src/HOL/Set.ML
 author paulson Tue Sep 15 15:04:07 1998 +0200 (1998-09-15) changeset 5490 85855f65d0c6 parent 5450 fe9d103464a4 child 5521 7970832271cc permissions -rw-r--r--
From Compl(A) to -A
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";
16 Goal "P(a) ==> a : {x. P(x)}";
17 by (Asm_simp_tac 1);
18 qed "CollectI";
20 Goal "a : {x. P(x)} ==> P(a)";
21 by (Asm_full_simp_tac 1);
22 qed "CollectD";
24 val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B";
25 by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
26 by (rtac Collect_mem_eq 1);
27 by (rtac Collect_mem_eq 1);
28 qed "set_ext";
30 val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
31 by (rtac (prem RS ext RS arg_cong) 1);
32 qed "Collect_cong";
34 val CollectE = make_elim CollectD;
40 section "Bounded quantifiers";
42 val prems = Goalw [Ball_def]
43     "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
44 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
45 qed "ballI";
47 Goalw [Ball_def] "[| ! x:A. P(x);  x:A |] ==> P(x)";
48 by (Blast_tac 1);
49 qed "bspec";
51 val major::prems = Goalw [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 Goalw [Bex_def] "[| P(x);  x:A |] ==> ? x:A. P(x)";
64 by (Blast_tac 1);
65 qed "bexI";
67 qed_goal "bexCI" Set.thy
68    "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)"
69  (fn prems=>
70   [ (rtac classical 1),
71     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
73 val major::prems = Goalw [Bex_def]
74     "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
75 by (rtac (major RS exE) 1);
76 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
77 qed "bexE";
82 (*Trival rewrite rule*)
83 Goal "(! x:A. P) = ((? x. x:A) --> P)";
84 by (simp_tac (simpset() addsimps [Ball_def]) 1);
85 qed "ball_triv";
87 (*Dual form for existentials*)
88 Goal "(? x:A. P) = ((? x. x:A) & P)";
89 by (simp_tac (simpset() addsimps [Bex_def]) 1);
90 qed "bex_triv";
94 (** Congruence rules **)
96 val prems = Goal
97     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
98 \    (! x:A. P(x)) = (! x:B. Q(x))";
99 by (resolve_tac (prems RL [ssubst]) 1);
100 by (REPEAT (ares_tac [ballI,iffI] 1
101      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
102 qed "ball_cong";
104 val prems = Goal
105     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
106 \    (? x:A. P(x)) = (? x:B. Q(x))";
107 by (resolve_tac (prems RL [ssubst]) 1);
108 by (REPEAT (etac bexE 1
109      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
110 qed "bex_cong";
112 section "Subsets";
114 val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
115 by (REPEAT (ares_tac (prems @ [ballI]) 1));
116 qed "subsetI";
118 (*While (:) is not, its type must be kept
121 seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type))
122     ["Ball", "Bex"];
123 (*need UNION, INTER also?*)
125 (*Image: retain the type of the set being expressed*)
128 (*Rule in Modus Ponens style*)
129 Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
130 by (Blast_tac 1);
131 qed "subsetD";
133 (*The same, with reversed premises for use with etac -- cf rev_mp*)
134 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
135  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
137 (*Converts A<=B to x:A ==> x:B*)
138 fun impOfSubs th = th RSN (2, rev_subsetD);
140 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
141  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
143 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
144  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
146 (*Classical elimination rule*)
147 val major::prems = Goalw [subset_def]
148     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
149 by (rtac (major RS ballE) 1);
150 by (REPEAT (eresolve_tac prems 1));
151 qed "subsetCE";
153 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
154 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
159 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
160  (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
162 Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
163 by (Blast_tac 1);
164 qed "subset_trans";
167 section "Equality";
169 (*Anti-symmetry of the subset relation*)
170 Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
171 by (rtac set_ext 1);
172 by (blast_tac (claset() addIs [subsetD]) 1);
173 qed "subset_antisym";
174 val equalityI = subset_antisym;
178 (* Equality rules from ZF set theory -- are they appropriate here? *)
179 Goal "A = B ==> A<=(B::'a set)";
180 by (etac ssubst 1);
181 by (rtac subset_refl 1);
182 qed "equalityD1";
184 Goal "A = B ==> B<=(A::'a set)";
185 by (etac ssubst 1);
186 by (rtac subset_refl 1);
187 qed "equalityD2";
189 val prems = Goal
190     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
191 by (resolve_tac prems 1);
192 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
193 qed "equalityE";
195 val major::prems = Goal
196     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
197 by (rtac (major RS equalityE) 1);
198 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
199 qed "equalityCE";
201 (*Lemma for creating induction formulae -- for "pattern matching" on p
202   To make the induction hypotheses usable, apply "spec" or "bspec" to
203   put universal quantifiers over the free variables in p. *)
204 val prems = Goal
205     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
206 by (rtac mp 1);
207 by (REPEAT (resolve_tac (refl::prems) 1));
208 qed "setup_induction";
211 section "The universal set -- UNIV";
213 qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
