src/CCL/Set.ML

-(*  Title:      set/set
+(*  Title:      Set/Set.ML
     ID:         $Id$
-
-For set.thy.
-
-Modified version of
-    Title:      HOL/set
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1991  University of Cambridge
-
-For set.thy.  Set theory for higher-order logic.  A set is simply a predicate.
 *)
 
-open Set;
-
-val [prem] = goal Set.thy "[| P(a) |] ==> a : {x. P(x)}";
+val [prem] = goal (the_context ()) "[| P(a) |] ==> a : {x. P(x)}";
 by (rtac (mem_Collect_iff RS iffD2) 1);
 by (rtac prem 1);
 qed "CollectI";
 
-val prems = goal Set.thy "[| a : {x. P(x)} |] ==> P(a)";
+val prems = goal (the_context ()) "[| a : {x. P(x)} |] ==> P(a)";
 by (resolve_tac (prems RL [mem_Collect_iff  RS iffD1]) 1);
 qed "CollectD";
 
 val CollectE = make_elim CollectD;
 
-val [prem] = goal Set.thy "[| !!x. x:A <-> x:B |] ==> A = B";
+val [prem] = goal (the_context ()) "[| !!x. x:A <-> x:B |] ==> A = B";
 by (rtac (set_extension RS iffD2) 1);
 by (rtac (prem RS allI) 1);
 qed "set_ext";
 
 (*** Bounded quantifiers ***)
 
-val prems = goalw Set.thy [Ball_def]
+val prems = goalw (the_context ()) [Ball_def]
     "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)";
 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
 qed "ballI";
 
-val [major,minor] = goalw Set.thy [Ball_def]
+val [major,minor] = goalw (the_context ()) [Ball_def]
     "[| ALL x:A. P(x);  x:A |] ==> P(x)";
 by (rtac (minor RS (major RS spec RS mp)) 1);
 qed "bspec";
 
-val major::prems = goalw Set.thy [Ball_def]
+val major::prems = goalw (the_context ()) [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);
 
-val prems = goalw Set.thy [Bex_def]
+val prems = goalw (the_context ()) [Bex_def]
     "[| P(x);  x:A |] ==> EX x:A. P(x)";
 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
 qed "bexI";
 
-qed_goal "bexCI" Set.thy 
+qed_goal "bexCI" (the_context ())
    "[| EX x:A. ~P(x) ==> P(a);  a:A |] ==> EX x:A. P(x)"
  (fn prems=>
   [ (rtac classical 1),
     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
 
-val major::prems = goalw Set.thy [Bex_def]
+val major::prems = goalw (the_context ()) [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";
 
 (*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
-val prems = goal Set.thy
+val prems = goal (the_context ())
     "(ALL x:A. True) <-> True";
 by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
 qed "ball_rew";
 
 (** Congruence rules **)
 
-val prems = goal Set.thy
+val prems = goal (the_context ())
     "[| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==> \
 \    (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))";
 by (resolve_tac (prems RL [ssubst,iffD2]) 1);
 by (REPEAT (ares_tac [ballI,iffI] 1
      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
 qed "ball_cong";
 
-val prems = goal Set.thy
+val prems = goal (the_context ())
     "[| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==> \
 \    (EX x:A. P(x)) <-> (EX x:A'. P'(x))";
 by (resolve_tac (prems RL [ssubst,iffD2]) 1);
 by (REPEAT (etac bexE 1
      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
 qed "bex_cong";
 
 (*** Rules for subsets ***)
 
-val prems = goalw Set.thy [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
+val prems = goalw (the_context ()) [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
 by (REPEAT (ares_tac (prems @ [ballI]) 1));
 qed "subsetI";
 
 (*Rule in Modus Ponens style*)
-val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
+val major::prems = goalw (the_context ()) [subset_def] "[| A <= B;  c:A |] ==> c:B";
 by (rtac (major RS bspec) 1);
 by (resolve_tac prems 1);
 qed "subsetD";
 
