src/CCL/set.ML
changeset 0 a5a9c433f639
child 8 c3d2c6dcf3f0
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/CCL/set.ML	Thu Sep 16 12:20:38 1993 +0200
     1.3 @@ -0,0 +1,355 @@
     1.4 +(*  Title: 	set/set
     1.5 +    ID:         $Id$
     1.6 +
     1.7 +For set.thy.
     1.8 +
     1.9 +Modified version of
    1.10 +    Title: 	HOL/set
    1.11 +    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    1.12 +    Copyright   1991  University of Cambridge
    1.13 +
    1.14 +For set.thy.  Set theory for higher-order logic.  A set is simply a predicate.
    1.15 +*)
    1.16 +
    1.17 +open Set;
    1.18 +
    1.19 +val [prem] = goal Set.thy "[| P(a) |] ==> a : {x.P(x)}";
    1.20 +by (rtac (mem_Collect_iff RS iffD2) 1);
    1.21 +by (rtac prem 1);
    1.22 +val CollectI = result();
    1.23 +
    1.24 +val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)";
    1.25 +by (resolve_tac (prems RL [mem_Collect_iff  RS iffD1]) 1);
    1.26 +val CollectD = result();
    1.27 +
    1.28 +val [prem] = goal Set.thy "[| !!x. x:A <-> x:B |] ==> A = B";
    1.29 +by (rtac (set_extension RS iffD2) 1);
    1.30 +by (rtac (prem RS allI) 1);
    1.31 +val set_ext = result();
    1.32 +
    1.33 +val prems = goal Set.thy "[| !!x. P(x) <-> Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
    1.34 +by (REPEAT (ares_tac [set_ext,iffI,CollectI] 1 ORELSE
    1.35 +            eresolve_tac ([CollectD] RL (prems RL [iffD1,iffD2])) 1));
    1.36 +val Collect_cong = result();
    1.37 +
    1.38 +val CollectE = make_elim CollectD;
    1.39 +
    1.40 +(*** Bounded quantifiers ***)
    1.41 +
    1.42 +val prems = goalw Set.thy [Ball_def]
    1.43 +    "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)";
    1.44 +by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
    1.45 +val ballI = result();
    1.46 +
    1.47 +val [major,minor] = goalw Set.thy [Ball_def]
    1.48 +    "[| ALL x:A. P(x);  x:A |] ==> P(x)";
    1.49 +by (rtac (minor RS (major RS spec RS mp)) 1);
    1.50 +val bspec = result();
    1.51 +
    1.52 +val major::prems = goalw Set.thy [Ball_def]
    1.53 +    "[| ALL x:A. P(x);  P(x) ==> Q;  ~ x:A ==> Q |] ==> Q";
    1.54 +by (rtac (major RS spec RS impCE) 1);
    1.55 +by (REPEAT (eresolve_tac prems 1));
    1.56 +val ballE = result();
    1.57 +
    1.58 +(*Takes assumptions ALL x:A.P(x) and a:A; creates assumption P(a)*)
    1.59 +fun ball_tac i = etac ballE i THEN contr_tac (i+1);
    1.60 +
    1.61 +val prems = goalw Set.thy [Bex_def]
    1.62 +    "[| P(x);  x:A |] ==> EX x:A. P(x)";
    1.63 +by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
    1.64 +val bexI = result();
    1.65 +
    1.66 +val bexCI = prove_goal Set.thy 
    1.67 +   "[| EX x:A. ~P(x) ==> P(a);  a:A |] ==> EX x:A.P(x)"
    1.68 + (fn prems=>
    1.69 +  [ (rtac classical 1),
    1.70 +    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
    1.71 +
    1.72 +val major::prems = goalw Set.thy [Bex_def]
    1.73 +    "[| EX x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
    1.74 +by (rtac (major RS exE) 1);
    1.75 +by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
    1.76 +val bexE = result();
    1.77 +
    1.78 +(*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
    1.79 +val prems = goal Set.thy
    1.80 +    "(ALL x:A. True) <-> True";
    1.81 +by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
    1.82 +val ball_rew = result();
    1.83 +
    1.84 +(** Congruence rules **)
    1.85 +
    1.86 +val prems = goal Set.thy
    1.87 +    "[| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==> \
    1.88 +\    (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))";
    1.89 +by (resolve_tac (prems RL [ssubst,iffD2]) 1);
