blob: f54468db605ba7b85ad6bde6b19959eced826587 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
|
{-# OPTIONS --without-K --safe #-}
module Data.Hypergraph.Base where
open import Data.Castable using (IsCastable)
open import Data.Fin using (Fin)
open import Relation.Binary
using
( Rel
; IsDecTotalOrder
; IsStrictTotalOrder
; Tri
; Trichotomous
; _Respectsˡ_
; _Respectsʳ_
; _Respects_
)
open import Relation.Binary.Bundles using (DecTotalOrder; StrictTotalOrder)
open import Relation.Nullary using (¬_)
open import Data.Nat.Base using (ℕ; _<_)
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
open import Data.Nat.Properties using (<-irrefl; <-trans; <-resp₂-≡; <-cmp)
open import Level using (Level; suc; _⊔_)
import Data.Vec.Base as VecBase
import Data.Vec.Relation.Binary.Equality.Cast as VecCast
import Relation.Binary.PropositionalEquality as ≡
import Relation.Binary.Properties.DecTotalOrder as DTOP
record HypergraphLabel {ℓ : Level} : Set (suc ℓ) where
field
Label : ℕ → Set ℓ
isCastable : IsCastable Label
-- _[_≈_] : (n : ℕ) → Rel (Label n) ℓ
_[_≤_] : (n : ℕ) → Rel (Label n) ℓ
isDecTotalOrder : (n : ℕ) → IsDecTotalOrder ≡._≡_ (n [_≤_])
decTotalOrder : (n : ℕ) → DecTotalOrder ℓ ℓ ℓ
decTotalOrder n = record
{ Carrier = Label n
; _≈_ = ≡._≡_
; _≤_ = n [_≤_]
; isDecTotalOrder = isDecTotalOrder n
}
open DTOP using (<-strictTotalOrder) renaming (_<_ to <)
_[_<_] : (n : ℕ) → Rel (Label n) ℓ
_[_<_] n = < (decTotalOrder n)
strictTotalOrder : (n : ℕ) → StrictTotalOrder ℓ ℓ ℓ
strictTotalOrder n = <-strictTotalOrder (decTotalOrder n)
open IsCastable isCastable public
module Edge (HL : HypergraphLabel) where
module HL = HypergraphLabel HL
open HL using (Label; cast; cast-is-id; cast-trans)
open VecBase using (Vec)
record Edge (v : ℕ) : Set where
field
{arity} : ℕ
label : Label arity
ports : Vec (Fin v) arity
open ≡ using (_≡_)
open VecCast using (_≈[_]_)
record ≈-Edge {n : ℕ} (E E′ : Edge n) : Set where
eta-equality
module E = Edge E
module E′ = Edge E′
field
≡arity : E.arity ≡ E′.arity
≡label : cast ≡arity E.label ≡ E′.label
≡ports : E.ports ≈[ ≡arity ] E′.ports
≈-Edge-refl : {v : ℕ} {x : Edge v} → ≈-Edge x x
≈-Edge-refl {_} {x} = record
{ ≡arity = ≡.refl
; ≡label = HL.≈-reflexive ≡.refl
; ≡ports = VecCast.≈-reflexive ≡.refl
}
where
open Edge x using (arity; label)
open DecTotalOrder (HL.decTotalOrder arity) using (module Eq)
≈-Edge-sym : {v : ℕ} {x y : Edge v} → ≈-Edge x y → ≈-Edge y x
≈-Edge-sym {_} {x} {y} x≈y = record
{ ≡arity = ≡.sym ≡arity
; ≡label = HL.≈-sym ≡label
; ≡ports = VecCast.≈-sym ≡ports
}
where
open ≈-Edge x≈y
open DecTotalOrder (HL.decTotalOrder E.arity) using (module Eq)
≈-Edge-trans : {v : ℕ} {i j k : Edge v} → ≈-Edge i j → ≈-Edge j k → ≈-Edge i k
≈-Edge-trans {_} {i} {j} {k} i≈j j≈k = record
{ ≡arity = ≡.trans i≈j.≡arity j≈k.≡arity
; ≡label = HL.≈-trans i≈j.≡label j≈k.≡label
; ≡ports = VecCast.≈-trans i≈j.≡ports j≈k.≡ports
}
where
module i≈j = ≈-Edge i≈j
module j≈k = ≈-Edge j≈k
open HL using (_[_<_])
data <-Edge {v : ℕ} : Edge v → Edge v → Set where
<-arity
: {x y : Edge v}
→ Edge.arity x < Edge.arity y
→ <-Edge x y
<-label
: {x y : Edge v}
(≡a : Edge.arity x ≡ Edge.arity y)
→ Edge.arity y [ cast ≡a (Edge.label x) < Edge.label y ]
→ <-Edge x y
<-Edge-irrefl : {v : ℕ} {x y : Edge v} → ≈-Edge x y → ¬ <-Edge x y
<-Edge-irrefl record { ≡arity = ≡a } (<-arity n<m) = <-irrefl ≡a n<m
<-Edge-irrefl record { ≡label = ≡l } (<-label _ (_ , x≉y)) = x≉y ≡l
