Lax Lean Archive

Homomorphism counts into a categorical product

Lax871432.CategoricalProductCounts · concepts/Lax871432/CategoricalProductCounts.lean · lax-871432

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    For simple graphs FF, G1G_1 and G2G_2,

    hom⁡(F,G1×G2)=hom⁡(F,G1)hom⁡(F,G2).\hom(F, G_1 \times G_2) = \hom(F, G_1) \hom(F, G_2).
    Concept map
    3 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    In the paper

    • page 3 of this submission's paper

    Lean source view on GitHub

    1import Lax871432.GraphProducts
    2import Lax871432.HomomorphismCounts
    3
    4/-!
    5---
    6title: Homomorphism counts into a categorical product
    7type: lemma
    8---
    9For simple graphs FF, G1G_1 and G2G_2,
    10hom⁡(F,G1×G2)=hom⁡(F,G1)hom⁡(F,G2).\hom(F, G_1 \times G_2) = \hom(F, G_1) \hom(F, G_2).
    11-/
    12
    13open Lax871432.GraphProducts Lax871432.HomomorphismCounts
    14
    15open scoped Lax871432.GraphProducts
    16
    17namespace Lax871432.CategoricalProductCounts
    18
    19/-- A homomorphism into a categorical product is a pair of homomorphisms into its two
    20factors. -/
    21axiom homCount_catProd_right {U V W : Type*} [Finite U] [Finite V] [Finite W]
    22 (F : SimpleGraph U) (G₁ : SimpleGraph V) (G₂ : SimpleGraph W) :
    23 homCount F (G₁ ×g G₂) = homCount F G₁ * homCount F G₂
    24
    25end Lax871432.CategoricalProductCounts
    26
    Show Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…