src/HOL/Set.ML
 author paulson Wed Mar 05 09:59:55 1997 +0100 (1997-03-05) changeset 2721 f08042e18c7d parent 2608 450c9b682a92 child 2858 1f3f5c44e159 permissions -rw-r--r--
New version of InterE, like its ZF counterpart
1 (*  Title:      HOL/set
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   1991  University of Cambridge
6 Set theory for higher-order logic.  A set is simply a predicate.
7 *)
9 open Set;
11 section "Relating predicates and sets";
15 goal Set.thy "!!a. P(a) ==> a : {x.P(x)}";
16 by (Asm_simp_tac 1);
17 qed "CollectI";
19 val prems = goal Set.thy "!!a. a : {x.P(x)} ==> P(a)";
20 by (Asm_full_simp_tac 1);
21 qed "CollectD";
23 val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B";
24 by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
25 by (rtac Collect_mem_eq 1);
26 by (rtac Collect_mem_eq 1);
27 qed "set_ext";
29 val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
30 by (rtac (prem RS ext RS arg_cong) 1);
31 qed "Collect_cong";
33 val CollectE = make_elim CollectD;
39 section "Bounded quantifiers";
41 val prems = goalw Set.thy [Ball_def]
42     "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
43 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
44 qed "ballI";
46 val [major,minor] = goalw Set.thy [Ball_def]
47     "[| ! x:A. P(x);  x:A |] ==> P(x)";
48 by (rtac (minor RS (major RS spec RS mp)) 1);
49 qed "bspec";
51 val major::prems = goalw Set.thy [Ball_def]
52     "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
53 by (rtac (major RS spec RS impCE) 1);
54 by (REPEAT (eresolve_tac prems 1));
55 qed "ballE";
57 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
58 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
63 val prems = goalw Set.thy [Bex_def]
64     "[| P(x);  x:A |] ==> ? x:A. P(x)";
65 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
66 qed "bexI";
68 qed_goal "bexCI" Set.thy
69    "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A.P(x)"
70  (fn prems=>
71   [ (rtac classical 1),
72     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
74 val major::prems = goalw Set.thy [Bex_def]
75     "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
76 by (rtac (major RS exE) 1);
77 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
78 qed "bexE";
83 (*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
84 goalw Set.thy [Ball_def] "(! x:A. True) = True";
85 by (Simp_tac 1);
86 qed "ball_True";
88 (*Dual form for existentials*)
89 goalw Set.thy [Bex_def] "(? x:A. False) = False";
90 by (Simp_tac 1);
91 qed "bex_False";
95 (** Congruence rules **)
97 val prems = goal Set.thy
98     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
99 \    (! x:A. P(x)) = (! x:B. Q(x))";
100 by (resolve_tac (prems RL [ssubst]) 1);
101 by (REPEAT (ares_tac [ballI,iffI] 1
102      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
103 qed "ball_cong";
105 val prems = goal Set.thy
106     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
107 \    (? x:A. P(x)) = (? x:B. Q(x))";
108 by (resolve_tac (prems RL [ssubst]) 1);
109 by (REPEAT (etac bexE 1
110      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
111 qed "bex_cong";
113 section "Subsets";
115 val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
116 by (REPEAT (ares_tac (prems @ [ballI]) 1));
117 qed "subsetI";
119 (*Rule in Modus Ponens style*)
120 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
121 by (rtac (major RS bspec) 1);
122 by (resolve_tac prems 1);
123 qed "subsetD";
125 (*The same, with reversed premises for use with etac -- cf rev_mp*)
126 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
127  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
129 (*Converts A<=B to x:A ==> x:B*)
130 fun impOfSubs th = th RSN (2, rev_subsetD);
132 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
133  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
135 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
136  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
138 (*Classical elimination rule*)
139 val major::prems = goalw Set.thy [subset_def]
140     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
141 by (rtac (major RS ballE) 1);
142 by (REPEAT (eresolve_tac prems 1));
143 qed "subsetCE";
145 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
146 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
151 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
152  (fn _=> [Fast_tac 1]);
154 val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
155 by (Fast_tac 1);
156 qed "subset_trans";
159 section "Equality";
161 (*Anti-symmetry of the subset relation*)
162 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
163 by (rtac (iffI RS set_ext) 1);
164 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
165 qed "subset_antisym";
166 val equalityI = subset_antisym;
170 (* Equality rules from ZF set theory -- are they appropriate here? *)
171 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
172 by (resolve_tac (prems RL [subst]) 1);
173 by (rtac subset_refl 1);
174 qed "equalityD1";
176 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
177 by (resolve_tac (prems RL [subst]) 1);
178 by (rtac subset_refl 1);
179 qed "equalityD2";
181 val prems = goal Set.thy
182     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
183 by (resolve_tac prems 1);
184 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
185 qed "equalityE";
187 val major::prems = goal Set.thy
188     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
189 by (rtac (major RS equalityE) 1);
190 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
191 qed "equalityCE";
193 (*Lemma for creating induction formulae -- for "pattern matching" on p
194   To make the induction hypotheses usable, apply "spec" or "bspec" to
195   put universal quantifiers over the free variables in p. *)
196 val prems = goal Set.thy
197     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
198 by (rtac mp 1);
199 by (REPEAT (resolve_tac (refl::prems) 1));
200 qed "setup_induction";
203 section "Set complement -- Compl";
205 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
206  (fn _ => [ (Fast_tac 1) ]);
210 val prems = goalw Set.thy [Compl_def]
211     "[| c:A ==> False |] ==> c : Compl(A)";
212 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
213 qed "ComplI";
215 (*This form, with negated conclusion, works well with the Classical prover.
