author | wenzelm |
Sun, 21 Mar 2010 15:57:40 +0100 | |
changeset 35847 | 19f1f7066917 |
parent 35355 | 613e133966ea |
child 35848 | 5443079512ea |
permissions | -rw-r--r-- |
27701 | 1 |
(* |
2 |
Title: Algebra/Congruence.thy |
|
3 |
Author: Clemens Ballarin, started 3 January 2008 |
|
4 |
Copyright: Clemens Ballarin |
|
5 |
*) |
|
6 |
||
7 |
theory Congruence imports Main begin |
|
8 |
||
9 |
section {* Objects *} |
|
10 |
||
27717
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27701
diff
changeset
|
11 |
subsection {* Structure with Carrier Set. *} |
27701 | 12 |
|
13 |
record 'a partial_object = |
|
14 |
carrier :: "'a set" |
|
15 |
||
27717
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27701
diff
changeset
|
16 |
|
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27701
diff
changeset
|
17 |
subsection {* Structure with Carrier and Equivalence Relation @{text eq} *} |
27701 | 18 |
|
19 |
record 'a eq_object = "'a partial_object" + |
|
20 |
eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".=\<index>" 50) |
|
21 |
||
35847 | 22 |
definition |
27701 | 23 |
elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<in>\<index>" 50) |
35847 | 24 |
where "x .\<in>\<^bsub>S\<^esub> A \<equiv> (\<exists>y \<in> A. x .=\<^bsub>S\<^esub> y)" |
27701 | 25 |
|
35847 | 26 |
definition |
27701 | 27 |
set_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.=}\<index>" 50) |
35847 | 28 |
where "A {.=}\<^bsub>S\<^esub> B == ((\<forall>x \<in> A. x .\<in>\<^bsub>S\<^esub> B) \<and> (\<forall>x \<in> B. x .\<in>\<^bsub>S\<^esub> A))" |
27701 | 29 |
|
35847 | 30 |
definition |
27701 | 31 |
eq_class_of :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set" ("class'_of\<index> _") |
35847 | 32 |
where "class_of\<^bsub>S\<^esub> x == {y \<in> carrier S. x .=\<^bsub>S\<^esub> y}" |
27701 | 33 |
|
35847 | 34 |
definition |
27701 | 35 |
eq_closure_of :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set" ("closure'_of\<index> _") |
35847 | 36 |
where "closure_of\<^bsub>S\<^esub> A == {y \<in> carrier S. y .\<in>\<^bsub>S\<^esub> A}" |
27701 | 37 |
|
35847 | 38 |
definition |
27701 | 39 |
eq_is_closed :: "_ \<Rightarrow> 'a set \<Rightarrow> bool" ("is'_closed\<index> _") |
35847 | 40 |
where "is_closed\<^bsub>S\<^esub> A == (A \<subseteq> carrier S \<and> closure_of\<^bsub>S\<^esub> A = A)" |
27701 | 41 |
|
35355
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
42 |
abbreviation |
27701 | 43 |
not_eq :: "_ \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".\<noteq>\<index>" 50) |
35355
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
44 |
where "x .\<noteq>\<^bsub>S\<^esub> y == ~(x .=\<^bsub>S\<^esub> y)" |
27701 | 45 |
|
35355
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
46 |
abbreviation |
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
47 |
not_elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<notin>\<index>" 50) |
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
48 |
where "x .\<notin>\<^bsub>S\<^esub> A == ~(x .\<in>\<^bsub>S\<^esub> A)" |
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
49 |
|
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
50 |
abbreviation |
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
51 |
set_not_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.\<noteq>}\<index>" 50) |
