Lax Lean Archive

Invariance and characterization of TAUT

Lax564036.TautologyInvariance · concepts/Lax564036/TautologyInvariance.lean · lax-564036

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    Being a DNF tautology is invariant under isomorphism of instances, and an instance is a yes-instance of TAUT exactly when it is a tautology.

    Concept map
    23 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 taut_iff proven

    2 tautology_iso 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 Lax904597.Machines
    8import Lax485149.Problems
    9import Lax485149.Complement
    10import Lax535992.ClassPTIME
    11import Lax564036.Hierarchy
    12import Lax564036.Difference
    13import Lax564036.Tautology
    14import Lax564036.ThreeDnfTautology
    15import Lax564036.SatUnsat
    16import Lax564036.QuantifiedBooleanFormulas
    17import Lax564036.AlternatingMachines
    18
    19/-!
    20---
    21title: Invariance and characterization of TAUT
    22type: lemma
    23---
    24Being a DNF tautology is invariant under isomorphism of instances, and an
    25instance is a yes-instance of TAUT exactly when it is a tautology.
    26-/
    27
    28namespace Lax564036.TautologyInvariance
    29
    30open FirstOrder FirstOrder.Language
    31open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    32open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    33open Lax485149.Problems Lax485149.Complement Lax535992.ClassPTIME
    34open Lax564036.Hierarchy Lax564036.Difference Lax564036.Tautology Lax564036.ThreeDnfTautology
    35open Lax564036.SatUnsat Lax564036.QuantifiedBooleanFormulas Lax564036.AlternatingMachines
    36
    37/-- Being a tautology is isomorphism-invariant. -/
    38axiom tautology_iso : ∀ {A B : Type} [sat.Structure A] [sat.Structure B],
    39 (A ≃[sat] B) → (Tautology A ↔ Tautology B)
    40
    41/-- The yes-instances of TAUT are exactly the tautologies. -/
    42axiom taut_iff : ∀ (A : Type) [sat.Structure A], TAUT A ↔ Tautology A
    43
    44end Lax564036.TautologyInvariance
    45
    Show ProofShow Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…