src/ZF/subset.ML

 Union and Intersection as lattice operations
 *)
 
 (*** cons ***)
 
-val cons_subsetI = prove_goal ZF.thy "[| a:C; B<=C |] ==> cons(a,B) <= C"
+qed_goal "cons_subsetI" ZF.thy "[| a:C; B<=C |] ==> cons(a,B) <= C"
  (fn prems=>
   [ (cut_facts_tac prems 1),
     (REPEAT (ares_tac [subsetI] 1
      ORELSE eresolve_tac  [consE,ssubst,subsetD] 1)) ]);
 
-val subset_consI = prove_goal ZF.thy "B <= cons(a,B)"
+qed_goal "subset_consI" ZF.thy "B <= cons(a,B)"
  (fn _=> [ (rtac subsetI 1), (etac consI2 1) ]);
 
 (*Useful for rewriting!*)
-val cons_subset_iff = prove_goal ZF.thy "cons(a,B)<=C <-> a:C & B<=C"
+qed_goal "cons_subset_iff" ZF.thy "cons(a,B)<=C <-> a:C & B<=C"
  (fn _=> [ (fast_tac upair_cs 1) ]);
 
 (*A safe special case of subset elimination, adding no new variables 
   [| cons(a,B) <= C; [| a : C; B <= C |] ==> R |] ==> R *)
 val cons_subsetE = standard (cons_subset_iff RS iffD1 RS conjE);
 
-val subset_empty_iff = prove_goal ZF.thy "A<=0 <-> A=0"
+qed_goal "subset_empty_iff" ZF.thy "A<=0 <-> A=0"
  (fn _=> [ (fast_tac (upair_cs addIs [equalityI]) 1) ]);
 
-val subset_cons_iff = prove_goal ZF.thy
+qed_goal "subset_cons_iff" ZF.thy
     "C<=cons(a,B) <-> C<=B | (a:C & C-{a} <= B)"
  (fn _=> [ (fast_tac upair_cs 1) ]);
 
 (*** succ ***)
 
-val subset_succI = prove_goal ZF.thy "i <= succ(i)"
+qed_goal "subset_succI" ZF.thy "i <= succ(i)"
  (fn _=> [ (rtac subsetI 1), (etac succI2 1) ]);
 
 (*But if j is an ordinal or is transitive, then i:j implies i<=j! 
   See ordinal/Ord_succ_subsetI*)
-val succ_subsetI = prove_goalw ZF.thy [succ_def]
+qed_goalw "succ_subsetI" ZF.thy [succ_def]
     "[| i:j;  i<=j |] ==> succ(i)<=j"
  (fn prems=>
   [ (REPEAT (ares_tac (prems@[cons_subsetI]) 1)) ]);
 
-val succ_subsetE = prove_goalw ZF.thy [succ_def] 
+qed_goalw "succ_subsetE" ZF.thy [succ_def] 
     "[| succ(i) <= j;  [| i:j;  i<=j |] ==> P \
 \    |] ==> P"
  (fn major::prems=>
   [ (rtac (major RS cons_subsetE) 1),
     (REPEAT (ares_tac prems 1)) ]);
 
 (*** singletons ***)
 
-val singleton_subsetI = prove_goal ZF.thy
+qed_goal "singleton_subsetI" ZF.thy
     "a:C ==> {a} <= C"
  (fn prems=>
   [ (REPEAT (resolve_tac (prems@[cons_subsetI,empty_subsetI]) 1)) ]);
 
-val singleton_subsetD = prove_goal ZF.thy
+qed_goal "singleton_subsetD" ZF.thy
     "{a} <= C  ==>  a:C"
  (fn prems=> [ (REPEAT (ares_tac (prems@[cons_subsetE]) 1)) ]);
 
 (*** Big Union -- least upper bound of a set  ***)
 
