src/HOL/Set.ML
author paulson
Wed Dec 24 10:02:30 1997 +0100 (1997-12-24)
changeset 4477 b3e5857d8d99
parent 4469 399868bf8444
child 4510 a37f515a0b25
permissions -rw-r--r--
New Auto_tac (by Oheimb), and new syntax (without parens), and 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 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*)
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goal Set.thy "(! 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 Set.thy "(? 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 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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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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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]);		(*Blast_tac would try order_refl and fail*)
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val prems = goal Set.thy "!!B. [| 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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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 "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 Set.thy [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 Set.thy [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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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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qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
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 (fn _ => [ (Blast_tac 1) ]);
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goalw Set.thy [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 Set.thy [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 thy "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 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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   316
paulson@2499
   317
AddSIs [ComplI];
paulson@2499
   318
AddSEs [ComplE];
paulson@1640
   319
clasohm@923
   320
nipkow@1548
   321
section "Binary union -- Un";
clasohm@923
   322
paulson@2499
   323
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
paulson@2891
   324
 (fn _ => [ Blast_tac 1 ]);
paulson@2499
   325
paulson@2499
   326
Addsimps [Un_iff];
paulson@2499
   327
paulson@2499
   328
goal Set.thy "!!c. c:A ==> c : A Un B";
paulson@2499
   329
by (Asm_simp_tac 1);
clasohm@923
   330
qed "UnI1";
clasohm@923
   331
paulson@2499
   332
goal Set.thy "!!c. c:B ==> c : A Un B";
paulson@2499
   333
by (Asm_simp_tac 1);
clasohm@923
   334
qed "UnI2";
clasohm@923
   335
clasohm@923
   336
(*Classical introduction rule: no commitment to A vs B*)
clasohm@923
   337
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
clasohm@923
   338
 (fn prems=>
paulson@2499
   339
  [ (Simp_tac 1),
paulson@2499
   340
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
clasohm@923
   341
clasohm@923
   342
val major::prems = goalw Set.thy [Un_def]
clasohm@923
   343
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   344
by (rtac (major RS CollectD RS disjE) 1);
clasohm@923
   345
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   346
qed "UnE";
clasohm@923
   347
paulson@2499
   348
AddSIs [UnCI];
paulson@2499
   349
AddSEs [UnE];
paulson@1640
   350
clasohm@923
   351
nipkow@1548
   352
section "Binary intersection -- Int";
clasohm@923
   353
paulson@2499
   354
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
paulson@2891
   355
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   356
paulson@2499
   357
Addsimps [Int_iff];
paulson@2499
   358
paulson@2499
   359
goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
paulson@2499
   360
by (Asm_simp_tac 1);
clasohm@923
   361
qed "IntI";
clasohm@923
   362
paulson@2499
   363
goal Set.thy "!!c. c : A Int B ==> c:A";
paulson@2499
   364
by (Asm_full_simp_tac 1);
clasohm@923
   365
qed "IntD1";
clasohm@923
   366
paulson@2499
   367
goal Set.thy "!!c. c : A Int B ==> c:B";
paulson@2499
   368
by (Asm_full_simp_tac 1);
clasohm@923
   369
qed "IntD2";
clasohm@923
   370
clasohm@923
   371
val [major,minor] = goal Set.thy
clasohm@923
   372
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
clasohm@923
   373
by (rtac minor 1);
clasohm@923
   374
by (rtac (major RS IntD1) 1);
clasohm@923
   375
by (rtac (major RS IntD2) 1);
clasohm@923
   376
qed "IntE";
clasohm@923
   377
paulson@2499
   378
AddSIs [IntI];
paulson@2499
