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Tautology of DNF formulas

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

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    Natural Language Statement

    Definition

    An instance is a structure over the vocabulary of CNF instances of the NP core, read here as a formula in disjunctive normal form: its clauses are terms, conjunctions of literals. It is a tautology when every assignment of truth values to its elements satisfies all the literals of some term. TAUT is the decision problem of the structures isomorphic to a tautology.

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    1import Mathlib.ModelTheory.Semantics
    2import Lax904597.Problems
    3import Lax485149.Problems
    4import Lax904597.Sat
    5
    6/-!
    7---
    8title: Tautology of DNF formulas
    9type: definition
    10---
    11An instance is a structure over the vocabulary of CNF instances of the NP
    12core, read here as a formula in disjunctive normal form: its clauses are
    13terms, conjunctions of literals. It is a tautology when every assignment of
    14truth values to its elements satisfies all the literals of some term. TAUT
    15is the decision problem of the structures isomorphic to a tautology.
    16-/
    17
    18namespace Lax564036.Tautology
    19
    20open Lax904597.Problems Lax485149.Problems
    21
    22open Lax904597.Sat
    23
    24open FirstOrder
    25
    26open Language Structure BoundedFormula
    27
    28section Taut
    29
    30variable (A : Type) [sat.Structure A]
    31
    32/-- A CNF instance of the NP core, read as a formula in disjunctive normal form,
    33is a *tautology* when every assignment of truth values to its elements makes
    34some term true – that is, satisfies every literal of that term. (Elements that
    35are not variables of the formula may be assigned arbitrarily; they are harmless
    36since no term mentions them.) -/
    37def Tautology : Prop :=
    38 ∀ ν : A → Prop, ∃ c : A, RelMap satIsClause ![c] ∧
    39 ∀ x : A, (RelMap satPosIn ![c, x] → ν x) ∧ (RelMap satNegIn ![c, x] → ¬ν x)
    40
    41end Taut
    42
    43/-- TAUT: is the instance, read as a formula in disjunctive normal form, a
    44tautology? -/
    45def TAUT : DecisionProblem sat := DecisionProblem.ofPred fun A _ => Tautology A
    46
    47end Lax564036.Tautology
    48
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