src/HOL/Set.ML
author paulson
Thu Jan 09 10:23:39 1997 +0100 (1997-01-09)
changeset 2499 0bc87b063447
parent 2031 03a843f0f447
child 2608 450c9b682a92
permissions -rw-r--r--
Tidying of proofs. New theorems are enterred immediately into the
relevant clasets or simpsets.
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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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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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AddIffs [mem_Collect_eq];
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goal Set.thy "!!a. P(a) ==> a : {x.P(x)}";
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by (Asm_simp_tac 1);
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qed "CollectI";
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val prems = goal Set.thy "!!a. a : {x.P(x)} ==> P(a)";
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by (Asm_full_simp_tac 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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AddSIs [CollectI];
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AddSEs [CollectE];
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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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AddSIs [ballI];
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AddEs  [ballE];
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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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AddIs  [bexI];
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AddSEs [bexE];
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(*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
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goalw Set.thy [Ball_def] "(! x:A. True) = True";
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by (Simp_tac 1);
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qed "ball_True";
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(*Dual form for existentials*)
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goalw Set.thy [Bex_def] "(? x:A. False) = False";
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by (Simp_tac 1);
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qed "bex_False";
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Addsimps [ball_True, bex_False];
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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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(*Converts A<=B to x:A ==> x:B*)
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fun impOfSubs th = th RSN (2, rev_subsetD);
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qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
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 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
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qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
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 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, 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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AddSIs [subsetI];
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AddEs  [subsetD, subsetCE];
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qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
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 (fn _=> [Fast_tac 1]);
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val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
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by (Fast_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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AddSIs [equalityI];
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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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qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
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 (fn _ => [ (Fast_tac 1) ]);
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Addsimps [Compl_iff];
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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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AddSIs [ComplI];
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AddSEs [ComplE];
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section "Binary union -- Un";
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qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
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 (fn _ => [ Fast_tac 1 ]);
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Addsimps [Un_iff];
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goal Set.thy "!!c. c:A ==> c : A Un B";
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by (Asm_simp_tac 1);
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qed "UnI1";
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goal Set.thy "!!c. c:B ==> c : A Un B";
