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Invariance and characterization of QSAT

Lax134656.QsatInvariance · concepts/Lax134656/QsatInvariance.lean · lax-134656

proven

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    Natural Language Statement

    Lemma

    Being a well-formed true quantified Boolean formula is invariant under isomorphism of instances, and an instance is a yes-instance of QSAT exactly when it is well formed and true.

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

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 qsat_iff proven

    2 qsatHolds_iso proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Machines
    7import Lax485149.Problems
    8import Lax485149.Complement
    9import Lax535992.InflationaryFixedPoint
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax564036.Hierarchy
    13import Lax134656.SecondOrderTransitiveClosure
    14import Lax134656.OrderFreeTransitiveClosure
    15import Lax134656.PartialFixedPoint
    16import Lax134656.Qsat
    17import Lax134656.SuccinctReach
    18import Lax134656.SpaceBoundedMachines
    19import Lax134656.ClassPSPACE
    20
    21/-!
    22---
    23title: Invariance and characterization of QSAT
    24type: lemma
    25---
    26Being a well-formed true quantified Boolean formula is invariant under
    27isomorphism of instances, and an instance is a yes-instance of QSAT exactly
    28when it is well formed and true.
    29-/
    30
    31namespace Lax134656.QsatInvariance
    32
    33open FirstOrder FirstOrder.Language
    34open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    35open Lax904597.Classes Lax904597.Machines
    36open Lax485149.Problems Lax485149.Complement
    37open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    38open Lax564036.Hierarchy
    39open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    40open Lax134656.PartialFixedPoint
    41open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    42
    43/-- Being well formed and true is isomorphism-invariant. -/
    44axiom qsatHolds_iso : ∀ {A B : Type} [qsat.Structure A] [qsat.Structure B],
    45 (A ≃[qsat] B) → (QsatHolds A ↔ QsatHolds B)
    46
    47/-- The yes-instances are exactly the instances with the defining property. -/
    48axiom qsat_iff : ∀ (A : Type) [qsat.Structure A], QSAT A ↔ QsatHolds A
    49
    50end Lax134656.QsatInvariance
    51
    Show ProofShow Proof
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    From Mathlib

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