    1.90 +by (REPEAT (ares_tac [ballI,iffI] 1
    1.91 +     ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
    1.92 +val ball_cong = result();
    1.93 +
    1.94 +val prems = goal Set.thy
    1.95 +    "[| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==> \
    1.96 +\    (EX x:A. P(x)) <-> (EX x:A'. P'(x))";
    1.97 +by (resolve_tac (prems RL [ssubst,iffD2]) 1);
    1.98 +by (REPEAT (etac bexE 1
    1.99 +     ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
   1.100 +val bex_cong = result();
   1.101 +
   1.102 +(*** Rules for subsets ***)
   1.103 +
   1.104 +val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
   1.105 +by (REPEAT (ares_tac (prems @ [ballI]) 1));
   1.106 +val subsetI = result();
   1.107 +
   1.108 +(*Rule in Modus Ponens style*)
   1.109 +val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
   1.110 +by (rtac (major RS bspec) 1);
   1.111 +by (resolve_tac prems 1);
   1.112 +val subsetD = result();
   1.113 +
   1.114 +(*Classical elimination rule*)
   1.115 +val major::prems = goalw Set.thy [subset_def] 
   1.116 +    "[| A <= B;  ~(c:A) ==> P;  c:B ==> P |] ==> P";
   1.117 +by (rtac (major RS ballE) 1);
   1.118 +by (REPEAT (eresolve_tac prems 1));
   1.119 +val subsetCE = result();
   1.120 +
   1.121 +(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
   1.122 +fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
   1.123 +
   1.124 +val subset_refl = prove_goal Set.thy "A <= A"
   1.125 + (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]);
   1.126 +
   1.127 +goal Set.thy "!!A B C. [| A<=B;  B<=C |] ==> A<=C";
   1.128 +br subsetI 1;
   1.129 +by (REPEAT (eresolve_tac [asm_rl, subsetD] 1));
   1.130 +val subset_trans = result();
   1.131 +
   1.132 +
   1.133 +(*** Rules for equality ***)
   1.134 +
   1.135 +(*Anti-symmetry of the subset relation*)
   1.136 +val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = B";
   1.137 +by (rtac (iffI RS set_ext) 1);
   1.138 +by (REPEAT (ares_tac (prems RL [subsetD]) 1));
   1.139 +val subset_antisym = result();
   1.140 +val equalityI = subset_antisym;
   1.141 +
   1.142 +(* Equality rules from ZF set theory -- are they appropriate here? *)
   1.143 +val prems = goal Set.thy "A = B ==> A<=B";
   1.144 +by (resolve_tac (prems RL [subst]) 1);
   1.145 +by (rtac subset_refl 1);
   1.146 +val equalityD1 = result();
   1.147 +
   1.148 +val prems = goal Set.thy "A = B ==> B<=A";
   1.149 +by (resolve_tac (prems RL [subst]) 1);
   1.150 +by (rtac subset_refl 1);
   1.151 +val equalityD2 = result();
   1.152 +
   1.153 +val prems = goal Set.thy
   1.154 +    "[| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P";
   1.155 +by (resolve_tac prems 1);
   1.156 +by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
   1.157 +val equalityE = result();
   1.158 +
   1.159 +val major::prems = goal Set.thy
   1.160 +    "[| A = B;  [| c:A; c:B |] ==> P;  [| ~ c:A; ~ c:B |] ==> P |]  ==>  P";
   1.161 +by (rtac (major RS equalityE) 1);
   1.162 +by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
   1.163 +val equalityCE = result();
   1.164 +
   1.165 +(*Lemma for creating induction formulae -- for "pattern matching" on p
   1.166 +  To make the induction hypotheses usable, apply "spec" or "bspec" to
   1.167 +  put universal quantifiers over the free variables in p. *)
   1.168 +val prems = goal Set.thy 
   1.169 +    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
   1.170 +by (rtac mp 1);
   1.171 +by (REPEAT (resolve_tac (refl::prems) 1));
   1.172 +val setup_induction = result();
   1.173 +
   1.174 +goal Set.thy "{x.x:A} = A";
   1.175 +by (REPEAT (ares_tac [equalityI,subsetI,CollectI] 1  ORELSE eresolve_tac [CollectD] 1));
   1.176 +val trivial_set = result();
   1.177 +
