src/HOL/Set.ML
 author clasohm Tue, 30 Jan 1996 15:24:36 +0100 changeset 1465 5d7a7e439cec parent 923 ff1574a81019 child 1531 e5eb247ad13c permissions -rw-r--r--
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 1465 5d7a7e439cec expanded tabs clasohm parents: 923 diff changeset 1 (* Title: HOL/set 923 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 2 ID: \$Id\$ 1465 5d7a7e439cec expanded tabs clasohm parents: 923 diff changeset 3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 923 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 4 Copyright 1991 University of Cambridge ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 5 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 6 For set.thy. Set theory for higher-order logic. A set is simply a predicate. ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 7 *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 8 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 9 open Set; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 10 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 11 val [prem] = goal Set.thy "[| P(a) |] ==> a : {x.P(x)}"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 12 by (rtac (mem_Collect_eq RS ssubst) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 13 by (rtac prem 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 14 qed "CollectI"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 15 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 16 val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 17 by (resolve_tac (prems RL [mem_Collect_eq RS subst]) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 18 qed "CollectD"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 19 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 20 val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 21 by (rtac (prem RS ext RS arg_cong RS box_equals) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 22 by (rtac Collect_mem_eq 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 23 by (rtac Collect_mem_eq 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 24 qed "set_ext"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 25 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 26 val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 27 by (rtac (prem RS ext RS arg_cong) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 28 qed "Collect_cong"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 29 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 30 val CollectE = make_elim CollectD; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 31 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 32 (*** Bounded quantifiers ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 33 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 34 val prems = goalw Set.thy [Ball_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 35 "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 36 by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 37 qed "ballI"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 38 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 39 val [major,minor] = goalw Set.thy [Ball_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 40 "[| ! x:A. P(x); x:A |] ==> P(x)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 41 by (rtac (minor RS (major RS spec RS mp)) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 42 qed "bspec"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 43 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 44 val major::prems = goalw Set.thy [Ball_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 45 "[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 46 by (rtac (major RS spec RS impCE) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 47 by (REPEAT (eresolve_tac prems 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 48 qed "ballE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 49 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 50 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 51 fun ball_tac i = etac ballE i THEN contr_tac (i+1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 52 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 53 val prems = goalw Set.thy [Bex_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 54 "[| P(x); x:A |] ==> ? x:A. P(x)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 55 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 56 qed "bexI"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 57 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 58 qed_goal "bexCI" Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 59 "[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A.P(x)" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 60 (fn prems=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 61 [ (rtac classical 1), ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 62 (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 63 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 64 val major::prems = goalw Set.thy [Bex_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 65 "[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 66 by (rtac (major RS exE) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 67 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 68 qed "bexE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 69 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 70 (*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 71 val prems = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 72 "(! x:A. True) = True"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 73 by (REPEAT (ares_tac [TrueI,ballI,iffI] 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 74 qed "ball_rew"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 75 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 76 (** Congruence rules **) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 77 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 78 val prems = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 79 "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 80 \ (! x:A. P(x)) = (! x:B. Q(x))"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 81 by (resolve_tac (prems RL [ssubst]) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 82 by (REPEAT (ares_tac [ballI,iffI] 1 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 83 ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 84 qed "ball_cong"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 85 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 86 val prems = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 87 "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 88 \ (? x:A. P(x)) = (? x:B. Q(x))"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 89 by (resolve_tac (prems RL [ssubst]) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 90 by (REPEAT (etac bexE 1 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 91 ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 92 qed "bex_cong"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 93 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 94 (*** Subsets ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 95 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 96 val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 97 by (REPEAT (ares_tac (prems @ [ballI]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 98 qed "subsetI"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 99 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 100 (*Rule in Modus Ponens style*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 101 val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 102 by (rtac (major RS bspec) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 103 by (resolve_tac prems 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 104 qed "subsetD"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 105 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 106 (*The same, with reversed premises for use with etac -- cf rev_mp*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 107 qed_goal "rev_subsetD" Set.thy "[| c:A; A <= B |] ==> c:B" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 108 (fn prems=> [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 109 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 110 (*Classical elimination rule*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 111 val major::prems = goalw Set.thy [subset_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 112 "[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 113 by (rtac (major RS ballE) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 114 by (REPEAT (eresolve_tac prems 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 115 qed "subsetCE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 116 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 117 (*Takes assumptions A<=B; c:A and creates the assumption c:B *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 118 fun set_mp_tac i = etac subsetCE i THEN mp_tac i; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 119 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 120 qed_goal "subset_refl" Set.thy "A <= (A::'a set)" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 121 (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 122 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 123 val prems = goal Set.thy "[| A<=B; B<=C |] ==> A<=(C::'a set)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 124 by (cut_facts_tac prems 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 125 by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 126 qed "subset_trans"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 127 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 128 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 129 (*** Equality ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 130 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 131 (*Anti-symmetry of the subset relation*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 132 val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 133 by (rtac (iffI RS set_ext) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 134 by (REPEAT (ares_tac (prems RL [subsetD]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 135 qed "subset_antisym"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 136 val equalityI = subset_antisym; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 137 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 138 (* Equality rules from ZF set theory -- are they appropriate here? *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 139 val prems = goal Set.thy "A = B ==> A<=(B::'a set)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 140 by (resolve_tac (prems RL [subst]) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 141 by (rtac subset_refl 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 142 qed "equalityD1"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 143 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 144 val prems = goal Set.thy "A = B ==> B<=(A::'a set)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 145 by (resolve_tac (prems RL [subst]) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 146 by (rtac subset_refl 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 147 qed "equalityD2"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 148 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 149 val prems = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 150 "[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 151 by (resolve_tac prems 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 152 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 153 qed "equalityE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 154 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 155 val major::prems = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 156 "[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 157 by (rtac (major RS equalityE) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 158 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 159 qed "equalityCE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 160 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 161 (*Lemma for creating induction formulae -- for "pattern matching" on p ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 162 To make the induction hypotheses usable, apply "spec" or "bspec" to ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 163 put universal quantifiers over the free variables in p. *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 164 val prems = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 165 "[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 166 by (rtac mp 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 167 by (REPEAT (resolve_tac (refl::prems) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 168 qed "setup_induction"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 169 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 170 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 171 (*** Set complement -- Compl ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 172 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 173 val prems = goalw Set.thy [Compl_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 174 "[| c:A ==> False |] ==> c : Compl(A)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 175 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 176 qed "ComplI"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 177 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 178 (*This form, with negated conclusion, works well with the Classical prover. ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 179 Negated assumptions behave like formulae on the right side of the notional ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 180 turnstile...*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 181 val major::prems = goalw Set.thy [Compl_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 182 "[| c : Compl(A) |] ==> c~:A"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 183 by (rtac (major RS CollectD) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 184 qed "ComplD"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 185 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 186 val ComplE = make_elim ComplD; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 187 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 188 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 189 (*** Binary union -- Un ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 190 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 191 val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 192 by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 193 qed "UnI1"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 194 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 195 val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 196 by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 197 qed "UnI2"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 198 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 199 (*Classical introduction rule: no commitment to A vs B*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 200 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 201 (fn prems=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 202 [ (rtac classical 1), ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 203 (REPEAT (ares_tac (prems@[UnI1,notI]) 1)), ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 204 (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 205 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 206 val major::prems = goalw Set.thy [Un_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 207 "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 208 by (rtac (major RS CollectD RS disjE) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 209 by (REPEAT (eresolve_tac prems 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 210 qed "UnE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 211 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 212 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 213 (*** Binary intersection -- Int ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 214 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 215 val prems = goalw Set.thy [Int_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 216 "[| c:A; c:B |] ==> c : A Int B"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 217 by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 218 qed "IntI"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 219 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 220 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 221 by (rtac (major RS CollectD RS conjunct1) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 222 qed "IntD1"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 