-
Notifications
You must be signed in to change notification settings - Fork 6
/
translate_functions.agda.html
227 lines (210 loc) · 48.9 KB
/
translate_functions.agda.html
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
219
220
221
222
223
224
225
226
227
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN">
<!-- Created by htmlize-1.55 in css mode. -->
<html>
<head>
<title>translate_functions.agda</title>
<style type="text/css">
<!--
body {
color: #655370;
background-color: #fbf8ef;
}
.agda2-highlight-bound-variable {
}
.agda2-highlight-datatype {
/* agda2-highlight-datatype-face */
color: #0000cd;
}
.agda2-highlight-field {
/* agda2-highlight-field-face */
color: #ee1289;
}
.agda2-highlight-function {
/* agda2-highlight-function-face */
color: #0000cd;
}
.agda2-highlight-inductive-constructor {
/* agda2-highlight-inductive-constructor-face */
color: #008b00;
}
.agda2-highlight-keyword {
/* agda2-highlight-keyword-face */
color: #cd6600;
}
.agda2-highlight-module {
/* agda2-highlight-module-face */
color: #a020f0;
}
.agda2-highlight-number {
/* agda2-highlight-number-face */
color: #a020f0;
}
.agda2-highlight-operator {
}
.agda2-highlight-postulate {
/* agda2-highlight-postulate-face */
color: #0000cd;
}
.agda2-highlight-primitive {
/* agda2-highlight-primitive-face */
color: #0000cd;
}
.agda2-highlight-primitive-type {
/* agda2-highlight-primitive-type-face */
color: #0000cd;
}
.agda2-highlight-record {
/* agda2-highlight-record-face */
color: #0000cd;
}
.agda2-highlight-symbol {
/* agda2-highlight-symbol-face */
color: #404040;
}
.comment {
/* font-lock-comment-face */
color: #2aa1ae;
background-color: #ecf3ec;
}
.hl-line {
/* hl-line */
background-color: #efeae9;
}
a {
color: inherit;
background-color: inherit;
font: inherit;
text-decoration: inherit;
}
a:hover {
text-decoration: underline;
}
-->
</style>
</head>
<body>
<pre>
<span class="comment"><span class="hl-line">-- [[file:~/thesis-proposal/thesis-proposal.org::*Missing%20Features][Missing Features:1]]</span></span><span class="hl-line">
</span><span class="agda2-highlight-keyword">open</span> <span class="agda2-highlight-keyword">import</span> <span class="agda2-highlight-module">Relation.Binary.PropositionalEquality</span>
<span class="agda2-highlight-keyword">open</span> <span class="agda2-highlight-module">≡-Reasoning</span>
<span class="comment">-- Z-notation for sums</span>
<span class="agda2-highlight-keyword">open</span> <span class="agda2-highlight-keyword">import</span> <span class="agda2-highlight-module">Level</span>
<span class="agda2-highlight-keyword">open</span> <span class="agda2-highlight-keyword">import</span> <span class="agda2-highlight-module">Data.Product</span> <span class="agda2-highlight-keyword">using</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-record">Σ</span> <span class="agda2-highlight-symbol">;</span> <span class="agda2-highlight-field">proj₁</span> <span class="agda2-highlight-symbol">;</span> <span class="agda2-highlight-field">proj₂</span> <span class="agda2-highlight-symbol">;</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">_×_</span></span> <span class="agda2-highlight-symbol">;</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-inductive-constructor">_,_</span></span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-function">Σ∶•</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-bound-variable">b</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-postulate">Level</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span> <span class="agda2-highlight-bound-variable">a</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">B</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-primitive-type">Set</span> <span class="agda2-highlight-bound-variable">b</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-primitive-type">Set</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-primitive">⊔</span></span> <span class="agda2-highlight-bound-variable">b</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-function">Σ∶•</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-record">Σ</span>
<span class="agda2-highlight-keyword">infix</span> <span class="agda2-highlight-number">-666</span> <span class="agda2-highlight-function">Σ∶•</span>
<span class="agda2-highlight-keyword">syntax</span> <span class="agda2-highlight-function">Σ∶•</span> <span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-symbol">(λ</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-function">Σ</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-function">∶</span> <span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-function">•</span> <span class="agda2-highlight-bound-variable">B</span>
<span class="comment">-- Missing Features:1 ends here</span>
<span class="comment">-- [[file:~/thesis-proposal/thesis-proposal.org::*Missing%20Features][Missing Features:2]]</span>
<span class="comment">-- One extreme: Completely bundled up</span>
<span class="agda2-highlight-keyword">record</span> <span class="agda2-highlight-record">Semigroup0</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set₁</span> <span class="agda2-highlight-keyword">where</span>
<span class="agda2-highlight-keyword">field</span>
<span class="agda2-highlight-field">Carrier</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-field">_⨾_</span></span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-field">Carrier</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-field">Carrier</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-field">Carrier</span>
<span class="agda2-highlight-field">assoc</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">y</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">z</span><span class="agda2-highlight-symbol">)</span>
<span class="comment">-- Missing Features:2 ends here</span>