   1.178 +(*** Rules for binary union -- Un ***)
   1.179 +
   1.180 +val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
   1.181 +by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
   1.182 +val UnI1 = result();
   1.183 +
   1.184 +val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
   1.185 +by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
   1.186 +val UnI2 = result();
   1.187 +
   1.188 +(*Classical introduction rule: no commitment to A vs B*)
   1.189 +val UnCI = prove_goal Set.thy "(~c:B ==> c:A) ==> c : A Un B"
   1.190 + (fn prems=>
   1.191 +  [ (rtac classical 1),
   1.192 +    (REPEAT (ares_tac (prems@[UnI1,notI]) 1)),
   1.193 +    (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]);
   1.194 +
   1.195 +val major::prems = goalw Set.thy [Un_def]
   1.196 +    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
   1.197 +by (rtac (major RS CollectD RS disjE) 1);
   1.198 +by (REPEAT (eresolve_tac prems 1));
   1.199 +val UnE = result();
   1.200 +
   1.201 +
   1.202 +(*** Rules for small intersection -- Int ***)
   1.203 +
   1.204 +val prems = goalw Set.thy [Int_def]
   1.205 +    "[| c:A;  c:B |] ==> c : A Int B";
   1.206 +by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
   1.207 +val IntI = result();
   1.208 +
   1.209 +val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
   1.210 +by (rtac (major RS CollectD RS conjunct1) 1);
   1.211 +val IntD1 = result();
   1.212 +
   1.213 +val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
   1.214 +by (rtac (major RS CollectD RS conjunct2) 1);
   1.215 +val IntD2 = result();
   1.216 +
   1.217 +val [major,minor] = goal Set.thy
   1.218 +    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
   1.219 +by (rtac minor 1);
   1.220 +by (rtac (major RS IntD1) 1);
   1.221 +by (rtac (major RS IntD2) 1);
   1.222 +val IntE = result();
   1.223 +
   1.224 +
   1.225 +(*** Rules for set complement -- Compl ***)
   1.226 +
   1.227 +val prems = goalw Set.thy [Compl_def]
   1.228 +    "[| c:A ==> False |] ==> c : Compl(A)";
   1.229 +by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
   1.230 +val ComplI = result();
   1.231 +
   1.232 +(*This form, with negated conclusion, works well with the Classical prover.
   1.233 +  Negated assumptions behave like formulae on the right side of the notional
   1.234 +  turnstile...*)
   1.235 +val major::prems = goalw Set.thy [Compl_def]
   1.236 +    "[| c : Compl(A) |] ==> ~c:A";
   1.237 +by (rtac (major RS CollectD) 1);
   1.238 +val ComplD = result();
   1.239 +
   1.240 +val ComplE = make_elim ComplD;
   1.241 +
   1.242 +
   1.243 +(*** Empty sets ***)
   1.244 +
   1.245 +goalw Set.thy [empty_def] "{x.False} = {}";
   1.246 +br refl 1;
   1.247 +val empty_eq = result();
   1.248 +
   1.249 +val [prem] = goalw Set.thy [empty_def] "a : {} ==> P";
   1.250 +by (rtac (prem RS CollectD RS FalseE) 1);
   1.251 +val emptyD = result();
   1.252 +
   1.253 +val emptyE = make_elim emptyD;
   1.254 +
   1.255 +val [prem] = goal Set.thy "~ A={} ==> (EX x.x:A)";
   1.256 +br (prem RS swap) 1;
   1.257 +br equalityI 1;
   1.258 +by (ALLGOALS (fast_tac (FOL_cs addSIs [subsetI] addSEs [emptyD])));
   1.259 +val not_emptyD = result();
   1.260 +
   1.261 +(*** Singleton sets ***)
   1.262 +
   1.263 +goalw Set.thy [singleton_def] "a : {a}";
   1.264 +by (rtac CollectI 1);
   1.265 +by (rtac refl 1);
   1.266 +val singletonI = result();
   1.267 +
   1.268 +val [major] = goalw Set.thy [singleton_def] "b : {a} ==> b=a"; 
   1.269 +by (rtac (major RS CollectD) 1);
   1.270 +val singletonD = result();
   1.271 +
   1.272 +val singletonE = make_elim singletonD;
   1.273 +
   1.274 +(*** Unions of families ***)
   1.275 +
