src/HOL/Set.ML
author paulson
Fri Aug 14 12:03:01 1998 +0200 (1998-08-14)
changeset 5318 72bf8039b53f
parent 5316 7a8975451a89
child 5336 721bf1a13f1a
permissions -rw-r--r--
expandshort
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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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Addsimps [Collect_mem_eq];
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AddIffs  [mem_Collect_eq];
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Goal "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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Goal "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 "[| !!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 "[| !!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 [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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Goalw [Ball_def] "[| ! x:A. P(x);  x:A |] ==> P(x)";
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by (Blast_tac 1);
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qed "bspec";
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val major::prems = Goalw [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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Goalw [Bex_def] "[| P(x);  x:A |] ==> ? x:A. P(x)";
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by (Blast_tac 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 [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*)
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Goal "(! x:A. P) = ((? x. x:A) --> P)";
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by (simp_tac (simpset() addsimps [Ball_def]) 1);
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qed "ball_triv";
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(*Dual form for existentials*)
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Goal "(? x:A. P) = ((? x. x:A) & P)";
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by (simp_tac (simpset() addsimps [Bex_def]) 1);
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qed "bex_triv";
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Addsimps [ball_triv, bex_triv];
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(** Congruence rules **)
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val prems = Goal
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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
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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 [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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Blast.overloaded ("op <=", domain_type); (*The <= relation is overloaded*)
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(*While (:) is not, its type must be kept
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  for overloading of = to work.*)
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Blast.overloaded ("op :", domain_type);
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seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type))
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    ["Ball", "Bex"];
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(*need UNION, INTER also?*)
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(*Image: retain the type of the set being expressed*)
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Blast.overloaded ("op ``", domain_type o domain_type);
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(*Rule in Modus Ponens style*)
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Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
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by (Blast_tac 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 [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]);		(*Blast_tac would try order_refl and fail*)
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Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
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by (Blast_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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Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
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by (rtac set_ext 1);
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by (blast_tac (claset() addIs [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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Goal "A = B ==> A<=(B::'a set)";
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by (etac ssubst 1);
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by (rtac subset_refl 1);
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qed "equalityD1";
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Goal "A = B ==> B<=(A::'a set)";
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by (etac ssubst 1);
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by (rtac subset_refl 1);
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qed "equalityD2";
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val prems = Goal
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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
