File: Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean

full_name: SimpleGraph.Finsubgraph
code: /-- The subtype of `G.subgraph` comprising those subgraphs with finite vertex sets. -/
abbrev Finsubgraph (G : SimpleGraph V) 
goal: Type u

full_name: SimpleGraph.FinsubgraphHom
code: /-- A graph homomorphism from a finite subgraph of G to F. -/
abbrev FinsubgraphHom (G' : G.Finsubgraph) (F : SimpleGraph W) 
goal: Type (max u v)

full_name: SimpleGraph.«_aux_Mathlib_Combinatorics_SimpleGraph_Finsubgraph___macroRules_SimpleGraph_term_→fg__1»
code: local infixl:50 " →fg " => FinsubgraphHom
goal: Lean.Macro

full_name: SimpleGraph._aux_Mathlib_Combinatorics_SimpleGraph_Finsubgraph___unexpand_SimpleGraph_FinsubgraphHom_1
code: local infixl:50 " →fg " => FinsubgraphHom
goal: Lean.PrettyPrinter.Unexpander

full_name: SimpleGraph.Finsubgraph.instOrderBot
code: instance : OrderBot G.Finsubgraph 
goal: OrderBot G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instSup
code: instance : Sup G.Finsubgraph 
goal: Sup G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instInf
code: instance : Inf G.Finsubgraph 
goal: Inf G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instSDiff
code: instance instSDiff : SDiff G.Finsubgraph 
goal: SDiff G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.coe_bot
code: @[simp, norm_cast] lemma coe_bot : (⊥ : G.Finsubgraph) = (⊥ : G.Subgraph) 
goal: ↑⊥ = ⊥

full_name: SimpleGraph.Finsubgraph.coe_sup
code: @[simp, norm_cast]
lemma coe_sup (G₁ G₂ : G.Finsubgraph) : ↑(G₁ ⊔ G₂) = (G₁ ⊔ G₂ : G.Subgraph) 
goal: ↑(G₁ ⊔ G₂) = ↑G₁ ⊔ ↑G₂

full_name: SimpleGraph.Finsubgraph.coe_inf
code: @[simp, norm_cast]
lemma coe_inf (G₁ G₂ : G.Finsubgraph) : ↑(G₁ ⊓ G₂) = (G₁ ⊓ G₂ : G.Subgraph) 
goal: ↑(G₁ ⊓ G₂) = ↑G₁ ⊓ ↑G₂

full_name: SimpleGraph.Finsubgraph.coe_sdiff
code: @[simp, norm_cast]
lemma coe_sdiff (G₁ G₂ : G.Finsubgraph) : ↑(G₁ \ G₂) = (G₁ \ G₂ : G.Subgraph) 
goal: ↑(G₁ \ G₂) = ↑G₁ \ ↑G₂

full_name: SimpleGraph.Finsubgraph.instGeneralizedCoheytingAlgebra
code: instance instGeneralizedCoheytingAlgebra : GeneralizedCoheytingAlgebra G.Finsubgraph 
goal: GeneralizedCoheytingAlgebra G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instTop
code: instance instTop : Top G.Finsubgraph 
goal: Top G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instHasCompl
code: instance instHasCompl : HasCompl G.Finsubgraph 
goal: HasCompl G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instHNot
code: instance instHNot : HNot G.Finsubgraph 
goal: HNot G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instHImp
code: instance instHImp : HImp G.Finsubgraph 
goal: HImp G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instSupSet
code: instance instSupSet : SupSet G.Finsubgraph 
goal: SupSet G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.instInfSet
code: instance instInfSet : InfSet G.Finsubgraph 
goal: InfSet G.Finsubgraph

full_name: SimpleGraph.Finsubgraph.coe_top
code: @[simp, norm_cast] lemma coe_top : (⊤ : G.Finsubgraph) = (⊤ : G.Subgraph) 
goal: ↑⊤ = ⊤

full_name: SimpleGraph.Finsubgraph.coe_compl
code: @[simp, norm_cast] lemma coe_compl (G' : G.Finsubgraph) : ↑(G'ᶜ) = (G'ᶜ : G.Subgraph) 
goal: ↑G'ᶜ = (↑G')ᶜ

full_name: SimpleGraph.Finsubgraph.coe_hnot
code: @[simp, norm_cast] lemma coe_hnot (G' : G.Finsubgraph) : ↑(￢G') = (￢G' : G.Subgraph) 
goal: ↑(￢G') = ￢↑G'

