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The values of the number problems

Lax366625.NumbersValue · concepts/Lax366625/NumbersValue.lean · lax-366625

proven

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    Natural Language Statement

    Lemma

    The numbers written by a circuit, by unit propagation on a Horn formula, and by a deterministic Turing machine are invariant under isomorphism of instances, so the value of each problem on an instance is the number it writes.

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

    This concept declares 6 statements. Each proof establishes one of them relative to its assumptions.

    1 circuitNumber_count_iso proven

    2 circuitNumber_eq proven

    3 dtmNumber_count_iso proven

    4 dtmNumber_eq proven

    5 hornNumber_count_iso proven

    6 hornNumber_eq 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 Lax535992.HornSat
    9import Lax535992.CircuitValue
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax366625.CountingProblems
    13import Lax366625.CountingClasses
    14import Lax366625.WitnessCounting
    15import Lax366625.SecondOrderCounting
    16import Lax366625.QuantitativeLogic
    17import Lax366625.CountingSat
    18import Lax366625.MachineNumbers
    19import Lax366625.CountingRuns
    20import Lax366625.NumberedCircuits
    21import Lax366625.HornNumbers
    22
    23/-!
    24---
    25title: The values of the number problems
    26type: lemma
    27---
    28The numbers written by a circuit, by unit propagation on a Horn formula, and
    29by a deterministic Turing machine are invariant under isomorphism of
    30instances, so the value of each problem on an instance is the number it
    31writes.
    32-/
    33
    34namespace Lax366625.NumbersValue
    35
    36open FirstOrder FirstOrder.Language
    37open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    38open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    39open Lax535992.ClassPTIME
    40open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    41open Lax366625.SecondOrderCounting
    42open Lax366625.QuantitativeLogic Lax366625.CountingSat Lax366625.MachineNumbers
    43open Lax366625.CountingRuns
    44open Lax366625.NumberedCircuits Lax366625.HornNumbers
    45
    46/-- The number written by a circuit is isomorphism-invariant. -/
    47axiom circuitNumber_count_iso : ∀ {A B : Type} [numCircuit.Structure A] [numCircuit.Structure B],
    48 (A ≃[numCircuit] B) → circuitNumber A = circuitNumber B
    49
    50/-- The value of the problem is the number it is defined by. -/
    51axiom circuitNumber_eq : ∀ (A : Type) [numCircuit.Structure A], CircuitNumber A = circuitNumber A
    52
    53/-- The number written by unit propagation is isomorphism-invariant. -/
    54axiom hornNumber_count_iso : ∀ {A B : Type} [satOut.Structure A] [satOut.Structure B],
    55 (A ≃[satOut] B) → hornNumber A = hornNumber B
    56
    57/-- The value of the problem is the number it is defined by. -/
    58axiom hornNumber_eq : ∀ (A : Type) [satOut.Structure A], HornNumber A = hornNumber A
    59
    60/-- The number written by a machine is isomorphism-invariant. -/
    61axiom dtmNumber_count_iso : ∀ {A B : Type} [turingOut.Structure A] [turingOut.Structure B],
    62 (A ≃[turingOut] B) → machineNumber A = machineNumber B
    63
    64/-- The value of the problem is the number it is defined by. -/
    65axiom dtmNumber_eq : ∀ (A : Type) [turingOut.Structure A], DTMNumber A = machineNumber A
    66
    67end Lax366625.NumbersValue
    68
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