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by (Asm_simp_tac 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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  [ (Simp_tac 1),
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    (REPEAT (ares_tac (prems@[disjCI]) 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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AddSIs [UnCI];
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AddSEs [UnE];
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section "Binary intersection -- Int";
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qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
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 (fn _ => [ (Fast_tac 1) ]);
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Addsimps [Int_iff];
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goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
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by (Asm_simp_tac 1);
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qed "IntI";
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goal Set.thy "!!c. c : A Int B ==> c:A";
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by (Asm_full_simp_tac 1);
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qed "IntD1";
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goal Set.thy "!!c. c : A Int B ==> c:B";
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by (Asm_full_simp_tac 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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AddSIs [IntI];
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AddSEs [IntE];
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section "Set difference";
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qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
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 (fn _ => [ (Fast_tac 1) ]);
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Addsimps [Diff_iff];
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qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
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 (fn _=> [ Asm_simp_tac 1 ]);
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qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
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 (fn _=> [ (Asm_full_simp_tac 1) ]);
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qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
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 (fn _=> [ (Asm_full_simp_tac 1) ]);
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qed_goal "DiffE" Set.thy "[| 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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AddSIs [DiffI];
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AddSEs [DiffE];
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section "The empty set -- {}";
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qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
paulson@2499
   316
 (fn _ => [ (Fast_tac 1) ]);
paulson@2499
   317
paulson@2499
   318
Addsimps [empty_iff];
paulson@2499
   319
paulson@2499
   320
qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
paulson@2499
   321
 (fn _ => [Full_simp_tac 1]);
paulson@2499
   322
paulson@2499
   323
AddSEs [emptyE];
clasohm@923
   324
clasohm@923
   325
qed_goal "empty_subsetI" Set.thy "{} <= A"
paulson@2499
   326
 (fn _ => [ (Fast_tac 1) ]);
clasohm@923
   327
clasohm@923
   328
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
paulson@2499
   329
 (fn [prem]=>
paulson@2499
   330
  [ (fast_tac (!claset addIs [prem RS FalseE]) 1) ]);
clasohm@923
   331
paulson@2499
   332
qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
paulson@2499
   333
 (fn _ => [ (Fast_tac 1) ]);
paulson@1640
   334
paulson@1816
   335
goal Set.thy "Ball {} P = True";
paulson@1816
   336
by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
paulson@1816
   337
qed "ball_empty";
paulson@1816
   338
paulson@1816
   339
goal Set.thy "Bex {} P = False";
paulson@1816
   340
by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
paulson@1816
   341
qed "bex_empty";
paulson@1816
   342
Addsimps [ball_empty, bex_empty];
paulson@1816
   343
clasohm@923
   344
nipkow@1548
   345
section "Augmenting a set -- insert";
clasohm@923
   346
paulson@2499
   347
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
paulson@2499
   348
 (fn _ => [Fast_tac 1]);
paulson@2499
   349
paulson@2499
   350
Addsimps [insert_iff];
clasohm@923
   351
paulson@2499
   352
qed_goal "insertI1" Set.thy "a : insert a B"
paulson@2499
   353
 (fn _ => [Simp_tac 1]);
paulson@2499
   354
paulson@2499
   355
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
paulson@2499
   356
 (fn _=> [Asm_simp_tac 1]);
clasohm@923
   357
clasohm@923
   358
qed_goalw "insertE" Set.thy [insert_def]
clasohm@923
   359
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
clasohm@923
   360
 (fn major::prems=>
clasohm@923
   361
  [ (rtac (major RS UnE) 1),
clasohm@923
   362
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
clasohm@923
   363
clasohm@923
   364
(*Classical introduction rule*)
clasohm@923
   365
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
