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
|
mod! BARE-NAT {
[ NzNat Zero < Nat ]
op 0 : -> Zero
op s : Nat -> NzNat
}
mod! SIMPLE-NAT {
protecting (BARE-NAT)
op _+_ : Nat Nat -> Nat
eq N:Nat + s(M:Nat) = s(N + M) .
eq N:Nat + 0 = N . eq 0 + N:Nat = N .
}
mod! TIMES-NAT {
protecting (SIMPLE-NAT)
op _*_ : Nat Nat -> Nat
vars M N : Nat
eq 0 * N = 0 .
eq N * 0 = 0 .
eq N * s(M) = (N * M) + N .
}
-- --------------------------------------------
-- MONOIDS
-- --------------------------------------------
mod* MON {
[ Elt ]
op null : -> Elt
op _;_ : Elt Elt -> Elt {assoc idr: null}
}
mod* CMON {
protecting(MON)
op _;_ : Elt Elt -> Elt {comm}
}
view bare-nat from TRIV to BARE-NAT { sort Elt -> Nat }
view plus from MON to SIMPLE-NAT {
sort Elt -> Nat,
op _;_ -> _+_,
op null -> 0
}
view times from MON to TIMES-NAT {
sort Elt -> Nat,
op _;_ -> _*_,
op null -> s(0)
}
view dual from MON to MON {
op X:Elt ; Y:Elt -> Y:Elt ; X:Elt
}
mod* MON* (Y :: TRIV) {
op _;_ : Elt Elt -> Elt {assoc}
op null : -> Elt
eq m:Elt ; null = m . eq null ; m:Elt = m .
}
view dual* from MON* to MON* {
op X:Elt ; Y:Elt -> Y:Elt ; X:Elt
}
view plus* from MON* to SIMPLE-NAT {
sort Elt -> Nat,
op _;_ -> _+_,
op null -> 0
}
view times* from MON* to TIMES-NAT {
sort Elt -> Nat,
op _;_ -> _*_,
op null -> s(0)
}
view bnat+ from MON*(bare-nat) to SIMPLE-NAT {
op _;_ -> _+_,
op null -> 0
}
-- -----------------------------------------------------------------
-- MONOIDS with POWERS
-- -----------------------------------------------------------------
mod* MON-POW (POWER :: MON, M :: MON)
{
op _^_ : Elt.M Elt.POWER -> Elt.M
vars m m' : Elt.M
vars p p' : Elt.POWER
eq (m ; m')^ p = (m ^ p) ; (m' ^ p) .
eq m ^ (p ; p') = (m ^ p) ; (m ^ p') .
eq m ^ null = null .
}
open MON-POW(plus,plus) * { op _^_ -> _*_ } .
ops m m' n n' : -> Nat .
-- LEMMA:
op _+_ : Nat Nat -> Nat { assoc }
-- the following should be true
red (m + m') * (n + n') == (m * n) + (m * n') + (m' * n) + (m' * n') .
close
view ^as* from MON-POW(plus,plus) to TIMES-NAT {
op _^_ -> _*_
}
mod* NAT^ (M :: MON-POW(plus,plus)) { }
open NAT^(^as*) .
red s(s(s(0))) * s(s(s(s(0)))) . -- it should be 12
close
mod* MON-POW-NAT {
protecting(MON-POW(POWER <= plus))
eq m:Elt.M ^ s(0) = m .
}
open MON-POW-NAT .
ops m m' : -> Elt.M .
ops n n' : -> Nat .
-- this should be true
red (m ; m') ^ (n + n') == (m ^ n) ; (m ^ n') ; (m' ^ n) ; (m' ^ n') .
close
-- this should be another version of TIMES-NAT
-- by getting the multiplication as power of the sum!
mod* NAT-TIMES {
protecting(MON-POW-NAT(M <= plus) * { op _^_ -> _*_ })
}
open NAT-TIMES .
