blob: 6904bfb80e443e229adf4a81310593ab75cc99d4 (
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
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
|
------------------------------------------------------------------------
-- Agda Helium
--
-- Definitions of ordered algebraic structures like monoids and rings
-- (packed in records together with sets, operations, etc.)
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Helium.Algebra.Ordered.StrictTotal.Bundles where
import Algebra.Bundles as Unordered
open import Algebra.Core
open import Data.Sum using (_⊎_)
open import Function using (flip)
open import Helium.Algebra.Core
open import Helium.Algebra.Bundles using
(RawAlmostGroup; AlmostGroup; AlmostAbelianGroup)
open import Helium.Algebra.Ordered.StrictTotal.Structures
open import Level using (suc; _⊔_)
open import Relation.Binary
open import Relation.Nullary as N
record RawMagma c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infixl 7 _∙_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
infix 4 _≉_ _≤_ _>_ _≥_
_≉_ : Rel Carrier _
x ≉ y = N.¬ x ≈ y
_≤_ : Rel Carrier _
x ≤ y = x < y ⊎ x ≈ y
_>_ : Rel Carrier _
_>_ = flip _<_
_≥_ : Rel Carrier _
_≥_ = flip _≤_
record Magma c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infixl 7 _∙_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
isMagma : IsMagma _≈_ _<_ _∙_
open IsMagma isMagma public
rawMagma : RawMagma _ _ _
rawMagma = record { _≈_ = _≈_; _<_ = _<_; _∙_ = _∙_ }
open RawMagma rawMagma public
using (_≉_; _≤_; _>_; _≥_)
record Semigroup c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infixl 7 _∙_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
isSemigroup : IsSemigroup _≈_ _<_ _∙_
open IsSemigroup isSemigroup public
magma : Magma c ℓ₁ ℓ₂
magma = record { isMagma = isMagma }
open Magma magma public
using (_≉_; _≤_; _>_; _≥_; rawMagma)
record RawMonoid c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infixl 7 _∙_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
ε : Carrier
rawMagma : RawMagma c ℓ₁ ℓ₂
rawMagma = record { _≈_ = _≈_; _<_ = _<_; _∙_ = _∙_ }
open RawMagma rawMagma public
using (_≉_; _≤_; _>_; _≥_)
record Monoid c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infixl 7 _∙_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
ε : Carrier
isMonoid : IsMonoid _≈_ _<_ _∙_ ε
open IsMonoid isMonoid public
semigroup : Semigroup _ _ _
semigroup = record { isSemigroup = isSemigroup }
open Semigroup semigroup public
using (_≉_; _≤_; _>_; _≥_; rawMagma; magma)
rawMonoid : RawMonoid _ _ _
rawMonoid = record { _≈_ = _≈_; _<_ = _<_; _∙_ = _∙_; ε = ε}
record RawGroup c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 8 _⁻¹
infixl 7 _∙_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
ε : Carrier
_⁻¹ : Op₁ Carrier
rawMonoid : RawMonoid c ℓ₁ ℓ₂
rawMonoid = record { _≈_ = _≈_; _<_ = _<_; _∙_ = _∙_; ε = ε }
open RawMonoid rawMonoid public
using (_≉_; _≤_; _>_; _≥_; rawMagma)
record Group c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 8 _⁻¹
infixl 7 _∙_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
ε : Carrier
_⁻¹ : Op₁ Carrier
isGroup : IsGroup _≈_ _<_ _∙_ ε _⁻¹
open IsGroup isGroup public
rawGroup : RawGroup _ _ _
rawGroup = record { _≈_ = _≈_; _<_ = _<_; _∙_ = _∙_; ε = ε; _⁻¹ = _⁻¹}
monoid : Monoid _ _ _
monoid = record { isMonoid = isMonoid }
open Monoid monoid public
using (_≉_; _≤_; _>_; _≥_; rawMagma; magma; semigroup; rawMonoid)
record AbelianGroup c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 8 _⁻¹
infixl 7 _∙_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
ε : Carrier
_⁻¹ : Op₁ Carrier
isAbelianGroup : IsAbelianGroup _≈_ _<_ _∙_ ε _⁻¹
open IsAbelianGroup isAbelianGroup public
group : Group _ _ _
group = record { isGroup = isGroup }
open Group group public
using
( _≉_; _≤_; _>_; _≥_
; magma; semigroup; monoid
; rawMagma; rawMonoid; rawGroup
)
record RawRing c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier
0# : Carrier
1# : Carrier
