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The Ehrenfeucht–Fraïssé method

Lax945089.EhrenfeuchtMethodology · concepts/Lax945089/EhrenfeuchtMethodology.lean · lax-945089

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    Natural Language Statement

    Theorem

    If two structures over a relational vocabulary are nn-round equivalent, they satisfy the same first-order sentences of quantifier depth at most nn. The inexpressibility results of this submission are contrapositives of it.

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

    Each proof establishes this claim relative to its assumptions.

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    1import Mathlib.ModelTheory.Order
    2import Mathlib.Order.Defs.LinearOrder
    3import Lax904597.Problems
    4import Lax904597.Interpretations
    5import Lax904597.Relativized
    6import Lax904597.SecondOrder
    7import Lax904597.Classes
    8import Lax485149.Problems
    9import Lax485149.FirstOrderDefinability
    10import Lax485149.TransitiveClosure
    11import Lax485149.DeterministicTransitiveClosure
    12import Lax485149.ClassNL
    13import Lax485149.ClassL
    14import Lax535992.InflationaryFixedPoint
    15import Lax535992.ClassPTIME
    16import Lax134656.PartialFixedPoint
    17import Lax895169.ArithmeticLogic
    18import Lax945089.OrderFreeFirstOrder
    19import Lax945089.EhrenfeuchtGames
    20import Lax945089.PebbleGames
    21import Lax945089.Even
    22import Lax945089.Parity
    23import Lax945089.TransitiveClosureReductions
    24
    25/-!
    26---
    27title: The Ehrenfeucht–Fraïssé method
    28type: theorem
    29---
    30If two structures over a relational vocabulary are nn-round equivalent,
    31they satisfy the same first-order sentences of quantifier depth at most
    32nn. The inexpressibility results of this submission are contrapositives
    33of it.
    34-/
    35
    36namespace Lax945089.EhrenfeuchtMethodology
    37
    38open FirstOrder FirstOrder.Language
    39open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    40open Lax904597.Classes
    41open Lax485149.Problems Lax485149.FirstOrderDefinability Lax485149.TransitiveClosure
    42open Lax485149.DeterministicTransitiveClosure Lax485149.ClassNL Lax485149.ClassL
    43open Lax535992.InflationaryFixedPoint Lax535992.ClassPTIME
    44open Lax134656.PartialFixedPoint Lax895169.ArithmeticLogic
    45open Lax945089.OrderFreeFirstOrder Lax945089.EhrenfeuchtGames Lax945089.PebbleGames
    46open Lax945089.Even Lax945089.Parity
    47open Lax945089.TransitiveClosureReductions
    48
    49/-- `n`-round equivalent structures satisfy the same sentences of depth at most `n`. -/
    50axiom realize_sentence_of_efEquiv :
    51 ∀ {L : Language.{0, 0}} {M N : Type} [L.Structure M] [L.Structure N] {n : ℕ} [L.IsRelational],
    52 EFEquiv L M N n → ∀ (φ : L.Sentence), qdepth φ ≤ n → (M ⊨ φ ↔ N ⊨ φ)
    53
    54end Lax945089.EhrenfeuchtMethodology
    55
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