 (*Classical elimination rule*)
-val major::prems = goalw Set.thy [subset_def] 
+val major::prems = goalw (the_context ()) [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;
 
-qed_goal "subset_refl" Set.thy "A <= A"
+qed_goal "subset_refl" (the_context ()) "A <= A"
  (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]);
 
 Goal "[| A<=B;  B<=C |] ==> A<=C";
 by (rtac subsetI 1);
 by (REPEAT (eresolve_tac [asm_rl, subsetD] 1));

 
 
 (*** Rules for equality ***)
 
 (*Anti-symmetry of the subset relation*)
-val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = B";
+val prems = goal (the_context ()) "[| A <= B;  B <= A |] ==> A = B";
 by (rtac (iffI RS set_ext) 1);
 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
 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";
+val prems = goal (the_context ()) "A = B ==> A<=B";
 by (resolve_tac (prems RL [subst]) 1);
 by (rtac subset_refl 1);
 qed "equalityD1";
 
-val prems = goal Set.thy "A = B ==> B<=A";
+val prems = goal (the_context ()) "A = B ==> B<=A";
 by (resolve_tac (prems RL [subst]) 1);
 by (rtac subset_refl 1);
 qed "equalityD2";
 
-val prems = goal Set.thy
+val prems = goal (the_context ())
     "[| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P";
 by (resolve_tac prems 1);
 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
 qed "equalityE";
 
-val major::prems = goal Set.thy
+val major::prems = goal (the_context ())
     "[| 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";
 
 (*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 
+val prems = goal (the_context ())
     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
 by (rtac mp 1);
 by (REPEAT (resolve_tac (refl::prems) 1));
 qed "setup_induction";
 

 by (REPEAT (ares_tac [equalityI,subsetI,CollectI] 1  ORELSE etac CollectD 1));
 qed "trivial_set";
 
 (*** Rules for binary union -- Un ***)
 
-val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
+val prems = goalw (the_context ()) [Un_def] "c:A ==> c : A Un B";
 by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
 qed "UnI1";
 
-val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
+val prems = goalw (the_context ()) [Un_def] "c:B ==> c : A Un B";
 by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
 qed "UnI2";
 
 (*Classical introduction rule: no commitment to A vs B*)
-qed_goal "UnCI" Set.thy "(~c:B ==> c:A) ==> c : A Un B"
+qed_goal "UnCI" (the_context ()) "(~c:B ==> c:A) ==> c : A Un B"
  (fn prems=>
   [ (rtac classical 1),
     (REPEAT (ares_tac (prems@[UnI1,notI]) 1)),
     (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]);
 
-val major::prems = goalw Set.thy [Un_def]
+val major::prems = goalw (the_context ()) [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";
 
 
 (*** Rules for small intersection -- Int ***)
 
-val prems = goalw Set.thy [Int_def]
+val prems = goalw (the_context ()) [Int_def]
     "[| c:A;  c:B |] ==> c : A Int B";
 by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
 qed "IntI";
 
-val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
+val [major] = goalw (the_context ()) [Int_def] "c : A Int B ==> c:A";
 by (rtac (major RS CollectD RS conjunct1) 1);
 qed "IntD1";
 
-val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
+val [major] = goalw (the_context ()) [Int_def] "c : A Int B ==> c:B";
 by (rtac (major RS CollectD RS conjunct2) 1);
 qed "IntD2";
 
-val [major,minor] = goal Set.thy
+val [major,minor] = goal (the_context ())
     "[| 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";
 
 
 (*** Rules for set complement -- Compl ***)
 