   379
AddSEs [IntE];
clasohm@923
   380
nipkow@1548
   381
section "Set difference";
clasohm@923
   382
paulson@2499
   383
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
paulson@2891
   384
 (fn _ => [ (Blast_tac 1) ]);
clasohm@923
   385
paulson@2499
   386
Addsimps [Diff_iff];
paulson@2499
   387
paulson@2499
   388
qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
paulson@2499
   389
 (fn _=> [ Asm_simp_tac 1 ]);
clasohm@923
   390
paulson@2499
   391
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
paulson@2499
   392
 (fn _=> [ (Asm_full_simp_tac 1) ]);
clasohm@923
   393
paulson@2499
   394
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
paulson@2499
   395
 (fn _=> [ (Asm_full_simp_tac 1) ]);
paulson@2499
   396
paulson@2499
   397
qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
clasohm@923
   398
 (fn prems=>
clasohm@923
   399
  [ (resolve_tac prems 1),
clasohm@923
   400
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
clasohm@923
   401
paulson@2499
   402
AddSIs [DiffI];
paulson@2499
   403
AddSEs [DiffE];
clasohm@923
   404
clasohm@923
   405
nipkow@1548
   406
section "Augmenting a set -- insert";
clasohm@923
   407
paulson@2499
   408
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
paulson@2891
   409
 (fn _ => [Blast_tac 1]);
paulson@2499
   410
paulson@2499
   411
Addsimps [insert_iff];
clasohm@923
   412
paulson@2499
   413
qed_goal "insertI1" Set.thy "a : insert a B"
paulson@2499
   414
 (fn _ => [Simp_tac 1]);
paulson@2499
   415
paulson@2499
   416
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
paulson@2499
   417
 (fn _=> [Asm_simp_tac 1]);
clasohm@923
   418
clasohm@923
   419
qed_goalw "insertE" Set.thy [insert_def]
clasohm@923
   420
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
clasohm@923
   421
 (fn major::prems=>
clasohm@923
   422
  [ (rtac (major RS UnE) 1),
clasohm@923
   423
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
clasohm@923
   424
clasohm@923
   425
(*Classical introduction rule*)
clasohm@923
   426
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
paulson@2499
   427
 (fn prems=>
paulson@2499
   428
  [ (Simp_tac 1),
paulson@2499
   429
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
paulson@2499
   430
paulson@2499
   431
AddSIs [insertCI]; 
paulson@2499
   432
AddSEs [insertE];
clasohm@923
   433
nipkow@1548
   434
section "Singletons, using insert";
clasohm@923
   435
clasohm@923
   436
qed_goal "singletonI" Set.thy "a : {a}"
clasohm@923
   437
 (fn _=> [ (rtac insertI1 1) ]);
clasohm@923
   438
paulson@2499
   439
goal Set.thy "!!a. b : {a} ==> b=a";
paulson@2891
   440
by (Blast_tac 1);
clasohm@923
   441
qed "singletonD";
clasohm@923
   442
oheimb@1776
   443
bind_thm ("singletonE", make_elim singletonD);
oheimb@1776
   444
paulson@2499
   445
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
paulson@2891
   446
(fn _ => [Blast_tac 1]);
clasohm@923
   447
paulson@2499
   448
goal Set.thy "!!a b. {a}={b} ==> a=b";
wenzelm@4089
   449
by (blast_tac (claset() addEs [equalityE]) 1);
clasohm@923
   450
qed "singleton_inject";
clasohm@923
   451
paulson@2858
   452
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
paulson@2858
   453
AddSIs [singletonI];   
paulson@2499
   454
AddSDs [singleton_inject];
paulson@3718
   455
AddSEs [singletonE];
paulson@2499
   456
wenzelm@3842
   457
goal Set.thy "{x. x=a} = {a}";
wenzelm@4423
   458
by (Blast_tac 1);
nipkow@3582
   459
qed "singleton_conv";
nipkow@3582
   460
Addsimps [singleton_conv];
nipkow@1531
   461
nipkow@1531
   462
nipkow@1548
   463
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
clasohm@923
   464
paulson@2499
   465
goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
paulson@2891
   466
by (Blast_tac 1);
paulson@2499
   467
qed "UN_iff";
paulson@2499
   468
paulson@2499
   469
Addsimps [UN_iff];
paulson@2499
   470
clasohm@923
   471
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@2499
   472
goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
paulson@4477
   473
by Auto_tac;
clasohm@923
   474
qed "UN_I";
clasohm@923
   475
clasohm@923
   476
val major::prems = goalw Set.thy [UNION_def]
clasohm@923
   477
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
clasohm@923
   478
by (rtac (major RS CollectD RS bexE) 1);
clasohm@923
   479
by (REPEAT (ares_tac prems 1));
clasohm@923
   480
qed "UN_E";
clasohm@923
   481
paulson@2499
   482
AddIs  [UN_I];
paulson@2499