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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 
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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 "The universal set -- UNIV";
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qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
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  (fn _ => [rtac CollectI 1, rtac TrueI 1]);
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Addsimps [UNIV_I];
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AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
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qed_goal "subset_UNIV" Set.thy "A <= UNIV"
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  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
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(** Eta-contracting these two rules (to remove P) causes them to be ignored
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    because of their interaction with congruence rules. **)
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Goalw [Ball_def] "Ball UNIV P = All P";
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by (Simp_tac 1);
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qed "ball_UNIV";
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Goalw [Bex_def] "Bex UNIV P = Ex P";
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by (Simp_tac 1);
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qed "bex_UNIV";
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Addsimps [ball_UNIV, bex_UNIV];
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section "The empty set -- {}";
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qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
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 (fn _ => [ (Blast_tac 1) ]);
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Addsimps [empty_iff];
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qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
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 (fn _ => [Full_simp_tac 1]);
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AddSEs [emptyE];
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qed_goal "empty_subsetI" Set.thy "{} <= A"
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 (fn _ => [ (Blast_tac 1) ]);
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(*One effect is to delete the ASSUMPTION {} <= A*)
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AddIffs [empty_subsetI];
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qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
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 (fn [prem]=>
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  [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
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(*Use for reasoning about disjointness: A Int B = {} *)
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qed_goal "equals0E" Set.thy "!!a. [| A={};  a:A |] ==> P"
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 (fn _ => [ (Blast_tac 1) ]);
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AddEs [equals0E, sym RS equals0E];
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Goalw [Ball_def] "Ball {} P = True";
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by (Simp_tac 1);
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qed "ball_empty";
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Goalw [Bex_def] "Bex {} P = False";
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by (Simp_tac 1);
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qed "bex_empty";
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Addsimps [ball_empty, bex_empty];
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Goal "UNIV ~= {}";
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by (blast_tac (claset() addEs [equalityE]) 1);
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qed "UNIV_not_empty";
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AddIffs [UNIV_not_empty];
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section "The Powerset operator -- Pow";
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qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
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 (fn _ => [ (Asm_simp_tac 1) ]);
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AddIffs [Pow_iff]; 
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qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
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 (fn _ => [ (etac CollectI 1) ]);
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qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
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 (fn _=> [ (etac CollectD 1) ]);
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val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
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val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
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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 _ => [ (Blast_tac 1) ]);
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Addsimps [Compl_iff];
