# HG changeset patch
# User wenzelm
# Date 1004306352 -3600
# Node ID 0a3dace545c521646a369bf23e37fcd700de71b9
# Parent  a394d3e9c8bbd8e415f1e7cd21df8e2618f52e28
converted theory "Set";

diff -r a394d3e9c8bb -r 0a3dace545c5 src/HOL/Set.ML
--- a/src/HOL/Set.ML	Sun Oct 28 22:58:39 2001 +0100
+++ b/src/HOL/Set.ML	Sun Oct 28 22:59:12 2001 +0100
@@ -1,845 +1,26 @@
-(*  Title:      HOL/Set.ML
-    ID:         $Id$
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1991  University of Cambridge
 
-Set theory for higher-order logic.  A set is simply a predicate.
-*)
-
-section "Relating predicates and sets";
-
-Addsimps [Collect_mem_eq];
-AddIffs  [mem_Collect_eq];
-
-Goal "P(a) ==> a : {x. P(x)}";
-by (Asm_simp_tac 1);
-qed "CollectI";
-
-Goal "a : {x. P(x)} ==> P(a)";
-by (Asm_full_simp_tac 1);
-qed "CollectD";
-
-val [prem] = Goal "(!!x. (x:A) = (x:B)) ==> A = B";
-by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
-by (rtac Collect_mem_eq 1);
-by (rtac Collect_mem_eq 1);
-qed "set_ext";
-
-val [prem] = Goal "(!!x. P(x)=Q(x)) ==> {x. P(x)} = {x. Q(x)}";
-by (rtac (prem RS ext RS arg_cong) 1);
-qed "Collect_cong";
-
-bind_thm ("CollectE", make_elim CollectD);
-
-AddSIs [CollectI];
-AddSEs [CollectE];
-
-
-section "Bounded quantifiers";
-
-val prems = Goalw [Ball_def]
-    "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)";
-by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
-qed "ballI";
-
-bind_thms ("strip", [impI, allI, ballI]);
-
-Goalw [Ball_def] "[| ALL x:A. P(x);  x:A |] ==> P(x)";
-by (Blast_tac 1);
-qed "bspec";
-
-val major::prems = Goalw [Ball_def]
-    "[| ALL x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
-by (rtac (major RS spec RS impCE) 1);
-by (REPEAT (eresolve_tac prems 1));
-qed "ballE";
-
-(*Takes assumptions ALL x:A.P(x) and a:A; creates assumption P(a)*)
-fun ball_tac i = etac ballE i THEN contr_tac (i+1);
-
-AddSIs [ballI];
-AddEs  [ballE];
-AddXDs [bspec];
-(* gives better instantiation for bound: *)
-claset_ref() := claset() addbefore ("bspec", datac bspec 1);
-
-(*Normally the best argument order: P(x) constrains the choice of x:A*)
-Goalw [Bex_def] "[| P(x);  x:A |] ==> EX x:A. P(x)";
-by (Blast_tac 1);
-qed "bexI";
-
-(*The best argument order when there is only one x:A*)
-Goalw [Bex_def] "[| x:A;  P(x) |] ==> EX x:A. P(x)";
-by (Blast_tac 1);
-qed "rev_bexI";
-
-val prems = Goal 
-   "[| ALL x:A. ~P(x) ==> P(a);  a:A |] ==> EX x:A. P(x)";
-by (rtac classical 1);
-by (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ;
-qed "bexCI";
-
-val major::prems = Goalw [Bex_def]
-    "[| EX x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
-by (rtac (major RS exE) 1);
-by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
-qed "bexE";
-
-AddIs  [bexI];
-AddSEs [bexE];
-
-(*Trival rewrite rule*)
-Goal "(ALL x:A. P) = ((EX x. x:A) --> P)";
-by (simp_tac (simpset() addsimps [Ball_def]) 1);
-qed "ball_triv";
-
-(*Dual form for existentials*)
-Goal "(EX x:A. P) = ((EX x. x:A) & P)";
-by (simp_tac (simpset() addsimps [Bex_def]) 1);
-qed "bex_triv";
-
-Addsimps [ball_triv, bex_triv];
-
-Goal "(EX x:A. x=a) = (a:A)";
-by(Blast_tac 1);
-qed "bex_triv_one_point1";
-
-Goal "(EX x:A. a=x) = (a:A)";
-by(Blast_tac 1);
-qed "bex_triv_one_point2";
-
-Goal "(EX x:A. x=a & P x) = (a:A & P a)";
-by(Blast_tac 1);
-qed "bex_one_point1";
-
-Goal "(EX x:A. a=x & P x) = (a:A & P a)";
-by(Blast_tac 1);
-qed "bex_one_point2";
-
-Goal "(ALL x:A. x=a --> P x) = (a:A --> P a)";
-by(Blast_tac 1);
-qed "ball_one_point1";
-
-Goal "(ALL x:A. a=x --> P x) = (a:A --> P a)";
-by(Blast_tac 1);
-qed "ball_one_point2";
-
-Addsimps [bex_triv_one_point1,bex_triv_one_point2,
-          bex_one_point1,bex_one_point2,
-          ball_one_point1,ball_one_point2];
-
-let
-val ex_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
-    ("EX x:A. P(x) & Q(x)",HOLogic.boolT)
-
-val prove_bex_tac = rewrite_goals_tac [Bex_def] THEN
-                    Quantifier1.prove_one_point_ex_tac;
-
-val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
-
-val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
-    ("ALL x:A. P(x) --> Q(x)",HOLogic.boolT)
-
-val prove_ball_tac = rewrite_goals_tac [Ball_def] THEN
-                     Quantifier1.prove_one_point_all_tac;
-
-val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
-
-val defBEX_regroup = mk_simproc "defined BEX" [ex_pattern] rearrange_bex;
-val defBALL_regroup = mk_simproc "defined BALL" [all_pattern] rearrange_ball;
-in
-
-Addsimprocs [defBALL_regroup,defBEX_regroup]
-
-end;
-
-(** Congruence rules **)
-
-val prems = Goalw [Ball_def]
-    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
-\    (ALL x:A. P(x)) = (ALL x:B. Q(x))";
-by (asm_simp_tac (simpset() addsimps prems) 1);
-qed "ball_cong";
-
-val prems = Goalw [Bex_def]
-    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
-\    (EX x:A. P(x)) = (EX x:B. Q(x))";
-by (asm_simp_tac (simpset() addcongs [conj_cong] addsimps prems) 1);
-qed "bex_cong";
-
-Addcongs [ball_cong,bex_cong];
-
-section "Subsets";
-
-val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
-by (REPEAT (ares_tac (prems @ [ballI]) 1));
-qed "subsetI";
-
-(*Map the type ('a set => anything) to just 'a.
-  For overloading constants whose first argument has type "'a set" *)
-fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
-
-(*While (:) is not, its type must be kept
-  for overloading of = to work.*)
-Blast.overloaded ("op :", domain_type);
-
-overload_1st_set "Ball";		(*need UNION, INTER also?*)
-overload_1st_set "Bex";
-
-(*Image: retain the type of the set being expressed*)
-Blast.overloaded ("image", domain_type);
-
-(*Rule in Modus Ponens style*)
-Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
-by (Blast_tac 1);
-qed "subsetD";
-AddXIs [subsetD];
-
-(*The same, with reversed premises for use with etac -- cf rev_mp*)
-Goal "[| c:A;  A <= B |] ==> c:B";
-by (REPEAT (ares_tac [subsetD] 1)) ;
-qed "rev_subsetD";
-AddXIs [rev_subsetD];
-
-(*Converts A<=B to x:A ==> x:B*)
-fun impOfSubs th = th RSN (2, rev_subsetD);
-
-(*Classical elimination rule*)
-val major::prems = Goalw [subset_def] 
-    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
-by (rtac (major RS ballE) 1);
-by (REPEAT (eresolve_tac prems 1));
-qed "subsetCE";
-
-(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
-fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
-
-AddSIs [subsetI];
-AddEs  [subsetD, subsetCE];
-
-Goal "[| A <= B; c ~: B |] ==> c ~: A";
-by (Blast_tac 1);
-qed "contra_subsetD";
-
-Goal "A <= (A::'a set)";
-by (Fast_tac 1);
-qed "subset_refl";
-
-Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
-by (Blast_tac 1);
-qed "subset_trans";
-
-
-section "Equality";
-
-(*Anti-symmetry of the subset relation*)
-Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
-by (rtac set_ext 1);
-by (blast_tac (claset() addIs [subsetD]) 1);
-qed "subset_antisym";
-bind_thm ("equalityI", subset_antisym);
-
-AddSIs [equalityI];
-
-(* Equality rules from ZF set theory -- are they appropriate here? *)
-Goal "A = B ==> A<=(B::'a set)";
-by (etac ssubst 1);
-by (rtac subset_refl 1);
-qed "equalityD1";
-
-Goal "A = B ==> B<=(A::'a set)";
-by (etac ssubst 1);
-by (rtac subset_refl 1);
-qed "equalityD2";
-
-(*Be careful when adding this to the claset as subset_empty is in the simpset:
-  A={} goes to {}<=A and A<={} and then back to A={} !*)
-val prems = Goal
-    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
-by (resolve_tac prems 1);
-by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
-qed "equalityE";
-
-val major::prems = Goal
-    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
-by (rtac (major RS equalityE) 1);
-by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
-qed "equalityCE";
-
-AddEs [equalityCE];
-
-(*Lemma for creating induction formulae -- for "pattern matching" on p
-  To make the induction hypotheses usable, apply "spec" or "bspec" to
-  put universal quantifiers over the free variables in p. *)
-val prems = Goal 
-    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
-by (rtac mp 1);
-by (REPEAT (resolve_tac (refl::prems) 1));
-qed "setup_induction";
-
-Goal "A = B ==> (x : A) = (x : B)";
-by (Asm_simp_tac 1);
-qed "eqset_imp_iff";
-
-
-section "The universal set -- UNIV";
-
-Goalw [UNIV_def] "x : UNIV";
-by (rtac CollectI 1);
-by (rtac TrueI 1);
-qed "UNIV_I";
-
-Addsimps [UNIV_I];
-AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
-
-Goal "EX x. x : UNIV";
-by (Simp_tac 1);
-qed "UNIV_witness";
-AddXIs [UNIV_witness];
-
-Goal "A <= UNIV";
-by (rtac subsetI 1);
-by (rtac UNIV_I 1);
-qed "subset_UNIV";
-
-(** Eta-contracting these two rules (to remove P) causes them to be ignored
-    because of their interaction with congruence rules. **)
-
-Goalw [Ball_def] "Ball UNIV P = All P";
-by (Simp_tac 1);
-qed "ball_UNIV";
-
-Goalw [Bex_def] "Bex UNIV P = Ex P";
-by (Simp_tac 1);
-qed "bex_UNIV";
-Addsimps [ball_UNIV, bex_UNIV];
-
-
-section "The empty set -- {}";
-
-Goalw [empty_def] "(c : {}) = False";
-by (Blast_tac 1) ;
-qed "empty_iff";
-
-Addsimps [empty_iff];
-
-Goal "a:{} ==> P";
-by (Full_simp_tac 1);
-qed "emptyE";
-
-AddSEs [emptyE];
-
-Goal "{} <= A";
-by (Blast_tac 1) ;
-qed "empty_subsetI";
-
-(*One effect is to delete the ASSUMPTION {} <= A*)
-AddIffs [empty_subsetI];
-
-val [prem]= Goal "[| !!y. y:A ==> False |] ==> A={}";
-by (blast_tac (claset() addIs [prem RS FalseE]) 1) ;
-qed "equals0I";
-
-(*Use for reasoning about disjointness: A Int B = {} *)
-Goal "A={} ==> a ~: A";
-by (Blast_tac 1) ;
-qed "equals0D";
-
-Goalw [Ball_def] "Ball {} P = True";
-by (Simp_tac 1);
-qed "ball_empty";
-
-Goalw [Bex_def] "Bex {} P = False";
-by (Simp_tac 1);
-qed "bex_empty";
-Addsimps [ball_empty, bex_empty];
-
-Goal "UNIV ~= {}";
-by (blast_tac (claset() addEs [equalityE]) 1);
-qed "UNIV_not_empty";
-AddIffs [UNIV_not_empty];
-
-
-
-section "The Powerset operator -- Pow";
-
-Goalw [Pow_def] "(A : Pow(B)) = (A <= B)";
-by (Asm_simp_tac 1);
-qed "Pow_iff";
-
-AddIffs [Pow_iff]; 
-
-Goalw [Pow_def] "A <= B ==> A : Pow(B)";
-by (etac CollectI 1);
-qed "PowI";
-
-Goalw [Pow_def] "A : Pow(B)  ==>  A<=B";
-by (etac CollectD 1);
-qed "PowD";
-
-
-bind_thm ("Pow_bottom", empty_subsetI RS PowI);        (* {}: Pow(B) *)
-bind_thm ("Pow_top", subset_refl RS PowI);             (* A : Pow(A) *)
-
-
-section "Set complement";
-
-Goalw [Compl_def] "(c : -A) = (c~:A)";
-by (Blast_tac 1);
-qed "Compl_iff";
-
-Addsimps [Compl_iff];
-
-val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
-by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
-qed "ComplI";
-
-(*This form, with negated conclusion, works well with the Classical prover.
-  Negated assumptions behave like formulae on the right side of the notional
-  turnstile...*)
-Goalw [Compl_def] "c : -A ==> c~:A";
-by (etac CollectD 1);
-qed "ComplD";
-
-bind_thm ("ComplE", make_elim ComplD);
-
-AddSIs [ComplI];
-AddSEs [ComplE];
-
-
-section "Binary union -- Un";
-
-Goalw [Un_def] "(c : A Un B) = (c:A | c:B)";
-by (Blast_tac 1);
-qed "Un_iff";
-Addsimps [Un_iff];
-
-Goal "c:A ==> c : A Un B";
-by (Asm_simp_tac 1);
-qed "UnI1";
+(* legacy ML bindings *)
 
