src/HOL/Set.ML
author paulson
Thu Apr 04 11:41:35 1996 +0200 (1996-04-04)
changeset 1640 581165679095
parent 1618 372880456b5b
child 1760 6f41a494f3b1
permissions -rw-r--r--
Added more _iff rewrites for Compl, Un, Int
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(*  Title:      HOL/set
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1991  University of Cambridge
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For set.thy.  Set theory for higher-order logic.  A set is simply a predicate.
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*)
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open Set;
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section "Relating predicates and sets";
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val [prem] = goal Set.thy "P(a) ==> a : {x.P(x)}";
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by (rtac (mem_Collect_eq RS ssubst) 1);
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by (rtac prem 1);
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qed "CollectI";
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val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)";
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by (resolve_tac (prems RL [mem_Collect_eq  RS subst]) 1);
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qed "CollectD";
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val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B";
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by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
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by (rtac Collect_mem_eq 1);
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by (rtac Collect_mem_eq 1);
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qed "set_ext";
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val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
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by (rtac (prem RS ext RS arg_cong) 1);
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qed "Collect_cong";
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val CollectE = make_elim CollectD;
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section "Bounded quantifiers";
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val prems = goalw Set.thy [Ball_def]
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    "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
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by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
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qed "ballI";
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val [major,minor] = goalw Set.thy [Ball_def]
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    "[| ! x:A. P(x);  x:A |] ==> P(x)";
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by (rtac (minor RS (major RS spec RS mp)) 1);
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qed "bspec";
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val major::prems = goalw Set.thy [Ball_def]
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    "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
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by (rtac (major RS spec RS impCE) 1);
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by (REPEAT (eresolve_tac prems 1));
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qed "ballE";
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(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
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fun ball_tac i = etac ballE i THEN contr_tac (i+1);
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val prems = goalw Set.thy [Bex_def]
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    "[| P(x);  x:A |] ==> ? x:A. P(x)";
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by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
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qed "bexI";
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qed_goal "bexCI" Set.thy 
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   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A.P(x)"
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 (fn prems=>
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  [ (rtac classical 1),
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    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
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val major::prems = goalw Set.thy [Bex_def]
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    "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
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by (rtac (major RS exE) 1);
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by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
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qed "bexE";
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(*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
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goal Set.thy "(! x:A. True) = True";
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by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
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qed "ball_rew";
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Addsimps [ball_rew];
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(** Congruence rules **)
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val prems = goal Set.thy
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    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
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\    (! x:A. P(x)) = (! x:B. Q(x))";
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by (resolve_tac (prems RL [ssubst]) 1);
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by (REPEAT (ares_tac [ballI,iffI] 1
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     ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
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qed "ball_cong";
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val prems = goal Set.thy
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    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
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\    (? x:A. P(x)) = (? x:B. Q(x))";
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by (resolve_tac (prems RL [ssubst]) 1);
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by (REPEAT (etac bexE 1
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     ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
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qed "bex_cong";
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section "Subsets";
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val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
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by (REPEAT (ares_tac (prems @ [ballI]) 1));
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qed "subsetI";
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(*Rule in Modus Ponens style*)
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val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
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by (rtac (major RS bspec) 1);
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by (resolve_tac prems 1);
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qed "subsetD";
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(*The same, with reversed premises for use with etac -- cf rev_mp*)
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qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
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 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
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(*Classical elimination rule*)
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val major::prems = goalw Set.thy [subset_def] 
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    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
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by (rtac (major RS ballE) 1);
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by (REPEAT (eresolve_tac prems 1));
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qed "subsetCE";
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(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
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fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
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qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
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 (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]);
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val prems = goal Set.thy "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
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by (cut_facts_tac prems 1);
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by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1));
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qed "subset_trans";
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section "Equality";
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(*Anti-symmetry of the subset relation*)
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val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
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by (rtac (iffI RS set_ext) 1);
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by (REPEAT (ares_tac (prems RL [subsetD]) 1));
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qed "subset_antisym";
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val equalityI = subset_antisym;
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(* Equality rules from ZF set theory -- are they appropriate here? *)
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val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
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by (resolve_tac (prems RL [subst]) 1);
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by (rtac subset_refl 1);
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qed "equalityD1";
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val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
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by (resolve_tac (prems RL [subst]) 1);
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by (rtac subset_refl 1);
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qed "equalityD2";
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val prems = goal Set.thy
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    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
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by (resolve_tac prems 1);
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by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
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qed "equalityE";
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val major::prems = goal Set.thy
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    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
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by (rtac (major RS equalityE) 1);
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by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
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qed "equalityCE";
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(*Lemma for creating induction formulae -- for "pattern matching" on p
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  To make the induction hypotheses usable, apply "spec" or "bspec" to
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  put universal quantifiers over the free variables in p. *)
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val prems = goal Set.thy 
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    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
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by (rtac mp 1);
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by (REPEAT (resolve_tac (refl::prems) 1));
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qed "setup_induction";
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section "Set complement -- Compl";
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val prems = goalw Set.thy [Compl_def]
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    "[| c:A ==> False |] ==> c : Compl(A)";
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by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
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qed "ComplI";
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(*This form, with negated conclusion, works well with the Classical prover.