214   (fn _ => [rtac CollectI 1, rtac TrueI 1]);
217 AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
219 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
220   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
222 (** Eta-contracting these two rules (to remove P) causes them to be ignored
223     because of their interaction with congruence rules. **)
225 Goalw [Ball_def] "Ball UNIV P = All P";
226 by (Simp_tac 1);
227 qed "ball_UNIV";
229 Goalw [Bex_def] "Bex UNIV P = Ex P";
230 by (Simp_tac 1);
231 qed "bex_UNIV";
235 section "The empty set -- {}";
237 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
238  (fn _ => [ (Blast_tac 1) ]);
242 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
243  (fn _ => [Full_simp_tac 1]);
247 qed_goal "empty_subsetI" Set.thy "{} <= A"
248  (fn _ => [ (Blast_tac 1) ]);
250 (*One effect is to delete the ASSUMPTION {} <= A*)
253 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
254  (fn [prem]=>
255   [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
257 (*Use for reasoning about disjointness: A Int B = {} *)
258 qed_goal "equals0D" Set.thy "!!a. A={} ==> a ~: A"
259  (fn _ => [ (Blast_tac 1) ]);
261 AddDs [equals0D, sym RS equals0D];
263 Goalw [Ball_def] "Ball {} P = True";
264 by (Simp_tac 1);
265 qed "ball_empty";
267 Goalw [Bex_def] "Bex {} P = False";
268 by (Simp_tac 1);
269 qed "bex_empty";
272 Goal "UNIV ~= {}";
273 by (blast_tac (claset() addEs [equalityE]) 1);
274 qed "UNIV_not_empty";
279 section "The Powerset operator -- Pow";
281 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
282  (fn _ => [ (Asm_simp_tac 1) ]);
286 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
287  (fn _ => [ (etac CollectI 1) ]);
289 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
290  (fn _=> [ (etac CollectD 1) ]);
292 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
293 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
296 section "Set complement -- Compl";
298 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : -A) = (c~:A)"
299  (fn _ => [ (Blast_tac 1) ]);
303 val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
304 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
305 qed "ComplI";
307 (*This form, with negated conclusion, works well with the Classical prover.
308   Negated assumptions behave like formulae on the right side of the notional
309   turnstile...*)
310 Goalw [Compl_def] "c : -A ==> c~:A";
311 by (etac CollectD 1);
312 qed "ComplD";
314 val ComplE = make_elim ComplD;
320 section "Binary union -- Un";
322 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
323  (fn _ => [ Blast_tac 1 ]);
327 Goal "c:A ==> c : A Un B";
328 by (Asm_simp_tac 1);
329 qed "UnI1";
331 Goal "c:B ==> c : A Un B";
332 by (Asm_simp_tac 1);
333 qed "UnI2";
335 (*Classical introduction rule: no commitment to A vs B*)
336 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
337  (fn prems=>
338   [ (Simp_tac 1),
339     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
341 val major::prems = Goalw [Un_def]
342     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
343 by (rtac (major RS CollectD RS disjE) 1);
344 by (REPEAT (eresolve_tac prems 1));
345 qed "UnE";
351 section "Binary intersection -- Int";
353 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
354  (fn _ => [ (Blast_tac 1) ]);
358 Goal "[| c:A;  c:B |] ==> c : A Int B";
359 by (Asm_simp_tac 1);
360 qed "IntI";
362 Goal "c : A Int B ==> c:A";
363 by (Asm_full_simp_tac 1);
364 qed "IntD1";
366 Goal "c : A Int B ==> c:B";
367 by (Asm_full_simp_tac 1);
368 qed "IntD2";
370 val [major,minor] = Goal
371     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
372 by (rtac minor 1);
373 by (rtac (major RS IntD1) 1);
374 by (rtac (major RS IntD2) 1);
375 qed "IntE";
380 section "Set difference";
382 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
383  (fn _ => [ (Blast_tac 1) ]);
387 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
388  (fn _=> [ Asm_simp_tac 1 ]);
390 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
391  (fn _=> [ (Asm_full_simp_tac 1) ]);
393 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
394  (fn _=> [ (Asm_full_simp_tac 1) ]);
396 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
397  (fn prems=>
398   [ (resolve_tac prems 1),
399     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
405 section "Augmenting a set -- insert";
407 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
408  (fn _ => [Blast_tac 1]);
412 qed_goal "insertI1" Set.thy "a : insert a B"
413  (fn _ => [Simp_tac 1]);
415 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
416  (fn _=> [Asm_simp_tac 1]);
418 qed_goalw "insertE" Set.thy [insert_def]
419     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
420  (fn major::prems=>
421   [ (rtac (major RS UnE) 1),
422     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
424 (*Classical introduction rule*)
425 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
426  (fn prems=>
427   [ (Simp_tac 1),
428     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
433 section "Singletons, using insert";
435 qed_goal "singletonI" Set.thy "a : {a}"
436  (fn _=> [ (rtac insertI1 1) ]);
438 Goal "b : {a} ==> b=a";
439 by (Blast_tac 1);
440 qed "singletonD";
442 bind_thm ("singletonE", make_elim singletonD);
444 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
445 (fn _ => [Blast_tac 1]);
447 Goal "{a}={b} ==> a=b";
448 by (blast_tac (claset() addEs [equalityE]) 1);
449 qed "singleton_inject";
451 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
