src/ZF/upair.thy
 author wenzelm Tue Sep 01 22:32:58 2015 +0200 (2015-09-01) changeset 61076 bdc1e2f0a86a parent 60770 240563fbf41d child 61798 27f3c10b0b50 permissions -rw-r--r--
eliminated \<Colon>;
1 (*  Title:      ZF/upair.thy
2     Author:     Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
3     Copyright   1993  University of Cambridge
5 Observe the order of dependence:
6     Upair is defined in terms of Replace
7     \<union> is defined in terms of Upair and \<Union>(similarly for Int)
8     cons is defined in terms of Upair and Un
9     Ordered pairs and descriptions are defined using cons ("set notation")
10 *)
12 section\<open>Unordered Pairs\<close>
14 theory upair
15 imports ZF
16 keywords "print_tcset" :: diag
17 begin
19 ML_file "Tools/typechk.ML"
21 lemma atomize_ball [symmetric, rulify]:
22      "(!!x. x \<in> A ==> P(x)) == Trueprop (\<forall>x\<in>A. P(x))"
23 by (simp add: Ball_def atomize_all atomize_imp)
26 subsection\<open>Unordered Pairs: constant @{term Upair}\<close>
28 lemma Upair_iff [simp]: "c \<in> Upair(a,b) \<longleftrightarrow> (c=a | c=b)"
29 by (unfold Upair_def, blast)
31 lemma UpairI1: "a \<in> Upair(a,b)"
32 by simp
34 lemma UpairI2: "b \<in> Upair(a,b)"
35 by simp
37 lemma UpairE: "[| a \<in> Upair(b,c);  a=b ==> P;  a=c ==> P |] ==> P"
38 by (simp, blast)
40 subsection\<open>Rules for Binary Union, Defined via @{term Upair}\<close>
42 lemma Un_iff [simp]: "c \<in> A \<union> B \<longleftrightarrow> (c \<in> A | c \<in> B)"
44 apply (blast intro: UpairI1 UpairI2 elim: UpairE)
45 done
47 lemma UnI1: "c \<in> A ==> c \<in> A \<union> B"
48 by simp
50 lemma UnI2: "c \<in> B ==> c \<in> A \<union> B"
51 by simp
53 declare UnI1 [elim?]  UnI2 [elim?]
55 lemma UnE [elim!]: "[| c \<in> A \<union> B;  c \<in> A ==> P;  c \<in> B ==> P |] ==> P"
56 by (simp, blast)
58 (*Stronger version of the rule above*)
59 lemma UnE': "[| c \<in> A \<union> B;  c \<in> A ==> P;  [| c \<in> B;  c\<notin>A |] ==> P |] ==> P"
60 by (simp, blast)
62 (*Classical introduction rule: no commitment to A vs B*)
63 lemma UnCI [intro!]: "(c \<notin> B ==> c \<in> A) ==> c \<in> A \<union> B"
64 by (simp, blast)
66 subsection\<open>Rules for Binary Intersection, Defined via @{term Upair}\<close>
68 lemma Int_iff [simp]: "c \<in> A \<inter> B \<longleftrightarrow> (c \<in> A & c \<in> B)"
69 apply (unfold Int_def)
70 apply (blast intro: UpairI1 UpairI2 elim: UpairE)
71 done
73 lemma IntI [intro!]: "[| c \<in> A;  c \<in> B |] ==> c \<in> A \<inter> B"
74 by simp
76 lemma IntD1: "c \<in> A \<inter> B ==> c \<in> A"
77 by simp
79 lemma IntD2: "c \<in> A \<inter> B ==> c \<in> B"
80 by simp
82 lemma IntE [elim!]: "[| c \<in> A \<inter> B;  [| c \<in> A; c \<in> B |] ==> P |] ==> P"
83 by simp
86 subsection\<open>Rules for Set Difference, Defined via @{term Upair}\<close>
88 lemma Diff_iff [simp]: "c \<in> A-B \<longleftrightarrow> (c \<in> A & c\<notin>B)"
89 by (unfold Diff_def, blast)
91 lemma DiffI [intro!]: "[| c \<in> A;  c \<notin> B |] ==> c \<in> A - B"
92 by simp
94 lemma DiffD1: "c \<in> A - B ==> c \<in> A"
95 by simp
97 lemma DiffD2: "c \<in> A - B ==> c \<notin> B"
98 by simp