full_name: SimpleGraph.Finsubgraph.coe_himp
code: @[simp, norm_cast]
lemma coe_himp (G₁ G₂ : G.Finsubgraph) : ↑(G₁ ⇨ G₂) = (G₁ ⇨ G₂ : G.Subgraph) 
goal: ↑(G₁ ⇨ G₂) = ↑G₁ ⇨ ↑G₂

full_name: SimpleGraph.Finsubgraph.coe_sSup
code: @[simp, norm_cast]
lemma coe_sSup (s : Set G.Finsubgraph) : sSup s = (⨆ G ∈ s, G : G.Subgraph) 
goal: ↑(sSup s) = ⨆ G_1 ∈ s, ↑G_1

full_name: SimpleGraph.Finsubgraph.coe_sInf
code: @[simp, norm_cast]
lemma coe_sInf (s : Set G.Finsubgraph) : sInf s = (⨅ G ∈ s, G : G.Subgraph) 
goal: ↑(sInf s) = ⨅ G_1 ∈ s, ↑G_1

full_name: SimpleGraph.Finsubgraph.coe_iSup
code: @[simp, norm_cast]
lemma coe_iSup {ι : Sort*} (f : ι → G.Finsubgraph) : ⨆ i, f i = (⨆ i, f i : G.Subgraph) 
goal: ↑(⨆ i, f i) = ⨆ i, ↑(f i)

full_name: SimpleGraph.Finsubgraph.coe_iInf
code: @[simp, norm_cast]
lemma coe_iInf {ι : Sort*} (f : ι → G.Finsubgraph) : ⨅ i, f i = (⨅ i, f i : G.Subgraph) 
goal: ↑(⨅ i, f i) = ⨅ i, ↑(f i)

full_name: SimpleGraph.Finsubgraph.instCompletelyDistribLattice
code: instance instCompletelyDistribLattice : CompletelyDistribLattice G.Finsubgraph 
goal: CompletelyDistribLattice G.Finsubgraph

full_name: SimpleGraph.singletonFinsubgraph
code: /-- The finite subgraph of G generated by a single vertex. -/
def singletonFinsubgraph (v : V) : G.Finsubgraph 
goal: G.Finsubgraph

full_name: SimpleGraph.finsubgraphOfAdj
code: /-- The finite subgraph of G generated by a single edge. -/
def finsubgraphOfAdj {u v : V} (e : G.Adj u v) : G.Finsubgraph 
goal: G.Finsubgraph

full_name: SimpleGraph.singletonFinsubgraph_le_adj_left
code: theorem singletonFinsubgraph_le_adj_left {u v : V} {e : G.Adj u v} :
    singletonFinsubgraph u ≤ finsubgraphOfAdj e 
goal: SimpleGraph.singletonFinsubgraph u ≤ SimpleGraph.finsubgraphOfAdj e

full_name: SimpleGraph.singletonFinsubgraph_le_adj_right
code: theorem singletonFinsubgraph_le_adj_right {u v : V} {e : G.Adj u v} :
    singletonFinsubgraph v ≤ finsubgraphOfAdj e 
goal: SimpleGraph.singletonFinsubgraph v ≤ SimpleGraph.finsubgraphOfAdj e

full_name: SimpleGraph.FinsubgraphHom.restrict
code: /-- Given a homomorphism from a subgraph to `F`, construct its restriction to a sub-subgraph. -/
def FinsubgraphHom.restrict {G' G'' : G.Finsubgraph} (h : G'' ≤ G') (f : G' →fg F) : G'' →fg F 
goal: SimpleGraph.FinsubgraphHom G'' F

full_name: SimpleGraph.finsubgraphHomFunctor
code: /-- The inverse system of finite homomorphisms. -/
def finsubgraphHomFunctor (G : SimpleGraph V) (F : SimpleGraph W) :
    G.Finsubgraphᵒᵖ ⥤ Type max u v 
goal: G.Finsubgraphᵒᵖ ⥤ Type (max u v)

full_name: SimpleGraph.nonempty_hom_of_forall_finite_subgraph_hom
code: /-- If every finite subgraph of a graph `G` has a homomorphism to a finite graph `F`, then there is
a homomorphism from the whole of `G` to `F`. -/
theorem nonempty_hom_of_forall_finite_subgraph_hom [Finite W]
    (h : ∀ G' : G.Subgraph, G'.verts.Finite → G'.coe →g F) : Nonempty (G →g F) 
goal: Nonempty (G →g F)