+-rawGroup : RawGroup c ℓ₁ ℓ₂
+-rawGroup = record { _≈_ = _≈_; _<_ = _<_; _∙_ = _+_; ε = 0#; _⁻¹ = -_ }
open RawGroup +-rawGroup public
using ( _≉_; _≤_; _>_; _≥_ )
renaming
( rawMagma to +-rawMagma
; rawMonoid to +-rawMonoid
)
*-rawMonoid : Unordered.RawMonoid c ℓ₁
*-rawMonoid = record { _≈_ = _≈_; _∙_ = _*_; ε = 1# }
open Unordered.RawMonoid *-rawMonoid public
using ()
renaming ( rawMagma to *-rawMagma )
record Ring c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier
0# : Carrier
1# : Carrier
isRing : IsRing _≈_ _<_ _+_ _*_ -_ 0# 1#
open IsRing isRing public
rawRing : RawRing c ℓ₁ ℓ₂
rawRing = record
{ _≈_ = _≈_
; _<_ = _<_
; _+_ = _+_
; _*_ = _*_
; -_ = -_
; 0# = 0#
; 1# = 1#
}
+-abelianGroup : AbelianGroup _ _ _
+-abelianGroup = record { isAbelianGroup = +-isAbelianGroup }
open AbelianGroup +-abelianGroup public
using ( _≉_; _≤_; _>_; _≥_ )
renaming
( rawMagma to +-rawMagma
; rawMonoid to +-rawMonoid
; rawGroup to +-rawGroup
; magma to +-magma
; semigroup to +-semigroup
; monoid to +-monoid
; group to +-group
)
*-monoid : Unordered.Monoid _ _
*-monoid = record { isMonoid = *-isMonoid }
open Unordered.Monoid *-monoid public
using ()
renaming
( rawMagma to *-rawMagma
; rawMonoid to *-rawMonoid
; magma to *-magma
; semigroup to *-semigroup
)
record CommutativeRing c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier
0# : Carrier
1# : Carrier
isCommutativeRing : IsCommutativeRing _≈_ _<_ _+_ _*_ -_ 0# 1#
open IsCommutativeRing isCommutativeRing public
ring : Ring _ _ _
ring = record { isRing = isRing }
open Ring ring public
using
( _≉_; _≤_; _>_; _≥_
; +-rawMagma; +-rawMonoid; +-rawGroup
; +-magma; +-semigroup; +-monoid; +-group; +-abelianGroup
; *-rawMagma; *-rawMonoid
; *-magma; *-semigroup; *-monoid
)
record RawField c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 9 _⁻¹
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier
0# : Carrier
1# : Carrier
_⁻¹ : AlmostOp₁ _≈_ 0#
infixl 7 _/_
_/_ : AlmostOp₂ _≈_ 0#
x / y≉0 = x * y≉0 ⁻¹
rawRing : RawRing c ℓ₁ ℓ₂
rawRing = record
{ _≈_ = _≈_
; _<_ = _<_
; _+_ = _+_
; _*_ = _*_
; -_ = -_
; 0# = 0#
; 1# = 1#
}
open RawRing rawRing public
using
( _≉_; _≤_; _>_; _≥_
; +-rawMagma; +-rawMonoid; +-rawGroup
; *-rawMagma; *-rawMonoid
)
*-rawAlmostGroup : RawAlmostGroup c ℓ₁
*-rawAlmostGroup = record
{ _≈_ = _≈_
; _∙_ = _*_
; 0# = 0#
; 1# = 1#
; _⁻¹ = _⁻¹
}
record DivisionRing c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 9 _⁻¹
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier
0# : Carrier
1# : Carrier
_⁻¹ : AlmostOp₁ _≈_ 0#
isDivisionRing : IsDivisionRing _≈_ _<_ _+_ _*_ -_ 0# 1# _⁻¹
open IsDivisionRing isDivisionRing public
rawField : RawField c ℓ₁ ℓ₂
rawField = record
{ _≈_ = _≈_
; _<_ = _<_
; _+_ = _+_
; _*_ = _*_
; -_ = -_
; 0# = 0#
; 1# = 1#
; _⁻¹ = _⁻¹
}
ring : Ring c ℓ₁ ℓ₂
ring = record { isRing = isRing }
open Ring ring public
using
( _≉_; _≤_; _>_; _≥_
; rawRing
; +-rawMagma; +-rawMonoid; +-rawGroup
; +-magma; +-semigroup; +-monoid; +-group; +-abelianGroup
; *-rawMagma; *-rawMonoid
; *-magma; *-semigroup; *-monoid
)
*-almostGroup : AlmostGroup _ _
*-almostGroup = record { isAlmostGroup = *-isAlmostGroup }
open AlmostGroup *-almostGroup public
using ()
renaming (rawAlmostGroup to *-rawAlmostGroup)
record Field c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infix 9 _⁻¹
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_ _<_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_<_ : Rel Carrier ℓ₂
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier
0# : Carrier
1# : Carrier
_⁻¹ : AlmostOp₁ _≈_ 0#
isField : IsField _≈_ _<_ _+_ _*_ -_ 0# 1# _⁻¹
open IsField isField public
divisionRing : DivisionRing c ℓ₁ ℓ₂
divisionRing = record { isDivisionRing = isDivisionRing }
open DivisionRing divisionRing
using
( _≉_; _≤_; _>_; _≥_
; +-rawMagma; +-rawMonoid; +-rawGroup
; +-magma; +-semigroup; +-monoid; +-group; +-abelianGroup
; *-rawMagma; *-rawMonoid; *-rawAlmostGroup
; *-magma; *-semigroup; *-monoid; *-almostGroup
)
|
