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Agreement with Lax functional equivalence

Lax825442.FunctionalEquivalenceConnection · concepts/Lax825442/FunctionalEquivalenceConnection.lean · lax-825442

proven

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    Natural Language Statement

    Theorem

    Mutual nondecreasing functional bounds agree with the existing Lax definition Lax153141.GraphParameters.FunctionallyEquivalentLax153141.GraphParameters.FunctionallyEquivalent, which permits arbitrary natural-valued bounding functions.

    For an arbitrary bounding function ff, its upper envelope h(n)=max⁡{f(0),…,f(n)}h(n)=\max\{f(0),\ldots,f(n)\} is nondecreasing and satisfies f(n)≤h(n)f(n)\le h(n). Replacing both bounding functions by these envelopes proves the agreement.

    Concept map
    4 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax825442.Equivalent
    2
    3/-!
    4---
    5title: Agreement with Lax functional equivalence
    6type: theorem
    7---
    8Mutual nondecreasing functional bounds agree with the existing Lax definition
    9`Lax153141.GraphParameters.FunctionallyEquivalent`, which permits arbitrary
    10natural-valued bounding functions.
    11
    12For an arbitrary bounding function ff, its upper envelope
    13h(n)=max⁡{f(0),…,f(n)}h(n)=\max\{f(0),\ldots,f(n)\} is nondecreasing and satisfies f(n)≤h(n)f(n)\le h(n).
    14Replacing both bounding functions by these envelopes proves the agreement.
    15-/
    16
    17namespace Lax825442.FunctionalEquivalenceConnection
    18
    19open Lax153141.GraphParameters
    20
    21/-- Mutual monotone bounds agree with the registered Lax definition. -/
    22axiom functionallyEquivalent_iff (p q : GraphParam) :
    23 (FunctionallyEquivalent p q) ↔ (Lax825442.Equivalent.Equivalent p q)
    24
    25end Lax825442.FunctionalEquivalenceConnection
    26
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