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  Negated assumptions behave like formulae on the right side of the notional
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  turnstile...*)
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val major::prems = goalw Set.thy [Compl_def]
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    "[| c : Compl(A) |] ==> c~:A";
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by (rtac (major RS CollectD) 1);
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qed "ComplD";
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val ComplE = make_elim ComplD;
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qed_goal "Compl_iff" Set.thy "(c : Compl(A)) = (c~:A)"
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 (fn _ => [ (fast_tac (HOL_cs addSIs [ComplI] addSEs [ComplE]) 1) ]);
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section "Binary union -- Un";
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val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
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by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
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qed "UnI1";
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val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
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by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
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qed "UnI2";
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(*Classical introduction rule: no commitment to A vs B*)
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qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
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 (fn prems=>
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  [ (rtac classical 1),
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    (REPEAT (ares_tac (prems@[UnI1,notI]) 1)),
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    (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]);
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val major::prems = goalw Set.thy [Un_def]
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    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
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by (rtac (major RS CollectD RS disjE) 1);
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by (REPEAT (eresolve_tac prems 1));
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qed "UnE";
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qed_goal "Un_iff" Set.thy "(c : A Un B) = (c:A | c:B)"
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 (fn _ => [ (fast_tac (HOL_cs addSIs [UnCI] addSEs [UnE]) 1) ]);
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section "Binary intersection -- Int";
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val prems = goalw Set.thy [Int_def]
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    "[| c:A;  c:B |] ==> c : A Int B";
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by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
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qed "IntI";
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val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
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by (rtac (major RS CollectD RS conjunct1) 1);
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qed "IntD1";
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val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
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by (rtac (major RS CollectD RS conjunct2) 1);
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qed "IntD2";
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val [major,minor] = goal Set.thy
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    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
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by (rtac minor 1);
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by (rtac (major RS IntD1) 1);
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by (rtac (major RS IntD2) 1);
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qed "IntE";
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qed_goal "Int_iff" Set.thy "(c : A Int B) = (c:A & c:B)"
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 (fn _ => [ (fast_tac (HOL_cs addSIs [IntI] addSEs [IntE]) 1) ]);
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section "Set difference";
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qed_goalw "DiffI" Set.thy [set_diff_def]
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    "[| c : A;  c ~: B |] ==> c : A - B"
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 (fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]);
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qed_goalw "DiffD1" Set.thy [set_diff_def]
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    "c : A - B ==> c : A"
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 (fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]);
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qed_goalw "DiffD2" Set.thy [set_diff_def]
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    "[| c : A - B;  c : B |] ==> P"
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 (fn [major,minor]=>
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     [rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]);
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qed_goal "DiffE" Set.thy
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    "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
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 (fn prems=>
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  [ (resolve_tac prems 1),
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    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
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qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)"
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 (fn _ => [ (fast_tac (HOL_cs addSIs [DiffI] addSEs [DiffE]) 1) ]);
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section "The empty set -- {}";
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qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P"
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 (fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]);
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qed_goal "empty_subsetI" Set.thy "{} <= A"
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 (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]);
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qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
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 (fn prems=>
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  [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1 
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      ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]);
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qed_goal "equals0D" Set.thy "[| A={};  a:A |] ==> P"
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 (fn [major,minor]=>
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  [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]);
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qed_goal "empty_iff" Set.thy "(c : {}) = False"
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 (fn _ => [ (fast_tac (HOL_cs addSEs [emptyE]) 1) ]);
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section "Augmenting a set -- insert";
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qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B"
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 (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]);
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qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B"
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 (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]);
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qed_goalw "insertE" Set.thy [insert_def]
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    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
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 (fn major::prems=>
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  [ (rtac (major RS UnE) 1),
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    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
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qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)"
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 (fn _ => [fast_tac (HOL_cs addIs [insertI1,insertI2] addSEs [insertE]) 1]);
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(*Classical introduction rule*)
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qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
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 (fn [prem]=>
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  [ (rtac (disjCI RS (insert_iff RS iffD2)) 1),
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    (etac prem 1) ]);
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section "Singletons, using insert";
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qed_goal "singletonI" Set.thy "a : {a}"
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 (fn _=> [ (rtac insertI1 1) ]);
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qed_goal "singletonE" Set.thy "[| a: {b};  a=b ==> P |] ==> P"
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 (fn major::prems=>
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  [ (rtac (major RS insertE) 1),
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    (REPEAT (eresolve_tac (prems @ [emptyE]) 1)) ]);
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goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a";
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by (fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1);
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qed "singletonD";
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val singletonE = make_elim singletonD;