-Goal "c:B ==> c : A Un B";
-by (Asm_simp_tac 1);
-qed "UnI2";
-
-AddXEs [UnI1, UnI2];
-
-
-(*Classical introduction rule: no commitment to A vs B*)
-
-val prems = Goal "(c~:B ==> c:A) ==> c : A Un B";
-by (Simp_tac 1);
-by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
-qed "UnCI";
-
-val major::prems = Goalw [Un_def]
-    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
-by (rtac (major RS CollectD RS disjE) 1);
-by (REPEAT (eresolve_tac prems 1));
-qed "UnE";
-
-AddSIs [UnCI];
-AddSEs [UnE];
-
-
-section "Binary intersection -- Int";
-
-Goalw [Int_def] "(c : A Int B) = (c:A & c:B)";
-by (Blast_tac 1);
-qed "Int_iff";
-Addsimps [Int_iff];
-
-Goal "[| c:A;  c:B |] ==> c : A Int B";
-by (Asm_simp_tac 1);
-qed "IntI";
-
-Goal "c : A Int B ==> c:A";
-by (Asm_full_simp_tac 1);
-qed "IntD1";
-
-Goal "c : A Int B ==> c:B";
-by (Asm_full_simp_tac 1);
-qed "IntD2";
-
-val [major,minor] = Goal
-    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
-by (rtac minor 1);
-by (rtac (major RS IntD1) 1);
-by (rtac (major RS IntD2) 1);
-qed "IntE";
-
-AddSIs [IntI];
-AddSEs [IntE];
-
-section "Set difference";
-
-Goalw [set_diff_def] "(c : A-B) = (c:A & c~:B)";
-by (Blast_tac 1);
-qed "Diff_iff";
-Addsimps [Diff_iff];
-
-Goal "[| c : A;  c ~: B |] ==> c : A - B";
-by (Asm_simp_tac 1) ;
-qed "DiffI";
-
-Goal "c : A - B ==> c : A";
-by (Asm_full_simp_tac 1) ;
-qed "DiffD1";
-
-Goal "[| c : A - B;  c : B |] ==> P";
-by (Asm_full_simp_tac 1) ;
-qed "DiffD2";
-
-val prems = Goal "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P";
-by (resolve_tac prems 1);
-by (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ;
-qed "DiffE";
-
-AddSIs [DiffI];
-AddSEs [DiffE];
-
-
-section "Augmenting a set -- insert";
-
-Goalw [insert_def] "(a : insert b A) = (a=b | a:A)";
-by (Blast_tac 1);
-qed "insert_iff";
-Addsimps [insert_iff];
-
-Goal "a : insert a B";
-by (Simp_tac 1);
-qed "insertI1";
-
-Goal "!!a. a : B ==> a : insert b B";
-by (Asm_simp_tac 1);
-qed "insertI2";
-
-val major::prems = Goalw [insert_def]
-    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P";
-by (rtac (major RS UnE) 1);
-by (REPEAT (eresolve_tac (prems @ [CollectE]) 1));
-qed "insertE";
-
-(*Classical introduction rule*)
-val prems = Goal "(a~:B ==> a=b) ==> a: insert b B";
-by (Simp_tac 1);
-by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
-qed "insertCI";
-
-AddSIs [insertCI]; 
-AddSEs [insertE];
-
-Goal "(A <= insert x B) = (if x:A then A-{x} <= B else A<=B)";
-by Auto_tac; 
-qed "subset_insert_iff";
-
-section "Singletons, using insert";
-
-Goal "a : {a}";
-by (rtac insertI1 1) ;
-qed "singletonI";
-
-Goal "b : {a} ==> b=a";
-by (Blast_tac 1);
-qed "singletonD";
-
-bind_thm ("singletonE", make_elim singletonD);
-
-Goal "(b : {a}) = (b=a)";
-by (Blast_tac 1);
-qed "singleton_iff";
-
-Goal "{a}={b} ==> a=b";
-by (blast_tac (claset() addEs [equalityE]) 1);
-qed "singleton_inject";
-
-(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
-AddSIs [singletonI];   
-AddSDs [singleton_inject];
-AddSEs [singletonE];
-
-Goal "({b} = insert a A) = (a = b & A <= {b})";
-by (blast_tac (claset() addSEs [equalityE]) 1);
-qed "singleton_insert_inj_eq";
-
-Goal "(insert a A = {b}) = (a = b & A <= {b})";
-by (blast_tac (claset() addSEs [equalityE]) 1);
-qed "singleton_insert_inj_eq'";
-
-AddIffs [singleton_insert_inj_eq, singleton_insert_inj_eq'];
-
-Goal "A <= {x} ==> A={} | A = {x}";
-by (Fast_tac 1);
-qed "subset_singletonD";
-
-Goal "{x. x=a} = {a}";
-by (Blast_tac 1);
-qed "singleton_conv";
-Addsimps [singleton_conv];
-
-Goal "{x. a=x} = {a}";
-by (Blast_tac 1);
-qed "singleton_conv2";
-Addsimps [singleton_conv2];
-
-Goal "[| A - {x} <= B; x : A |] ==> A <= insert x B"; 
-by(Blast_tac 1);
-qed "diff_single_insert";
-
-
-section "Unions of families -- UNION x:A. B(x) is Union(B`A)";
-
-Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
-by (Blast_tac 1);
-qed "UN_iff";
-
-Addsimps [UN_iff];
-
-(*The order of the premises presupposes that A is rigid; b may be flexible*)
-Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
-by Auto_tac;
-qed "UN_I";
-
-val major::prems = Goalw [UNION_def]
-    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
-by (rtac (major RS CollectD RS bexE) 1);
-by (REPEAT (ares_tac prems 1));
-qed "UN_E";
-
-AddIs  [UN_I];
-AddSEs [UN_E];
-
-val prems = Goalw [UNION_def]
-    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
-\    (UN x:A. C(x)) = (UN x:B. D(x))";
-by (asm_simp_tac (simpset() addsimps prems) 1);
-qed "UN_cong";
-Addcongs [UN_cong];
-
-
-section "Intersections of families -- INTER x:A. B(x) is Inter(B`A)";
-
-Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
-by Auto_tac;
-qed "INT_iff";
-
-Addsimps [INT_iff];
-
-val prems = Goalw [INTER_def]
-    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
-by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
-qed "INT_I";
-
-Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
-by Auto_tac;
-qed "INT_D";
-
-(*"Classical" elimination -- by the Excluded Middle on a:A *)
-val major::prems = Goalw [INTER_def]
-    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
-by (rtac (major RS CollectD RS ballE) 1);
-by (REPEAT (eresolve_tac prems 1));
-qed "INT_E";
-
-AddSIs [INT_I];
-AddEs  [INT_D, INT_E];
-
-val prems = Goalw [INTER_def]
-    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
-\    (INT x:A. C(x)) = (INT x:B. D(x))";
-by (asm_simp_tac (simpset() addsimps prems) 1);
-qed "INT_cong";
-Addcongs [INT_cong];
-
-
-section "Union";
-
-Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
-by (Blast_tac 1);
-qed "Union_iff";
-
-Addsimps [Union_iff];
-
-(*The order of the premises presupposes that C is rigid; A may be flexible*)
-Goal "[| X:C;  A:X |] ==> A : Union(C)";
-by Auto_tac;
-qed "UnionI";
-
-val major::prems = Goalw [Union_def]
-    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
-by (rtac (major RS UN_E) 1);
-by (REPEAT (ares_tac prems 1));
-qed "UnionE";
-
-AddIs  [UnionI];
