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Invariance and characterization of deterministic reachability

Lax485149.DeterministicReachabilityInvariance · concepts/Lax485149/DeterministicReachabilityInvariance.lean · lax-485149

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    Natural Language Statement

    Lemma

    Reachability along forced edges is invariant under isomorphism of graphs, and a graph is a yes-instance of REACHd exactly when some marked target is reachable from some marked source along forced edges.

    Concept map
    20 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 detReachable_iso proven

    2 reachd_iff proven

    3 unreachd_iff proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax485149.Problems
    8import Lax485149.Complement
    9import Lax485149.SecondOrderAtoms
    10import Lax485149.KromFragment
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.FirstOrderDefinability
    14import Lax485149.HeadAutomata
    15import Lax485149.Reachability
    16import Lax485149.DeterministicReachability
    17import Lax485149.TwoSat
    18import Lax485149.ClassNL
    19import Lax485149.ClassL
    20
    21/-!
    22---
    23title: Invariance and characterization of deterministic reachability
    24type: lemma
    25---
    26Reachability along forced edges is invariant under isomorphism of graphs,
    27and a graph is a yes-instance of REACHd exactly when some marked target is
    28reachable from some marked source along forced edges.
    29-/
    30
    31namespace Lax485149.DeterministicReachabilityInvariance
    32
    33open FirstOrder FirstOrder.Language
    34open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    35open Lax904597.Classes Lax904597.Sat
    36open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    37open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    38open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    39open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    40
    41/-- Deterministic reachability is isomorphism-invariant. -/
    42axiom detReachable_iso : ∀ {A B : Type} [stGraph.Structure A] [stGraph.Structure B],
    43 (A ≃[stGraph] B) → (DetReachable A ↔ DetReachable B)
    44
    45/-- The yes-instances of REACHd are exactly the graphs where a marked target is
    46reachable from a marked source along forced arcs. -/
    47axiom reachd_iff : ∀ (A : Type) [stGraph.Structure A], REACHd A ↔ DetReachable A
    48
    49/-- The yes-instances of UNREACHd are exactly the graphs where no marked target
    50is reachable from a marked source along forced arcs. -/
    51axiom unreachd_iff : ∀ (A : Type) [stGraph.Structure A], UNREACHd A ↔ ¬DetReachable A
    52
    53end Lax485149.DeterministicReachabilityInvariance
    54
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