216   Negated assumptions behave like formulae on the right side of the notional
217   turnstile...*)
218 val major::prems = goalw Set.thy [Compl_def]
219     "c : Compl(A) ==> c~:A";
220 by (rtac (major RS CollectD) 1);
221 qed "ComplD";
223 val ComplE = make_elim ComplD;
229 section "Binary union -- Un";
231 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
232  (fn _ => [ Fast_tac 1 ]);
236 goal Set.thy "!!c. c:A ==> c : A Un B";
237 by (Asm_simp_tac 1);
238 qed "UnI1";
240 goal Set.thy "!!c. c:B ==> c : A Un B";
241 by (Asm_simp_tac 1);
242 qed "UnI2";
244 (*Classical introduction rule: no commitment to A vs B*)
245 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
246  (fn prems=>
247   [ (Simp_tac 1),
248     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
250 val major::prems = goalw Set.thy [Un_def]
251     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
252 by (rtac (major RS CollectD RS disjE) 1);
253 by (REPEAT (eresolve_tac prems 1));
254 qed "UnE";
260 section "Binary intersection -- Int";
262 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
263  (fn _ => [ (Fast_tac 1) ]);
267 goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
268 by (Asm_simp_tac 1);
269 qed "IntI";
271 goal Set.thy "!!c. c : A Int B ==> c:A";
272 by (Asm_full_simp_tac 1);
273 qed "IntD1";
275 goal Set.thy "!!c. c : A Int B ==> c:B";
276 by (Asm_full_simp_tac 1);
277 qed "IntD2";
279 val [major,minor] = goal Set.thy
280     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
281 by (rtac minor 1);
282 by (rtac (major RS IntD1) 1);
283 by (rtac (major RS IntD2) 1);
284 qed "IntE";
289 section "Set difference";
291 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
292  (fn _ => [ (Fast_tac 1) ]);
296 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
297  (fn _=> [ Asm_simp_tac 1 ]);
299 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
300  (fn _=> [ (Asm_full_simp_tac 1) ]);
302 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
303  (fn _=> [ (Asm_full_simp_tac 1) ]);
305 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
306  (fn prems=>
307   [ (resolve_tac prems 1),
308     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
313 section "The empty set -- {}";
315 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
316  (fn _ => [ (Fast_tac 1) ]);
320 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
321  (fn _ => [Full_simp_tac 1]);
325 qed_goal "empty_subsetI" Set.thy "{} <= A"
326  (fn _ => [ (Fast_tac 1) ]);
328 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
329  (fn [prem]=>
330   [ (fast_tac (!claset addIs [prem RS FalseE]) 1) ]);
332 qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
333  (fn _ => [ (Fast_tac 1) ]);
335 goal Set.thy "Ball {} P = True";
336 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
337 qed "ball_empty";
339 goal Set.thy "Bex {} P = False";
340 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
341 qed "bex_empty";
344 goalw Set.thy [Ball_def] "(!x:A.False) = (A = {})";
345 by(Fast_tac 1);
346 qed "ball_False";
349 (* The dual is probably not helpful:
350 goal Set.thy "(? x:A.True) = (A ~= {})";
351 by(Fast_tac 1);
352 qed "bex_True";
354 *)
357 section "Augmenting a set -- insert";
359 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
360  (fn _ => [Fast_tac 1]);
364 qed_goal "insertI1" Set.thy "a : insert a B"
365  (fn _ => [Simp_tac 1]);
367 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
368  (fn _=> [Asm_simp_tac 1]);
370 qed_goalw "insertE" Set.thy [insert_def]
371     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
372  (fn major::prems=>
373   [ (rtac (major RS UnE) 1),
374     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
376 (*Classical introduction rule*)
377 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
378  (fn prems=>
379   [ (Simp_tac 1),
380     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
385 section "Singletons, using insert";
387 qed_goal "singletonI" Set.thy "a : {a}"
388  (fn _=> [ (rtac insertI1 1) ]);
390 goal Set.thy "!!a. b : {a} ==> b=a";
391 by (Fast_tac 1);
392 qed "singletonD";
394 bind_thm ("singletonE", make_elim singletonD);
396 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
397 (fn _ => [Fast_tac 1]);
399 goal Set.thy "!!a b. {a}={b} ==> a=b";
400 by (fast_tac (!claset addEs [equalityE]) 1);
401 qed "singleton_inject";
406 section "The universal set -- UNIV";
408 qed_goal "UNIV_I" Set.thy "x : UNIV"
409   (fn _ => [rtac ComplI 1, etac emptyE 1]);
411 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
412   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
415 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