456 Goal "{x. x=a} = {a}";
457 by (Blast_tac 1);
458 qed "singleton_conv";
462 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
464 Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
465 by (Blast_tac 1);
466 qed "UN_iff";
470 (*The order of the premises presupposes that A is rigid; b may be flexible*)
471 Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
472 by Auto_tac;
473 qed "UN_I";
475 val major::prems = Goalw [UNION_def]
476     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
477 by (rtac (major RS CollectD RS bexE) 1);
478 by (REPEAT (ares_tac prems 1));
479 qed "UN_E";
484 val prems = Goal
485     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
486 \    (UN x:A. C(x)) = (UN x:B. D(x))";
487 by (REPEAT (etac UN_E 1
488      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
489                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
490 qed "UN_cong";
493 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
495 Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
496 by Auto_tac;
497 qed "INT_iff";
501 val prems = Goalw [INTER_def]
502     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
503 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
504 qed "INT_I";
506 Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
507 by Auto_tac;
508 qed "INT_D";
510 (*"Classical" elimination -- by the Excluded Middle on a:A *)
511 val major::prems = Goalw [INTER_def]
512     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
513 by (rtac (major RS CollectD RS ballE) 1);
514 by (REPEAT (eresolve_tac prems 1));
515 qed "INT_E";
520 val prems = Goal
521     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
522 \    (INT x:A. C(x)) = (INT x:B. D(x))";
523 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
524 by (REPEAT (dtac INT_D 1
525      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
526 qed "INT_cong";
529 section "Union";
531 Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
532 by (Blast_tac 1);
533 qed "Union_iff";
537 (*The order of the premises presupposes that C is rigid; A may be flexible*)
538 Goal "[| X:C;  A:X |] ==> A : Union(C)";
539 by Auto_tac;
540 qed "UnionI";
542 val major::prems = Goalw [Union_def]
543     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
544 by (rtac (major RS UN_E) 1);
545 by (REPEAT (ares_tac prems 1));
546 qed "UnionE";
552 section "Inter";
554 Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
555 by (Blast_tac 1);
556 qed "Inter_iff";
560 val prems = Goalw [Inter_def]
561     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
562 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
563 qed "InterI";
565 (*A "destruct" rule -- every X in C contains A as an element, but
566   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
567 Goal "[| A : Inter(C);  X:C |] ==> A:X";
568 by Auto_tac;
569 qed "InterD";
571 (*"Classical" elimination rule -- does not require proving X:C *)
572 val major::prems = Goalw [Inter_def]
573     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
574 by (rtac (major RS INT_E) 1);
575 by (REPEAT (eresolve_tac prems 1));
576 qed "InterE";
582 (*** Image of a set under a function ***)
584 (*Frequently b does not have the syntactic form of f(x).*)
585 Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
586 by (Blast_tac 1);
587 qed "image_eqI";
590 bind_thm ("imageI", refl RS image_eqI);
592 (*The eta-expansion gives variable-name preservation.*)
593 val major::prems = Goalw [image_def]
594     "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P";
595 by (rtac (major RS CollectD RS bexE) 1);
596 by (REPEAT (ares_tac prems 1));
597 qed "imageE";
602 Goal "f``(A Un B) = f``A Un f``B";
603 by (Blast_tac 1);
604 qed "image_Un";
606 Goal "(z : f``A) = (EX x:A. z = f x)";
607 by (Blast_tac 1);
608 qed "image_iff";
610 (*This rewrite rule would confuse users if made default.*)
611 Goal "(f``A <= B) = (ALL x:A. f(x): B)";
612 by (Blast_tac 1);
613 qed "image_subset_iff";
615 (*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
616   many existing proofs.*)
617 val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
618 by (blast_tac (claset() addIs prems) 1);
619 qed "image_subsetI";
622 (*** Range of a function -- just a translation for image! ***)
624 Goal "b=f(x) ==> b : range(f)";
625 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
626 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
628 bind_thm ("rangeI", UNIV_I RS imageI);
630 val [major,minor] = Goal
631     "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P";
632 by (rtac (major RS imageE) 1);
633 by (etac minor 1);
634 qed "rangeE";
637 (*** Set reasoning tools ***)
640 (** Rewrite rules for boolean case-splitting: faster than
642 **)
644 bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
645 bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
647 (*Split ifs on either side of the membership relation.
648 	Not for Addsimps -- can cause goals to blow up!*)
649 bind_thm ("split_if_mem1",
650     read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
651 bind_thm ("split_if_mem2",
652     read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
654 val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
655 		  split_if_mem1, split_if_mem2];
659 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
660                  mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
662 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
664 simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
665                     setmksimps (mksimps mksimps_pairs);