-val prems = goalw Set.thy [Compl_def]
+val prems = goalw (the_context ()) [Compl_def]
     "[| c:A ==> False |] ==> c : Compl(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...*)
-val major::prems = goalw Set.thy [Compl_def]
+val major::prems = goalw (the_context ()) [Compl_def]
     "[| c : Compl(A) |] ==> ~c:A";
 by (rtac (major RS CollectD) 1);
 qed "ComplD";
 
 val ComplE = make_elim ComplD;

 
 Goalw [empty_def] "{x. False} = {}";
 by (rtac refl 1);
 qed "empty_eq";
 
-val [prem] = goalw Set.thy [empty_def] "a : {} ==> P";
+val [prem] = goalw (the_context ()) [empty_def] "a : {} ==> P";
 by (rtac (prem RS CollectD RS FalseE) 1);
 qed "emptyD";
 
 val emptyE = make_elim emptyD;
 
-val [prem] = goal Set.thy "~ A={} ==> (EX x. x:A)";
+val [prem] = goal (the_context ()) "~ A={} ==> (EX x. x:A)";
 by (rtac (prem RS swap) 1);
 by (rtac equalityI 1);
 by (ALLGOALS (fast_tac (FOL_cs addSIs [subsetI] addSEs [emptyD])));
 qed "not_emptyD";
 

 Goalw [singleton_def] "a : {a}";
 by (rtac CollectI 1);
 by (rtac refl 1);
 qed "singletonI";
 
-val [major] = goalw Set.thy [singleton_def] "b : {a} ==> b=a"; 
+val [major] = goalw (the_context ()) [singleton_def] "b : {a} ==> b=a";
 by (rtac (major RS CollectD) 1);
 qed "singletonD";
 
 val singletonE = make_elim singletonD;
 
 (*** Unions of families ***)
 
 (*The order of the premises presupposes that A is rigid; b may be flexible*)
-val prems = goalw Set.thy [UNION_def]
+val prems = goalw (the_context ()) [UNION_def]
     "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
 by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
 qed "UN_I";
 
-val major::prems = goalw Set.thy [UNION_def]
+val major::prems = goalw (the_context ()) [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";
 
-val prems = goal Set.thy
+val prems = goal (the_context ())
     "[| 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] @ 
+     ORELSE ares_tac ([UN_I,equalityI,subsetI] @
                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
 qed "UN_cong";
 
 (*** Intersections of families -- INTER x:A. B(x) is Inter(B)``A ) *)
 
-val prems = goalw Set.thy [INTER_def]
+val prems = goalw (the_context ()) [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";
 
-val major::prems = goalw Set.thy [INTER_def]
+val major::prems = goalw (the_context ()) [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);
 qed "INT_D";
 
 (*"Classical" elimination rule -- does not require proving X:C *)
-val major::prems = goalw Set.thy [INTER_def]
+val major::prems = goalw (the_context ()) [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";
 
-val prems = goal Set.thy
+val prems = goal (the_context ())
     "[| 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));
 qed "INT_cong";
 
 (*** Rules for Unions ***)
 
 (*The order of the premises presupposes that C is rigid; A may be flexible*)
-val prems = goalw Set.thy [Union_def]
+val prems = goalw (the_context ()) [Union_def]
     "[| X:C;  A:X |] ==> A : Union(C)";
 by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
 qed "UnionI";
 
-val major::prems = goalw Set.thy [Union_def]
+val major::prems = goalw (the_context ()) [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";
 
 (*** Rules for Inter ***)
 
-val prems = goalw Set.thy [Inter_def]
+val prems = goalw (the_context ()) [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". *)
-val major::prems = goalw Set.thy [Inter_def]
+val major::prems = goalw (the_context ()) [Inter_def]
     "[| A : Inter(C);  X:C |] ==> A:X";
 by (rtac (major RS INT_D) 1);
 by (resolve_tac prems 1);
 qed "InterD";
 
 (*"Classical" elimination rule -- does not require proving X:C *)
-val major::prems = goalw Set.thy [Inter_def]
+val major::prems = goalw (the_context ()) [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));
 qed "InterE";