<-Edge-trans : {v : ℕ} {i j k : Edge v} → <-Edge i j → <-Edge j k → <-Edge i k
<-Edge-trans (<-arity i<j) (<-arity j<k) = <-arity (<-trans i<j j<k)
<-Edge-trans (<-arity i<j) (<-label ≡.refl j<k) = <-arity i<j
<-Edge-trans (<-label ≡.refl i<j) (<-arity j<k) = <-arity j<k
<-Edge-trans {_} {i} (<-label ≡.refl i<j) (<-label ≡.refl j<k)
= <-label ≡.refl (<-label-trans i<j (<-respˡ-≈ (HL.≈-reflexive ≡.refl) j<k))
where
open DTOP (HL.decTotalOrder (Edge.arity i)) using (<-respˡ-≈) renaming (<-trans to <-label-trans)
<-Edge-respˡ-≈ : {v : ℕ} {y : Edge v} → (λ x → <-Edge x y) Respects ≈-Edge
<-Edge-respˡ-≈ ≈x (<-arity x₁<y) = <-arity (proj₂ <-resp₂-≡ ≡arity x₁<y)
where
open ≈-Edge ≈x using (≡arity)
<-Edge-respˡ-≈ {_} {y} record { ≡arity = ≡.refl ; ≡label = ≡.refl } (<-label ≡.refl x₁<y)
= <-label ≡.refl (<-respˡ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) x₁<y)
where
module y = Edge y
open DTOP (HL.decTotalOrder y.arity) using (<-respˡ-≈)
<-Edge-respʳ-≈ : {v : ℕ} {x : Edge v} → <-Edge x Respects ≈-Edge
<-Edge-respʳ-≈ record { ≡arity = ≡a } (<-arity x<y₁) = <-arity (proj₁ <-resp₂-≡ ≡a x<y₁)
<-Edge-respʳ-≈ {_} {x} record { ≡arity = ≡.refl ; ≡label = ≡.refl } (<-label ≡.refl x<y₁)
= <-label ≡.refl (<-respʳ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) x<y₁)
where
module x = Edge x
open DTOP (HL.decTotalOrder x.arity) using (<-respʳ-≈)
open Tri
open ≈-Edge
tri : {v : ℕ} → Trichotomous (≈-Edge {v}) (<-Edge {v})
tri x y with <-cmp x.arity y.arity
where
module x = Edge x
module y = Edge y
tri x y | tri< x<y x≢y y≮x = tri< (<-arity x<y) (λ x≡y → x≢y (≡arity x≡y)) ¬y<x
where
¬y<x : ¬ <-Edge y x
¬y<x (<-arity y<x) = y≮x y<x
¬y<x (<-label ≡a y<x) = x≢y (≡.sym ≡a)
tri x y | tri≈ x≮y ≡.refl y≮x = compare-label
where
module x = Edge x
module y = Edge y
open StrictTotalOrder (HL.strictTotalOrder x.arity) using (compare)
import Relation.Binary.Properties.DecTotalOrder
open DTOP (HL.decTotalOrder x.arity) using (<-respˡ-≈)
compare-label : Tri (<-Edge x y) (≈-Edge x y) (<-Edge y x)
compare-label with compare x.label y.label
... | tri< x<y x≢y y≮x′ = tri<
(<-label ≡.refl (<-respˡ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) x<y))
(λ x≡y → x≢y (≡.trans (≡.sym (HL.≈-reflexive ≡.refl)) (≡label x≡y)))
¬y<x
where
¬y<x : ¬ <-Edge y x
¬y<x (<-arity y<x) = y≮x y<x
¬y<x (<-label ≡a y<x) = y≮x′ (<-respˡ-≈ (HL.≈-reflexive ≡.refl) y<x)
... | tri≈ x≮y x≡y y≮x = compare-ports
where
compare-ports : Tri (<-Edge x y) (≈-Edge x y) (<-Edge y x)
compare-ports = ?
... | tri> x≮y′ x≢y y<x = tri>
¬x<y
(λ x≡y → x≢y (≡.trans (≡.sym (HL.≈-reflexive ≡.refl)) (≡label x≡y)))
(<-label ≡.refl (<-respˡ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) y<x))
where
¬x<y : ¬ <-Edge x y
¬x<y (<-arity x<y) = x≮y x<y
¬x<y (<-label ≡a x<y) = x≮y′ (<-respˡ-≈ (HL.≈-reflexive ≡.refl) x<y)
tri x y | tri> x≮y x≢y y<x = tri> ¬x<y (λ x≡y → x≢y (≡arity x≡y)) (<-arity y<x)
where
¬x<y : ¬ <-Edge x y
¬x<y (<-arity x<y) = x≮y x<y
¬x<y (<-label ≡a x<y) = x≢y ≡a
isStrictTotalOrder : {v : ℕ} → IsStrictTotalOrder (≈-Edge {v}) (<-Edge {v})
isStrictTotalOrder = record
{ isStrictPartialOrder = record
{ isEquivalence = record
{ refl = ≈-Edge-refl
; sym = ≈-Edge-sym
; trans = ≈-Edge-trans
}
; irrefl = <-Edge-irrefl
; trans = <-Edge-trans
; <-resp-≈ = <-Edge-respʳ-≈ , <-Edge-respˡ-≈
}
; compare = tri
}
module HypergraphList (HL : HypergraphLabel) where
open import Data.List.Base using (List)
open Edge HL using (Edge)
record Hypergraph (v : ℕ) : Set where
field
edges : List (Edge v)
|