417 goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
418 by (Fast_tac 1);
419 qed "UN_iff";
423 (*The order of the premises presupposes that A is rigid; b may be flexible*)
424 goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
425 by (Auto_tac());
426 qed "UN_I";
428 val major::prems = goalw Set.thy [UNION_def]
429     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
430 by (rtac (major RS CollectD RS bexE) 1);
431 by (REPEAT (ares_tac prems 1));
432 qed "UN_E";
437 val prems = goal Set.thy
438     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
439 \    (UN x:A. C(x)) = (UN x:B. D(x))";
440 by (REPEAT (etac UN_E 1
441      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
442                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
443 qed "UN_cong";
446 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
448 goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
449 by (Auto_tac());
450 qed "INT_iff";
454 val prems = goalw Set.thy [INTER_def]
455     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
456 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
457 qed "INT_I";
459 goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
460 by (Auto_tac());
461 qed "INT_D";
463 (*"Classical" elimination -- by the Excluded Middle on a:A *)
464 val major::prems = goalw Set.thy [INTER_def]
465     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
466 by (rtac (major RS CollectD RS ballE) 1);
467 by (REPEAT (eresolve_tac prems 1));
468 qed "INT_E";
473 val prems = goal Set.thy
474     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
475 \    (INT x:A. C(x)) = (INT x:B. D(x))";
476 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
477 by (REPEAT (dtac INT_D 1
478      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
479 qed "INT_cong";
482 section "Unions over a type; UNION1(B) = Union(range(B))";
484 goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
485 by (Simp_tac 1);
486 by (Fast_tac 1);
487 qed "UN1_iff";
491 (*The order of the premises presupposes that A is rigid; b may be flexible*)
492 goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
493 by (Auto_tac());
494 qed "UN1_I";
496 val major::prems = goalw Set.thy [UNION1_def]
497     "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
498 by (rtac (major RS UN_E) 1);
499 by (REPEAT (ares_tac prems 1));
500 qed "UN1_E";
506 section "Intersections over a type; INTER1(B) = Inter(range(B))";
508 goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
509 by (Simp_tac 1);
510 by (Fast_tac 1);
511 qed "INT1_iff";
515 val prems = goalw Set.thy [INTER1_def]
516     "(!!x. b: B(x)) ==> b : (INT x. B(x))";
517 by (REPEAT (ares_tac (INT_I::prems) 1));
518 qed "INT1_I";
520 goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
521 by (Asm_full_simp_tac 1);
522 qed "INT1_D";
528 section "Union";
530 goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
531 by (Fast_tac 1);
532 qed "Union_iff";
536 (*The order of the premises presupposes that C is rigid; A may be flexible*)
537 goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
538 by (Auto_tac());
539 qed "UnionI";
541 val major::prems = goalw Set.thy [Union_def]
542     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
543 by (rtac (major RS UN_E) 1);
544 by (REPEAT (ares_tac prems 1));
545 qed "UnionE";
551 section "Inter";
553 goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
554 by (Fast_tac 1);
555 qed "Inter_iff";
559 val prems = goalw Set.thy [Inter_def]
560     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
561 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
562 qed "InterI";
564 (*A "destruct" rule -- every X in C contains A as an element, but
565   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
566 goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
567 by (Auto_tac());
568 qed "InterD";
570 (*"Classical" elimination rule -- does not require proving X:C *)
571 val major::prems = goalw Set.thy [Inter_def]
572     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
573 by (rtac (major RS INT_E) 1);
574 by (REPEAT (eresolve_tac prems 1));
575 qed "InterE";
581 section "The Powerset operator -- Pow";
583 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
584  (fn _ => [ (Asm_simp_tac 1) ]);
588 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
589  (fn _ => [ (etac CollectI 1) ]);
591 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
592  (fn _=> [ (etac CollectD 1) ]);
594 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
595 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
599 (*** Set reasoning tools ***)