223 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 224 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 225 by (rtac (major RS CollectD RS conjunct2) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 226 qed "IntD2"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 227 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 228 val [major,minor] = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 229 "[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 230 by (rtac minor 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 231 by (rtac (major RS IntD1) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 232 by (rtac (major RS IntD2) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 233 qed "IntE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 234 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 235 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 236 (*** Set difference ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 237 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 238 qed_goalw "DiffI" Set.thy [set_diff_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 239 "[| c : A; c ~: B |] ==> c : A - B" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 240 (fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 241 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 242 qed_goalw "DiffD1" Set.thy [set_diff_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 243 "c : A - B ==> c : A" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 244 (fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 245 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 246 qed_goalw "DiffD2" Set.thy [set_diff_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 247 "[| c : A - B; c : B |] ==> P" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 248 (fn [major,minor]=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 249 [rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 250 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 251 qed_goal "DiffE" Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 252 "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 253 (fn prems=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 254 [ (resolve_tac prems 1), ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 255 (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 256 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 257 qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 258 (fn _ => [ (fast_tac (HOL_cs addSIs [DiffI] addSEs [DiffE]) 1) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 259 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 260 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 261 (*** The empty set -- {} ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 262 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 263 qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 264 (fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 265 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 266 qed_goal "empty_subsetI" Set.thy "{} <= A" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 267 (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 268 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 269 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 270 (fn prems=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 271 [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 272 ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 273 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 274 qed_goal "equals0D" Set.thy "[| A={}; a:A |] ==> P" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 275 (fn [major,minor]=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 276 [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 277 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 278 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 279 (*** Augmenting a set -- insert ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 280 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 281 qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 282 (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 283 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 284 qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 285 (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 286 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 287 qed_goalw "insertE" Set.thy [insert_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 288 "[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 289 (fn major::prems=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 290 [ (rtac (major RS UnE) 1), ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 291 (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 292 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 293 qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 294 (fn _ => [fast_tac (HOL_cs addIs [insertI1,insertI2] addSEs [insertE]) 1]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 295 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 296 (*Classical introduction rule*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 297 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 298 (fn [prem]=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 299 [ (rtac (disjCI RS (insert_iff RS iffD2)) 1), ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 300 (etac prem 1) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 301 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 302 (*** Singletons, using insert ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 303 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 304 qed_goal "singletonI" Set.thy "a : {a}" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 305 (fn _=> [ (rtac insertI1 1) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 306 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 307 qed_goal "singletonE" Set.thy "[| a: {b}; a=b ==> P |] ==> P" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 308 (fn major::prems=> ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 309 [ (rtac (major RS insertE) 1), ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 310 (REPEAT (eresolve_tac (prems @ [emptyE]) 1)) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 311 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 312 goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 313 by(fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 314 qed "singletonD"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 315 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 316 val singletonE = make_elim singletonD; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 317 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 318 val [major] = goal Set.thy "{a}={b} ==> a=b"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 319 by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 320 by (rtac singletonI 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 321 qed "singleton_inject"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 322 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 323 (*** Unions of families -- UNION x:A. B(x) is Union(B``A) ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 324 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 325 (*The order of the premises presupposes that A is rigid; b may be flexible*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 326 val prems = goalw Set.thy [UNION_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 327 "[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 328 by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 329 qed "UN_I"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 330 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 331 val major::prems = goalw Set.thy [UNION_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 332 "[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 333 by (rtac (major RS CollectD RS bexE) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 334 by (REPEAT (ares_tac prems 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 335 qed "UN_E"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 336 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 337 val prems = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 338 "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 339 \ (UN x:A. C(x)) = (UN x:B. D(x))"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 340 by (REPEAT (etac UN_E 1 