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Conditioning preserves bounded costs and total mass

Lax263307.ConditionedCosts · concepts/Lax263307/ConditionedCosts.lean · lax-263307

proven

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    Natural Language Statement

    Theorem

    Let y∈Rr(G,k)y\in R_r(G,k) and let ∣A∣≤r|A|\le r with yA>0y_A>0. The conditioned costs xv=yA∪{v}/yAx_v=y_{A\cup\{v\}}/y_A lie in [0,1][0,1], sum to kk, and equal one for every v∈Av\in A. Whenever xv>0x_v>0, the enlarged conditioning set still has positive moment.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

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    1import Lax263307.LinearRelaxation
    2
    3/-!
    4---
    5title: Conditioning preserves bounded costs and total mass
    6type: theorem
    7---
    8Let y∈Rr(G,k)y\in R_r(G,k) and let ∣A∣≤r|A|\le r with yA>0y_A>0.
    9The conditioned costs xv=yA∪{v}/yAx_v=y_{A\cup\{v\}}/y_A lie in [0,1][0,1],
    10sum to kk, and equal one for every v∈Av\in A.
    11Whenever xv>0x_v>0, the enlarged conditioning set still has positive moment.
    12
    13# Attribution
    14
    15Bonnet–Chang, Section 4.1, the identities preceding Lemma 12.
    16-/
    17
    18namespace Lax263307.ConditionedCosts
    19
    20open LinearRelaxation
    21
    22/-- Basic invariants of the conditioning operation. -/
    23axiom conditioning_invariants {n : ℕ} (G : SimpleGraph (Fin n)) (k r : ℕ)
    24 (y : MomentVector n r) (feasible : IsFeasible G k r y)
    25 (A : Finset (Fin n)) (size : A.card ≤ r) (positive : 0 < value y A) :
    26 (∀ v, 0 ≤ conditioned y A v ∧ conditioned y A v ≤ 1) ∧
    27 (∑ v : Fin n, conditioned y A v) = k ∧
    28 (∀ v ∈ A, conditioned y A v = 1) ∧
    29 (∀ v, 0 < conditioned y A v → 0 < value y (insert v A))
    30
    31end Lax263307.ConditionedCosts
    32
    Show Proof

    Attribution

    Bonnet–Chang, Section 4.1, the identities preceding Lemma 12.

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