src/CCL/Set.ML
 author paulson Fri Feb 16 17:24:51 1996 +0100 (1996-02-16) changeset 1511 09354d37a5ab parent 1459 d12da312eff4 child 3837 d7f033c74b38 permissions -rw-r--r--
Elimination of fully-functorial style.
Type tactic changed to a type abbrevation (from a datatype).
Constructor tactic and function apply deleted.
1 (*  Title:      set/set
2     ID:         \$Id\$
4 For set.thy.
6 Modified version of
7     Title:      HOL/set
8     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
9     Copyright   1991  University of Cambridge
11 For set.thy.  Set theory for higher-order logic.  A set is simply a predicate.
12 *)
14 open Set;
16 val [prem] = goal Set.thy "[| P(a) |] ==> a : {x.P(x)}";
17 by (rtac (mem_Collect_iff RS iffD2) 1);
18 by (rtac prem 1);
19 qed "CollectI";
21 val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)";
22 by (resolve_tac (prems RL [mem_Collect_iff  RS iffD1]) 1);
23 qed "CollectD";
25 val CollectE = make_elim CollectD;
27 val [prem] = goal Set.thy "[| !!x. x:A <-> x:B |] ==> A = B";
28 by (rtac (set_extension RS iffD2) 1);
29 by (rtac (prem RS allI) 1);
30 qed "set_ext";
32 (*** Bounded quantifiers ***)
34 val prems = goalw Set.thy [Ball_def]
35     "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)";
36 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
37 qed "ballI";
39 val [major,minor] = goalw Set.thy [Ball_def]
40     "[| ALL x:A. P(x);  x:A |] ==> P(x)";
41 by (rtac (minor RS (major RS spec RS mp)) 1);
42 qed "bspec";
44 val major::prems = goalw Set.thy [Ball_def]
45     "[| ALL x:A. P(x);  P(x) ==> Q;  ~ x:A ==> Q |] ==> Q";
46 by (rtac (major RS spec RS impCE) 1);
47 by (REPEAT (eresolve_tac prems 1));
48 qed "ballE";
50 (*Takes assumptions ALL x:A.P(x) and a:A; creates assumption P(a)*)
51 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
53 val prems = goalw Set.thy [Bex_def]
54     "[| P(x);  x:A |] ==> EX x:A. P(x)";
55 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
56 qed "bexI";
58 qed_goal "bexCI" Set.thy
59    "[| EX x:A. ~P(x) ==> P(a);  a:A |] ==> EX x:A.P(x)"
60  (fn prems=>
61   [ (rtac classical 1),
62     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
64 val major::prems = goalw Set.thy [Bex_def]
65     "[| EX x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
66 by (rtac (major RS exE) 1);
67 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
68 qed "bexE";
70 (*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
71 val prems = goal Set.thy
72     "(ALL x:A. True) <-> True";
73 by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
74 qed "ball_rew";
76 (** Congruence rules **)
78 val prems = goal Set.thy
79     "[| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==> \
80 \    (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))";
81 by (resolve_tac (prems RL [ssubst,iffD2]) 1);
82 by (REPEAT (ares_tac [ballI,iffI] 1
83      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
84 qed "ball_cong";
86 val prems = goal Set.thy
87     "[| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==> \
88 \    (EX x:A. P(x)) <-> (EX x:A'. P'(x))";
89 by (resolve_tac (prems RL [ssubst,iffD2]) 1);
90 by (REPEAT (etac bexE 1
91      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
92 qed "bex_cong";
94 (*** Rules for subsets ***)
96 val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
97 by (REPEAT (ares_tac (prems @ [ballI]) 1));
98 qed "subsetI";
100 (*Rule in Modus Ponens style*)
101 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
102 by (rtac (major RS bspec) 1);
103 by (resolve_tac prems 1);
104 qed "subsetD";
106 (*Classical elimination rule*)
107 val major::prems = goalw Set.thy [subset_def]
108     "[| A <= B;  ~(c:A) ==> P;  c:B ==> P |] ==> P";
109 by (rtac (major RS ballE) 1);
110 by (REPEAT (eresolve_tac prems 1));