-val Union_subset_iff = prove_goal ZF.thy "Union(A) <= C <-> (ALL x:A. x <= C)"
- (fn _ => [ fast_tac upair_cs 1 ]);
-
-val Union_upper = prove_goal ZF.thy
+qed_goal "Union_subset_iff" ZF.thy "Union(A) <= C <-> (ALL x:A. x <= C)"
+ (fn _ => [ fast_tac upair_cs 1 ]);
+
+qed_goal "Union_upper" ZF.thy
     "B:A ==> B <= Union(A)"
  (fn prems=> [ (REPEAT (ares_tac (prems@[subsetI,UnionI]) 1)) ]);
 
-val Union_least = prove_goal ZF.thy
+qed_goal "Union_least" ZF.thy
     "[| !!x. x:A ==> x<=C |] ==> Union(A) <= C"
  (fn [prem]=>
   [ (rtac (ballI RS (Union_subset_iff RS iffD2)) 1),
     (etac prem 1) ]);
 
 (*** Union of a family of sets  ***)
 
 goal ZF.thy "A <= (UN i:I. B(i)) <-> A = (UN i:I. A Int B(i))";
 by (fast_tac (upair_cs addSIs [equalityI] addSEs [equalityE]) 1);
-val subset_UN_iff_eq = result();
-
-val UN_subset_iff = prove_goal ZF.thy
+qed "subset_UN_iff_eq";
+
+qed_goal "UN_subset_iff" ZF.thy
      "(UN x:A.B(x)) <= C <-> (ALL x:A. B(x) <= C)"
  (fn _ => [ fast_tac upair_cs 1 ]);
 
-val UN_upper = prove_goal ZF.thy
+qed_goal "UN_upper" ZF.thy
     "!!x A. x:A ==> B(x) <= (UN x:A.B(x))"
  (fn _ => [ etac (RepFunI RS Union_upper) 1 ]);
 
-val UN_least = prove_goal ZF.thy
+qed_goal "UN_least" ZF.thy
     "[| !!x. x:A ==> B(x)<=C |] ==> (UN x:A.B(x)) <= C"
  (fn [prem]=>
   [ (rtac (ballI RS (UN_subset_iff RS iffD2)) 1),
     (etac prem 1) ]);
 
 
 (*** Big Intersection -- greatest lower bound of a nonempty set ***)
 
-val Inter_subset_iff = prove_goal ZF.thy
+qed_goal "Inter_subset_iff" ZF.thy
      "!!a A. a: A  ==>  C <= Inter(A) <-> (ALL x:A. C <= x)"
  (fn _ => [ fast_tac upair_cs 1 ]);
 
-val Inter_lower = prove_goal ZF.thy "B:A ==> Inter(A) <= B"
+qed_goal "Inter_lower" ZF.thy "B:A ==> Inter(A) <= B"
  (fn prems=>
   [ (REPEAT (resolve_tac (prems@[subsetI]) 1
      ORELSE etac InterD 1)) ]);
 
-val Inter_greatest = prove_goal ZF.thy
+qed_goal "Inter_greatest" ZF.thy
     "[| a:A;  !!x. x:A ==> C<=x |] ==> C <= Inter(A)"
  (fn [prem1,prem2]=>
   [ (rtac ([prem1, ballI] MRS (Inter_subset_iff RS iffD2)) 1),
     (etac prem2 1) ]);
 
 (*** Intersection of a family of sets  ***)
 
-val INT_lower = prove_goal ZF.thy
+qed_goal "INT_lower" ZF.thy
     "x:A ==> (INT x:A.B(x)) <= B(x)"
  (fn [prem] =>
   [ rtac (prem RS RepFunI RS Inter_lower) 1 ]);
 
-val INT_greatest = prove_goal ZF.thy
+qed_goal "INT_greatest" ZF.thy
     "[| a:A;  !!x. x:A ==> C<=B(x) |] ==> C <= (INT x:A.B(x))"
  (fn [nonempty,prem] =>
   [ rtac (nonempty RS RepFunI RS Inter_greatest) 1,
     REPEAT (eresolve_tac [RepFunE, prem, ssubst] 1) ]);
 
 
 (*** Finite Union -- the least upper bound of 2 sets ***)
 