-AddSEs [UnionE];
-
-
-section "Inter";
-
-Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
-by (Blast_tac 1);
-qed "Inter_iff";
-
-Addsimps [Inter_iff];
-
-val prems = Goalw [Inter_def]
-    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
-by (REPEAT (ares_tac ([INT_I] @ prems) 1));
-qed "InterI";
-
-(*A "destruct" rule -- every X in C contains A as an element, but
-  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
-Goal "[| A : Inter(C);  X:C |] ==> A:X";
-by Auto_tac;
-qed "InterD";
-
-(*"Classical" elimination rule -- does not require proving X:C *)
-val major::prems = Goalw [Inter_def]
-    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
-by (rtac (major RS INT_E) 1);
-by (REPEAT (eresolve_tac prems 1));
-qed "InterE";
-
-AddSIs [InterI];
-AddEs  [InterD, InterE];
-
-
-(*** Image of a set under a function ***)
-
-(*Frequently b does not have the syntactic form of f(x).*)
-Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f`A";
-by (Blast_tac 1);
-qed "image_eqI";
-Addsimps [image_eqI];
-
-bind_thm ("imageI", refl RS image_eqI);
-
-(*This version's more effective when we already have the required x*)
-Goalw [image_def] "[| x:A;  b=f(x) |] ==> b : f`A";
-by (Blast_tac 1);
-qed "rev_image_eqI";
-
-(*The eta-expansion gives variable-name preservation.*)
-val major::prems = Goalw [image_def]
-    "[| b : (%x. f(x))`A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
-by (rtac (major RS CollectD RS bexE) 1);
-by (REPEAT (ares_tac prems 1));
-qed "imageE";
-
-AddIs  [image_eqI];
-AddSEs [imageE]; 
-
-Goal "f`(A Un B) = f`A Un f`B";
-by (Blast_tac 1);
-qed "image_Un";
-
-Goal "(z : f`A) = (EX x:A. z = f x)";
-by (Blast_tac 1);
-qed "image_iff";
-
-(*This rewrite rule would confuse users if made default.*)
-Goal "(f`A <= B) = (ALL x:A. f(x): B)";
-by (Blast_tac 1);
-qed "image_subset_iff";
-
-Goal "(B <= f ` A) = (? AA. AA <= A & B = f ` AA)";
-by Safe_tac;
-by  (Fast_tac 2);
-by (res_inst_tac [("x","{a. a : A & f a : B}")] exI 1);
-by (Fast_tac 1);
-qed "subset_image_iff";
-
-(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
-  many existing proofs.*)
-val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f`A <= B";
-by (blast_tac (claset() addIs prems) 1);
-qed "image_subsetI";
-
-(*** Range of a function -- just a translation for image! ***)
-
-Goal "b=f(x) ==> b : range(f)";
-by (EVERY1 [etac image_eqI, rtac UNIV_I]);
-bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
-
-bind_thm ("rangeI", UNIV_I RS imageI);
-
-val [major,minor] = Goal 
-    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
-by (rtac (major RS imageE) 1);
-by (etac minor 1);
-qed "rangeE";
-AddXEs [rangeE];
-
-
-(*** Set reasoning tools ***)
-
-
-(** Rewrite rules for boolean case-splitting: faster than 
-	addsplits[split_if]
-**)
-
-bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
-bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
-
-(*Split ifs on either side of the membership relation.
-	Not for Addsimps -- can cause goals to blow up!*)
-bind_thm ("split_if_mem1", inst "P" "%x. x : ?b" split_if);
-bind_thm ("split_if_mem2", inst "P" "%x. ?a : x" split_if);
-
-bind_thms ("split_ifs", [if_bool_eq_conj, split_if_eq1, split_if_eq2,
-			 split_if_mem1, split_if_mem2]);
-
-
-(*Each of these has ALREADY been added to simpset() above.*)
-bind_thms ("mem_simps", [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
-                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff]);
-
-(*Would like to add these, but the existing code only searches for the 
-  outer-level constant, which in this case is just "op :"; we instead need
-  to use term-nets to associate patterns with rules.  Also, if a rule fails to
-  apply, then the formula should be kept.
-  [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]), 
-   ("op Int", [IntD1,IntD2]),
-   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
- *)
-val mksimps_pairs =
-  [("Ball",[bspec])] @ mksimps_pairs;
-
-simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
-
-Addsimps[subset_UNIV, subset_refl];
-
-
-(*** The 'proper subset' relation (<) ***)
-
-Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
-by (Blast_tac 1);
-qed "psubsetI";
-AddSIs [psubsetI];
-
-Goalw [psubset_def]
-  "(A < insert x B) = (if x:B then A<B else if x:A then A-{x} < B else A<=B)";
-by (asm_simp_tac (simpset() addsimps [subset_insert_iff]) 1);
-by (Blast_tac 1); 
-qed "psubset_insert_iff";
-
-bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);
-
-bind_thm ("psubset_imp_subset", psubset_eq RS iffD1 RS conjunct1);
-
-Goal"[| (A::'a set) < B; B <= C |] ==> A < C";
-by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
-qed "psubset_subset_trans";
-
-Goal"[| (A::'a set) <= B; B < C|] ==> A < C";
-by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
-qed "subset_psubset_trans";
-
-Goalw [psubset_def] "A < B ==> EX b. b : (B - A)";
-by (Blast_tac 1);
-qed "psubset_imp_ex_mem";
-
-Goal "(!!x. x:A ==> P x) == Trueprop (ALL x:A. P x)";
-by (simp_tac (simpset () addsimps [Ball_def, thm "atomize_all", thm "atomize_imp"]) 1);
-qed "atomize_ball";
+structure Set =
+struct
+  val thy = the_context ();
+  val Ball_def = Ball_def;
+  val Bex_def = Bex_def;
+  val Collect_mem_eq = Collect_mem_eq;
+  val Compl_def = Compl_def;
+  val INTER_def = INTER_def;
+  val Int_def = Int_def;
+  val Inter_def = Inter_def;
+  val Pow_def = Pow_def;
+  val UNION_def = UNION_def;
+  val UNIV_def = UNIV_def;
+  val Un_def = Un_def;
+  val Union_def = Union_def;
+  val empty_def = empty_def;
+  val image_def = image_def;
+  val insert_def = insert_def;
+  val mem_Collect_eq = mem_Collect_eq;
+  val psubset_def = psubset_def;
+  val set_diff_def = set_diff_def;
+  val subset_def = subset_def;
+end;
diff -r a394d3e9c8bb -r 0a3dace545c5 src/HOL/Set.thy
--- a/src/HOL/Set.thy	Sun Oct 28 22:58:39 2001 +0100
+++ b/src/HOL/Set.thy	Sun Oct 28 22:59:12 2001 +0100
@@ -4,70 +4,71 @@
     Copyright   1993  University of Cambridge
 *)
 
