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Lax307052.Reachability

Bounded reachability

concepts/Lax307052/Reachability.lean · lax-307052

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    Definition

    A directed graph on NN vertices is given by its Boolean adjacency matrix. A walk may repeat vertices. The recursive reachability test divides the length bound in two and enumerates possible middle vertices.

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    1import Mathlib.Data.List.FinRange
    2import Mathlib.Logic.Relation
    3
    4/-!
    5---
    6title: Bounded reachability
    7type: definition
    8---
    9A directed graph on NN vertices is given by its Boolean adjacency matrix.
    10A walk may repeat vertices. The recursive reachability test divides the
    11length bound in two and enumerates possible middle vertices.
    12-/
    13
    14namespace Lax307052.Reachability
    15
    16abbrev Graph (N : ℕ) := Fin N → Fin N → Bool
    17
    18inductive Walk {α : Type} (R : α → α → Prop) : ℕ → α → α → Prop
    19 | nil (a : α) : Walk R 0 a a
    20 | tail {n : ℕ} {a b c : α} : Walk R n a b → R b c → Walk R (n + 1) a c
    21
    22def Within {α : Type} (R : α → α → Prop) (n : ℕ) (a b : α) : Prop :=
    23 ∃ k ≤ n, Walk R k a b
    24
    25def Reachable {N : ℕ} (G : Graph N) (a b : Fin N) : Prop :=
    26 Relation.ReflTransGen (fun u v => G u v = true) a b
    27
    28def search {N : ℕ} (G : Graph N) : ℕ → Fin N → Fin N → Bool
    29 | 0, a, b => decide (a = b) || G a b
    30 | k + 1, a, b => (List.finRange N).any fun m => search G k a m && search G k m b
    31
    32end Lax307052.Reachability
    33

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