   483
AddSEs [UN_E];
paulson@2499
   484
clasohm@923
   485
val prems = goal Set.thy
clasohm@923
   486
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   487
\    (UN x:A. C(x)) = (UN x:B. D(x))";
clasohm@923
   488
by (REPEAT (etac UN_E 1
clasohm@923
   489
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
clasohm@1465
   490
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
clasohm@923
   491
qed "UN_cong";
clasohm@923
   492
clasohm@923
   493
nipkow@1548
   494
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
clasohm@923
   495
paulson@2499
   496
goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
paulson@4477
   497
by Auto_tac;
paulson@2499
   498
qed "INT_iff";
paulson@2499
   499
paulson@2499
   500
Addsimps [INT_iff];
paulson@2499
   501
clasohm@923
   502
val prems = goalw Set.thy [INTER_def]
clasohm@923
   503
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
clasohm@923
   504
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
clasohm@923
   505
qed "INT_I";
clasohm@923
   506
paulson@2499
   507
goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
paulson@4477
   508
by Auto_tac;
clasohm@923
   509
qed "INT_D";
clasohm@923
   510
clasohm@923
   511
(*"Classical" elimination -- by the Excluded Middle on a:A *)
clasohm@923
   512
val major::prems = goalw Set.thy [INTER_def]
clasohm@923
   513
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
clasohm@923
   514
by (rtac (major RS CollectD RS ballE) 1);
clasohm@923
   515
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   516
qed "INT_E";
clasohm@923
   517
paulson@2499
   518
AddSIs [INT_I];
paulson@2499
   519
AddEs  [INT_D, INT_E];
paulson@2499
   520
clasohm@923
   521
val prems = goal Set.thy
clasohm@923
   522
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   523
\    (INT x:A. C(x)) = (INT x:B. D(x))";
clasohm@923
   524
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
clasohm@923
   525
by (REPEAT (dtac INT_D 1
clasohm@923
   526
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
clasohm@923
   527
qed "INT_cong";
clasohm@923
   528
clasohm@923
   529
nipkow@1548
   530
section "Union";
clasohm@923
   531
paulson@2499
   532
goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
paulson@2891
   533
by (Blast_tac 1);
paulson@2499
   534
qed "Union_iff";
paulson@2499
   535
paulson@2499
   536
Addsimps [Union_iff];
paulson@2499
   537
clasohm@923
   538
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@2499
   539
goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
paulson@4477
   540
by Auto_tac;
clasohm@923
   541
qed "UnionI";
clasohm@923
   542
clasohm@923
   543
val major::prems = goalw Set.thy [Union_def]
clasohm@923
   544
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
clasohm@923
   545
by (rtac (major RS UN_E) 1);
clasohm@923
   546
by (REPEAT (ares_tac prems 1));
clasohm@923
   547
qed "UnionE";
clasohm@923
   548
paulson@2499
   549
AddIs  [UnionI];
paulson@2499
   550
AddSEs [UnionE];
paulson@2499
   551
paulson@2499
   552
nipkow@1548
   553
section "Inter";
clasohm@923
   554
paulson@2499
   555
goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
paulson@2891
   556
by (Blast_tac 1);
paulson@2499
   557
qed "Inter_iff";
paulson@2499
   558
paulson@2499
   559
Addsimps [Inter_iff];
paulson@2499
   560
clasohm@923
   561
val prems = goalw Set.thy [Inter_def]
clasohm@923
   562
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
clasohm@923
   563
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
clasohm@923
   564
qed "InterI";
clasohm@923
   565
clasohm@923
   566
(*A "destruct" rule -- every X in C contains A as an element, but
clasohm@923
   567
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
paulson@2499
   568
goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
paulson@4477
   569
by Auto_tac;
clasohm@923
   570
qed "InterD";
clasohm@923
   571
clasohm@923
   572
(*"Classical" elimination rule -- does not require proving X:C *)
clasohm@923
   573
val major::prems = goalw Set.thy [Inter_def]
paulson@2721
   574
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
clasohm@923
   575
by (rtac (major RS INT_E) 1);
clasohm@923
   576
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   577
qed "InterE";
clasohm@923
   578
paulson@2499
   579
AddSIs [InterI];
paulson@2499
   580
AddEs  [InterD, InterE];
paulson@2499
   581
paulson@2499
   582
nipkow@2912
   583
(*** Image of a set under a function ***)