100 lemma DiffE [elim!]: "[| c \<in> A - B;  [| c \<in> A; c\<notin>B |] ==> P |] ==> P"
101 by simp
104 subsection\<open>Rules for @{term cons}\<close>
106 lemma cons_iff [simp]: "a \<in> cons(b,A) \<longleftrightarrow> (a=b | a \<in> A)"
107 apply (unfold cons_def)
108 apply (blast intro: UpairI1 UpairI2 elim: UpairE)
109 done
111 (*risky as a typechecking rule, but solves otherwise unconstrained goals of
112 the form x \<in> ?A*)
113 lemma consI1 [simp,TC]: "a \<in> cons(a,B)"
114 by simp
117 lemma consI2: "a \<in> B ==> a \<in> cons(b,B)"
118 by simp
120 lemma consE [elim!]: "[| a \<in> cons(b,A);  a=b ==> P;  a \<in> A ==> P |] ==> P"
121 by (simp, blast)
123 (*Stronger version of the rule above*)
124 lemma consE':
125     "[| a \<in> cons(b,A);  a=b ==> P;  [| a \<in> A;  a\<noteq>b |] ==> P |] ==> P"
126 by (simp, blast)
128 (*Classical introduction rule*)
129 lemma consCI [intro!]: "(a\<notin>B ==> a=b) ==> a \<in> cons(b,B)"
130 by (simp, blast)
132 lemma cons_not_0 [simp]: "cons(a,B) \<noteq> 0"
133 by (blast elim: equalityE)
135 lemmas cons_neq_0 = cons_not_0 [THEN notE]
137 declare cons_not_0 [THEN not_sym, simp]
140 subsection\<open>Singletons\<close>
142 lemma singleton_iff: "a \<in> {b} \<longleftrightarrow> a=b"
143 by simp
145 lemma singletonI [intro!]: "a \<in> {a}"
146 by (rule consI1)
148 lemmas singletonE = singleton_iff [THEN iffD1, elim_format, elim!]
151 subsection\<open>Descriptions\<close>
153 lemma the_equality [intro]:
154     "[| P(a);  !!x. P(x) ==> x=a |] ==> (THE x. P(x)) = a"
155 apply (unfold the_def)
156 apply (fast dest: subst)
157 done
159 (* Only use this if you already know EX!x. P(x) *)
160 lemma the_equality2: "[| EX! x. P(x);  P(a) |] ==> (THE x. P(x)) = a"
161 by blast
163 lemma theI: "EX! x. P(x) ==> P(THE x. P(x))"
164 apply (erule ex1E)
165 apply (subst the_equality)
166 apply (blast+)
167 done
169 (*No congruence rule is necessary: if @{term"\<forall>y.P(y)\<longleftrightarrow>Q(y)"} then
170   @{term "THE x.P(x)"}  rewrites to @{term "THE x.Q(x)"} *)
172 (*If it's "undefined", it's zero!*)
173 lemma the_0: "~ (EX! x. P(x)) ==> (THE x. P(x))=0"
174 apply (unfold the_def)
175 apply (blast elim!: ReplaceE)
176 done
178 (*Easier to apply than theI: conclusion has only one occurrence of P*)
179 lemma theI2:
180     assumes p1: "~ Q(0) ==> EX! x. P(x)"
181         and p2: "!!x. P(x) ==> Q(x)"
182     shows "Q(THE x. P(x))"
183 apply (rule classical)
184 apply (rule p2)
185 apply (rule theI)
186 apply (rule classical)
187 apply (rule p1)
188 apply (erule the_0 [THEN subst], assumption)
189 done
191 lemma the_eq_trivial [simp]: "(THE x. x = a) = a"
192 by blast
194 lemma the_eq_trivial2 [simp]: "(THE x. a = x) = a"
195 by blast
198 subsection\<open>Conditional Terms: @{text "if-then-else"}\<close>
200 lemma if_true [simp]: "(if True then a else b) = a"
201 by (unfold if_def, blast)
203 lemma if_false [simp]: "(if False then a else b) = b"
204 by (unfold if_def, blast)
206 (*Never use with case splitting, or if P is known to be true or false*)
207 lemma if_cong:
208     "[| P\<longleftrightarrow>Q;  Q ==> a=c;  ~Q ==> b=d |]