-Set = HOL +
+header {* Set theory for higher-order logic *}
+
+theory Set = HOL
+files ("subset.ML") ("equalities.ML") ("mono.ML"):
+
+text {* A set in HOL is simply a predicate. *}
 
 
-(** Core syntax **)
+subsection {* Basic syntax *}
 
 global
 
-types
-  'a set
-
-arities
-  set :: (term) term
-
-instance
-  set :: (term) {ord, minus}
-
-syntax
-  "op :"        :: ['a, 'a set] => bool             ("op :")
+typedecl 'a set
+arities set :: ("term") "term"
 
 consts
-  "{}"          :: 'a set                           ("{}")
-  UNIV          :: 'a set
-  insert        :: ['a, 'a set] => 'a set
-  Collect       :: ('a => bool) => 'a set               (*comprehension*)
-  Int           :: ['a set, 'a set] => 'a set       (infixl 70)
-  Un            :: ['a set, 'a set] => 'a set       (infixl 65)
-  UNION, INTER  :: ['a set, 'a => 'b set] => 'b set     (*general*)
-  Union, Inter  :: (('a set) set) => 'a set             (*of a set*)
-  Pow           :: 'a set => 'a set set                 (*powerset*)
-  Ball, Bex     :: ['a set, 'a => bool] => bool         (*bounded quantifiers*)
-  "image"       :: ['a => 'b, 'a set] => ('b set)   (infixr "`" 90)
-  (*membership*)
-  "op :"        :: ['a, 'a set] => bool             ("(_/ : _)" [50, 51] 50)
+  "{}"          :: "'a set"                             ("{}")
+  UNIV          :: "'a set"
+  insert        :: "'a => 'a set => 'a set"
+  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
+  Int           :: "'a set => 'a set => 'a set"          (infixl 70)
+  Un            :: "'a set => 'a set => 'a set"          (infixl 65)
+  UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
+  INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
+  Union         :: "'a set set => 'a set"                -- "union of a set"
+  Inter         :: "'a set set => 'a set"                -- "intersection of a set"
+  Pow           :: "'a set => 'a set set"                -- "powerset"
+  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
+  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
+  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
+
+syntax
+  "op :"        :: "'a => 'a set => bool"                ("op :")
+consts
+  "op :"        :: "'a => 'a set => bool"                ("(_/ : _)" [50, 51] 50)  -- "membership"
+
+local
+
+instance set :: ("term") ord ..
+instance set :: ("term") minus ..
 
 
-(** Additional concrete syntax **)
+subsection {* Additional concrete syntax *}
 
 syntax
-  range         :: ('a => 'b) => 'b set                 (*of function*)
-
-  (* Infix syntax for non-membership *)
+  range         :: "('a => 'b) => 'b set"             -- "of function"
 
-  "op ~:"       :: ['a, 'a set] => bool               ("op ~:")
-  "op ~:"       :: ['a, 'a set] => bool               ("(_/ ~: _)" [50, 51] 50)
-
+  "op ~:"       :: "'a => 'a set => bool"                 ("op ~:")  -- "non-membership"
+  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ ~: _)" [50, 51] 50)
 
-  "@Finset"     :: args => 'a set                     ("{(_)}")
-  "@Coll"       :: [pttrn, bool] => 'a set            ("(1{_./ _})")
-  "@SetCompr"   :: ['a, idts, bool] => 'a set         ("(1{_ |/_./ _})")
-
-  (* Big Intersection / Union *)
+  "@Finset"     :: "args => 'a set"                       ("{(_)}")
+  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
+  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
 
-  "@INTER1"     :: [pttrns, 'b set] => 'b set         ("(3INT _./ _)" 10)
-  "@UNION1"     :: [pttrns, 'b set] => 'b set         ("(3UN _./ _)" 10)
-  "@INTER"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3INT _:_./ _)" 10)
-  "@UNION"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3UN _:_./ _)" 10)
+  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" 10)
+  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" 10)
+  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" 10)
+  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" 10)
 
-  (* Bounded Quantifiers *)
-  "_Ball"       :: [pttrn, 'a set, bool] => bool      ("(3ALL _:_./ _)" [0, 0, 10] 10)
-  "_Bex"        :: [pttrn, 'a set, bool] => bool      ("(3EX _:_./ _)" [0, 0, 10] 10)
+  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
+  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
 
 syntax (HOL)
-  "_Ball"       :: [pttrn, 'a set, bool] => bool      ("(3! _:_./ _)" [0, 0, 10] 10)
-  "_Bex"        :: [pttrn, 'a set, bool] => bool      ("(3? _:_./ _)" [0, 0, 10] 10)
+  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
+  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
 
 translations
   "range f"     == "f`UNIV"
@@ -85,120 +86,811 @@
   "EX x:A. P"   == "Bex A (%x. P)"
 
 syntax ("" output)
-  "_setle"      :: ['a set, 'a set] => bool           ("op <=")
-  "_setle"      :: ['a set, 'a set] => bool           ("(_/ <= _)" [50, 51] 50)
-  "_setless"    :: ['a set, 'a set] => bool           ("op <")
-  "_setless"    :: ['a set, 'a set] => bool           ("(_/ < _)" [50, 51] 50)
+  "_setle"      :: "'a set => 'a set => bool"             ("op <=")
+  "_setle"      :: "'a set => 'a set => bool"             ("(_/ <= _)" [50, 51] 50)
+  "_setless"    :: "'a set => 'a set => bool"             ("op <")
+  "_setless"    :: "'a set => 'a set => bool"             ("(_/ < _)" [50, 51] 50)
 