paulson@2499
   366
 (fn prems=>
paulson@2499
   367
  [ (Simp_tac 1),
paulson@2499
   368
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
paulson@2499
   369
paulson@2499
   370
AddSIs [insertCI]; 
paulson@2499
   371
AddSEs [insertE];
clasohm@923
   372
nipkow@1548
   373
section "Singletons, using insert";
clasohm@923
   374
clasohm@923
   375
qed_goal "singletonI" Set.thy "a : {a}"
clasohm@923
   376
 (fn _=> [ (rtac insertI1 1) ]);
clasohm@923
   377
paulson@2499
   378
goal Set.thy "!!a. b : {a} ==> b=a";
paulson@2499
   379
by (Fast_tac 1);
clasohm@923
   380
qed "singletonD";
clasohm@923
   381
oheimb@1776
   382
bind_thm ("singletonE", make_elim singletonD);
oheimb@1776
   383
paulson@2499
   384
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
paulson@2499
   385
(fn _ => [Fast_tac 1]);
clasohm@923
   386
paulson@2499
   387
goal Set.thy "!!a b. {a}={b} ==> a=b";
paulson@2499
   388
by (fast_tac (!claset addEs [equalityE]) 1);
clasohm@923
   389
qed "singleton_inject";
clasohm@923
   390
paulson@2499
   391
AddSDs [singleton_inject];
paulson@2499
   392
nipkow@1531
   393
nipkow@1548
   394
section "The universal set -- UNIV";
nipkow@1531
   395
paulson@1882
   396
qed_goal "UNIV_I" Set.thy "x : UNIV"
paulson@1882
   397
  (fn _ => [rtac ComplI 1, etac emptyE 1]);
paulson@1882
   398
nipkow@1531
   399
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
paulson@1882
   400
  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
nipkow@1531
   401
nipkow@1531
   402
nipkow@1548
   403
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
clasohm@923
   404
paulson@2499
   405
goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
paulson@2499
   406
by (Fast_tac 1);
paulson@2499
   407
qed "UN_iff";
paulson@2499
   408
paulson@2499
   409
Addsimps [UN_iff];
paulson@2499
   410
clasohm@923
   411
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@2499
   412
goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
paulson@2499
   413
by (Auto_tac());
clasohm@923
   414
qed "UN_I";
clasohm@923
   415
clasohm@923
   416
val major::prems = goalw Set.thy [UNION_def]
clasohm@923
   417
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
clasohm@923
   418
by (rtac (major RS CollectD RS bexE) 1);
clasohm@923
   419
by (REPEAT (ares_tac prems 1));
clasohm@923
   420
qed "UN_E";
clasohm@923
   421
paulson@2499
   422
AddIs  [UN_I];
paulson@2499
   423
AddSEs [UN_E];
paulson@2499
   424
clasohm@923
   425
val prems = goal Set.thy
clasohm@923
   426
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   427
\    (UN x:A. C(x)) = (UN x:B. D(x))";
clasohm@923
   428
by (REPEAT (etac UN_E 1
clasohm@923
   429
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
clasohm@1465
   430
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
clasohm@923
   431
qed "UN_cong";
clasohm@923
   432
clasohm@923
   433
nipkow@1548
   434
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
clasohm@923
   435
paulson@2499
   436
goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
paulson@2499
   437
by (Auto_tac());
paulson@2499
   438
qed "INT_iff";
paulson@2499
   439
paulson@2499
   440
Addsimps [INT_iff];
paulson@2499
   441
clasohm@923
   442
val prems = goalw Set.thy [INTER_def]
clasohm@923
   443
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
clasohm@923
   444
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
clasohm@923
   445
qed "INT_I";
clasohm@923
   446
paulson@2499
   447
goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
paulson@2499
   448
by (Auto_tac());
clasohm@923
   449
qed "INT_D";
clasohm@923
   450
clasohm@923
   451
(*"Classical" elimination -- by the Excluded Middle on a:A *)
clasohm@923
   452
val major::prems = goalw Set.thy [INTER_def]
clasohm@923
   453
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
clasohm@923
   454
by (rtac (major RS CollectD RS ballE) 1);
clasohm@923
   455
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   456
qed "INT_E";
clasohm@923
   457
paulson@2499
   458
AddSIs [INT_I];
paulson@2499
   459
AddEs  [INT_D, INT_E];
paulson@2499
   460
clasohm@923
   461
val prems = goal Set.thy
clasohm@923
   462
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   463
\    (INT x:A. C(x)) = (INT x:B. D(x))";
clasohm@923
   464
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
clasohm@923
   465
by (REPEAT (dtac INT_D 1
clasohm@923
   466
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
clasohm@923
   467
qed "INT_cong";
clasohm@923
   468
clasohm@923
   469
nipkow@1548
   470
section "Unions over a type; UNION1(B) = Union(range(B))";
clasohm@923
   471
paulson@2499
   472
goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
paulson@2499
   473
by (Simp_tac 1);
paulson@2499
   474
by (Fast_tac 1);
paulson@2499
   475
qed "UN1_iff";
paulson@2499
   476
paulson@2499
   477
Addsimps [UN1_iff];
paulson@2499
   478
clasohm@923
   479
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@2499
   480
goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
paulson@2499
   481
by (Auto_tac());
clasohm@923
   482
qed "UN1_I";
clasohm@923
   483
clasohm@923
   484
val major::prems = goalw Set.thy [UNION1_def]
clasohm@923
   485
