First-Ordered Structures in Graph Theory

This file defines first-order languages, structures, and theories in graph theory.

Main Definitions

• FirstOrder.Language.graph is the language consisting of a single relation representing adjacency.
• SimpleGraph.structure is the first-order structure corresponding to a given simple graph.
• FirstOrder.Language.Theory.simpleGraph is the theory of simple graphs.
• FirstOrder.Language.simpleGraphOfStructure gives the simple graph corresponding to a model of the theory of simple graphs.

Simple Graphs

def FirstOrder.Language.graph : FirstOrder.Language

The language consisting of a single relation representing adjacency.

def FirstOrder.Language.adj : FirstOrder.Language.Relations FirstOrder.Language.graph 2

The symbol representing the adjacency relation.

def SimpleGraph.structure {V : Type w'} (G : SimpleGraph V) : FirstOrder.Language.Structure FirstOrder.Language.graph V

Any simple graph can be thought of as a structure in the language of graphs.

instance FirstOrder.Language.graph.instIsRelational : FirstOrder.Language.IsRelational FirstOrder.Language.graph

instance FirstOrder.Language.graph.instSubsingleton {n : ℕ} : Subsingleton (FirstOrder.Language.Relations FirstOrder.Language.graph n)

def FirstOrder.Language.Theory.simpleGraph : FirstOrder.Language.Theory FirstOrder.Language.graph

The theory of simple graphs.

@[simp]

theorem FirstOrder.Language.Theory.simpleGraph_model_iff {V : Type w'} [FirstOrder.Language.Structure FirstOrder.Language.graph V] : V ⊨ FirstOrder.Language.Theory.simpleGraph ↔ (Irreflexive fun x y => FirstOrder.Language.Structure.RelMap FirstOrder.Language.adj ![x, y]) ∧ Symmetric fun x y => FirstOrder.Language.Structure.RelMap FirstOrder.Language.adj ![x, y]

instance FirstOrder.Language.simpleGraph_model {V : Type w'} (G : SimpleGraph V) : V ⊨ FirstOrder.Language.Theory.simpleGraph

@[simp]

theorem FirstOrder.Language.simpleGraphOfStructure_Adj (V : Type w') [FirstOrder.Language.Structure FirstOrder.Language.graph V] [V ⊨ FirstOrder.Language.Theory.simpleGraph] (x : V) (y : V) : SimpleGraph.Adj (FirstOrder.Language.simpleGraphOfStructure V) x y = FirstOrder.Language.Structure.RelMap FirstOrder.Language.adj ![x, y]

def FirstOrder.Language.simpleGraphOfStructure (V : Type w') [FirstOrder.Language.Structure FirstOrder.Language.graph V] [V ⊨ FirstOrder.Language.Theory.simpleGraph] : SimpleGraph V

Any model of the theory of simple graphs represents a simple graph.

@[simp]

theorem SimpleGraph.simpleGraphOfStructure {V : Type w'} (G : SimpleGraph V) : FirstOrder.Language.simpleGraphOfStructure V = G

@[simp]

theorem FirstOrder.Language.structure_simpleGraphOfStructure {V : Type w'} [S : FirstOrder.Language.Structure FirstOrder.Language.graph V] [V ⊨ FirstOrder.Language.Theory.simpleGraph] : SimpleGraph.structure (FirstOrder.Language.simpleGraphOfStructure V) = S

theorem FirstOrder.Language.Theory.simpleGraph_isSatisfiable : FirstOrder.Language.Theory.IsSatisfiable FirstOrder.Language.Theory.simpleGraph