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val prems = Goalw [Compl_def] "[| 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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Goalw [Compl_def] "c : Compl(A) ==> c~:A";
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by (etac 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)"
paulson@2891
   325
 (fn _ => [ Blast_tac 1 ]);
paulson@2499
   326
paulson@2499
   327
Addsimps [Un_iff];
paulson@2499
   328
paulson@5143
   329
Goal "c:A ==> c : A Un B";
paulson@2499
   330
by (Asm_simp_tac 1);
clasohm@923
   331
qed "UnI1";
clasohm@923
   332
paulson@5143
   333
Goal "c:B ==> c : A Un B";
paulson@2499
   334
by (Asm_simp_tac 1);
clasohm@923
   335
qed "UnI2";
clasohm@923
   336
clasohm@923
   337
(*Classical introduction rule: no commitment to A vs B*)
clasohm@923
   338
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
clasohm@923
   339
 (fn prems=>
paulson@2499
   340
  [ (Simp_tac 1),
paulson@2499
   341
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
clasohm@923
   342
paulson@5316
   343
val major::prems = Goalw [Un_def]
clasohm@923
   344
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   345
by (rtac (major RS CollectD RS disjE) 1);
clasohm@923
   346
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   347
qed "UnE";
clasohm@923
   348
paulson@2499
   349
AddSIs [UnCI];
paulson@2499
   350
AddSEs [UnE];
paulson@1640
   351
clasohm@923
   352
nipkow@1548
   353
section "Binary intersection -- Int";
clasohm@923
   354
paulson@2499
   355
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
paulson@2891
   356
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   357
paulson@2499
   358
Addsimps [Int_iff];
paulson@2499
   359
paulson@5143
   360
Goal "[| c:A;  c:B |] ==> c : A Int B";
paulson@2499
   361
by (Asm_simp_tac 1);
clasohm@923
   362
qed "IntI";
clasohm@923
   363
paulson@5143
   364
Goal "c : A Int B ==> c:A";
paulson@2499
   365
by (Asm_full_simp_tac 1);
clasohm@923
   366
qed "IntD1";
clasohm@923
   367
paulson@5143
   368
Goal "c : A Int B ==> c:B";
paulson@2499
   369
by (Asm_full_simp_tac 1);
clasohm@923
   370
qed "IntD2";
clasohm@923
   371
paulson@5316
   372
val [major,minor] = Goal
clasohm@923
   373
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
clasohm@923
   374
by (rtac minor 1);
clasohm@923
   375
by (rtac (major RS IntD1) 1);
clasohm@923
   376
by (rtac (major RS IntD2) 1);
clasohm@923
   377
qed "IntE";
clasohm@923
   378
paulson@2499
   379
AddSIs [IntI];
paulson@2499
   380
AddSEs [IntE];
clasohm@923
   381
nipkow@1548
   382
section "Set difference";
clasohm@923
   383
paulson@2499
   384
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
paulson@2891
   385
 (fn _ => [ (Blast_tac 1) ]);
clasohm@923
   386
paulson@2499
   387
Addsimps [Diff_iff];
paulson@2499
   388
paulson@2499
   389
qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
paulson@2499
   390
 (fn _=> [ Asm_simp_tac 1 ]);
clasohm@923
   391
paulson@2499
   392
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
paulson@2499
   393
 (fn _=> [ (Asm_full_simp_tac 1) ]);
clasohm@923
   394
paulson@2499
   395
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
paulson@2499
   396
 (fn _=> [ (Asm_full_simp_tac 1) ]);
paulson@2499
   397
paulson@2499
   398
qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
clasohm@923
   399
 (fn prems=>
clasohm@923
   400
  [ (resolve_tac prems 1),
clasohm@923
   401
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
clasohm@923
   402
paulson@2499
   403
AddSIs [DiffI];
paulson@2499
   404
AddSEs [DiffE];
clasohm@923
   405
clasohm@923
   406
nipkow@1548
   407
section "Augmenting a set -- insert";
clasohm@923
   408
paulson@2499
   409
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
paulson@2891
   410
 (fn _ => [Blast_tac 1]);
paulson@2499
   411
paulson@2499
   412
Addsimps [insert_iff];
clasohm@923
   413
paulson@2499
   414
qed_goal "insertI1" Set.thy "a : insert a B"
paulson@2499
   415
 (fn _ => [Simp_tac 1]);
paulson@2499
   416
paulson@2499
   417
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
paulson@2499
   418
 (fn _=> [Asm_simp_tac 1]);
clasohm@923
   419
clasohm@923
   420
qed_goalw "insertE" Set.thy [insert_def]
clasohm@923
   421
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
clasohm@923
   422
 (fn major::prems=>
clasohm@923
   423
  [ (rtac (major RS UnE) 1),
clasohm@923
   424
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
clasohm@923
   425
clasohm@923
   426
(*Classical introduction rule*)
clasohm@923
   427
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
paulson@2499
   428
 (fn prems=>
paulson@2499
   429
  [ (Simp_tac 1),
paulson@2499
   430
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
paulson@2499
   431
paulson@2499
   432
AddSIs [insertCI]; 
paulson@2499
   433
AddSEs [insertE];
clasohm@923
   434
nipkow@1548
   435
section "Singletons, using insert";
clasohm@923
   436
clasohm@923
   437
qed_goal "singletonI" Set.thy "a : {a}"
clasohm@923
   438
 (fn _=> [ (rtac insertI1 1) ]);
clasohm@923
   439
paulson@5143
   440
Goal "b : {a} ==> b=a";
paulson@2891
   441
by (Blast_tac 1);
clasohm@923
   442
qed "singletonD";
clasohm@923
   443
