Set.ML

 open Set;
 
 val [prem] = goal Set.thy "[| P(a) |] ==> a : {x.P(x)}";
 by (rtac (mem_Collect_eq RS ssubst) 1);
 by (rtac prem 1);
-val CollectI = result();
+qed "CollectI";
 
 val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)";
 by (resolve_tac (prems RL [mem_Collect_eq  RS subst]) 1);
-val CollectD = result();
+qed "CollectD";
 
 val [prem] = goal Set.thy "[| !!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);
-val set_ext = result();
+qed "set_ext";
 
 val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
 by (rtac (prem RS ext RS arg_cong) 1);
-val Collect_cong = result();
+qed "Collect_cong";
 
 val CollectE = make_elim CollectD;
 
 (*** Bounded quantifiers ***)
 
 val prems = goalw Set.thy [Ball_def]
     "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
-val ballI = result();
+qed "ballI";
 
 val [major,minor] = goalw Set.thy [Ball_def]
     "[| ! x:A. P(x);  x:A |] ==> P(x)";
 by (rtac (minor RS (major RS spec RS mp)) 1);
-val bspec = result();
+qed "bspec";
 
 val major::prems = goalw Set.thy [Ball_def]
     "[| ! 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));
-val ballE = result();
+qed "ballE";
 
 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
 
 val prems = goalw Set.thy [Bex_def]
     "[| P(x);  x:A |] ==> ? x:A. P(x)";
 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
-val bexI = result();
+qed "bexI";
 
 val bexCI = prove_goal Set.thy 
    "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A.P(x)"
  (fn prems=>
   [ (rtac classical 1),

 
 val major::prems = goalw Set.thy [Bex_def]
     "[| ? 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));
-val bexE = result();
+qed "bexE";
 
 (*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
 val prems = goal Set.thy
     "(! x:A. True) = True";
 by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
-val ball_rew = result();
+qed "ball_rew";
 
 (** Congruence rules **)
 
 val prems = goal Set.thy
     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
 \    (! x:A. P(x)) = (! x:B. Q(x))";
 by (resolve_tac (prems RL [ssubst]) 1);
 by (REPEAT (ares_tac [ballI,iffI] 1
      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
-val ball_cong = result();
+qed "ball_cong";
 
 val prems = goal Set.thy
     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
 \    (? x:A. P(x)) = (? x:B. Q(x))";
 by (resolve_tac (prems RL [ssubst]) 1);
 by (REPEAT (etac bexE 1
      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
-val bex_cong = result();
+qed "bex_cong";
 
 (*** Subsets ***)
 
 val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
 by (REPEAT (ares_tac (prems @ [ballI]) 1));
-val subsetI = result();
+qed "subsetI";
 
 (*Rule in Modus Ponens style*)
 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
 by (rtac (major RS bspec) 1);
 by (resolve_tac prems 1);
-val subsetD = result();
+qed "subsetD";
 
 (*The same, with reversed premises for use with etac -- cf rev_mp*)
 val rev_subsetD = prove_goal Set.thy "[| c:A;  A <= B |] ==> c:B"
  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
 
 (*Classical elimination rule*)
 val major::prems = goalw Set.thy [subset_def] 
     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
 by (rtac (major RS ballE) 1);
 by (REPEAT (eresolve_tac prems 1));
-val subsetCE = result();
+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;
 
 val subset_refl = prove_goal Set.thy "A <= (A::'a set)"
  (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]);
 
 val prems = goal Set.thy "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
 by (cut_facts_tac prems 1);
 by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1));
-val subset_trans = result();
+qed "subset_trans";
 
 
 (*** Equality ***)
 
 (*Anti-symmetry of the subset relation*)
 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
 by (rtac (iffI RS set_ext) 1);
 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
-val subset_antisym = result();
+qed "subset_antisym";
 val equalityI = subset_antisym;
 
 (* Equality rules from ZF set theory -- are they appropriate here? *)
 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
 by (resolve_tac (prems RL [subst]) 1);
 by (rtac subset_refl 1);
-val equalityD1 = result();
+qed "equalityD1";
 
 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
 by (resolve_tac (prems RL [subst]) 1);
 by (rtac subset_refl 1);
-val equalityD2 = result();
+qed "equalityD2";
 
 val prems = goal Set.thy
     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
 by (resolve_tac prems 1);
 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
-val equalityE = result();
+qed "equalityE";
 
 val major::prems = goal Set.thy
     "[| 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));
-val equalityCE = result();
+qed "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 Set.thy 
     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
 by (rtac mp 1);
 by (REPEAT (resolve_tac (refl::prems) 1));
-val setup_induction = result();
+qed "setup_induction";
 
 
 (*** Set complement -- Compl ***)
 
 val prems = goalw Set.thy [Compl_def]
     "[| c:A ==> False |] ==> c : Compl(A)";
 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
-val ComplI = result();
+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...*)
 val major::prems = goalw Set.thy [Compl_def]
     "[| c : Compl(A) |] ==> c~:A";
 by (rtac (major RS CollectD) 1);
-val ComplD = result();
+qed "ComplD";
 
 val ComplE = make_elim ComplD;
 
 
 (*** Binary union -- Un ***)
 
 val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
 by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
-val UnI1 = result();
+qed "UnI1";
 
 val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
 by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
-val UnI2 = result();
+qed "UnI2";
 
