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Some graphs stuff (#98)

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Patrick Stevens
2020-02-15 13:41:51 +00:00
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parent 9de323c5e8
commit f41f5226b9
18 changed files with 275 additions and 39 deletions
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{-# OPTIONS --warning=error --safe --without-K #-}
open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Functions
open import Setoids.Setoids
open import Setoids.Subset
module Graphs.Definition where
record Graph {a b : _} (c : _) {V' : Set a} (V : Setoid {a} {b} V') : Set (a ⊔ b ⊔ lsuc c) where
field
_<->_ : Rel {a} {c} V'
noSelfRelation : (x : V') → x <-> x → False
symmetric : {x y : V'} → x <-> y → y <-> x
wellDefined : {x y r s : V'} → Setoid._∼_ V x y → Setoid._∼_ V r s → x <-> r → y <-> s
record GraphIso {a b c d e m : _} {V1' : Set a} {V2' : Set b} {V1 : Setoid {a} {c} V1'} {V2 : Setoid {b} {d} V2'} (G : Graph e V1) (H : Graph m V2) (f : V1' → V2') : Set (a ⊔ b ⊔ c ⊔ d ⊔ e ⊔ m) where
field
bij : SetoidBijection V1 V2 f
respects : (x y : V1') → Graph._<->_ G x y → Graph._<->_ H (f x) (f y)
respects' : (x y : V1') → Graph._<->_ H (f x) (f y) → Graph._<->_ G x y
record Subgraph {a b c d e : _} {V' : Set a} {V : Setoid {a} {b} V'} (G : Graph c V) {pred : V' → Set d} (sub : subset V pred) (_<->'_ : Rel {_} {e} (Sg V' pred)) : Set (a ⊔ b ⊔ c ⊔ d ⊔ e) where
field
inherits : {x y : Sg V' pred} → (x <->' y) → (Graph._<->_ G (underlying x) (underlying y))
symmetric : {x y : Sg V' pred} → (x <->' y) → (y <->' x)
wellDefined : {x y r s : Sg V' pred} → Setoid._∼_ (subsetSetoid V sub) x y → Setoid._∼_ (subsetSetoid V sub) r s → x <->' r → y <->' s
subgraphIsGraph : {a b c d e : _} {V' : Set a} {V : Setoid {a} {b} V'} {G : Graph c V} {pred : V' → Set d} {sub : subset V pred} {rel : Rel {_} {e} (Sg V' pred)} (H : Subgraph G sub rel) → Graph e (subsetSetoid V sub)
Graph._<->_ (subgraphIsGraph {rel = rel} H) = rel
Graph.noSelfRelation (subgraphIsGraph {G = G} H) (v , isSub) v=v = Graph.noSelfRelation G v (Subgraph.inherits H v=v)
Graph.symmetric (subgraphIsGraph H) v1=v2 = Subgraph.symmetric H v1=v2
Graph.wellDefined (subgraphIsGraph H) = Subgraph.wellDefined H