First-Order 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

inductive FirstOrder.Language.graphRel : ℕ → Type

The type of relations for the language of graphs, consisting of a single binary relation adj.

• adj : FirstOrder.Language.graphRel 2

instance FirstOrder.Language.instDecidableEqGraphRel {a✝ : ℕ} : DecidableEq (FirstOrder.Language.graphRel a✝)

Equations

• FirstOrder.Language.instDecidableEqGraphRel = FirstOrder.Language.decEqgraphRel✝

def FirstOrder.Language.graph : FirstOrder.Language

The language consisting of a single relation representing adjacency.

Equations

• FirstOrder.Language.graph = { Functions := fun (x : ℕ) => Empty, Relations := FirstOrder.Language.graphRel }

instance FirstOrder.Language.instgraphIsRelational : FirstOrder.Language.graph.IsRelational

Equations

• ⋯ = instIsEmptyEmpty

@[reducible, inline]

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

The symbol representing the adjacency relation.

Equations

• FirstOrder.Language.adj = FirstOrder.Language.graphRel.adj

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

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

Equations

• One or more equations did not get rendered due to their size.

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

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

The theory of simple graphs.

Equations

• FirstOrder.Language.Theory.simpleGraph = {FirstOrder.Language.adj.irreflexive, FirstOrder.Language.adj.symmetric}

@[simp]

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

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

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

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

Equations

• FirstOrder.Language.simpleGraphOfStructure V = { Adj := fun (x y : V) => FirstOrder.Language.Structure.RelMap FirstOrder.Language.adj ![x, y], symm := ⋯, loopless := ⋯ }

@[simp]

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

@[simp]

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

@[simp]

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

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