 syntax (symbols)
-  "_setle"      :: ['a set, 'a set] => bool           ("op \\<subseteq>")
-  "_setle"      :: ['a set, 'a set] => bool           ("(_/ \\<subseteq> _)" [50, 51] 50)
-  "_setless"    :: ['a set, 'a set] => bool           ("op \\<subset>")
-  "_setless"    :: ['a set, 'a set] => bool           ("(_/ \\<subset> _)" [50, 51] 50)
-  "op Int"      :: ['a set, 'a set] => 'a set         (infixl "\\<inter>" 70)
-  "op Un"       :: ['a set, 'a set] => 'a set         (infixl "\\<union>" 65)
-  "op :"        :: ['a, 'a set] => bool               ("op \\<in>")
-  "op :"        :: ['a, 'a set] => bool               ("(_/ \\<in> _)" [50, 51] 50)
-  "op ~:"       :: ['a, 'a set] => bool               ("op \\<notin>")
-  "op ~:"       :: ['a, 'a set] => bool               ("(_/ \\<notin> _)" [50, 51] 50)
-  "@UNION1"     :: [pttrns, 'b set] => 'b set         ("(3\\<Union>_./ _)" 10)
-  "@INTER1"     :: [pttrns, 'b set] => 'b set         ("(3\\<Inter>_./ _)" 10)
-  "@UNION"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3\\<Union>_\\<in>_./ _)" 10)
-  "@INTER"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3\\<Inter>_\\<in>_./ _)" 10)
-  Union         :: (('a set) set) => 'a set           ("\\<Union>_" [90] 90)
-  Inter         :: (('a set) set) => 'a set           ("\\<Inter>_" [90] 90)
-  "_Ball"       :: [pttrn, 'a set, bool] => bool      ("(3\\<forall>_\\<in>_./ _)" [0, 0, 10] 10)
-  "_Bex"        :: [pttrn, 'a set, bool] => bool      ("(3\\<exists>_\\<in>_./ _)" [0, 0, 10] 10)
+  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
+  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
+  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
+  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
+  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
+  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
+  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
+  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
+  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
+  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
+  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" 10)
+  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" 10)
+  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" 10)
+  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" 10)
+  Union         :: "'a set set => 'a set"                 ("\<Union>_" [90] 90)
+  Inter         :: "'a set set => 'a set"                 ("\<Inter>_" [90] 90)
+  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
+  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
 
 translations
-  "op \\<subseteq>" => "op <= :: [_ set, _ set] => bool"
-  "op \\<subset>" => "op <  :: [_ set, _ set] => bool"
-
+  "op \<subseteq>" => "op <= :: _ set => _ set => bool"
+  "op \<subset>" => "op <  :: _ set => _ set => bool"
 
 
-(** Rules and definitions **)
+typed_print_translation {*
+  let
+    fun le_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =
+          list_comb (Syntax.const "_setle", ts)
+      | le_tr' _ _ _ = raise Match;
+
+    fun less_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =
+          list_comb (Syntax.const "_setless", ts)
+      | less_tr' _ _ _ = raise Match;
+  in [("op <=", le_tr'), ("op <", less_tr')] end
+*}
 
-local
+text {*
+  \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
+  "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
+  only translated if @{text "[0..n] subset bvs(e)"}.
+*}
+
+parse_translation {*
+  let
+    val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
 
-rules
+    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
+      | nvars _ = 1;
+
+    fun setcompr_tr [e, idts, b] =
+      let
+        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
+        val P = Syntax.const "op &" $ eq $ b;
+        val exP = ex_tr [idts, P];
+      in Syntax.const "Collect" $ Abs ("", dummyT, exP) end;
+
+  in [("@SetCompr", setcompr_tr)] end;
+*}
 
-  (* Isomorphisms between Predicates and Sets *)
+print_translation {*
+  let
+    val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
+
+    fun setcompr_tr' [Abs (_, _, P)] =
+      let
+        fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
+          | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
+              if n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
+                ((0 upto (n - 1)) subset add_loose_bnos (e, 0, [])) then ()
+              else raise Match;
 
-  mem_Collect_eq    "(a : {x. P(x)}) = P(a)"
-  Collect_mem_eq    "{x. x:A} = A"
+        fun tr' (_ $ abs) =
+          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
+          in Syntax.const "@SetCompr" $ e $ idts $ Q end;
+      in check (P, 0); tr' P end;
+  in [("Collect", setcompr_tr')] end;
+*}
+
+
+subsection {* Rules and definitions *}
+
+text {* Isomorphisms between predicates and sets. *}
 
+axioms
+  mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
+  Collect_mem_eq [simp]: "{x. x:A} = A"
+
+defs
+  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
+  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
+
+defs (overloaded)
+  subset_def:   "A <= B         == ALL x:A. x:B"
+  psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B"
+  Compl_def:    "- A            == {x. ~x:A}"
 
 defs
-  Ball_def      "Ball A P       == ! x. x:A --> P(x)"
-  Bex_def       "Bex A P        == ? x. x:A & P(x)"
-  subset_def    "A <= B         == ! x:A. x:B"
-  psubset_def   "A < B          == (A::'a set) <= B & ~ A=B"
-  Compl_def     "- A            == {x. ~x:A}"
-  Un_def        "A Un B         == {x. x:A | x:B}"
-  Int_def       "A Int B        == {x. x:A & x:B}"
-  set_diff_def  "A - B          == {x. x:A & ~x:B}"
-  INTER_def     "INTER A B      == {y. ! x:A. y: B(x)}"
-  UNION_def     "UNION A B      == {y. ? x:A. y: B(x)}"
-  Inter_def     "Inter S        == (INT x:S. x)"
-  Union_def     "Union S        == (UN x:S. x)"
-  Pow_def       "Pow A          == {B. B <= A}"
-  empty_def     "{}             == {x. False}"
-  UNIV_def      "UNIV           == {x. True}"
-  insert_def    "insert a B     == {x. x=a} Un B"
-  image_def     "f`A           == {y. ? x:A. y=f(x)}"
+  Un_def:       "A Un B         == {x. x:A | x:B}"
+  Int_def:      "A Int B        == {x. x:A & x:B}"
+  set_diff_def: "A - B          == {x. x:A & ~x:B}"
+  INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
+  UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
+  Inter_def:    "Inter S        == (INT x:S. x)"
+  Union_def:    "Union S        == (UN x:S. x)"
+  Pow_def:      "Pow A          == {B. B <= A}"
+  empty_def:    "{}             == {x. False}"
+  UNIV_def:     "UNIV           == {x. True}"
+  insert_def:   "insert a B     == {x. x=a} Un B"
+  image_def:    "f`A            == {y. EX x:A. y = f(x)}"
+
+
+subsection {* Lemmas and proof tool setup *}
+
+subsubsection {* Relating predicates and sets *}
+
+lemma CollectI [intro!]: "P(a) ==> a : {x. P(x)}"
+  by simp
+
+lemma CollectD: "a : {x. P(x)} ==> P(a)"
+  by simp
+
+lemma set_ext: "(!!x. (x:A) = (x:B)) ==> A = B"
+proof -
+  case rule_context
+  show ?thesis
+    apply (rule prems [THEN ext, THEN arg_cong, THEN box_equals])
+     apply (rule Collect_mem_eq)
+    apply (rule Collect_mem_eq)
+    done
+qed
+
+lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
+  by simp
+
+lemmas CollectE [elim!] = CollectD [elim_format]
+
+
+subsubsection {* Bounded quantifiers *}
+
+lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
+  by (simp add: Ball_def)
+
+lemmas strip = impI allI ballI
+
+lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
+  by (simp add: Ball_def)
+
+lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
+  by (unfold Ball_def) blast
+
+text {*
+  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
+  @{prop "a:A"}; creates assumption @{prop "P a"}.
+*}
+
+ML {*
+  local val ballE = thm "ballE"
+  in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;
+*}
+
+text {*
+  Gives better instantiation for bound:
+*}
+
+ML_setup {*
+  claset_ref() := claset() addbefore ("bspec", datac (thm "bspec") 1);
+*}
+
+lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
+  -- {* Normally the best argument order: @{prop "P x"} constrains the
+    choice of @{prop "x:A"}. *}
+  by (unfold Bex_def) blast
+
+lemma rev_bexI: "x:A ==> P x ==> EX x:A. P x"
+  -- {* The best argument order when there is only one @{prop "x:A"}. *}
+  by (unfold Bex_def) blast
+
+lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
+  by (unfold Bex_def) blast
+
+lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
+  by (unfold Bex_def) blast
+
+lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
+  -- {* Trival rewrite rule. *}
+  by (simp add: Ball_def)
+
+lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
+  -- {* Dual form for existentials. *}
+  by (simp add: Bex_def)
+
+lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
+  by blast
+
+lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
+  by blast
+
+lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
+  by blast
+
+lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
+  by blast
+
+lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
+  by blast
+
+lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
+  by blast
+
+ML_setup {*
+  let
+    val Ball_def = thm "Ball_def";
+    val Bex_def = thm "Bex_def";
+
+    val ex_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
+      ("EX x:A. P x & Q x", HOLogic.boolT);
+
+    val prove_bex_tac =
+      rewrite_goals_tac [Bex_def] THEN Quantifier1.prove_one_point_ex_tac;
+    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
+
+    val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
+      ("ALL x:A. P x --> Q x", HOLogic.boolT);
+
+    val prove_ball_tac =
+      rewrite_goals_tac [Ball_def] THEN Quantifier1.prove_one_point_all_tac;
+    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
+
+    val defBEX_regroup = mk_simproc "defined BEX" [ex_pattern] rearrange_bex;
+    val defBALL_regroup = mk_simproc "defined BALL" [all_pattern] rearrange_ball;
+  in
+    Addsimprocs [defBALL_regroup, defBEX_regroup]
+  end;
+*}
+
+
+subsubsection {* Congruence rules *}
+
+lemma ball_cong [cong]:
+  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
+    (ALL x:A. P x) = (ALL x:B. Q x)"
+  by (simp add: Ball_def)
+
+lemma bex_cong [cong]:
+  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
+    (EX x:A. P x) = (EX x:B. Q x)"
+  by (simp add: Bex_def cong: conj_cong)
 