111 qed "subsetCE";
113 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
114 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
116 qed_goal "subset_refl" Set.thy "A <= A"
117  (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]);
119 goal Set.thy "!!A B C. [| A<=B;  B<=C |] ==> A<=C";
120 by (rtac subsetI 1);
121 by (REPEAT (eresolve_tac [asm_rl, subsetD] 1));
122 qed "subset_trans";
125 (*** Rules for equality ***)
127 (*Anti-symmetry of the subset relation*)
128 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = B";
129 by (rtac (iffI RS set_ext) 1);
130 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
131 qed "subset_antisym";
132 val equalityI = subset_antisym;
134 (* Equality rules from ZF set theory -- are they appropriate here? *)
135 val prems = goal Set.thy "A = B ==> A<=B";
136 by (resolve_tac (prems RL [subst]) 1);
137 by (rtac subset_refl 1);
138 qed "equalityD1";
140 val prems = goal Set.thy "A = B ==> B<=A";
141 by (resolve_tac (prems RL [subst]) 1);
142 by (rtac subset_refl 1);
143 qed "equalityD2";
145 val prems = goal Set.thy
146     "[| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P";
147 by (resolve_tac prems 1);
148 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
149 qed "equalityE";
151 val major::prems = goal Set.thy
152     "[| A = B;  [| c:A; c:B |] ==> P;  [| ~ c:A; ~ c:B |] ==> P |]  ==>  P";
153 by (rtac (major RS equalityE) 1);
154 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
155 qed "equalityCE";
157 (*Lemma for creating induction formulae -- for "pattern matching" on p
158   To make the induction hypotheses usable, apply "spec" or "bspec" to
159   put universal quantifiers over the free variables in p. *)
160 val prems = goal Set.thy
161     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
162 by (rtac mp 1);
163 by (REPEAT (resolve_tac (refl::prems) 1));
164 qed "setup_induction";
166 goal Set.thy "{x.x:A} = A";
167 by (REPEAT (ares_tac [equalityI,subsetI,CollectI] 1  ORELSE etac CollectD 1));
168 qed "trivial_set";
170 (*** Rules for binary union -- Un ***)
172 val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
173 by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
174 qed "UnI1";
176 val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
177 by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
178 qed "UnI2";
180 (*Classical introduction rule: no commitment to A vs B*)
181 qed_goal "UnCI" Set.thy "(~c:B ==> c:A) ==> c : A Un B"
182  (fn prems=>
183   [ (rtac classical 1),
184     (REPEAT (ares_tac (prems@[UnI1,notI]) 1)),
185     (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]);
187 val major::prems = goalw Set.thy [Un_def]
188     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
189 by (rtac (major RS CollectD RS disjE) 1);
190 by (REPEAT (eresolve_tac prems 1));
191 qed "UnE";
194 (*** Rules for small intersection -- Int ***)
196 val prems = goalw Set.thy [Int_def]
197     "[| c:A;  c:B |] ==> c : A Int B";
198 by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
199 qed "IntI";
201 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
202 by (rtac (major RS CollectD RS conjunct1) 1);
203 qed "IntD1";
205 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
206 by (rtac (major RS CollectD RS conjunct2) 1);
207 qed "IntD2";
209 val [major,minor] = goal Set.thy
210     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
211 by (rtac minor 1);
212 by (rtac (major RS IntD1) 1);
213 by (rtac (major RS IntD2) 1);
214 qed "IntE";
217 (*** Rules for set complement -- Compl ***)
219 val prems = goalw Set.thy [Compl_def]
220     "[| c:A ==> False |] ==> c : Compl(A)";
221 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
222 qed "ComplI";
224 (*This form, with negated conclusion, works well with the Classical prover.
