axiom product_space_dimension {K V W : Type} [Field K] [AddCommGroup V] [Module K V]
    28 [AddCommGroup W] [Module K W] [FiniteDimensional K V] [FiniteDimensional K W]
    29 (C : Submodule K V) (R : Submodule K W) :
    30 Module.finrank K (C.prod R) = Module.finrank K C+Module.finrank K R
    31
    32axiom gain_product {K V W : Type} [Field K] [AddCommGroup V] [Module K V]
    33 [AddCommGroup W] [Module K W] [FiniteDimensional K V] [FiniteDimensional K W]
    34 (U C : Submodule K V) (Z R : Submodule K W) :
    35 gain (U.prod Z) (C.prod R) = gain U C+gain Z R
    36
    37axiom span_after_product {K V W ι : Type} [Field K] [AddCommGroup V] [Module K V]
    38 [AddCommGroup W] [Module K W]
    39 (C : ι → Submodule K V) (R : ι → Submodule K W)
    40 (U : Submodule K V) (Z : Submodule K W) (l : List ι) :
    41 spanAfter (fun i => (C i).prod (R i)) (U.prod Z) l =
    42 (spanAfter C U l).prod (spanAfter R Z l)
    43
    44axiom sum_gain_product {K V W ι : Type} [Field K] [AddCommGroup V] [Module K V]
    45 [AddCommGroup W] [Module K W] [FiniteDimensional K V] [FiniteDimensional K W]
    46 (C : ι → Submodule K V) (R : ι → Submodule K W)
    47 (U : Submodule K V) (Z : Submodule K W) (l : List ι) :
    48 (l.map (fun i => gain (U.prod Z) ((C i).prod (R i)))).sum =
    49 (l.map (fun i => gain U (C i))).sum+(l.map (fun i => gain Z (R i))).sum
    50
    51axiom deficit_product {K V W ι : Type} [Field K] [AddCommGroup V] [Module K V]
    52 [AddCommGroup W] [Module K W] [FiniteDimensional K V] [FiniteDimensional K W]
    53 (C : ι → Submodule K V) (R : ι → Submodule K W)
    54 (U : Submodule K V) (Z : Submodule K W) (l : List ι) :
    55 deficit (fun i => (C i).prod (R i)) (U.prod Z) l = deficit C U l+deficit R Z l
    56
    57end Lax342547.CombinedSpans
    58
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