Lax Lean Archive

Hind–Oellermann deletion–contraction theorem

Lax17.HindOellermann · concepts/Lax17/HindOellermann.lean · lax-17

proven

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    Natural Language Statement

    Theorem

    Deleting or contracting a nonterminal edge preserves terminal element-connectivity.

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    2 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    1import Lax17.TerminalConnectivity
    2
    3/-!
    4---
    5title: Hind--Oellermann deletion--contraction theorem
    6type: theorem
    7---
    8Deleting or contracting a nonterminal edge preserves terminal
    9element-connectivity.
    10-/
    11
    12namespace Lax17.HindOellermann
    13
    14universe u
    15
    16/-- Hind--Oellermann deletion--contraction for terminal element
    17connectivity. -/
    18axiom hindOellermannDeletionContraction :
    19 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    20 (H : Lax17.TerminalConnectivity.EdgeIndexedGraph V)
    21 (terminals : Finset V) (k : ℕ) (e₀ : H.Edge),
    22 H.left e₀ ∉ terminals → H.right e₀ ∉ terminals →
    23 H.TerminalElementConnectedAtLeast terminals k →
    24 (H.deleteEdge e₀).TerminalElementConnectedAtLeast terminals k ∨
    25 ∃ (W : Type u) (_ : Fintype W) (_ : DecidableEq W)
    26 (K : Lax17.TerminalConnectivity.EdgeIndexedGraph W)
    27 (mapVertex : V → W),
    28 Nonempty (H.IsContraction e₀ K mapVertex) ∧
    29 K.TerminalElementConnectedAtLeast
    30 (Lax17.TerminalConnectivity.EdgeIndexedGraph.terminalImage
    31 mapVertex terminals) k
    32
    33end Lax17.HindOellermann
    34
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