209      ==> (if P then a else b) = (if Q then c else d)"
212 (*Prevents simplification of x and y \<in> faster and allows the execution
213   of functional programs. NOW THE DEFAULT.*)
214 lemma if_weak_cong: "P\<longleftrightarrow>Q ==> (if P then x else y) = (if Q then x else y)"
215 by simp
217 (*Not needed for rewriting, since P would rewrite to True anyway*)
218 lemma if_P: "P ==> (if P then a else b) = a"
219 by (unfold if_def, blast)
221 (*Not needed for rewriting, since P would rewrite to False anyway*)
222 lemma if_not_P: "~P ==> (if P then a else b) = b"
223 by (unfold if_def, blast)
225 lemma split_if [split]:
226      "P(if Q then x else y) \<longleftrightarrow> ((Q \<longrightarrow> P(x)) & (~Q \<longrightarrow> P(y)))"
227 by (case_tac Q, simp_all)
229 (** Rewrite rules for boolean case-splitting: faster than split_if [split]
230 **)
232 lemmas split_if_eq1 = split_if [of "%x. x = b"] for b
233 lemmas split_if_eq2 = split_if [of "%x. a = x"] for x
235 lemmas split_if_mem1 = split_if [of "%x. x \<in> b"] for b
236 lemmas split_if_mem2 = split_if [of "%x. a \<in> x"] for x
238 lemmas split_ifs = split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
240 (*Logically equivalent to split_if_mem2*)
241 lemma if_iff: "a: (if P then x else y) \<longleftrightarrow> P & a \<in> x | ~P & a \<in> y"
242 by simp
244 lemma if_type [TC]:
245     "[| P ==> a \<in> A;  ~P ==> b \<in> A |] ==> (if P then a else b): A"
246 by simp
248 (** Splitting IFs in the assumptions **)
250 lemma split_if_asm: "P(if Q then x else y) \<longleftrightarrow> (~((Q & ~P(x)) | (~Q & ~P(y))))"
251 by simp
253 lemmas if_splits = split_if split_if_asm
256 subsection\<open>Consequences of Foundation\<close>
258 (*was called mem_anti_sym*)
259 lemma mem_asym: "[| a \<in> b;  ~P ==> b \<in> a |] ==> P"
260 apply (rule classical)
261 apply (rule_tac A1 = "{a,b}" in foundation [THEN disjE])
262 apply (blast elim!: equalityE)+
263 done
265 (*was called mem_anti_refl*)
266 lemma mem_irrefl: "a \<in> a ==> P"
267 by (blast intro: mem_asym)
269 (*mem_irrefl should NOT be added to default databases:
270       it would be tried on most goals, making proofs slower!*)
272 lemma mem_not_refl: "a \<notin> a"
273 apply (rule notI)
274 apply (erule mem_irrefl)
275 done
277 (*Good for proving inequalities by rewriting*)
278 lemma mem_imp_not_eq: "a \<in> A ==> a \<noteq> A"
279 by (blast elim!: mem_irrefl)
281 lemma eq_imp_not_mem: "a=A ==> a \<notin> A"
282 by (blast intro: elim: mem_irrefl)
284 subsection\<open>Rules for Successor\<close>
286 lemma succ_iff: "i \<in> succ(j) \<longleftrightarrow> i=j | i \<in> j"
287 by (unfold succ_def, blast)
289 lemma succI1 [simp]: "i \<in> succ(i)"
292 lemma succI2: "i \<in> j ==> i \<in> succ(j)"
295 lemma succE [elim!]:
296     "[| i \<in> succ(j);  i=j ==> P;  i \<in> j ==> P |] ==> P"
297 apply (simp add: succ_iff, blast)
298 done
300 (*Classical introduction rule*)
301 lemma succCI [intro!]: "(i\<notin>j ==> i=j) ==> i \<in> succ(j)"
302 by (simp add: succ_iff, blast)
304 lemma succ_not_0 [simp]: "succ(n) \<noteq> 0"
305 by (blast elim!: equalityE)
307 lemmas succ_neq_0 = succ_not_0 [THEN notE, elim!]