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
29237
diff
changeset
|
52 |
where "A {.\<noteq>}\<^bsub>S\<^esub> B == ~(A {.=}\<^bsub>S\<^esub> B)" |
27701 | 53 |
|
54 |
locale equivalence = |
|
55 |
fixes S (structure) |
|
56 |
assumes refl [simp, intro]: "x \<in> carrier S \<Longrightarrow> x .= x" |
|
57 |
and sym [sym]: "\<lbrakk> x .= y; x \<in> carrier S; y \<in> carrier S \<rbrakk> \<Longrightarrow> y .= x" |
|
58 |
and trans [trans]: "\<lbrakk> x .= y; y .= z; x \<in> carrier S; y \<in> carrier S; z \<in> carrier S \<rbrakk> \<Longrightarrow> x .= z" |
|
59 |
||
27717
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27701
diff
changeset
|
60 |
(* Lemmas by Stephan Hohe *) |
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27701
diff
changeset
|
61 |
|
27701 | 62 |
lemma elemI: |
63 |
fixes R (structure) |
|
64 |
assumes "a' \<in> A" and "a .= a'" |
|
65 |
shows "a .\<in> A" |
|
66 |
unfolding elem_def |
|
67 |
using assms |
|
68 |
by fast |
|
69 |
||
70 |
lemma (in equivalence) elem_exact: |
|
71 |
assumes "a \<in> carrier S" and "a \<in> A" |
|
72 |
shows "a .\<in> A" |
|
73 |
using assms |
|
74 |
by (fast intro: elemI) |
|
75 |
||
76 |
lemma elemE: |
|
77 |
fixes S (structure) |
|
78 |
assumes "a .\<in> A" |
|
79 |
and "\<And>a'. \<lbrakk>a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P" |
|
80 |
shows "P" |
|
81 |
using assms |
|
82 |
unfolding elem_def |
|
83 |
by fast |
|
84 |
||
85 |
lemma (in equivalence) elem_cong_l [trans]: |
|
86 |
assumes cong: "a' .= a" |
|
87 |
and a: "a .\<in> A" |
|
88 |
and carr: "a \<in> carrier S" "a' \<in> carrier S" |
|
89 |
and Acarr: "A \<subseteq> carrier S" |
|
90 |
shows "a' .\<in> A" |
|
91 |
using a |
|
92 |
apply (elim elemE, intro elemI) |
|
93 |
proof assumption |
|
94 |
fix b |
|
95 |
assume bA: "b \<in> A" |
|
96 |
note [simp] = carr bA[THEN subsetD[OF Acarr]] |
|
97 |
note cong |
|
98 |
also assume "a .= b" |
|
99 |
finally show "a' .= b" by simp |
|
100 |
qed |
|
101 |
||
102 |
lemma (in equivalence) elem_subsetD: |
|
103 |
assumes "A \<subseteq> B" |
|
104 |
and aA: "a .\<in> A" |
|
105 |
shows "a .\<in> B" |
|
106 |
using assms |
|
107 |
by (fast intro: elemI elim: elemE dest: subsetD) |
|
108 |
||
109 |
lemma (in equivalence) mem_imp_elem [simp, intro]: |
|
110 |
"[| x \<in> A; x \<in> carrier S |] ==> x .\<in> A" |
|
111 |
unfolding elem_def by blast |
|
112 |
||
113 |
lemma set_eqI: |
|
114 |
fixes R (structure) |
|
115 |
assumes ltr: "\<And>a. a \<in> A \<Longrightarrow> a .\<in> B" |
|
116 |
and rtl: "\<And>b. b \<in> B \<Longrightarrow> b .\<in> A" |
|
117 |
shows "A {.=} B" |
|
118 |
unfolding set_eq_def |
|
119 |
by (fast intro: ltr rtl) |
|
120 |
||
121 |
lemma set_eqI2: |
|
122 |
fixes R (structure) |
|
123 |
assumes ltr: "\<And>a b. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a .= b" |
|
124 |
and rtl: "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b .= a" |
|
125 |
shows "A {.=} B" |
|
126 |
by (intro set_eqI, unfold elem_def) (fast intro: ltr rtl)+ |
|
127 |
||
128 |
lemma set_eqD1: |
|
129 |
fixes R (structure) |
|
130 |
assumes AA': "A {.=} A'" |
|
131 |
and "a \<in> A" |
|
132 |
shows "\<exists>a'\<in>A'. a .= a'" |
|
133 |
using assms |
|
134 |
unfolding set_eq_def elem_def |
|
135 |
by fast |
|
136 |
||
137 |
lemma set_eqD2: |
|
138 |
fixes R (structure) |
|
139 |