nipkow@2912
   584
nipkow@2912
   585
(*Frequently b does not have the syntactic form of f(x).*)
nipkow@2912
   586
val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
nipkow@2912
   587
by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
nipkow@2912
   588
qed "image_eqI";
nipkow@3909
   589
Addsimps [image_eqI];
nipkow@2912
   590
nipkow@2912
   591
bind_thm ("imageI", refl RS image_eqI);
nipkow@2912
   592
nipkow@2912
   593
(*The eta-expansion gives variable-name preservation.*)
nipkow@2912
   594
val major::prems = goalw thy [image_def]
wenzelm@3842
   595
    "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
nipkow@2912
   596
by (rtac (major RS CollectD RS bexE) 1);
nipkow@2912
   597
by (REPEAT (ares_tac prems 1));
nipkow@2912
   598
qed "imageE";
nipkow@2912
   599
nipkow@2912
   600
AddIs  [image_eqI];
nipkow@2912
   601
AddSEs [imageE]; 
nipkow@2912
   602
nipkow@2912
   603
goalw thy [o_def] "(f o g)``r = f``(g``r)";
paulson@2935
   604
by (Blast_tac 1);
nipkow@2912
   605
qed "image_compose";
nipkow@2912
   606
nipkow@2912
   607
goal thy "f``(A Un B) = f``A Un f``B";
paulson@2935
   608
by (Blast_tac 1);
nipkow@2912
   609
qed "image_Un";
nipkow@2912
   610
paulson@3960
   611
goal Set.thy "(z : f``A) = (EX x:A. z = f x)";
paulson@3960
   612
by (Blast_tac 1);
paulson@3960
   613
qed "image_iff";
paulson@3960
   614
nipkow@2912
   615
nipkow@2912
   616
(*** Range of a function -- just a translation for image! ***)
nipkow@2912
   617
nipkow@2912
   618
goal thy "!!b. b=f(x) ==> b : range(f)";
nipkow@2912
   619
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
nipkow@2912
   620
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
nipkow@2912
   621
nipkow@2912
   622
bind_thm ("rangeI", UNIV_I RS imageI);
nipkow@2912
   623
nipkow@2912
   624
val [major,minor] = goal thy 
wenzelm@3842
   625
    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
nipkow@2912
   626
by (rtac (major RS imageE) 1);
nipkow@2912
   627
by (etac minor 1);
nipkow@2912
   628
qed "rangeE";
nipkow@2912
   629
oheimb@1776
   630
oheimb@1776
   631
(*** Set reasoning tools ***)
oheimb@1776
   632
oheimb@1776
   633
paulson@3912
   634
(** Rewrite rules for boolean case-splitting: faster than 
nipkow@3919
   635
	addsplits[expand_if]
paulson@3912
   636
**)
paulson@3912
   637
paulson@3912
   638
bind_thm ("expand_if_eq1", read_instantiate [("P", "%x. x = ?b")] expand_if);
paulson@3912
   639
bind_thm ("expand_if_eq2", read_instantiate [("P", "%x. ?a = x")] expand_if);
paulson@3912
   640
paulson@3912
   641
bind_thm ("expand_if_mem1", 
paulson@3912
   642
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] expand_if);
paulson@3912
   643
bind_thm ("expand_if_mem2", 
paulson@3912
   644
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] expand_if);
paulson@3912
   645
paulson@3912
   646
val expand_ifs = [if_bool_eq, expand_if_eq1, expand_if_eq2,
paulson@3912
   647
		  expand_if_mem1, expand_if_mem2];
paulson@3912
   648
paulson@3912
   649
wenzelm@4089
   650
(*Each of these has ALREADY been added to simpset() above.*)
paulson@2024
   651
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
paulson@4159
   652
                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
oheimb@1776
   653
paulson@1937
   654
(*Not for Addsimps -- it can cause goals to blow up!*)
paulson@1937
   655
goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
wenzelm@4089
   656
by (simp_tac (simpset() addsplits [expand_if]) 1);
paulson@1937
   657
qed "mem_if";
paulson@1937
   658
oheimb@1776
   659
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
oheimb@1776
   660
wenzelm@4089
   661
simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
oheimb@1776
   662
                    setmksimps (mksimps mksimps_pairs);
nipkow@3222
   663
nipkow@3222
   664
Addsimps[subset_UNIV, empty_subsetI, subset_refl];
nipkow@3222
   665
nipkow@3222
   666
nipkow@3222
   667
(*** < ***)
nipkow@3222
   668
nipkow@3222
   669
goalw Set.thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
nipkow@3222
   670
by (Blast_tac 1);
nipkow@3222
   671
qed "psubsetI";
nipkow@3222
   672
nipkow@3222
   673
goalw Set.thy [psubset_def]
nipkow@3222
   674
    "!!x. A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
paulson@4477
   675
by Auto_tac;
nipkow@3222
   676
qed "psubset_insertD";
paulson@4059
   677
paulson@4059
   678
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);