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No order-free induction defines a linear order

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

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    Natural Language Statement

    Theorem

    On a bare set with at least as many elements as its variable budget, no binary relation defined by an order-free inflationary induction is a linear order: a transposition of the universe preserves every stage of the induction, so the relation is symmetric. The order that the logics of polynomial time are given cannot be built by an isomorphism-invariant induction.

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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: No order-free induction defines a linear order
    28type: theorem
    29---
    30On a bare set with at least as many elements as its variable budget, no
    31binary relation defined by an order-free inflationary induction is a linear
    32order: a transposition of the universe preserves every stage of the
    33induction, so the relation is symmetric. The order that the logics of
    34polynomial time are given cannot be built by an isomorphism-invariant
    35induction.
    36-/
    37
    38namespace Lax945089.NoDefinableOrder
    39
    40open FirstOrder FirstOrder.Language
    41open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    42open Lax904597.Classes
    43open Lax485149.Problems Lax485149.FirstOrderDefinability Lax485149.TransitiveClosure
    44open Lax485149.DeterministicTransitiveClosure Lax485149.ClassNL Lax485149.ClassL
    45open Lax535992.InflationaryFixedPoint Lax535992.ClassPTIME
    46open Lax134656.PartialFixedPoint Lax895169.ArithmeticLogic
    47open Lax945089.OrderFreeFirstOrder Lax945089.EhrenfeuchtGames Lax945089.PebbleGames
    48open Lax945089.Even Lax945089.Parity
    49open Lax945089.TransitiveClosureReductions
    50
    51/-- No binary variable of an order-free inflationary limit is a linear order on a
    52large enough bare set. -/
    53axiom not_isLinearOrder_inflLimit :
    54 ∀ {k : ℕ} (d : StepDef Language.empty) {i : d.B.ι}, StepDef.VarBound d k →
    55 ∀ (harity : d.B.arity i = 2) (A : Type) [Language.empty.Structure A] [Finite A], k ≤ Nat.card A →
    56 ¬IsLinearOrder A fun x y => d.inflLimit A i fun p => ![x, y] (Fin.cast harity p)
    57
    58end Lax945089.NoDefinableOrder
    59
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