 
-end
+subsubsection {* Subsets *}
+
+lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A <= B"
+  by (simp add: subset_def)
+
+text {*
+  \medskip Map the type @{text "'a set => anything"} to just @{typ
+  'a}; for overloading constants whose first argument has type @{typ
+  "'a set"}.
+*}
+
+ML {*
+  fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
+*}
+
+ML "
+  (* While (:) is not, its type must be kept
+    for overloading of = to work. *)
+  Blast.overloaded (\"op :\", domain_type);
+
+  overload_1st_set \"Ball\";            (*need UNION, INTER also?*)
+  overload_1st_set \"Bex\";
+
+  (*Image: retain the type of the set being expressed*)
+  Blast.overloaded (\"image\", domain_type);
+"
+
+lemma subsetD [elim]: "A <= B ==> c:A ==> c:B"
+  -- {* Rule in Modus Ponens style. *}
+  by (unfold subset_def) blast
+
+declare subsetD [intro?] -- FIXME
+
+lemma rev_subsetD: "c:A ==> A <= B ==> c:B"
+  -- {* The same, with reversed premises for use with @{text erule} --
+      cf @{text rev_mp}. *}
+  by (rule subsetD)
+
+declare rev_subsetD [intro?] -- FIXME
+
+text {*
+  \medskip Converts @{prop "A <= B"} to @{prop "x:A ==> x:B"}.
+*}
+
+ML {*
+  local val rev_subsetD = thm "rev_subsetD"
+  in fun impOfSubs th = th RSN (2, rev_subsetD) end;
+*}
+
+lemma subsetCE [elim]: "A <= B ==> (c~:A ==> P) ==> (c:B ==> P) ==> P"
+  -- {* Classical elimination rule. *}
+  by (unfold subset_def) blast
+
+text {*
+  \medskip Takes assumptions @{prop "A <= B"}; @{prop "c:A"} and
+  creates the assumption @{prop "c:B"}.
+*}
+
+ML {*
+  local val subsetCE = thm "subsetCE"
+  in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
+*}
+
+lemma contra_subsetD: "A <= B ==> c ~: B ==> c ~: A"
+  by blast
+
+lemma subset_refl: "A <= (A::'a set)"
+  by fast
+
+lemma subset_trans: "A <= B ==> B <= C ==> A <= (C::'a set)"
+  by blast
 
 
-ML
+subsubsection {* Equality *}
+
+lemma subset_antisym [intro!]: "A <= B ==> B <= A ==> A = (B::'a set)"
+  -- {* Anti-symmetry of the subset relation. *}
+  by (rule set_ext) (blast intro: subsetD)
+
+lemmas equalityI = subset_antisym
+
+text {*
+  \medskip Equality rules from ZF set theory -- are they appropriate
+  here?
+*}
+
+lemma equalityD1: "A = B ==> A <= (B::'a set)"
+  by (simp add: subset_refl)
+
+lemma equalityD2: "A = B ==> B <= (A::'a set)"
+  by (simp add: subset_refl)
+
+text {*
+  \medskip Be careful when adding this to the claset as @{text
+  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
+  <= A"} and @{prop "A <= {}"} and then back to @{prop "A = {}"}!
+*}
+
+lemma equalityE: "A = B ==> (A <= B ==> B <= (A::'a set) ==> P) ==> P"
+  by (simp add: subset_refl)
 
-local
+lemma equalityCE [elim]:
+    "A = B ==> (c:A ==> c:B ==> P) ==> (c~:A ==> c~:B ==> P) ==> P"
+  by blast
+
+text {*
+  \medskip Lemma for creating induction formulae -- for "pattern
+  matching" on @{text p}.  To make the induction hypotheses usable,
+  apply @{text spec} or @{text bspec} to put universal quantifiers over the free
+  variables in @{text p}.
+*}
+
+lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"
+  by simp
 
-(* Set inclusion *)
+lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
+  by simp
+
+
+subsubsection {* The universal set -- UNIV *}
+
+lemma UNIV_I [simp]: "x : UNIV"
+  by (simp add: UNIV_def)
+
+declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
+
+lemma UNIV_witness [intro?]: "EX x. x : UNIV"
+  by simp
+
+lemma subset_UNIV: "A <= UNIV"
+  by (rule subsetI) (rule UNIV_I)
 
-fun le_tr' _ (*op <=*) (Type ("fun", (Type ("set", _) :: _))) ts =
-      list_comb (Syntax.const "_setle", ts)
-  | le_tr' _ (*op <=*) _ _ = raise Match;
+text {*
+  \medskip Eta-contracting these two rules (to remove @{text P})
+  causes them to be ignored because of their interaction with
+  congruence rules.
+*}
+
+lemma ball_UNIV [simp]: "Ball UNIV P = All P"
+  by (simp add: Ball_def)
+
+lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
+  by (simp add: Bex_def)
+
+
+subsubsection {* The empty set *}
+
+lemma empty_iff [simp]: "(c : {}) = False"
+  by (simp add: empty_def)
+
+lemma emptyE [elim!]: "a : {} ==> P"
+  by simp
+
+lemma empty_subsetI [iff]: "{} <= A"
+    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
+  by blast
+
+lemma equals0I: "(!!y. y:A ==> False) ==> A = {}"
+  by blast
 
-fun less_tr' _ (*op <*) (Type ("fun", (Type ("set", _) :: _))) ts =
-      list_comb (Syntax.const "_setless", ts)
-  | less_tr' _ (*op <*) _ _ = raise Match;
+lemma equals0D: "A={} ==> a ~: A"
+    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
+  by blast
+
+lemma ball_empty [simp]: "Ball {} P = True"
+  by (simp add: Ball_def)
+
+lemma bex_empty [simp]: "Bex {} P = False"
+  by (simp add: Bex_def)
+
+lemma UNIV_not_empty [iff]: "UNIV ~= {}"
+  by (blast elim: equalityE)
+
+
+section {* The Powerset operator -- Pow *}
+
+lemma Pow_iff [iff]: "(A : Pow B) = (A <= B)"
+  by (simp add: Pow_def)
+
+lemma PowI: "A <= B ==> A : Pow B"
+  by (simp add: Pow_def)
+
+lemma PowD: "A : Pow B ==> A <= B"
+  by (simp add: Pow_def)
+
+lemma Pow_bottom: "{}: Pow B"
+  by simp
+
+lemma Pow_top: "A : Pow A"
+  by (simp add: subset_refl)
 
 
-(* Translates between { e | x1..xn. P} and {u. ? x1..xn. u=e & P}      *)
-(* {y. ? x1..xn. y = e & P} is only translated if [0..n] subset bvs(e) *)
+subsubsection {* Set complement *}
+
+lemma Compl_iff [simp]: "(c : -A) = (c~:A)"
+  by (unfold Compl_def) blast
+
+lemma ComplI [intro!]: "(c:A ==> False) ==> c : -A"
+  by (unfold Compl_def) blast
+
+text {*
+  \medskip This form, with negated conclusion, works well with the
+  Classical prover.  Negated assumptions behave like formulae on the
+  right side of the notional turnstile ... *}
+
+lemma ComplD: "c : -A ==> c~:A"
+  by (unfold Compl_def) blast
+
+lemmas ComplE [elim!] = ComplD [elim_format]
+
+
+subsubsection {* Binary union -- Un *}
 
-val ex_tr = snd(mk_binder_tr("EX ","Ex"));
+lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
+  by (unfold Un_def) blast
+
+lemma UnI1 [elim?]: "c:A ==> c : A Un B"
+  by simp
+
+lemma UnI2 [elim?]: "c:B ==> c : A Un B"
+  by simp
 
-fun nvars(Const("_idts",_) $ _ $ idts) = nvars(idts)+1
-  | nvars(_) = 1;
+text {*
+  \medskip Classical introduction rule: no commitment to @{prop A} vs
+  @{prop B}.
+*}
+
+lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
+  by auto
+
+lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
+  by (unfold Un_def) blast
+
+
+section {* Binary intersection -- Int *}
 