assumes AA': "A {.=} A'" |
|
140 |
and "a' \<in> A'" |
|
141 |
shows "\<exists>a\<in>A. a' .= a" |
|
142 |
using assms |
|
143 |
unfolding set_eq_def elem_def |
|
144 |
by fast |
|
145 |
||
146 |
lemma set_eqE: |
|
147 |
fixes R (structure) |
|
148 |
assumes AB: "A {.=} B" |
|
149 |
and r: "\<lbrakk>\<forall>a\<in>A. a .\<in> B; \<forall>b\<in>B. b .\<in> A\<rbrakk> \<Longrightarrow> P" |
|
150 |
shows "P" |
|
151 |
using AB |
|
152 |
unfolding set_eq_def |
|
153 |
by (blast dest: r) |
|
154 |
||
155 |
lemma set_eqE2: |
|
156 |
fixes R (structure) |
|
157 |
assumes AB: "A {.=} B" |
|
158 |
and r: "\<lbrakk>\<forall>a\<in>A. (\<exists>b\<in>B. a .= b); \<forall>b\<in>B. (\<exists>a\<in>A. b .= a)\<rbrakk> \<Longrightarrow> P" |
|
159 |
shows "P" |
|
160 |
using AB |
|
161 |
unfolding set_eq_def elem_def |
|
162 |
by (blast dest: r) |
|
163 |
||
164 |
lemma set_eqE': |
|
165 |
fixes R (structure) |
|
166 |
assumes AB: "A {.=} B" |
|
167 |
and aA: "a \<in> A" and bB: "b \<in> B" |
|
168 |
and r: "\<And>a' b'. \<lbrakk>a' \<in> A; b .= a'; b' \<in> B; a .= b'\<rbrakk> \<Longrightarrow> P" |
|
169 |
shows "P" |
|
170 |
proof - |
|
171 |
from AB aA |
|
172 |
have "\<exists>b'\<in>B. a .= b'" by (rule set_eqD1) |
|
173 |
from this obtain b' |
|
174 |
where b': "b' \<in> B" "a .= b'" by auto |
|
175 |
||
176 |
from AB bB |
|
177 |
have "\<exists>a'\<in>A. b .= a'" by (rule set_eqD2) |
|
178 |
from this obtain a' |
|
179 |
where a': "a' \<in> A" "b .= a'" by auto |
|
180 |
||
181 |
from a' b' |
|
182 |
show "P" by (rule r) |
|
183 |
qed |
|
184 |
||
185 |
lemma (in equivalence) eq_elem_cong_r [trans]: |
|
186 |
assumes a: "a .\<in> A" |
|
187 |
and cong: "A {.=} A'" |
|
188 |
and carr: "a \<in> carrier S" |
|
189 |
and Carr: "A \<subseteq> carrier S" "A' \<subseteq> carrier S" |
|
190 |
shows "a .\<in> A'" |
|
191 |
using a cong |
|
192 |
proof (elim elemE set_eqE) |
|
193 |
fix b |
|
194 |
assume bA: "b \<in> A" |
|
195 |
and inA': "\<forall>b\<in>A. b .\<in> A'" |
|
196 |
note [simp] = carr Carr Carr[THEN subsetD] bA |
|
197 |
assume "a .= b" |
|
198 |
also from bA inA' |
|
199 |
have "b .\<in> A'" by fast |
|
200 |
finally |
|
201 |
show "a .\<in> A'" by simp |
|
202 |
qed |
|
203 |
||
204 |
lemma (in equivalence) set_eq_sym [sym]: |
|
205 |
assumes "A {.=} B" |
|
206 |
and "A \<subseteq> carrier S" "B \<subseteq> carrier S" |
|
207 |
shows "B {.=} A" |
|
208 |
using assms |
|
209 |
unfolding set_eq_def elem_def |
|
210 |
by fast |
|
211 |
||
212 |
(* FIXME: the following two required in Isabelle 2008, not Isabelle 2007 *) |
|
27717
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27701
diff
changeset
|
213 |
(* alternatively, could declare lemmas [trans] = ssubst [where 'a = "'a set"] *) |
27701 | 214 |
|
215 |
lemma (in equivalence) equal_set_eq_trans [trans]: |
|
216 |
assumes AB: "A = B" and BC: "B {.=} C" |
|
217 |
shows "A {.=} C" |
|
218 |
using AB BC by simp |
|
219 |
||
220 |
lemma (in equivalence) set_eq_equal_trans [trans]: |
|
221 |
assumes AB: "A {.=} B" and BC: "B = C" |
|
222 |
shows "A {.=} C" |
|
223 |
using AB BC by simp |
|
224 |
||
27717
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27701
diff
changeset
|
225 |
|
27701 | 226 |
lemma (in equivalence) set_eq_trans [trans]: |
227 |
assumes AB: "A {.=} B" and BC: "B {.=} C" |
|
228 |