   1.276 +(*The order of the premises presupposes that A is rigid; b may be flexible*)
   1.277 +val prems = goalw Set.thy [UNION_def]
   1.278 +    "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
   1.279 +by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
   1.280 +val UN_I = result();
   1.281 +
   1.282 +val major::prems = goalw Set.thy [UNION_def]
   1.283 +    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
   1.284 +by (rtac (major RS CollectD RS bexE) 1);
   1.285 +by (REPEAT (ares_tac prems 1));
   1.286 +val UN_E = result();
   1.287 +
   1.288 +val prems = goal Set.thy
   1.289 +    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
   1.290 +\    (UN x:A. C(x)) = (UN x:B. D(x))";
   1.291 +by (REPEAT (etac UN_E 1
   1.292 +     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
   1.293 +		      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
   1.294 +val UN_cong = result();
   1.295 +
   1.296 +(*** Intersections of families -- INTER x:A. B(x) is Inter(B)``A ) *)
   1.297 +
   1.298 +val prems = goalw Set.thy [INTER_def]
   1.299 +    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
   1.300 +by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
   1.301 +val INT_I = result();
   1.302 +
   1.303 +val major::prems = goalw Set.thy [INTER_def]
   1.304 +    "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
   1.305 +by (rtac (major RS CollectD RS bspec) 1);
   1.306 +by (resolve_tac prems 1);
   1.307 +val INT_D = result();
   1.308 +
   1.309 +(*"Classical" elimination rule -- does not require proving X:C *)
   1.310 +val major::prems = goalw Set.thy [INTER_def]
   1.311 +    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  ~ a:A ==> R |] ==> R";
   1.312 +by (rtac (major RS CollectD RS ballE) 1);
   1.313 +by (REPEAT (eresolve_tac prems 1));
   1.314 +val INT_E = result();
   1.315 +
   1.316 +val prems = goal Set.thy
   1.317 +    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
   1.318 +\    (INT x:A. C(x)) = (INT x:B. D(x))";
   1.319 +by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
   1.320 +by (REPEAT (dtac INT_D 1
   1.321 +     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
   1.322 +val INT_cong = result();
   1.323 +
   1.324 +(*** Rules for Unions ***)
   1.325 +
   1.326 +(*The order of the premises presupposes that C is rigid; A may be flexible*)
   1.327 +val prems = goalw Set.thy [Union_def]
   1.328 +    "[| X:C;  A:X |] ==> A : Union(C)";
   1.329 +by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
   1.330 +val UnionI = result();
   1.331 +
   1.332 +val major::prems = goalw Set.thy [Union_def]
   1.333 +    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
   1.334 +by (rtac (major RS UN_E) 1);
   1.335 +by (REPEAT (ares_tac prems 1));
   1.336 +val UnionE = result();
   1.337 +
   1.338 +(*** Rules for Inter ***)
   1.339 +
   1.340 +val prems = goalw Set.thy [Inter_def]
   1.341 +    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
   1.342 +by (REPEAT (ares_tac ([INT_I] @ prems) 1));
   1.343 +val InterI = result();
   1.344 +
   1.345 +(*A "destruct" rule -- every X in C contains A as an element, but
   1.346 +  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
   1.347 +val major::prems = goalw Set.thy [Inter_def]
   1.348 +    "[| A : Inter(C);  X:C |] ==> A:X";
   1.349 +by (rtac (major RS INT_D) 1);
   1.350 +by (resolve_tac prems 1);
   1.351 +val InterD = result();
   1.352 +
   1.353 +(*"Classical" elimination rule -- does not require proving X:C *)
   1.354 +val major::prems = goalw Set.thy [Inter_def]
   1.355 +    "[| A : Inter(C);  A:X ==> R;  ~ X:C ==> R |] ==> R";
   1.356 +by (rtac (major RS INT_E) 1);
   1.357 +by (REPEAT (eresolve_tac prems 1));
   1.358 +val InterE = result();