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 341 ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 1465 5d7a7e439cec expanded tabs clasohm parents: 923 diff changeset 342 (prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); 923 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 343 qed "UN_cong"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 344 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 345 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 346 (*** Intersections of families -- INTER x:A. B(x) is Inter(B``A) *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 347 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 348 val prems = goalw Set.thy [INTER_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 349 "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 350 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 351 qed "INT_I"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 352 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 353 val major::prems = goalw Set.thy [INTER_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 354 "[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 355 by (rtac (major RS CollectD RS bspec) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 356 by (resolve_tac prems 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 357 qed "INT_D"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 358 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 359 (*"Classical" elimination -- by the Excluded Middle on a:A *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 360 val major::prems = goalw Set.thy [INTER_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 361 "[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 362 by (rtac (major RS CollectD RS ballE) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 363 by (REPEAT (eresolve_tac prems 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 364 qed "INT_E"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 365 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 366 val prems = goal Set.thy ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 367 "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 368 \ (INT x:A. C(x)) = (INT x:B. D(x))"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 369 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 370 by (REPEAT (dtac INT_D 1 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 371 ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 372 qed "INT_cong"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 373 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 374 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 375 (*** Unions over a type; UNION1(B) = Union(range(B)) ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 376 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 377 (*The order of the premises presupposes that A is rigid; b may be flexible*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 378 val prems = goalw Set.thy [UNION1_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 379 "b: B(x) ==> b: (UN x. B(x))"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 380 by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 381 qed "UN1_I"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 382 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 383 val major::prems = goalw Set.thy [UNION1_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 384 "[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 385 by (rtac (major RS UN_E) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 386 by (REPEAT (ares_tac prems 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 387 qed "UN1_E"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 388 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 389 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 390 (*** Intersections over a type; INTER1(B) = Inter(range(B)) *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 391 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 392 val prems = goalw Set.thy [INTER1_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 393 "(!!x. b: B(x)) ==> b : (INT x. B(x))"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 394 by (REPEAT (ares_tac (INT_I::prems) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 395 qed "INT1_I"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 396 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 397 val [major] = goalw Set.thy [INTER1_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 398 "b : (INT x. B(x)) ==> b: B(a)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 399 by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 400 qed "INT1_D"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 401 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 402 (*** Unions ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 403 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 404 (*The order of the premises presupposes that C is rigid; A may be flexible*) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 405 val prems = goalw Set.thy [Union_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 406 "[| X:C; A:X |] ==> A : Union(C)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 407 by (REPEAT (resolve_tac (prems @ [UN_I]) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 408 qed "UnionI"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 409 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 410 val major::prems = goalw Set.thy [Union_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 411 "[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 412 by (rtac (major RS UN_E) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 413 by (REPEAT (ares_tac prems 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 414 qed "UnionE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 415 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 416 (*** Inter ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 417 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 418 val prems = goalw Set.thy [Inter_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 419 "[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 420 by (REPEAT (ares_tac ([INT_I] @ prems) 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 421 qed "InterI"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 422 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 423 (*A "destruct" rule -- every X in C contains A as an element, but ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 424 A:X can hold when X:C does not! This rule is analogous to "spec". *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 425 val major::prems = goalw Set.thy [Inter_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 426 "[| A : Inter(C); X:C |] ==> A:X"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 427 by (rtac (major RS INT_D) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 428 by (resolve_tac prems 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 429 qed "InterD"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 430 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 431 (*"Classical" elimination rule -- does not require proving X:C *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 432 val major::prems = goalw Set.thy [Inter_def] ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 433 "[| A : Inter(C); A:X ==> R; X~:C ==> R |] ==> R"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 434 by (rtac (major RS INT_E) 1); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 435 by (REPEAT (eresolve_tac prems 1)); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 436 qed "InterE"; ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 437 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 438 (*** Powerset ***) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 439 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 440 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 441 (fn _ => [ (etac CollectI 1) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 442 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 443 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 444 (fn _=> [ (etac CollectD 1) ]); ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 445 ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 446 val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) ff1574a81019 new version of HOL with curried function application clasohm parents: diff changeset 447 val Pow_top = subset_refl RS PowI; (* A : Pow(A) *)