and carr: "A \<subseteq> carrier S" "B \<subseteq> carrier S" "C \<subseteq> carrier S" |
|
229 |
shows "A {.=} C" |
|
230 |
proof (intro set_eqI) |
|
231 |
fix a |
|
232 |
assume aA: "a \<in> A" |
|
233 |
with carr have "a \<in> carrier S" by fast |
|
234 |
note [simp] = carr this |
|
235 |
||
236 |
from aA |
|
237 |
have "a .\<in> A" by (simp add: elem_exact) |
|
238 |
also note AB |
|
239 |
also note BC |
|
240 |
finally |
|
241 |
show "a .\<in> C" by simp |
|
242 |
next |
|
243 |
fix c |
|
244 |
assume cC: "c \<in> C" |
|
245 |
with carr have "c \<in> carrier S" by fast |
|
246 |
note [simp] = carr this |
|
247 |
||
248 |
from cC |
|
249 |
have "c .\<in> C" by (simp add: elem_exact) |
|
250 |
also note BC[symmetric] |
|
251 |
also note AB[symmetric] |
|
252 |
finally |
|
253 |
show "c .\<in> A" by simp |
|
254 |
qed |
|
255 |
||
256 |
(* FIXME: generalise for insert *) |
|
257 |
||
258 |
(* |
|
259 |
lemma (in equivalence) set_eq_insert: |
|
260 |
assumes x: "x .= x'" |
|
261 |
and carr: "x \<in> carrier S" "x' \<in> carrier S" "A \<subseteq> carrier S" |
|
262 |
shows "insert x A {.=} insert x' A" |
|
263 |
unfolding set_eq_def elem_def |
|
264 |
apply rule |
|
265 |
apply rule |
|
266 |
apply (case_tac "xa = x") |
|
267 |
using x apply fast |
|
268 |
apply (subgoal_tac "xa \<in> A") prefer 2 apply fast |
|
269 |
apply (rule_tac x=xa in bexI) |
|
270 |
using carr apply (rule_tac refl) apply auto [1] |
|
271 |
apply safe |
|
272 |
*) |
|
273 |
||
274 |
lemma (in equivalence) set_eq_pairI: |
|
275 |
assumes xx': "x .= x'" |
|
276 |
and carr: "x \<in> carrier S" "x' \<in> carrier S" "y \<in> carrier S" |
|
277 |
shows "{x, y} {.=} {x', y}" |
|
278 |
unfolding set_eq_def elem_def |
|
279 |
proof safe |
|
280 |
have "x' \<in> {x', y}" by fast |
|
281 |
with xx' show "\<exists>b\<in>{x', y}. x .= b" by fast |
|
282 |
next |
|
283 |
have "y \<in> {x', y}" by fast |
|
284 |
with carr show "\<exists>b\<in>{x', y}. y .= b" by fast |
|
285 |
next |
|
286 |
have "x \<in> {x, y}" by fast |
|
287 |
with xx'[symmetric] carr |
|
288 |
show "\<exists>a\<in>{x, y}. x' .= a" by fast |
|
289 |
next |
|
290 |
have "y \<in> {x, y}" by fast |
|
291 |
with carr show "\<exists>a\<in>{x, y}. y .= a" by fast |
|
292 |
qed |
|
293 |
||
294 |
lemma (in equivalence) is_closedI: |
|
295 |
assumes closed: "!!x y. [| x .= y; x \<in> A; y \<in> carrier S |] ==> y \<in> A" |
|
296 |
and S: "A \<subseteq> carrier S" |
|
297 |
shows "is_closed A" |
|
298 |
unfolding eq_is_closed_def eq_closure_of_def elem_def |
|
299 |
using S |
|
300 |
by (blast dest: closed sym) |
|
301 |
||
302 |
lemma (in equivalence) closure_of_eq: |
|
303 |
"[| x .= x'; A \<subseteq> carrier S; x \<in> closure_of A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> closure_of A" |
|
304 |
unfolding eq_closure_of_def elem_def |
|
305 |
by (blast intro: trans sym) |
|
306 |
||
307 |
lemma (in equivalence) is_closed_eq [dest]: |
|
308 |
"[| x .= x'; x \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> A" |
|
309 |
unfolding eq_is_closed_def |
|
310 |
using closure_of_eq [where A = A] |
|
311 |
by simp |
|
312 |
||
313 |
lemma (in equivalence) is_closed_eq_rev [dest]: |
|
314 |
"[| x .= x'; x' \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x \<in> A" |
|
315 |
by (drule sym) (simp_all add: is_closed_eq) |
|
316 |
||
317 |