oheimb@1776
   444
bind_thm ("singletonE", make_elim singletonD);
oheimb@1776
   445
paulson@2499
   446
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
paulson@2891
   447
(fn _ => [Blast_tac 1]);
clasohm@923
   448
paulson@5143
   449
Goal "{a}={b} ==> a=b";
wenzelm@4089
   450
by (blast_tac (claset() addEs [equalityE]) 1);
clasohm@923
   451
qed "singleton_inject";
clasohm@923
   452
paulson@2858
   453
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
paulson@2858
   454
AddSIs [singletonI];   
paulson@2499
   455
AddSDs [singleton_inject];
paulson@3718
   456
AddSEs [singletonE];
paulson@2499
   457
wenzelm@5069
   458
Goal "{x. x=a} = {a}";
wenzelm@4423
   459
by (Blast_tac 1);
nipkow@3582
   460
qed "singleton_conv";
nipkow@3582
   461
Addsimps [singleton_conv];
nipkow@1531
   462
nipkow@1531
   463
nipkow@1548
   464
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
clasohm@923
   465
wenzelm@5069
   466
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
paulson@2891
   467
by (Blast_tac 1);
paulson@2499
   468
qed "UN_iff";
paulson@2499
   469
paulson@2499
   470
Addsimps [UN_iff];
paulson@2499
   471
clasohm@923
   472
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@5143
   473
Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
paulson@4477
   474
by Auto_tac;
clasohm@923
   475
qed "UN_I";
clasohm@923
   476
paulson@5316
   477
val major::prems = Goalw [UNION_def]
clasohm@923
   478
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
clasohm@923
   479
by (rtac (major RS CollectD RS bexE) 1);
clasohm@923
   480
by (REPEAT (ares_tac prems 1));
clasohm@923
   481
qed "UN_E";
clasohm@923
   482
paulson@2499
   483
AddIs  [UN_I];
paulson@2499
   484
AddSEs [UN_E];
paulson@2499
   485
paulson@5316
   486
val prems = Goal
clasohm@923
   487
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   488
\    (UN x:A. C(x)) = (UN x:B. D(x))";
clasohm@923
   489
by (REPEAT (etac UN_E 1
clasohm@923
   490
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
clasohm@1465
   491
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
clasohm@923
   492
qed "UN_cong";
clasohm@923
   493
clasohm@923
   494
nipkow@1548
   495
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
clasohm@923
   496
wenzelm@5069
   497
Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
paulson@4477
   498
by Auto_tac;
paulson@2499
   499
qed "INT_iff";
paulson@2499
   500
paulson@2499
   501
Addsimps [INT_iff];
paulson@2499
   502
paulson@5316
   503
val prems = Goalw [INTER_def]
clasohm@923
   504
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
clasohm@923
   505
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
clasohm@923
   506
qed "INT_I";
clasohm@923
   507
paulson@5143
   508
Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
paulson@4477
   509
by Auto_tac;
clasohm@923
   510
qed "INT_D";
clasohm@923
   511
clasohm@923
   512
(*"Classical" elimination -- by the Excluded Middle on a:A *)
paulson@5316
   513
val major::prems = Goalw [INTER_def]
clasohm@923
   514
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
clasohm@923
   515
by (rtac (major RS CollectD RS ballE) 1);
clasohm@923
   516
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   517
qed "INT_E";
clasohm@923
   518
paulson@2499
   519
AddSIs [INT_I];
paulson@2499
   520
AddEs  [INT_D, INT_E];
paulson@2499
   521
paulson@5316
   522
val prems = Goal
clasohm@923
   523
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   524
\    (INT x:A. C(x)) = (INT x:B. D(x))";
clasohm@923
   525
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
clasohm@923
   526
by (REPEAT (dtac INT_D 1
clasohm@923
   527
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
clasohm@923
   528
qed "INT_cong";
clasohm@923
   529
clasohm@923
   530
nipkow@1548
   531
section "Union";
clasohm@923
   532
wenzelm@5069
   533
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
paulson@2891
   534
by (Blast_tac 1);
paulson@2499
   535
qed "Union_iff";
paulson@2499
   536
paulson@2499
   537
Addsimps [Union_iff];
paulson@2499
   538
clasohm@923
   539
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@5143
   540
Goal "[| X:C;  A:X |] ==> A : Union(C)";
paulson@4477
   541
by Auto_tac;
clasohm@923
   542
qed "UnionI";
clasohm@923
   543
paulson@5316
   544
val major::prems = Goalw [Union_def]
clasohm@923
   545
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
clasohm@923
   546
by (rtac (major RS UN_E) 1);
clasohm@923
   547
by (REPEAT (ares_tac prems 1));
clasohm@923
   548
qed "UnionE";
clasohm@923
   549
paulson@2499
   550
AddIs  [UnionI];
paulson@2499
   551
AddSEs [UnionE];
paulson@2499
   552
paulson@2499
   553
nipkow@1548
   554
section "Inter";
clasohm@923
   555
wenzelm@5069
   556
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
paulson@2891
   557
by (Blast_tac 1);
paulson@2499
   558
qed "Inter_iff";
paulson@2499
   559
paulson@2499
   560
Addsimps [Inter_iff];
paulson@2499
   561
paulson@5316
   562
val prems = Goalw [Inter_def]
clasohm@923
   563
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
clasohm@923
   564
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
clasohm@923
   565
qed "InterI";
clasohm@923
   566
clasohm@923
   567
(*A "destruct" rule -- every X in C contains A as an element, but