<span class="comment">-- [[file:~/thesis-proposal/thesis-proposal.org::*Missing%20Features][Missing Features:3]]</span>
<span class="comment">-- ‘Typeclass’ on a given Carrier</span>
<span class="comment">-- Alternatively: Carrier is known as runtime.</span>
<span class="agda2-highlight-keyword">record</span> <span class="agda2-highlight-record">Semigroup1</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span><span class="agda2-highlight-symbol">):</span> <span class="agda2-highlight-primitive-type">Set₁</span> <span class="agda2-highlight-keyword">where</span>
<span class="agda2-highlight-keyword">field</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-field">_⨾_</span></span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">Carrier</span>
<span class="agda2-highlight-field">assoc</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">y</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">z</span><span class="agda2-highlight-symbol">)</span>
<span class="comment">-- Missing Features:3 ends here</span>
<span class="comment">-- [[file:~/thesis-proposal/thesis-proposal.org::*Missing%20Features][Missing Features:4]]</span>
<span class="comment">-- Two items known at run time --c.f., “IsSemi” above.</span>
<span class="agda2-highlight-keyword">record</span> <span class="agda2-highlight-record">Semigroup2</span>
<span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾_</span></span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">Carrier</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span> <span class="agda2-highlight-keyword">where</span>
<span class="agda2-highlight-keyword">field</span>
<span class="agda2-highlight-field">assoc</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾</span></span> <span class="agda2-highlight-bound-variable">y</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾</span></span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾</span></span> <span class="agda2-highlight-bound-variable">z</span><span class="agda2-highlight-symbol">)</span>
<span class="comment">-- Missing Features:4 ends here</span>
<span class="comment">-- [[file:~/thesis-proposal/thesis-proposal.org::*Missing%20Features][Missing Features:5]]</span>
<span class="comment">-- A value of “Semigroup3 C op pf” is trivially the empty record, if any,</span>
<span class="comment">-- provided ‘pf’ is a proof that ‘C’ forms a semigroup with ‘op’.</span>
<span class="comment">-- This type is usualy written “Σ C ∶ Set • Σ _⨾_ ∶ C → C → C • Σ assoc ∶ ⋯”.</span>
<span class="agda2-highlight-keyword">record</span> <span class="agda2-highlight-record">Semigroup3</span>
<span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾_</span></span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">Carrier</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">Carrier</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">assoc</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾</span></span> <span class="agda2-highlight-bound-variable">y</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾</span></span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾</span></span> <span class="agda2-highlight-bound-variable">z</span><span class="agda2-highlight-symbol">))</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span> <span class="agda2-highlight-keyword">where</span>
<span class="comment">-- no fields</span>
<span class="comment">-- Missing Features:5 ends here</span>
<span class="comment">-- [[file:~/thesis-proposal/thesis-proposal.org::*Missing%20Features][Missing Features:6]]</span>
<span class="agda2-highlight-function">Surjection</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀{</span><span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-bound-variable">B</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-primitive-type">Set</span>
<span class="agda2-highlight-function">Surjection</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">A</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-symbol">∀</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">b</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-function">Σ</span> <span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-function">∶</span> <span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-function">•</span> <span class="agda2-highlight-bound-variable">b</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a</span>
<span class="comment">-- (Σ a ∶ A • P a) ≈ { (a, proof) ❙ a ∈ A ∧ pf is a proof of P(a) }</span>
<span class="agda2-highlight-function">Injection</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀{</span><span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-bound-variable">B</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-primitive-type">Set</span>
<span class="agda2-highlight-function">Injection</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">A</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-symbol">∀</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-bound-variable">y</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">y</span>
<span class="comment">-- Missing Features:6 ends here</span>
<span class="comment">-- [[file:~/thesis-proposal/thesis-proposal.org::*Missing%20Features][Missing Features:7]]</span>
<span class="agda2-highlight-function">translate1</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀{</span><span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-function">Surjection</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-function">Injection</span> <span class="agda2-highlight-bound-variable">f</span>
<span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-record">Semigroup1</span> <span class="agda2-highlight-bound-variable">A</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-record">Semigroup1</span> <span class="agda2-highlight-bound-variable">B</span>