309 declare succ_not_0 [THEN not_sym, simp]
310 declare sym [THEN succ_neq_0, elim!]
312 (* @{term"succ(c) \<subseteq> B ==> c \<in> B"} *)
313 lemmas succ_subsetD = succI1 [THEN [2] subsetD]
315 (* @{term"succ(b) \<noteq> b"} *)
316 lemmas succ_neq_self = succI1 [THEN mem_imp_not_eq, THEN not_sym]
318 lemma succ_inject_iff [simp]: "succ(m) = succ(n) \<longleftrightarrow> m=n"
319 by (blast elim: mem_asym elim!: equalityE)
321 lemmas succ_inject = succ_inject_iff [THEN iffD1, dest!]
324 subsection\<open>Miniscoping of the Bounded Universal Quantifier\<close>
326 lemma ball_simps1:
327      "(\<forall>x\<in>A. P(x) & Q)   \<longleftrightarrow> (\<forall>x\<in>A. P(x)) & (A=0 | Q)"
328      "(\<forall>x\<in>A. P(x) | Q)   \<longleftrightarrow> ((\<forall>x\<in>A. P(x)) | Q)"
329      "(\<forall>x\<in>A. P(x) \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P(x)) \<longrightarrow> Q)"
330      "(~(\<forall>x\<in>A. P(x))) \<longleftrightarrow> (\<exists>x\<in>A. ~P(x))"
331      "(\<forall>x\<in>0.P(x)) \<longleftrightarrow> True"
332      "(\<forall>x\<in>succ(i).P(x)) \<longleftrightarrow> P(i) & (\<forall>x\<in>i. P(x))"
333      "(\<forall>x\<in>cons(a,B).P(x)) \<longleftrightarrow> P(a) & (\<forall>x\<in>B. P(x))"
334      "(\<forall>x\<in>RepFun(A,f). P(x)) \<longleftrightarrow> (\<forall>y\<in>A. P(f(y)))"
335      "(\<forall>x\<in>\<Union>(A).P(x)) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P(x))"
336 by blast+
338 lemma ball_simps2:
339      "(\<forall>x\<in>A. P & Q(x))   \<longleftrightarrow> (A=0 | P) & (\<forall>x\<in>A. Q(x))"
340      "(\<forall>x\<in>A. P | Q(x))   \<longleftrightarrow> (P | (\<forall>x\<in>A. Q(x)))"
341      "(\<forall>x\<in>A. P \<longrightarrow> Q(x)) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q(x)))"
342 by blast+
344 lemma ball_simps3:
345      "(\<forall>x\<in>Collect(A,Q).P(x)) \<longleftrightarrow> (\<forall>x\<in>A. Q(x) \<longrightarrow> P(x))"
346 by blast+
348 lemmas ball_simps [simp] = ball_simps1 ball_simps2 ball_simps3
350 lemma ball_conj_distrib:
351     "(\<forall>x\<in>A. P(x) & Q(x)) \<longleftrightarrow> ((\<forall>x\<in>A. P(x)) & (\<forall>x\<in>A. Q(x)))"
352 by blast
355 subsection\<open>Miniscoping of the Bounded Existential Quantifier\<close>
357 lemma bex_simps1:
358      "(\<exists>x\<in>A. P(x) & Q) \<longleftrightarrow> ((\<exists>x\<in>A. P(x)) & Q)"
359      "(\<exists>x\<in>A. P(x) | Q) \<longleftrightarrow> (\<exists>x\<in>A. P(x)) | (A\<noteq>0 & Q)"