clasohm@923
   568
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
paulson@5143
   569
Goal "[| A : Inter(C);  X:C |] ==> A:X";
paulson@4477
   570
by Auto_tac;
clasohm@923
   571
qed "InterD";
clasohm@923
   572
clasohm@923
   573
(*"Classical" elimination rule -- does not require proving X:C *)
paulson@5316
   574
val major::prems = Goalw [Inter_def]
paulson@2721
   575
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
clasohm@923
   576
by (rtac (major RS INT_E) 1);
clasohm@923
   577
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   578
qed "InterE";
clasohm@923
   579
paulson@2499
   580
AddSIs [InterI];
paulson@2499
   581
AddEs  [InterD, InterE];
paulson@2499
   582
paulson@2499
   583
nipkow@2912
   584
(*** Image of a set under a function ***)
nipkow@2912
   585
nipkow@2912
   586
(*Frequently b does not have the syntactic form of f(x).*)
paulson@5316
   587
Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
paulson@5316
   588
by (Blast_tac 1);
nipkow@2912
   589
qed "image_eqI";
nipkow@3909
   590
Addsimps [image_eqI];
nipkow@2912
   591
nipkow@2912
   592
bind_thm ("imageI", refl RS image_eqI);
nipkow@2912
   593
nipkow@2912
   594
(*The eta-expansion gives variable-name preservation.*)
paulson@5316
   595
val major::prems = Goalw [image_def]
wenzelm@3842
   596
    "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
nipkow@2912
   597
by (rtac (major RS CollectD RS bexE) 1);
nipkow@2912
   598
by (REPEAT (ares_tac prems 1));
nipkow@2912
   599
qed "imageE";
nipkow@2912
   600
nipkow@2912
   601
AddIs  [image_eqI];
nipkow@2912
   602
AddSEs [imageE]; 
nipkow@2912
   603
wenzelm@5069
   604
Goal "f``(A Un B) = f``A Un f``B";
paulson@2935
   605
by (Blast_tac 1);
nipkow@2912
   606
qed "image_Un";
nipkow@2912
   607
wenzelm@5069
   608
Goal "(z : f``A) = (EX x:A. z = f x)";
paulson@3960
   609
by (Blast_tac 1);
paulson@3960
   610
qed "image_iff";
paulson@3960
   611
paulson@4523
   612
(*This rewrite rule would confuse users if made default.*)
wenzelm@5069
   613
Goal "(f``A <= B) = (ALL x:A. f(x): B)";
paulson@4523
   614
by (Blast_tac 1);
paulson@4523
   615
qed "image_subset_iff";
paulson@4523
   616
paulson@4523
   617
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
paulson@4523
   618
  many existing proofs.*)
paulson@5316
   619
val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
paulson@4510
   620
by (blast_tac (claset() addIs prems) 1);
paulson@4510
   621
qed "image_subsetI";
paulson@4510
   622
nipkow@2912
   623
nipkow@2912
   624
(*** Range of a function -- just a translation for image! ***)
nipkow@2912
   625
paulson@5143
   626
Goal "b=f(x) ==> b : range(f)";
nipkow@2912
   627
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
nipkow@2912
   628
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
nipkow@2912
   629
nipkow@2912
   630
bind_thm ("rangeI", UNIV_I RS imageI);
nipkow@2912
   631
paulson@5316
   632
val [major,minor] = Goal 
wenzelm@3842
   633
    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
nipkow@2912
   634
by (rtac (major RS imageE) 1);
nipkow@2912
   635
by (etac minor 1);
nipkow@2912
   636
qed "rangeE";
nipkow@2912
   637
oheimb@1776
   638
oheimb@1776
   639
(*** Set reasoning tools ***)
oheimb@1776
   640
oheimb@1776
   641
paulson@3912
   642
(** Rewrite rules for boolean case-splitting: faster than 
nipkow@4830
   643
	addsplits[split_if]
paulson@3912
   644
**)
paulson@3912
   645
nipkow@4830
   646
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
nipkow@4830
   647
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
paulson@3912
   648
paulson@5237
   649
(*Split ifs on either side of the membership relation.
paulson@5237
   650
	Not for Addsimps -- can cause goals to blow up!*)
nipkow@4830
   651
bind_thm ("split_if_mem1", 
nipkow@4830
   652
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
nipkow@4830
   653
bind_thm ("split_if_mem2", 
nipkow@4830
   654
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
paulson@3912
   655
nipkow@4830
   656
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
nipkow@4830
   657
		  split_if_mem1, split_if_mem2];
paulson@3912
   658
paulson@3912
   659
wenzelm@4089
   660
(*Each of these has ALREADY been added to simpset() above.*)
paulson@2024
   661
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
paulson@4159
   662
                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
oheimb@1776
   663
oheimb@1776
   664
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
oheimb@1776
   665
wenzelm@4089
   666
simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
oheimb@1776
   667
                    setmksimps (mksimps mksimps_pairs);
nipkow@3222
   668
paulson@5256
   669
Addsimps[subset_UNIV, subset_refl];
nipkow@3222
   670
nipkow@3222
   671
nipkow@3222
   672
(*** < ***)
nipkow@3222
   673
wenzelm@5069
   674
Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
nipkow@3222
   675
by (Blast_tac 1);
nipkow@3222
   676
qed "psubsetI";
nipkow@3222
   677
paulson@5148
   678
Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
paulson@4477
   679
by Auto_tac;
nipkow@3222
   680
qed "psubset_insertD";
paulson@4059
   681
paulson@4059
   682
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);