-fun setcompr_tr[e,idts,b] =
-  let val eq = Syntax.const("op =") $ Bound(nvars(idts)) $ e
-      val P = Syntax.const("op &") $ eq $ b
-      val exP = ex_tr [idts,P]
-  in Syntax.const("Collect") $ Abs("",dummyT,exP) end;
+lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
+  by (unfold Int_def) blast
+
+lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
+  by simp
+
+lemma IntD1: "c : A Int B ==> c:A"
+  by simp
+
+lemma IntD2: "c : A Int B ==> c:B"
+  by simp
+
+lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
+  by simp
+
+
+section {* Set difference *}
+
+lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
+  by (unfold set_diff_def) blast
 
-val ex_tr' = snd(mk_binder_tr' ("Ex","DUMMY"));
+lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
+  by simp
+
+lemma DiffD1: "c : A - B ==> c : A"
+  by simp
+
+lemma DiffD2: "c : A - B ==> c : B ==> P"
+  by simp
+
+lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
+  by simp
+
+
+subsubsection {* Augmenting a set -- insert *}
+
+lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
+  by (unfold insert_def) blast
+
+lemma insertI1: "a : insert a B"
+  by simp
+
+lemma insertI2: "a : B ==> a : insert b B"
+  by simp
 
-fun setcompr_tr'[Abs(_,_,P)] =
-  let fun ok(Const("Ex",_)$Abs(_,_,P),n) = ok(P,n+1)
-        | ok(Const("op &",_) $ (Const("op =",_) $ Bound(m) $ e) $ P, n) =
-            if n>0 andalso m=n andalso not(loose_bvar1(P,n)) andalso
-               ((0 upto (n-1)) subset add_loose_bnos(e,0,[]))
-            then () else raise Match
+lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
+  by (unfold insert_def) blast
+
+lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
+  -- {* Classical introduction rule. *}
+  by auto
+
+lemma subset_insert_iff: "(A <= insert x B) = (if x:A then A - {x} <= B else A <= B)"
+  by auto
+
+
+subsubsection {* Singletons, using insert *}
+
+lemma singletonI [intro!]: "a : {a}"
+    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
+  by (rule insertI1)
+
+lemma singletonD: "b : {a} ==> b = a"
+  by blast
+
+lemmas singletonE [elim!] = singletonD [elim_format]
+
+lemma singleton_iff: "(b : {a}) = (b = a)"
+  by blast
+
+lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
+  by blast
+
+lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A <= {b})"
+  by blast
+
+lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A <= {b})"
+  by blast
+
+lemma subset_singletonD: "A <= {x} ==> A={} | A = {x}"
+  by fast
+
+lemma singleton_conv [simp]: "{x. x = a} = {a}"
+  by blast
+
+lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
+  by blast
 
-      fun tr'(_ $ abs) =
-        let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr'[abs]
-        in Syntax.const("@SetCompr") $ e $ idts $ Q end
-  in ok(P,0); tr'(P) end;
+lemma diff_single_insert: "A - {x} <= B ==> x : A ==> A <= insert x B"
+  by blast
+
+
+subsubsection {* Unions of families *}
+
+text {*
+  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
+*}
+
+lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
+  by (unfold UNION_def) blast
+
+lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
+  -- {* The order of the premises presupposes that @{term A} is rigid;
+    @{term b} may be flexible. *}
+  by auto
+
+lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
+  by (unfold UNION_def) blast
 
-in
+lemma UN_cong [cong]:
+    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
+  by (simp add: UNION_def)
+
+
+subsubsection {* Intersections of families *}
+
+text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
+
+lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
+  by (unfold INTER_def) blast
 
-val parse_translation = [("@SetCompr", setcompr_tr)];
-val print_translation = [("Collect", setcompr_tr')];
-val typed_print_translation = [("op <=", le_tr'), ("op <", less_tr')];
+lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
+  by (unfold INTER_def) blast
+
+lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
+  by auto
+
+lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
+  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
+  by (unfold INTER_def) blast
+
+lemma INT_cong [cong]:
+    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
+  by (simp add: INTER_def)
 
 
-end;
+subsubsection {* Union *}
+
+lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
+  by (unfold Union_def) blast
+
+lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
+  -- {* The order of the premises presupposes that @{term C} is rigid;
+    @{term A} may be flexible. *}
+  by auto
+
+lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
+  by (unfold Union_def) blast
+
+
+subsubsection {* Inter *}
+
+lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
+  by (unfold Inter_def) blast
+
+lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
+  by (simp add: Inter_def)
+
+text {*
+  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
+  contains @{term A} as an element, but @{prop "A:X"} can hold when
+  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
+*}
+
+lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
+  by auto
+
+lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
+  -- {* ``Classical'' elimination rule -- does not require proving
+    @{prop "X:C"}. *}
+  by (unfold Inter_def) blast
+
+text {*
+  \medskip Image of a set under a function.  Frequently @{term b} does
+  not have the syntactic form of @{term "f x"}.
+*}
+
+lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
+  by (unfold image_def) blast
+
+lemma imageI: "x : A ==> f x : f ` A"
+  by (rule image_eqI) (rule refl)
+
+lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
+  -- {* This version's more effective when we already have the
+    required @{term x}. *}
+  by (unfold image_def) blast
+
+lemma imageE [elim!]:
+  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
+  -- {* The eta-expansion gives variable-name preservation. *}
+  by (unfold image_def) blast
+
+lemma image_Un: "f`(A Un B) = f`A Un f`B"
+  by blast
+
+lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
+  by blast
+
+lemma image_subset_iff: "(f`A <= B) = (ALL x:A. f x: B)"
+  -- {* This rewrite rule would confuse users if made default. *}
+  by blast
+
+lemma subset_image_iff: "(B <= f ` A) = (EX AA. AA <= A & B = f ` AA)"
+  apply safe
+   prefer 2 apply fast
+  apply (rule_tac x = "{a. a : A & f a : B}" in exI)
+  apply fast
+  done
+
+lemma image_subsetI: "(!!x. x:A ==> f x : B) ==> f`A <= B"
+  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
+    @{text hypsubst}, but breaks too many existing proofs. *}
+  by blast
+
+text {*
+  \medskip Range of a function -- just a translation for image!
+*}
+
+lemma range_eqI: "b = f x ==> b : range f"
+  by simp
+
+lemma rangeI: "f x : range f"
+  by simp
+
+lemma rangeE [elim?]: "b : range (%x. f x) ==> (!!x. b = f x ==> P) ==> P"
+  by blast
+
+
+subsubsection {* Set reasoning tools *}
+
+text {*
+  Rewrite rules for boolean case-splitting: faster than @{text
+  "split_if [split]"}.
+*}
+
+lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
+  by (rule split_if)
+
+lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
+  by (rule split_if)
+
+text {*
+  Split ifs on either side of the membership relation.  Not for @{text
+  "[simp]"} -- can cause goals to blow up!
+*}
+
+lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
+  by (rule split_if)
+
+lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
+  by (rule split_if)
+
+lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
+
+lemmas mem_simps =
+  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
+  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
+  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
+
+(*Would like to add these, but the existing code only searches for the
+  outer-level constant, which in this case is just "op :"; we instead need
+  to use term-nets to associate patterns with rules.  Also, if a rule fails to
+  apply, then the formula should be kept.
+  [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
+   ("op Int", [IntD1,IntD2]),
+   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
+ *)
+
+ML_setup {*
+  val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
+  simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
+*}
+
+declare subset_UNIV [simp] subset_refl [simp]
+
+
+subsubsection {* The ``proper subset'' relation *}
+
+lemma psubsetI [intro!]: "(A::'a set) <= B ==> A ~= B ==> A < B"
+  by (unfold psubset_def) blast
+
+lemma psubset_insert_iff:
+  "(A < insert x B) = (if x:B then A < B else if x:A then A - {x} < B else A <= B)"
+  apply (simp add: psubset_def subset_insert_iff)
+  apply blast
+  done
+
+lemma psubset_eq: "((A::'a set) < B) = (A <= B & A ~= B)"
+  by (simp only: psubset_def)
+
+lemma psubset_imp_subset: "(A::'a set) < B ==> A <= B"
+  by (simp add: psubset_eq)
+
+lemma psubset_subset_trans: "(A::'a set) < B ==> B <= C ==> A < C"
+  by (auto simp add: psubset_eq)
+
+lemma subset_psubset_trans: "(A::'a set) <= B ==> B < C ==> A < C"
+  by (auto simp add: psubset_eq)
+
+lemma psubset_imp_ex_mem: "A < B ==> EX b. b : (B - A)"
+  by (unfold psubset_def) blast
+
+lemma atomize_ball:
+    "(!!x. x:A ==> P x) == Trueprop (ALL x:A. P x)"
+  by (simp only: Ball_def atomize_all atomize_imp)
+
+declare atomize_ball [symmetric, rulify]
+
+
+subsection {* Further set-theory lemmas *}
+
+use "subset.ML"
+use "equalities.ML"
+use "mono.ML"
+
+end
diff -r a394d3e9c8bb -r 0a3dace545c5 src/HOL/Typedef.thy
--- a/src/HOL/Typedef.thy	Sun Oct 28 22:58:39 2001 +0100
+++ b/src/HOL/Typedef.thy	Sun Oct 28 22:59:12 2001 +0100
@@ -3,18 +3,15 @@
     Author:     Markus Wenzel, TU Munich
 *)
 