<span class="agda2-highlight-function">translate1</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-bound-variable">inj</span> <span class="agda2-highlight-bound-variable">AS</span> <span class="agda2-highlight-symbol">=</span>
<span class="agda2-highlight-keyword">let</span>
<span class="agda2-highlight-keyword">open</span> <span class="agda2-highlight-module">Semigroup1</span> <span class="agda2-highlight-bound-variable">AS</span>
<span class="comment">-- x ⨾′ y is obtained by applying f to the ⨾-composition of the pre-images of x and y.</span>
<span class="agda2-highlight-keyword">infix</span> <span class="agda2-highlight-number">5</span> _⨾′_
<span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾′_</span></span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-symbol">λ</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-keyword">let</span> <span class="agda2-highlight-bound-variable">a0</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-field">proj₁</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-bound-variable">x</span><span class="agda2-highlight-symbol">);</span> <span class="agda2-highlight-bound-variable">a1</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-field">proj₁</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-bound-variable">y</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-keyword">in</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a0</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a1</span><span class="agda2-highlight-symbol">)</span>
<span class="comment">-- f distributes over ⨾ turning it into ⨾′.</span>
<span class="agda2-highlight-bound-variable">factor</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a′</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-bound-variable">factor</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">=</span>
<span class="agda2-highlight-keyword">let</span> <span class="agda2-highlight-bound-variable">𝒶</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-inductive-constructor">,</span></span> <span class="agda2-highlight-bound-variable">m</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-bound-variable">𝒶′</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-inductive-constructor">,</span></span> <span class="agda2-highlight-bound-variable">w</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-keyword">in</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">begin</span></span>
<span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a′</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-inductive-constructor">refl</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">𝒶</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">𝒶′</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-function">cong</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-function">cong₂</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">_⨾_</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">inj</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-function">sym</span> <span class="agda2-highlight-bound-variable">m</span><span class="agda2-highlight-symbol">))</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">inj</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-function">sym</span> <span class="agda2-highlight-bound-variable">w</span><span class="agda2-highlight-symbol">)))</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">∎</span></span>
<span class="agda2-highlight-bound-variable">distribute</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-datatype">≡</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a′</span>
<span class="agda2-highlight-bound-variable">distribute</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-function">sym</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">factor</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">a′</span><span class="agda2-highlight-symbol">})</span>
<span class="agda2-highlight-keyword">in</span> <span class="comment">-- Bundle up ⨾′ along with a proof of associtivity </span>
<span class="agda2-highlight-keyword">record</span> <span class="agda2-highlight-symbol">{</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">_⨾_</span></span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾′_</span></span><span class="agda2-highlight-symbol">;</span> <span class="agda2-highlight-field">assoc</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-symbol">λ</span> <span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-bound-variable">z</span> <span class="agda2-highlight-symbol">→</span>
<span class="agda2-highlight-keyword">let</span>
<span class="comment">-- Obtain f-pre-images</span>
<span class="agda2-highlight-bound-variable">a₀</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-inductive-constructor">,</span></span> <span class="agda2-highlight-bound-variable">x≈fa₀</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-bound-variable">x</span>
<span class="agda2-highlight-bound-variable">a₁</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-inductive-constructor">,</span></span> <span class="agda2-highlight-bound-variable">y≈fa₁</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-bound-variable">y</span>
<span class="agda2-highlight-bound-variable">a₂</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-inductive-constructor">,</span></span> <span class="agda2-highlight-bound-variable">z≈fa₂</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-bound-variable">z</span>
<span class="agda2-highlight-keyword">in</span>
<span class="comment">{- Tersely, we rewrite along the pre-images,
factor f, perform the associativity of ⨾,
then distribute f and rewrite along the pre-images.
