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Compact real-capacity flows and a maximizing flow

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

proven

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    Natural Language Statement

    Lemma

    Skew real flows satisfy directed capacity bounds and interior conservation. Their feasible set is closed and bounded in a finite product, so a maximum source divergence exists for every nonnegative finite capacity network.

    Concept map
    1 concept
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    Proven claimThis conceptDescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 feasible_closed proven

    2 feasible_compact proven

    3 maximal_flow proven

    4 zero_feasible proven

    Lean source view on GitHub

    1import Mathlib.Topology.Instances.Real.Lemmas
    2import Mathlib.Analysis.Convex.StdSimplex
    3import Mathlib.Data.Fintype.BigOperators
    4import Mathlib.Analysis.Convex.Function
    5
    6/-!
    7---
    8title: Compact real-capacity flows and a maximizing flow
    9type: lemma
    10---
    11Skew real flows satisfy directed capacity bounds and interior conservation. Their feasible set is closed and bounded in a finite product, so a maximum source divergence exists for every nonnegative finite capacity network.
    12-/
    13
    14namespace Lax342547.RealFlows
    15
    16open scoped BigOperators Topology
    17
    18noncomputable def divergence {V : Type} [Fintype V] (f : V → V → ℝ) (a : V) : ℝ := ∑ b, f a b
    19
    20def Feasible {V : Type} [Fintype V] (c f : V → V → ℝ) (s t : V) : Prop :=
    21 (∀ a b, f a b = -f b a) ∧ (∀ a b, f a b ≤ c a b) ∧
    22 (∀ a, a ≠ s → a ≠ t → divergence f a = 0)
    23
    24def Residual {V : Type} (c f : V → V → ℝ) (a b : V) : Prop := f a b < c a b
    25
    26axiom zero_feasible {V : Type} [Fintype V] (c : V → V → ℝ) (s t : V)
    27 (hc : ∀ a b, 0 ≤ c a b) : Feasible c (fun _ _ => 0) s t
    28
    29axiom feasible_closed {V : Type} [Fintype V] (c : V → V → ℝ) (s t : V) :
    30 IsClosed {f | Feasible c f s t}
    31
    32axiom feasible_compact {V : Type} [Fintype V] (c : V → V → ℝ) (s t : V) :
    33 IsCompact {f | Feasible c f s t}
    34
    35axiom maximal_flow {V : Type} [Fintype V] (c : V → V → ℝ) (s t : V)
    36 (hc : ∀ a b, 0 ≤ c a b) :
    37 ∃ f, Feasible c f s t ∧ ∀ g, Feasible c g s t → divergence g s ≤ divergence f s
    38
    39end Lax342547.RealFlows
    40
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