-header {* Set-theory lemmas and HOL type definitions *}
+header {* HOL type definitions *}
 
 theory Typedef = Set
-files "subset.ML" "equalities.ML" "mono.ML" ("Tools/typedef_package.ML"):
+files ("Tools/typedef_package.ML"):
 
-(*belongs to theory Set*)
-declare atomize_ball [symmetric, rulify]
-
-(* Courtesy of Stephan Merz *)
 lemma Least_mono: 
   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
     ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
+    -- {* Courtesy of Stephan Merz *}
   apply clarify
   apply (erule_tac P = "%x. x : S" in LeastI2)
    apply fast
diff -r a394d3e9c8bb -r 0a3dace545c5 src/HOL/subset.ML
--- a/src/HOL/subset.ML	Sun Oct 28 22:58:39 2001 +0100
+++ b/src/HOL/subset.ML	Sun Oct 28 22:59:12 2001 +0100
@@ -7,6 +7,163 @@
 operations.
 *)
 
+(* legacy ML bindings *)
+
+val Ball_def = thm "Ball_def";
+val Bex_def = thm "Bex_def";
+val Collect_mem_eq = thm "Collect_mem_eq";
+val Compl_def = thm "Compl_def";
+val INTER_def = thm "INTER_def";
+val Int_def = thm "Int_def";
+val Inter_def = thm "Inter_def";
+val Pow_def = thm "Pow_def";
+val UNION_def = thm "UNION_def";
+val UNIV_def = thm "UNIV_def";
+val Un_def = thm "Un_def";
+val Union_def = thm "Union_def";
+val empty_def = thm "empty_def";
+val image_def = thm "image_def";
+val insert_def = thm "insert_def";
+val mem_Collect_eq = thm "mem_Collect_eq";
+val psubset_def = thm "psubset_def";
+val set_diff_def = thm "set_diff_def";
+val subset_def = thm "subset_def";
+val CollectI = thm "CollectI";
+val CollectD = thm "CollectD";
+val set_ext = thm "set_ext";
+val Collect_cong = thm "Collect_cong";
+val CollectE = thm "CollectE";
+val ballI = thm "ballI";
+val strip = thms "strip";
+val bspec = thm "bspec";
+val ballE = thm "ballE";
+val bexI = thm "bexI";
+val rev_bexI = thm "rev_bexI";
+val bexCI = thm "bexCI";
+val bexE = thm "bexE";
+val ball_triv = thm "ball_triv";
+val bex_triv = thm "bex_triv";
+val bex_triv_one_point1 = thm "bex_triv_one_point1";
+val bex_triv_one_point2 = thm "bex_triv_one_point2";
+val bex_one_point1 = thm "bex_one_point1";
+val bex_one_point2 = thm "bex_one_point2";
+val ball_one_point1 = thm "ball_one_point1";
+val ball_one_point2 = thm "ball_one_point2";
+val ball_cong = thm "ball_cong";
+val bex_cong = thm "bex_cong";
+val subsetI = thm "subsetI";
+val subsetD = thm "subsetD";
+val rev_subsetD = thm "rev_subsetD";
+val subsetCE = thm "subsetCE";
+val contra_subsetD = thm "contra_subsetD";
+val subset_refl = thm "subset_refl";
+val subset_trans = thm "subset_trans";
+val subset_antisym = thm "subset_antisym";
+val equalityI = thm "equalityI";
+val equalityD1 = thm "equalityD1";
+val equalityD2 = thm "equalityD2";
+val equalityE = thm "equalityE";
+val equalityCE = thm "equalityCE";
+val setup_induction = thm "setup_induction";
+val eqset_imp_iff = thm "eqset_imp_iff";
+val UNIV_I = thm "UNIV_I";
+val UNIV_witness = thm "UNIV_witness";
+val subset_UNIV = thm "subset_UNIV";
+val ball_UNIV = thm "ball_UNIV";
+val bex_UNIV = thm "bex_UNIV";
+val empty_iff = thm "empty_iff";
+val emptyE = thm "emptyE";
+val empty_subsetI = thm "empty_subsetI";
+val equals0I = thm "equals0I";
+val equals0D = thm "equals0D";
+val ball_empty = thm "ball_empty";
+val bex_empty = thm "bex_empty";
+val UNIV_not_empty = thm "UNIV_not_empty";
+val Pow_iff = thm "Pow_iff";
+val PowI = thm "PowI";
+val PowD = thm "PowD";
+val Pow_bottom = thm "Pow_bottom";
+val Pow_top = thm "Pow_top";
+val Compl_iff = thm "Compl_iff";
+val ComplI = thm "ComplI";
+val ComplD = thm "ComplD";
+val ComplE = thm "ComplE";
+val Un_iff = thm "Un_iff";
+val UnI1 = thm "UnI1";
+val UnI2 = thm "UnI2";
+val UnCI = thm "UnCI";
+val UnE = thm "UnE";
+val Int_iff = thm "Int_iff";
+val IntI = thm "IntI";
+val IntD1 = thm "IntD1";
+val IntD2 = thm "IntD2";
+val IntE = thm "IntE";
+val Diff_iff = thm "Diff_iff";
+val DiffI = thm "DiffI";
+val DiffD1 = thm "DiffD1";
+val DiffD2 = thm "DiffD2";
+val DiffE = thm "DiffE";
+val insert_iff = thm "insert_iff";
+val insertI1 = thm "insertI1";
+val insertI2 = thm "insertI2";
+val insertE = thm "insertE";
+val insertCI = thm "insertCI";
+val subset_insert_iff = thm "subset_insert_iff";
+val singletonI = thm "singletonI";
+val singletonD = thm "singletonD";
+val singletonE = thm "singletonE";
+val singleton_iff = thm "singleton_iff";
+val singleton_inject = thm "singleton_inject";
+val singleton_insert_inj_eq = thm "singleton_insert_inj_eq";
+val singleton_insert_inj_eq' = thm "singleton_insert_inj_eq'";
+val subset_singletonD = thm "subset_singletonD";
+val singleton_conv = thm "singleton_conv";
+val singleton_conv2 = thm "singleton_conv2";
+val diff_single_insert = thm "diff_single_insert";
+val UN_iff = thm "UN_iff";
+val UN_I = thm "UN_I";
+val UN_E = thm "UN_E";
+val UN_cong = thm "UN_cong";
+val INT_iff = thm "INT_iff";
+val INT_I = thm "INT_I";
+val INT_D = thm "INT_D";
+val INT_E = thm "INT_E";
+val INT_cong = thm "INT_cong";
+val Union_iff = thm "Union_iff";
+val UnionI = thm "UnionI";
+val UnionE = thm "UnionE";
+val Inter_iff = thm "Inter_iff";
+val InterI = thm "InterI";
+val InterD = thm "InterD";
+val InterE = thm "InterE";
+val image_eqI = thm "image_eqI";
+val imageI = thm "imageI";
+val rev_image_eqI = thm "rev_image_eqI";
+val imageE = thm "imageE";
+val image_Un = thm "image_Un";
+val image_iff = thm "image_iff";
+val image_subset_iff = thm "image_subset_iff";
+val subset_image_iff = thm "subset_image_iff";
+val image_subsetI = thm "image_subsetI";
+val range_eqI = thm "range_eqI";
+val rangeI = thm "rangeI";
+val rangeE = thm "rangeE";
+val split_if_eq1 = thm "split_if_eq1";
+val split_if_eq2 = thm "split_if_eq2";
+val split_if_mem1 = thm "split_if_mem1";
+val split_if_mem2 = thm "split_if_mem2";
+val split_ifs = thms "split_ifs";
+val mem_simps = thms "mem_simps";
+val psubsetI = thm "psubsetI";
+val psubset_insert_iff = thm "psubset_insert_iff";
+val psubset_eq = thm "psubset_eq";
+val psubset_imp_subset = thm "psubset_imp_subset";
+val psubset_subset_trans = thm "psubset_subset_trans";
+val subset_psubset_trans = thm "subset_psubset_trans";
+val psubset_imp_ex_mem = thm "psubset_imp_ex_mem";
+val atomize_ball = thm "atomize_ball";
+
+
 (*** insert ***)
 
 Goal "B <= insert a B";