360      "(\<exists>x\<in>A. P(x) \<longrightarrow> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P(x)) \<longrightarrow> (A\<noteq>0 & Q))"
361      "(\<exists>x\<in>0.P(x)) \<longleftrightarrow> False"
362      "(\<exists>x\<in>succ(i).P(x)) \<longleftrightarrow> P(i) | (\<exists>x\<in>i. P(x))"
363      "(\<exists>x\<in>cons(a,B).P(x)) \<longleftrightarrow> P(a) | (\<exists>x\<in>B. P(x))"
364      "(\<exists>x\<in>RepFun(A,f). P(x)) \<longleftrightarrow> (\<exists>y\<in>A. P(f(y)))"
365      "(\<exists>x\<in>\<Union>(A).P(x)) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y.  P(x))"
366      "(~(\<exists>x\<in>A. P(x))) \<longleftrightarrow> (\<forall>x\<in>A. ~P(x))"
367 by blast+
369 lemma bex_simps2:
370      "(\<exists>x\<in>A. P & Q(x)) \<longleftrightarrow> (P & (\<exists>x\<in>A. Q(x)))"
371      "(\<exists>x\<in>A. P | Q(x)) \<longleftrightarrow> (A\<noteq>0 & P) | (\<exists>x\<in>A. Q(x))"
372      "(\<exists>x\<in>A. P \<longrightarrow> Q(x)) \<longleftrightarrow> ((A=0 | P) \<longrightarrow> (\<exists>x\<in>A. Q(x)))"
373 by blast+
375 lemma bex_simps3:
376      "(\<exists>x\<in>Collect(A,Q).P(x)) \<longleftrightarrow> (\<exists>x\<in>A. Q(x) & P(x))"
377 by blast
379 lemmas bex_simps [simp] = bex_simps1 bex_simps2 bex_simps3
381 lemma bex_disj_distrib:
382     "(\<exists>x\<in>A. P(x) | Q(x)) \<longleftrightarrow> ((\<exists>x\<in>A. P(x)) | (\<exists>x\<in>A. Q(x)))"
383 by blast
386 (** One-point rule for bounded quantifiers: see HOL/Set.ML **)
388 lemma bex_triv_one_point1 [simp]: "(\<exists>x\<in>A. x=a) \<longleftrightarrow> (a \<in> A)"
389 by blast
391 lemma bex_triv_one_point2 [simp]: "(\<exists>x\<in>A. a=x) \<longleftrightarrow> (a \<in> A)"
392 by blast
394 lemma bex_one_point1 [simp]: "(\<exists>x\<in>A. x=a & P(x)) \<longleftrightarrow> (a \<in> A & P(a))"
395 by blast
397 lemma bex_one_point2 [simp]: "(\<exists>x\<in>A. a=x & P(x)) \<longleftrightarrow> (a \<in> A & P(a))"
398 by blast
400 lemma ball_one_point1 [simp]: "(\<forall>x\<in>A. x=a \<longrightarrow> P(x)) \<longleftrightarrow> (a \<in> A \<longrightarrow> P(a))"
401 by blast
403 lemma ball_one_point2 [simp]: "(\<forall>x\<in>A. a=x \<longrightarrow> P(x)) \<longleftrightarrow> (a \<in> A \<longrightarrow> P(a))"
404 by blast
407 subsection\<open>Miniscoping of the Replacement Operator\<close>
409 text\<open>These cover both @{term Replace} and @{term Collect}\<close>
410 lemma Rep_simps [simp]:
