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Greedy mode space exposure

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

proven

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    Natural Language Statement

    Lemma

    Finite greedy ordering of actual subspaces has nonnegative decreasing span increments; tails retain the greedy property and their increment sum equals the dimension increase.

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

    This concept declares 12 statements. Each proof establishes one of them relative to its assumptions.

    1 exists_greedy_order proven

    2 gain_antitone proven

    3 gain_le_rank proven

    4 gain_nonneg proven

    5 greedy_decreasing proven

    6 greedy_drop proven

    7 increments_drop proven

    8 increments_le proven

    9 increments_length proven

    10 increments_nonneg proven

    11 increments_sum proven

    12 span_after_eq proven

    Lean source view on GitHub

    1import Lax342547.SpanDeficits
    2import Mathlib.Data.Finset.Max
    3import Mathlib.Data.Real.Basic
    4
    5/-!
    6---
    7title: Greedy mode space exposure
    8type: lemma
    9---
    10Finite greedy ordering of actual subspaces has nonnegative decreasing span increments; tails retain the greedy property and their increment sum equals the dimension increase.
    11-/
    12
    13namespace Lax342547.GreedySpans
    14
    15open scoped BigOperators
    16
    17noncomputable def gain {K V : Type} [Field K] [AddCommGroup V] [Module K V]
    18 [FiniteDimensional K V] (W S : Submodule K V) : ℝ :=
    19 (Module.finrank K (W ⊔ S : Submodule K V) : ℝ)-Module.finrank K W
    20
    21def Greedy {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    22 [FiniteDimensional K V] (S : ι → Submodule K V) : Submodule K V → List ι → Prop
    23 | _, [] => True
    24 | W, i::l => (∀ j ∈ l, gain W (S j) ≤ gain W (S i)) ∧ Greedy S (W ⊔ S i) l
    25
    26axiom gain_nonneg {K V : Type} [Field K] [AddCommGroup V] [Module K V]
    27 [FiniteDimensional K V] (W S : Submodule K V) : 0 ≤ gain W S
    28
    29axiom gain_le_rank {K V : Type} [Field K] [AddCommGroup V] [Module K V]
    30 [FiniteDimensional K V] (W S : Submodule K V) : gain W S ≤ Module.finrank K S
    31
    32axiom gain_antitone {K V : Type} [Field K] [AddCommGroup V] [Module K V]
    33 [FiniteDimensional K V] {W U : Submodule K V} (hWU : W ≤ U) (S : Submodule K V) :
    34 gain U S ≤ gain W S
    35
    36axiom exists_greedy_order {K V ι : Type} [Field K] [DecidableEq ι] [AddCommGroup V] [Module K V]
    37 [FiniteDimensional K V] (S : ι → Submodule K V) (I : Finset ι) (W : Submodule K V) :
    38 ∃ l : List ι, l.Nodup ∧ l.toFinset = I ∧ Greedy S W l
    39
    40noncomputable def spanAfter {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    41 (S : ι → Submodule K V) : Submodule K V → List ι → Submodule K V
    42 | W, [] => W
    43 | W, i::l => spanAfter S (W ⊔ S i) l
    44
    45noncomputable def increments {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    46 [FiniteDimensional K V] (S : ι → Submodule K V) : Submodule K V → List ι → List ℝ
    47 | _, [] => []
    48 | W, i::l => gain W (S i)::increments S (W ⊔ S i) l
    49
    50axiom increments_length {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    51 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) :
    52 (increments S W l).length = l.length
    53
    54axiom span_after_eq {K V ι : Type} [Field K] [DecidableEq ι] [AddCommGroup V] [Module K V]
    55 (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) :
    56 spanAfter S W l = W ⊔ l.toFinset.sup S
    57
    58axiom increments_sum {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    59 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) :
    60 (increments S W l).sum = (Module.finrank K (spanAfter S W l) : ℝ)-Module.finrank K W
    61
    62axiom increments_le {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    63 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) (a : ℝ)
    64 (h : ∀ i ∈ l, gain W (S i) ≤ a) : ∀ x ∈ increments S W l, x ≤ a
    65
    66axiom increments_nonneg {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    67 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) :
    68 ∀ x ∈ increments S W l, 0 ≤ x
    69
    70axiom greedy_decreasing {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    71 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι)
    72 (h : Greedy S W l) : (increments S W l).Pairwise (fun a b => b ≤ a)
    73
    74axiom greedy_drop {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    75 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι)
    76 (h : Greedy S W l) (n : ℕ) : Greedy S (spanAfter S W (l.take n)) (l.drop n)
    77
    78axiom increments_drop {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    79 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) (n : ℕ) :
    80 (increments S W l).drop n = increments S (spanAfter S W (l.take n)) (l.drop n)
    81
    82end Lax342547.GreedySpans
    83
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