lemma closure_of_closed [simp, intro]: |
|
318 |
fixes S (structure) |
|
319 |
shows "closure_of A \<subseteq> carrier S" |
|
320 |
unfolding eq_closure_of_def |
|
321 |
by fast |
|
322 |
||
323 |
lemma closure_of_memI: |
|
324 |
fixes S (structure) |
|
325 |
assumes "a .\<in> A" |
|
326 |
and "a \<in> carrier S" |
|
327 |
shows "a \<in> closure_of A" |
|
328 |
unfolding eq_closure_of_def |
|
329 |
using assms |
|
330 |
by fast |
|
331 |
||
332 |
lemma closure_ofI2: |
|
333 |
fixes S (structure) |
|
334 |
assumes "a .= a'" |
|
335 |
and "a' \<in> A" |
|
336 |
and "a \<in> carrier S" |
|
337 |
shows "a \<in> closure_of A" |
|
338 |
unfolding eq_closure_of_def elem_def |
|
339 |
using assms |
|
340 |
by fast |
|
341 |
||
342 |
lemma closure_of_memE: |
|
343 |
fixes S (structure) |
|
344 |
assumes p: "a \<in> closure_of A" |
|
345 |
and r: "\<lbrakk>a \<in> carrier S; a .\<in> A\<rbrakk> \<Longrightarrow> P" |
|
346 |
shows "P" |
|
347 |
proof - |
|
348 |
from p |
|
349 |
have acarr: "a \<in> carrier S" |
|
350 |
and "a .\<in> A" |
|
351 |
by (simp add: eq_closure_of_def)+ |
|
352 |
thus "P" by (rule r) |
|
353 |
qed |
|
354 |
||
355 |
lemma closure_ofE2: |
|
356 |
fixes S (structure) |
|
357 |
assumes p: "a \<in> closure_of A" |
|
358 |
and r: "\<And>a'. \<lbrakk>a \<in> carrier S; a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P" |
|
359 |
shows "P" |
|
360 |
proof - |
|
361 |
from p have acarr: "a \<in> carrier S" by (simp add: eq_closure_of_def) |
|
362 |
||
363 |
from p have "\<exists>a'\<in>A. a .= a'" by (simp add: eq_closure_of_def elem_def) |
|
364 |
from this obtain a' |
|
365 |
where "a' \<in> A" and "a .= a'" by auto |
|
366 |
||
367 |
from acarr and this |
|
368 |
show "P" by (rule r) |
|
369 |
qed |
|
370 |
||
371 |
(* |
|
372 |
lemma (in equivalence) classes_consistent: |
|
373 |
assumes Acarr: "A \<subseteq> carrier S" |
|
374 |
shows "is_closed (closure_of A)" |
|
375 |
apply (blast intro: elemI elim elemE) |
|
376 |
using assms |
|
377 |
apply (intro is_closedI closure_of_memI, simp) |
|
378 |
apply (elim elemE closure_of_memE) |
|
379 |
proof - |
|
380 |
fix x a' a'' |
|
381 |
assume carr: "x \<in> carrier S" "a' \<in> carrier S" |
|
382 |
assume a''A: "a'' \<in> A" |
|
383 |
with Acarr have "a'' \<in> carrier S" by fast |
|
384 |
note [simp] = carr this Acarr |
|
385 |
||
386 |
assume "x .= a'" |
|
387 |
also assume "a' .= a''" |
|
388 |
also from a''A |
|
389 |
have "a'' .\<in> A" by (simp add: elem_exact) |
|
390 |
finally show "x .\<in> A" by simp |
|
391 |
qed |
|
392 |
*) |
|
393 |
(* |
|
394 |
lemma (in equivalence) classes_small: |
|
395 |
assumes "is_closed B" |
|
396 |
and "A \<subseteq> B" |
|
397 |
shows "closure_of A \<subseteq> B" |
|
398 |
using assms |
|
399 |
by (blast dest: is_closedD2 elem_subsetD elim: closure_of_memE) |
|
400 |
||
401 |
lemma (in equivalence) classes_eq: |
|
402 |
assumes "A \<subseteq> carrier S" |
|
403 |
shows "A {.=} closure_of A" |
|
404 |
using assms |
|
405 |
by (blast intro: set_eqI elem_exact closure_of_memI elim: closure_of_memE) |
|
406 |
||
407 |
lemma (in equivalence) complete_classes: |
|
408 |
assumes c: "is_closed A" |
|
409 |
shows "A = closure_of A" |
|
410 |
using assms |
|
411 |
by (blast intro: closure_of_memI elem_exact dest: is_closedD1 is_closedD2 closure_of_memE) |
|
412 |
*) |
|
413 |
||
414 |
end |