411      "{x. y \<in> 0, R(x,y)} = 0"
412      "{x \<in> 0. P(x)} = 0"
413      "{x \<in> A. Q} = (if Q then A else 0)"
414      "RepFun(0,f) = 0"
415      "RepFun(succ(i),f) = cons(f(i), RepFun(i,f))"
416      "RepFun(cons(a,B),f) = cons(f(a), RepFun(B,f))"
417 by (simp_all, blast+)
420 subsection\<open>Miniscoping of Unions\<close>
422 lemma UN_simps1:
423      "(\<Union>x\<in>C. cons(a, B(x))) = (if C=0 then 0 else cons(a, \<Union>x\<in>C. B(x)))"
424      "(\<Union>x\<in>C. A(x) \<union> B')   = (if C=0 then 0 else (\<Union>x\<in>C. A(x)) \<union> B')"
425      "(\<Union>x\<in>C. A' \<union> B(x))   = (if C=0 then 0 else A' \<union> (\<Union>x\<in>C. B(x)))"
426      "(\<Union>x\<in>C. A(x) \<inter> B')  = ((\<Union>x\<in>C. A(x)) \<inter> B')"
427      "(\<Union>x\<in>C. A' \<inter> B(x))  = (A' \<inter> (\<Union>x\<in>C. B(x)))"
428      "(\<Union>x\<in>C. A(x) - B')    = ((\<Union>x\<in>C. A(x)) - B')"
429      "(\<Union>x\<in>C. A' - B(x))    = (if C=0 then 0 else A' - (\<Inter>x\<in>C. B(x)))"
431 apply (blast intro!: equalityI )+
432 done
434 lemma UN_simps2:
435       "(\<Union>x\<in>\<Union>(A). B(x)) = (\<Union>y\<in>A. \<Union>x\<in>y. B(x))"
436       "(\<Union>z\<in>(\<Union>x\<in>A. B(x)). C(z)) = (\<Union>x\<in>A. \<Union>z\<in>B(x). C(z))"
437       "(\<Union>x\<in>RepFun(A,f). B(x))     = (\<Union>a\<in>A. B(f(a)))"
438 by blast+
440 lemmas UN_simps [simp] = UN_simps1 UN_simps2
442 text\<open>Opposite of miniscoping: pull the operator out\<close>
444 lemma UN_extend_simps1:
445      "(\<Union>x\<in>C. A(x)) \<union> B   = (if C=0 then B else (\<Union>x\<in>C. A(x) \<union> B))"
446      "((\<Union>x\<in>C. A(x)) \<inter> B) = (\<Union>x\<in>C. A(x) \<inter> B)"
447      "((\<Union>x\<in>C. A(x)) - B) = (\<Union>x\<in>C. A(x) - B)"
448 apply simp_all
449 apply blast+
450 done
452 lemma UN_extend_simps2:
453      "cons(a, \<Union>x\<in>C. B(x)) = (if C=0 then {a} else (\<Union>x\<in>C. cons(a, B(x))))"
454      "A \<union> (\<Union>x\<in>C. B(x))   = (if C=0 then A else (\<Union>x\<in>C. A \<union> B(x)))"
455      "(A \<inter> (\<Union>x\<in>C. B(x))) = (\<Union>x\<in>C. A \<inter> B(x))"
456      "A - (\<Inter>x\<in>C. B(x))    = (if C=0 then A else (\<Union>x\<in>C. A - B(x)))"
457      "(\<Union>y\<in>A. \<Union>x\<in>y. B(x)) = (\<Union>x\<in>\<Union>(A). B(x))"
458      "(\<Union>a\<in>A. B(f(a))) = (\<Union>x\<in>RepFun(A,f). B(x))"
460 apply (blast intro!: equalityI)+
461 done