 (*Classical introduction rule: no commitment to A vs B*)
 val UnCI = prove_goal Set.thy "(c~:B ==> c:A) ==> c : A Un B"
  (fn prems=>
   [ (rtac classical 1),

 
 val major::prems = goalw Set.thy [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));
-val UnE = result();
+qed "UnE";
 
 
 (*** Binary intersection -- Int ***)
 
 val prems = goalw Set.thy [Int_def]
     "[| c:A;  c:B |] ==> c : A Int B";
 by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
-val IntI = result();
+qed "IntI";
 
 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
 by (rtac (major RS CollectD RS conjunct1) 1);
-val IntD1 = result();
+qed "IntD1";
 
 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
 by (rtac (major RS CollectD RS conjunct2) 1);
-val IntD2 = result();
+qed "IntD2";
 
 val [major,minor] = goal Set.thy
     "[| 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);
-val IntE = result();
+qed "IntE";
 
 
 (*** Set difference ***)
 
 val DiffI = prove_goalw Set.thy [set_diff_def]

   [ (rtac (major RS insertE) 1),
     (REPEAT (eresolve_tac (prems @ [emptyE]) 1)) ]);
 
 goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a";
 by(fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1);
-val singletonD = result();
+qed "singletonD";
 
 val singletonE = make_elim singletonD;
 
 val [major] = goal Set.thy "{a}={b} ==> a=b";
 by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1);
 by (rtac singletonI 1);
-val singleton_inject = result();
+qed "singleton_inject";
 
 (*** Unions of families -- UNION x:A. B(x) is Union(B``A)  ***)
 
 (*The order of the premises presupposes that A is rigid; b may be flexible*)
 val prems = goalw Set.thy [UNION_def]
     "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
 by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
-val UN_I = result();
+qed "UN_I";
 
 val major::prems = goalw Set.thy [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));
-val UN_E = result();
+qed "UN_E";
 
 val prems = goal Set.thy
     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
 \    (UN x:A. C(x)) = (UN x:B. D(x))";
 by (REPEAT (etac UN_E 1
      ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
 		      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
-val UN_cong = result();
+qed "UN_cong";
 
 
 (*** Intersections of families -- INTER x:A. B(x) is Inter(B``A) *)
 
 val prems = goalw Set.thy [INTER_def]
     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
-val INT_I = result();
+qed "INT_I";
 
 val major::prems = goalw Set.thy [INTER_def]
     "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
 by (rtac (major RS CollectD RS bspec) 1);
 by (resolve_tac prems 1);
-val INT_D = result();
+qed "INT_D";
 
 (*"Classical" elimination -- by the Excluded Middle on a:A *)
 val major::prems = goalw Set.thy [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));
-val INT_E = result();
+qed "INT_E";
 
 val prems = goal Set.thy
     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
 \    (INT x:A. C(x)) = (INT x:B. D(x))";
 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
 by (REPEAT (dtac INT_D 1
      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
-val INT_cong = result();
+qed "INT_cong";
 
 
 (*** Unions over a type; UNION1(B) = Union(range(B)) ***)
 
 (*The order of the premises presupposes that A is rigid; b may be flexible*)
 val prems = goalw Set.thy [UNION1_def]
     "b: B(x) ==> b: (UN x. B(x))";
 by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1));
-val UN1_I = result();
+qed "UN1_I";
 
 val major::prems = goalw Set.thy [UNION1_def]
     "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
 by (rtac (major RS UN_E) 1);
 by (REPEAT (ares_tac prems 1));
-val UN1_E = result();
+qed "UN1_E";
 
 
 (*** Intersections over a type; INTER1(B) = Inter(range(B)) *)
 
 val prems = goalw Set.thy [INTER1_def]
     "(!!x. b: B(x)) ==> b : (INT x. B(x))";
 by (REPEAT (ares_tac (INT_I::prems) 1));
-val INT1_I = result();
+qed "INT1_I";
 
 val [major] = goalw Set.thy [INTER1_def]
     "b : (INT x. B(x)) ==> b: B(a)";
 by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1);
-val INT1_D = result();
+qed "INT1_D";
 
 (*** Unions ***)
 
 (*The order of the premises presupposes that C is rigid; A may be flexible*)
 val prems = goalw Set.thy [Union_def]
     "[| X:C;  A:X |] ==> A : Union(C)";
 by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
-val UnionI = result();
+qed "UnionI";
 
 val major::prems = goalw Set.thy [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));
-val UnionE = result();
+qed "UnionE";
 
 (*** Inter ***)
 
 val prems = goalw Set.thy [Inter_def]
     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
-val InterI = result();
+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". *)
 val major::prems = goalw Set.thy [Inter_def]
     "[| A : Inter(C);  X:C |] ==> A:X";
 by (rtac (major RS INT_D) 1);
 by (resolve_tac prems 1);
-val InterD = result();
+qed "InterD";
 
 (*"Classical" elimination rule -- does not require proving X:C *)
 val major::prems = goalw Set.thy [Inter_def]
     "[| A : Inter(C);  A:X ==> R;  X~:C ==> R |] ==> R";
 by (rtac (major RS INT_E) 1);
 by (REPEAT (eresolve_tac prems 1));
-val InterE = result();
+qed "InterE";
 
 (*** Powerset ***)
 
 val PowI = prove_goalw Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
  (fn _ => [ (etac CollectI 1) ]);