--- a/src/CCL/Set.ML Sat Sep 17 14:02:31 2005 +0200
+++ b/src/CCL/Set.ML Sat Sep 17 17:35:26 2005 +0200
@@ -1,47 +1,36 @@
-(* 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));
@@ -50,32 +39,32 @@
(*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);
@@ -83,7 +72,7 @@
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);
@@ -93,18 +82,18 @@
(*** 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));
@@ -113,7 +102,7 @@
(*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";
@@ -125,30 +114,30 @@
(*** 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));
@@ -157,7 +146,7 @@
(*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));
@@ -169,22 +158,22 @@
(*** 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));
@@ -193,20 +182,20 @@
(*** 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);
@@ -216,7 +205,7 @@
(*** 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";
@@ -224,7 +213,7 @@
(*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";
@@ -238,13 +227,13 @@
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])));
@@ -257,7 +246,7 @@
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";
@@ -266,46 +255,46 @@
(*** 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]));
@@ -316,12 +305,12 @@
(*** 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));
@@ -329,21 +318,21 @@
(*** 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));