463 lemma UN_UN_extend:
464      "(\<Union>x\<in>A. \<Union>z\<in>B(x). C(z)) = (\<Union>z\<in>(\<Union>x\<in>A. B(x)). C(z))"
465 by blast
467 lemmas UN_extend_simps = UN_extend_simps1 UN_extend_simps2 UN_UN_extend
470 subsection\<open>Miniscoping of Intersections\<close>
472 lemma INT_simps1:
473      "(\<Inter>x\<in>C. A(x) \<inter> B) = (\<Inter>x\<in>C. A(x)) \<inter> B"
474      "(\<Inter>x\<in>C. A(x) - B)   = (\<Inter>x\<in>C. A(x)) - B"
475      "(\<Inter>x\<in>C. A(x) \<union> B)  = (if C=0 then 0 else (\<Inter>x\<in>C. A(x)) \<union> B)"
476 by (simp_all add: Inter_def, blast+)
478 lemma INT_simps2:
479      "(\<Inter>x\<in>C. A \<inter> B(x)) = A \<inter> (\<Inter>x\<in>C. B(x))"
480      "(\<Inter>x\<in>C. A - B(x))   = (if C=0 then 0 else A - (\<Union>x\<in>C. B(x)))"
481      "(\<Inter>x\<in>C. cons(a, B(x))) = (if C=0 then 0 else cons(a, \<Inter>x\<in>C. B(x)))"
482      "(\<Inter>x\<in>C. A \<union> B(x))  = (if C=0 then 0 else A \<union> (\<Inter>x\<in>C. B(x)))"
484 apply (blast intro!: equalityI)+
485 done
487 lemmas INT_simps [simp] = INT_simps1 INT_simps2
489 text\<open>Opposite of miniscoping: pull the operator out\<close>
492 lemma INT_extend_simps1:
493      "(\<Inter>x\<in>C. A(x)) \<inter> B = (\<Inter>x\<in>C. A(x) \<inter> B)"
494      "(\<Inter>x\<in>C. A(x)) - B = (\<Inter>x\<in>C. A(x) - B)"
495      "(\<Inter>x\<in>C. A(x)) \<union> B  = (if C=0 then B else (\<Inter>x\<in>C. A(x) \<union> B))"
496 apply (simp_all add: Inter_def, blast+)
497 done
499 lemma INT_extend_simps2:
500      "A \<inter> (\<Inter>x\<in>C. B(x)) = (\<Inter>x\<in>C. A \<inter> B(x))"
501      "A - (\<Union>x\<in>C. B(x))   = (if C=0 then A else (\<Inter>x\<in>C. A - B(x)))"
502      "cons(a, \<Inter>x\<in>C. B(x)) = (if C=0 then {a} else (\<Inter>x\<in>C. cons(a, B(x))))"
503      "A \<union> (\<Inter>x\<in>C. B(x))  = (if C=0 then A else (\<Inter>x\<in>C. A \<union> B(x)))"
505 apply (blast intro!: equalityI)+
506 done
508 lemmas INT_extend_simps = INT_extend_simps1 INT_extend_simps2
511 subsection\<open>Other simprules\<close>
514 (*** Miniscoping: pushing in big Unions, Intersections, quantifiers, etc. ***)
516 lemma misc_simps [simp]:
517      "0 \<union> A = A"
518      "A \<union> 0 = A"
519      "0 \<inter> A = 0"
520      "A \<inter> 0 = 0"
521      "0 - A = 0"
522      "A - 0 = A"
523      "\<Union>(0) = 0"
524      "\<Union>(cons(b,A)) = b \<union> \<Union>(A)"
525      "\<Inter>({b}) = b"
526 by blast+
528 end