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FO(IFP) ⊆ FO(PFP)

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

proven

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    Natural Language Statement

    Theorem

    Every problem definable with inflationary fixed points is definable with partial fixed points, on ordered structures and without an order: disjoining each variable's own atom onto its step formula turns an inflationary induction into a partial one with the same stages. And an order-free definition is an ordered one that does not use the order, for both logics.

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

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 ifpDefinable_pfpDefinable proven

    2 ifpDefinableFree_ifpDefinable proven

    3 ifpDefinableFree_pfpDefinableFree proven

    4 pfpDefinableFree_pfpDefinable 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: FO(IFP) ⊆ FO(PFP)
    24type: theorem
    25---
    26Every problem definable with inflationary fixed points is definable with
    27partial fixed points, on ordered structures and without an order:
    28disjoining each variable's own atom onto its step formula turns an
    29inflationary induction into a partial one with the same stages. And an
    30order-free definition is an ordered one that does not use the order, for
    31both logics.
    32-/
    33
    34namespace Lax134656.InflationaryInPartial
    35
    36open FirstOrder FirstOrder.Language
    37open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    38open Lax904597.Classes Lax904597.Machines
    39open Lax485149.Problems Lax485149.Complement
    40open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    41open Lax564036.Hierarchy
    42open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    43open Lax134656.PartialFixedPoint
    44open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    45
    46/-- Every FO(≤, IFP) definable problem is FO(≤, PFP) definable. -/
    47axiom ifpDefinable_pfpDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    48 IFPDefinable P → PFPDefinable P
    49
    50/-- Every order-free FO(IFP) definable problem is order-free FO(PFP) definable. -/
    51axiom ifpDefinableFree_pfpDefinableFree :
    52 ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    53 IFPDefinableFree P → PFPDefinableFree P
    54
    55/-- Every order-free FO(IFP) definable problem is FO(≤, IFP) definable. -/
    56axiom ifpDefinableFree_ifpDefinable :
    57 ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    58 IFPDefinableFree P → IFPDefinable P
    59
    60/-- Every order-free FO(PFP) definable problem is FO(≤, PFP) definable. -/
    61axiom pfpDefinableFree_pfpDefinable :
    62 ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    63 PFPDefinableFree P → PFPDefinable P
    64
    65end Lax134656.InflationaryInPartial
    66
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