--- 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;