-- LEMMA:
eq M:Nat * s(N:Nat) = (M * N) + M .
red s(s(s(0))) * s(s(s(s(0)))) . -- it should be 12
close
view nat-times from MON to NAT-TIMES {
sort Elt -> Nat,
op null -> s(0),
op _;_ -> _*_
}
mod* NAT-POW {
protecting(MON-POW-NAT(M <= nat-times))
}
open NAT-POW .
-- LEMMA:
eq M:Nat * s(N:Nat) = (M * N) + M . -- (1)
-- LEMMA:
eq M:Nat ^ s(N:Nat) = (M ^ N) * M . -- (2)
red s(s(s(0))) ^ s(s(s(s(0)))) . -- it should be 81
close
-- ----------------------------------------------
-- SEMIRINGS
-- ----------------------------------------------
-- commutative monoids with powers
mod* CMON-POW (POWER :: MON, M :: CMON)
{
op _^_ : Elt.M Elt.POWER -> Elt.M
vars m m' : Elt.M
vars p p' : Elt.POWER
eq (m ; m')^ p = (m ^ p) ; (m' ^ p) .
eq m ^ (p ; p') = (m ^ p) ; (m ^ p') .
eq m ^ null = null .
}
module* SRNG
{
protecting (CMON-POW(MON, CMON) *
{ sort Elt -> Srng,
op (_;_) -> _+_,
op null -> 0,
op _^_ -> _*_ }
)
}
open SRNG .
ops m n m' n' : -> Srng .
-- the following should be true
red (m + n) * (m' + n') == (n * n') + (n * m') + (m * n') + (m * m') .
close
-- ----------------------------------------------------------------
-- full parameterised version of MONOIDS with POWERS
-- ----------------------------------------------------------------
mod* MON*-POW (POWER :: MON*, M :: MON*)
{
op _^_ : Elt.M Elt.POWER -> Elt.M
vars m m' : Elt.M
vars p p' : Elt.POWER
eq (m ; m')^ p = (m ^ p) ; (m' ^ p) .
eq m ^ (p ; p') = (m ^ p) ; (m ^ p') .
eq m ^ null = null .
}
** describe MON*-POW(POWER <= plus*, M <= dual*)
open MON*-POW(plus*,plus*) * { op _^_ -> _*_ } .
ops m m' n n' : -> Nat .
** we need associativity and commutativity of _+_ here
op _+_ : Nat Nat -> Nat { assoc comm }
-- the following should be true
red (m + m') * (n + n') == (m * n) + (m * n') + (m' * n) + (m' * n') .
close
-- ------------------------------------------------------------
-- ------------------------------------------------------------
--> proof that *dual* is indeed a specification morphism
mod* MONOID (X :: MON) { }
open MONOID(dual) .
ops x y z : -> Elt .
op _|_ : Elt Elt -> Elt .
eq X:Elt | Y:Elt = Y ; X .
red (x | y) | z == x | (y | z) .
red x | null == x .
red null | x == x .
close
view star from MON to INT {
sort Elt -> Int,
op null -> 0,
op X:Elt ; Y:Elt -> X:Int + Y:Int - X * Y
}
--> proof that *star* is indeed a specification morphism
open MONOID(star) .
ops x y z : -> Int .
vars X Y Z : Int .
op _;_ : Int Int -> Int .
eq X ; Y = X + Y - X * Y .
-- lemmas:
op _+_ : Int Int -> Int {assoc comm idr: 0}
op _*_ : Int Int -> Int {assoc comm idr: 1}
eq X + 0 = X .
eq X * 0 = 0 .
eq X * (Y + Z) = (X * Y) + (X * Z) .
eq X * (- Y) = - (X * Y) .
eq - - X = X .
eq - (X + Y) = (- X) + (- Y) .
-- proof
red (x ; y) ; z == x ; (y ; z) .
red x ; 0 == x .
red 0 ; x == x .
close
--
eof
--
$Id: monoid.mod,v 1.1.1.1 2003-06-19 08:30:10 sawada Exp $
|