-val Un_subset_iff = prove_goal ZF.thy "A Un B <= C <-> A <= C & B <= C"
- (fn _ => [ fast_tac upair_cs 1 ]);
-
-val Un_upper1 = prove_goal ZF.thy "A <= A Un B"
+qed_goal "Un_subset_iff" ZF.thy "A Un B <= C <-> A <= C & B <= C"
+ (fn _ => [ fast_tac upair_cs 1 ]);
+
+qed_goal "Un_upper1" ZF.thy "A <= A Un B"
  (fn _ => [ (REPEAT (ares_tac [subsetI,UnI1] 1)) ]);
 
-val Un_upper2 = prove_goal ZF.thy "B <= A Un B"
+qed_goal "Un_upper2" ZF.thy "B <= A Un B"
  (fn _ => [ (REPEAT (ares_tac [subsetI,UnI2] 1)) ]);
 
-val Un_least = prove_goal ZF.thy "!!A B C. [| A<=C;  B<=C |] ==> A Un B <= C"
+qed_goal "Un_least" ZF.thy "!!A B C. [| A<=C;  B<=C |] ==> A Un B <= C"
  (fn _ =>
   [ (rtac (Un_subset_iff RS iffD2) 1),
     (REPEAT (ares_tac [conjI] 1)) ]);
 
 (*** Finite Intersection -- the greatest lower bound of 2 sets *)
 
-val Int_subset_iff = prove_goal ZF.thy "C <= A Int B <-> C <= A & C <= B"
- (fn _ => [ fast_tac upair_cs 1 ]);
-
-val Int_lower1 = prove_goal ZF.thy "A Int B <= A"
+qed_goal "Int_subset_iff" ZF.thy "C <= A Int B <-> C <= A & C <= B"
+ (fn _ => [ fast_tac upair_cs 1 ]);
+
+qed_goal "Int_lower1" ZF.thy "A Int B <= A"
  (fn _ => [ (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1)) ]);
 
-val Int_lower2 = prove_goal ZF.thy "A Int B <= B"
+qed_goal "Int_lower2" ZF.thy "A Int B <= B"
  (fn _ => [ (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1)) ]);
 
-val Int_greatest = prove_goal ZF.thy
+qed_goal "Int_greatest" ZF.thy
                              "!!A B C. [| C<=A;  C<=B |] ==> C <= A Int B"
  (fn prems=>
   [ (rtac (Int_subset_iff RS iffD2) 1),
     (REPEAT (ares_tac [conjI] 1)) ]);
 
 (*** Set difference *)
 
-val Diff_subset = prove_goal ZF.thy "A-B <= A"
+qed_goal "Diff_subset" ZF.thy "A-B <= A"
  (fn _ => [ (REPEAT (ares_tac [subsetI] 1 ORELSE etac DiffE 1)) ]);
 
-val Diff_contains = prove_goal ZF.thy
+qed_goal "Diff_contains" ZF.thy
     "[| C<=A;  C Int B = 0 |] ==> C <= A-B"
  (fn prems=>
   [ (cut_facts_tac prems 1),
     (rtac subsetI 1),
     (REPEAT (ares_tac [DiffI,IntI,notI] 1
      ORELSE eresolve_tac [subsetD,equals0D] 1)) ]);
 
 (** Collect **)
 
-val Collect_subset = prove_goal ZF.thy "Collect(A,P) <= A"
+qed_goal "Collect_subset" ZF.thy "Collect(A,P) <= A"
  (fn _ => [ (REPEAT (ares_tac [subsetI] 1 ORELSE etac CollectD1 1)) ]);
 
 (** RepFun **)
 
 val prems = goal ZF.thy "[| !!x. x:A ==> f(x): B |] ==> {f(x). x:A} <= B";
 by (rtac subsetI 1);
 by (etac RepFunE 1);
 by (etac ssubst 1);
 by (eresolve_tac prems 1);
-val RepFun_subset = result();
+qed "RepFun_subset";
 
 (*A more powerful claset for subset reasoning*)
 val subset_cs = subset0_cs 
   addSIs [subset_refl,cons_subsetI,subset_consI,Union_least,UN_least,Un_least,
 	  Inter_greatest,Int_greatest,RepFun_subset]