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val [major] = goal Set.thy "{a}={b} ==> a=b";
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by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1);
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by (rtac singletonI 1);
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qed "singleton_inject";
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section "The universal set -- UNIV";
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qed_goal "subset_UNIV" Set.thy "A <= UNIV"
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  (fn _ => [rtac subsetI 1, rtac ComplI 1, etac emptyE 1]);
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section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
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(*The order of the premises presupposes that A is rigid; b may be flexible*)
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val prems = goalw Set.thy [UNION_def]
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    "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
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by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
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qed "UN_I";
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val major::prems = goalw Set.thy [UNION_def]
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    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
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by (rtac (major RS CollectD RS bexE) 1);
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by (REPEAT (ares_tac prems 1));
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qed "UN_E";
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val prems = goal Set.thy
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    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
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\    (UN x:A. C(x)) = (UN x:B. D(x))";
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by (REPEAT (etac UN_E 1
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     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
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                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
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qed "UN_cong";
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section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
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val prems = goalw Set.thy [INTER_def]
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    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
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by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
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qed "INT_I";
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val major::prems = goalw Set.thy [INTER_def]
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    "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
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by (rtac (major RS CollectD RS bspec) 1);
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by (resolve_tac prems 1);
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qed "INT_D";
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(*"Classical" elimination -- by the Excluded Middle on a:A *)
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val major::prems = goalw Set.thy [INTER_def]
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    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
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by (rtac (major RS CollectD RS ballE) 1);
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by (REPEAT (eresolve_tac prems 1));
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qed "INT_E";
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   385
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val prems = goal Set.thy
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    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
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\    (INT x:A. C(x)) = (INT x:B. D(x))";
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   389
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
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   390
by (REPEAT (dtac INT_D 1
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     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
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   392
qed "INT_cong";
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   393
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   394
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   395
section "Unions over a type; UNION1(B) = Union(range(B))";
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   396
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(*The order of the premises presupposes that A is rigid; b may be flexible*)
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val prems = goalw Set.thy [UNION1_def]
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   399
    "b: B(x) ==> b: (UN x. B(x))";
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   400
by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1));
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   401
qed "UN1_I";
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   402
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   403
val major::prems = goalw Set.thy [UNION1_def]
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   404
    "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
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   405
by (rtac (major RS UN_E) 1);
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   406
by (REPEAT (ares_tac prems 1));
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   407
qed "UN1_E";
clasohm@923
   408
clasohm@923
   409
nipkow@1548
   410
section "Intersections over a type; INTER1(B) = Inter(range(B))";
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   411
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   412
val prems = goalw Set.thy [INTER1_def]
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   413
    "(!!x. b: B(x)) ==> b : (INT x. B(x))";
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   414
by (REPEAT (ares_tac (INT_I::prems) 1));
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   415
qed "INT1_I";
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   416
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   417
val [major] = goalw Set.thy [INTER1_def]
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   418
    "b : (INT x. B(x)) ==> b: B(a)";
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   419
by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1);
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   420
qed "INT1_D";
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   421
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   422
section "Union";
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   423
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   424
(*The order of the premises presupposes that C is rigid; A may be flexible*)
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   425
val prems = goalw Set.thy [Union_def]
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   426
    "[| X:C;  A:X |] ==> A : Union(C)";
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   427
by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
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   428
qed "UnionI";
clasohm@923
   429
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   430
val major::prems = goalw Set.thy [Union_def]
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   431
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
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   432
by (rtac (major RS UN_E) 1);
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   433
by (REPEAT (ares_tac prems 1));
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   434
qed "UnionE";
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   435
nipkow@1548
   436
section "Inter";
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   437
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   438
val prems = goalw Set.thy [Inter_def]
clasohm@923
   439
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
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   440
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
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   441
qed "InterI";
clasohm@923
   442
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   443
(*A "destruct" rule -- every X in C contains A as an element, but
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  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
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   445
val major::prems = goalw Set.thy [Inter_def]
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   446
    "[| A : Inter(C);  X:C |] ==> A:X";
clasohm@923
   447
by (rtac (major RS INT_D) 1);
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   448
by (resolve_tac prems 1);
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   449
qed "InterD";
clasohm@923
   450
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   451
(*"Classical" elimination rule -- does not require proving X:C *)
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   452
val major::prems = goalw Set.thy [Inter_def]
clasohm@923
   453
    "[| A : Inter(C);  A:X ==> R;  X~:C ==> R |] ==> R";
clasohm@923
   454
by (rtac (major RS INT_E) 1);
clasohm@923
   455
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   456
qed "InterE";
clasohm@923
   457
nipkow@1548
   458
section "The Powerset operator -- Pow";
clasohm@923
   459
clasohm@923
   460
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
clasohm@923
   461
 (fn _ => [ (etac CollectI 1) ]);
clasohm@923
   462
clasohm@923
   463
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
clasohm@923
   464
 (fn _=> [ (etac CollectD 1) ]);
clasohm@923
   465
clasohm@923
   466
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
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   467
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)