-}</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">begin</span></span>
<span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">y</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">z</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-function">cong₂</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾′_</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-function">cong₂</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾′_</span></span> <span class="agda2-highlight-bound-variable">x≈fa₀</span> <span class="agda2-highlight-bound-variable">y≈fa₁</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-bound-variable">z≈fa₂</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₀</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₁</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₂</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-function">cong</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₂</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-bound-variable">factor</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a₀</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a₁</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₂</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-bound-variable">factor</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">((</span><span class="agda2-highlight-bound-variable">a₀</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a₁</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a₂</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-function">cong</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-field">assoc</span> <span class="agda2-highlight-symbol">_</span> <span class="agda2-highlight-symbol">_</span> <span class="agda2-highlight-symbol">_)</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a₀</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a₁</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a₂</span><span class="agda2-highlight-symbol">))</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-bound-variable">distribute</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₀</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">a₁</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">⨾</span></span> <span class="agda2-highlight-bound-variable">a₂</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-function">cong</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₀</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′_</span></span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-bound-variable">distribute</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₀</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₁</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">a₂</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">≡⟨</span></span> <span class="agda2-highlight-function">sym</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-function">cong₂</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾′_</span></span> <span class="agda2-highlight-bound-variable">x≈fa₀</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-function">cong₂</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">_⨾′_</span></span> <span class="agda2-highlight-bound-variable">y≈fa₁</span> <span class="agda2-highlight-bound-variable">z≈fa₂</span><span class="agda2-highlight-symbol">))</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">⟩</span></span>
<span class="agda2-highlight-bound-variable">x</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">y</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-bound-variable">⨾′</span></span> <span class="agda2-highlight-bound-variable">z</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-operator"><span class="agda2-highlight-function">∎</span></span>
<span class="agda2-highlight-symbol">}</span>
<span class="comment">-- Missing Features:7 ends here</span>
<span class="comment">-- [[file:~/thesis-proposal/thesis-proposal.org::*Missing%20Features][Missing Features:8]]</span>
<span class="agda2-highlight-function">translate0</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-symbol">∀{</span><span class="agda2-highlight-bound-variable">B</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-primitive-type">Set</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">AS</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-record">Semigroup0</span><span class="agda2-highlight-symbol">)</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-field">Semigroup0.Carrier</span> <span class="agda2-highlight-bound-variable">AS</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-function">Surjection</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-function">Injection</span> <span class="agda2-highlight-bound-variable">f</span>
<span class="agda2-highlight-symbol">→</span> <span class="agda2-highlight-record">Semigroup0</span>
<span class="agda2-highlight-function">translate0</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-bound-variable">B</span><span class="agda2-highlight-symbol">}</span> <span class="agda2-highlight-bound-variable">AS</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-bound-variable">inj</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-keyword">record</span> <span class="agda2-highlight-symbol">{</span> <span class="agda2-highlight-field">Carrier</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-bound-variable">B</span> <span class="agda2-highlight-symbol">;</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-field">_⨾_</span></span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">_⨾_</span></span> <span class="agda2-highlight-symbol">;</span> <span class="agda2-highlight-field">assoc</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-function">assoc</span> <span class="agda2-highlight-symbol">}</span>
<span class="agda2-highlight-keyword">where</span>
<span class="comment">-- Repackage ‘AS’ from a ‘Semigroup0’ to a ‘Semigroup1’</span>
<span class="comment">-- only to immediatley unpack it, so that its contents</span>
<span class="comment">-- are available to be repacked above as a ‘Semigroup0’.</span>
<span class="agda2-highlight-function">pack</span> <span class="agda2-highlight-symbol">:</span> <span class="agda2-highlight-record">Semigroup1</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-field">Semigroup0.Carrier</span> <span class="agda2-highlight-bound-variable">AS</span><span class="agda2-highlight-symbol">)</span>
<span class="agda2-highlight-function">pack</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-keyword">let</span> <span class="agda2-highlight-keyword">open</span> <span class="agda2-highlight-module">Semigroup0</span> <span class="agda2-highlight-bound-variable">AS</span>
<span class="agda2-highlight-keyword">in</span> <span class="agda2-highlight-keyword">record</span> <span class="agda2-highlight-symbol">{</span><span class="agda2-highlight-operator"><span class="agda2-highlight-field">_⨾_</span></span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-operator"><span class="agda2-highlight-function">_⨾_</span></span><span class="agda2-highlight-symbol">;</span> <span class="agda2-highlight-field">assoc</span> <span class="agda2-highlight-symbol">=</span> <span class="agda2-highlight-function">assoc</span> <span class="agda2-highlight-symbol">}</span>
<span class="agda2-highlight-keyword">open</span> <span class="agda2-highlight-module">Semigroup1</span> <span class="agda2-highlight-symbol">(</span><span class="agda2-highlight-function">translate1</span> <span class="agda2-highlight-bound-variable">f</span> <span class="agda2-highlight-bound-variable">surj</span> <span class="agda2-highlight-bound-variable">inj</span> <span class="agda2-highlight-function">pack</span><span class="agda2-highlight-symbol">)</span>
<span class="comment">-- Missing Features:8 ends here</span>
</pre>
</body>
</html>