225   Negated assumptions behave like formulae on the right side of the notional
226   turnstile...*)
227 val major::prems = goalw Set.thy [Compl_def]
228     "[| c : Compl(A) |] ==> ~c:A";
229 by (rtac (major RS CollectD) 1);
230 qed "ComplD";
232 val ComplE = make_elim ComplD;
235 (*** Empty sets ***)
237 goalw Set.thy [empty_def] "{x.False} = {}";
238 by (rtac refl 1);
239 qed "empty_eq";
241 val [prem] = goalw Set.thy [empty_def] "a : {} ==> P";
242 by (rtac (prem RS CollectD RS FalseE) 1);
243 qed "emptyD";
245 val emptyE = make_elim emptyD;
247 val [prem] = goal Set.thy "~ A={} ==> (EX x.x:A)";
248 by (rtac (prem RS swap) 1);
249 by (rtac equalityI 1);
251 qed "not_emptyD";
253 (*** Singleton sets ***)
255 goalw Set.thy [singleton_def] "a : {a}";
256 by (rtac CollectI 1);
257 by (rtac refl 1);
258 qed "singletonI";
260 val [major] = goalw Set.thy [singleton_def] "b : {a} ==> b=a";
261 by (rtac (major RS CollectD) 1);
262 qed "singletonD";
264 val singletonE = make_elim singletonD;
266 (*** Unions of families ***)
268 (*The order of the premises presupposes that A is rigid; b may be flexible*)
269 val prems = goalw Set.thy [UNION_def]
270     "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
271 by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
272 qed "UN_I";
274 val major::prems = goalw Set.thy [UNION_def]
275     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
276 by (rtac (major RS CollectD RS bexE) 1);
277 by (REPEAT (ares_tac prems 1));
278 qed "UN_E";
280 val prems = goal Set.thy
281     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
282 \    (UN x:A. C(x)) = (UN x:B. D(x))";
283 by (REPEAT (etac UN_E 1
284      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
285                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
286 qed "UN_cong";
288 (*** Intersections of families -- INTER x:A. B(x) is Inter(B)``A ) *)
290 val prems = goalw Set.thy [INTER_def]
291     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
292 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
293 qed "INT_I";
295 val major::prems = goalw Set.thy [INTER_def]
296     "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
297 by (rtac (major RS CollectD RS bspec) 1);
298 by (resolve_tac prems 1);
299 qed "INT_D";
301 (*"Classical" elimination rule -- does not require proving X:C *)
302 val major::prems = goalw Set.thy [INTER_def]
303     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  ~ a:A ==> R |] ==> R";
304 by (rtac (major RS CollectD RS ballE) 1);
305 by (REPEAT (eresolve_tac prems 1));
306 qed "INT_E";
308 val prems = goal Set.thy
309     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
310 \    (INT x:A. C(x)) = (INT x:B. D(x))";
311 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
312 by (REPEAT (dtac INT_D 1
313      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
314 qed "INT_cong";
316 (*** Rules for Unions ***)
318 (*The order of the premises presupposes that C is rigid; A may be flexible*)
319 val prems = goalw Set.thy [Union_def]
320     "[| X:C;  A:X |] ==> A : Union(C)";
321 by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
322 qed "UnionI";
324 val major::prems = goalw Set.thy [Union_def]
325     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
326 by (rtac (major RS UN_E) 1);
327 by (REPEAT (ares_tac prems 1));
328 qed "UnionE";
330 (*** Rules for Inter ***)
332 val prems = goalw Set.thy [Inter_def]
333     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
334 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
335 qed "InterI";
337 (*A "destruct" rule -- every X in C contains A as an element, but
338   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
339 val major::prems = goalw Set.thy [Inter_def]
340     "[| A : Inter(C);  X:C |] ==> A:X";
341 by (rtac (major RS INT_D) 1);
342 by (resolve_tac prems 1);
343 qed "InterD";
345 (*"Classical" elimination rule -- does not require proving X:C *)
346 val major::prems = goalw Set.thy [Inter_def]
347     "[| A : Inter(C);  A:X ==> R;  ~ X:C ==> R |] ==> R";
348 by (rtac (major RS INT_E) 1);
349 by (REPEAT (eresolve_tac prems 1));
350 qed "InterE";