    "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
clasohm@923
   486
by (rtac (major RS UN_E) 1);
clasohm@923
   487
by (REPEAT (ares_tac prems 1));
clasohm@923
   488
qed "UN1_E";
clasohm@923
   489
paulson@2499
   490
AddIs  [UN1_I];
paulson@2499
   491
AddSEs [UN1_E];
paulson@2499
   492
clasohm@923
   493
nipkow@1548
   494
section "Intersections over a type; INTER1(B) = Inter(range(B))";
clasohm@923
   495
paulson@2499
   496
goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
paulson@2499
   497
by (Simp_tac 1);
paulson@2499
   498
by (Fast_tac 1);
paulson@2499
   499
qed "INT1_iff";
paulson@2499
   500
paulson@2499
   501
Addsimps [INT1_iff];
paulson@2499
   502
clasohm@923
   503
val prems = goalw Set.thy [INTER1_def]
clasohm@923
   504
    "(!!x. b: B(x)) ==> b : (INT x. B(x))";
clasohm@923
   505
by (REPEAT (ares_tac (INT_I::prems) 1));
clasohm@923
   506
qed "INT1_I";
clasohm@923
   507
paulson@2499
   508
goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
paulson@2499
   509
by (Asm_full_simp_tac 1);
clasohm@923
   510
qed "INT1_D";
clasohm@923
   511
paulson@2499
   512
AddSIs [INT1_I]; 
paulson@2499
   513
AddDs  [INT1_D];
paulson@2499
   514
paulson@2499
   515
nipkow@1548
   516
section "Union";
clasohm@923
   517
paulson@2499
   518
goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
paulson@2499
   519
by (Fast_tac 1);
paulson@2499
   520
qed "Union_iff";
paulson@2499
   521
paulson@2499
   522
Addsimps [Union_iff];
paulson@2499
   523
clasohm@923
   524
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@2499
   525
goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
paulson@2499
   526
by (Auto_tac());
clasohm@923
   527
qed "UnionI";
clasohm@923
   528
clasohm@923
   529
val major::prems = goalw Set.thy [Union_def]
clasohm@923
   530
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
clasohm@923
   531
by (rtac (major RS UN_E) 1);
clasohm@923
   532
by (REPEAT (ares_tac prems 1));
clasohm@923
   533
qed "UnionE";
clasohm@923
   534
paulson@2499
   535
AddIs  [UnionI];
paulson@2499
   536
AddSEs [UnionE];
paulson@2499
   537
paulson@2499
   538
nipkow@1548
   539
section "Inter";
clasohm@923
   540
paulson@2499
   541
goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
paulson@2499
   542
by (Fast_tac 1);
paulson@2499
   543
qed "Inter_iff";
paulson@2499
   544
paulson@2499
   545
Addsimps [Inter_iff];
paulson@2499
   546
clasohm@923
   547
val prems = goalw Set.thy [Inter_def]
clasohm@923
   548
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
clasohm@923
   549
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
clasohm@923
   550
qed "InterI";
clasohm@923
   551
clasohm@923
   552
(*A "destruct" rule -- every X in C contains A as an element, but
clasohm@923
   553
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
paulson@2499
   554
goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
paulson@2499
   555
by (Auto_tac());
clasohm@923
   556
qed "InterD";
clasohm@923
   557
clasohm@923
   558
(*"Classical" elimination rule -- does not require proving X:C *)
clasohm@923
   559
val major::prems = goalw Set.thy [Inter_def]
clasohm@923
   560
    "[| A : Inter(C);  A:X ==> R;  X~:C ==> R |] ==> R";
clasohm@923
   561
by (rtac (major RS INT_E) 1);
clasohm@923
   562
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   563
qed "InterE";
clasohm@923
   564
paulson@2499
   565
AddSIs [InterI];
paulson@2499
   566
AddEs  [InterD, InterE];
paulson@2499
   567
paulson@2499
   568
nipkow@1548
   569
section "The Powerset operator -- Pow";
clasohm@923
   570
paulson@2499
   571
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
paulson@2499
   572
 (fn _ => [ (Asm_simp_tac 1) ]);
paulson@2499
   573
paulson@2499
   574
AddIffs [Pow_iff]; 
paulson@2499
   575
clasohm@923
   576
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
clasohm@923
   577
 (fn _ => [ (etac CollectI 1) ]);
clasohm@923
   578
clasohm@923
   579
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
clasohm@923
   580
 (fn _=> [ (etac CollectD 1) ]);
clasohm@923
   581
clasohm@923
   582
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
clasohm@923
   583
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
oheimb@1776
   584
oheimb@1776
   585
oheimb@1776
   586
oheimb@1776
   587
(*** Set reasoning tools ***)
oheimb@1776
   588
oheimb@1776
   589
paulson@2499
   590
(*Each of these has ALREADY been added to !simpset above.*)
paulson@2024
   591
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
paulson@2499
   592
                 mem_Collect_eq, 
paulson@2499
   593
		 UN_iff, UN1_iff, Union_iff, 
paulson@2499
   594
		 INT_iff, INT1_iff, Inter_iff];
oheimb@1776
   595
paulson@1937
   596
(*Not for Addsimps -- it can cause goals to blow up!*)
paulson@1937
   597
goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
paulson@1937
   598
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
paulson@1937
   599
qed "mem_if";
paulson@1937
   600
oheimb@1776
   601
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
oheimb@1776
   602
paulson@2499
   603
simpset := !simpset addcongs [ball_cong,bex_cong]
oheimb@1776
   604
                    setmksimps (mksimps mksimps_pairs);