diff -r 3a8d722fd3ff -r 16c4ea954511 subset.ML --- a/subset.ML Fri Nov 11 10:35:03 1994 +0100 +++ b/subset.ML Mon Nov 21 17:50:34 1994 +0100 @@ -17,35 +17,35 @@ val prems = goal Set.thy "B:A ==> B <= Union(A)"; by (REPEAT (ares_tac (prems@[subsetI,UnionI]) 1)); -val Union_upper = result(); +qed "Union_upper"; val [prem] = goal Set.thy "[| !!X. X:A ==> X<=C |] ==> Union(A) <= C"; br subsetI 1; by (REPEAT (eresolve_tac [asm_rl, UnionE, prem RS subsetD] 1)); -val Union_least = result(); +qed "Union_least"; (** General union **) val prems = goal Set.thy "a:A ==> B(a) <= (UN x:A. B(x))"; by (REPEAT (ares_tac (prems@[UN_I RS subsetI]) 1)); -val UN_upper = result(); +qed "UN_upper"; val [prem] = goal Set.thy "[| !!x. x:A ==> B(x)<=C |] ==> (UN x:A. B(x)) <= C"; br subsetI 1; by (REPEAT (eresolve_tac [asm_rl, UN_E, prem RS subsetD] 1)); -val UN_least = result(); +qed "UN_least"; goal Set.thy "B(a) <= (UN x. B(x))"; by (REPEAT (ares_tac [UN1_I RS subsetI] 1)); -val UN1_upper = result(); +qed "UN1_upper"; val [prem] = goal Set.thy "[| !!x. B(x)<=C |] ==> (UN x. B(x)) <= C"; br subsetI 1; by (REPEAT (eresolve_tac [asm_rl, UN1_E, prem RS subsetD] 1)); -val UN1_least = result(); +qed "UN1_least"; (*** Big Intersection -- greatest lower bound of a set ***) @@ -53,67 +53,67 @@ val prems = goal Set.thy "B:A ==> Inter(A) <= B"; br subsetI 1; by (REPEAT (resolve_tac prems 1 ORELSE etac InterD 1)); -val Inter_lower = result(); +qed "Inter_lower"; val [prem] = goal Set.thy "[| !!X. X:A ==> C<=X |] ==> C <= Inter(A)"; br (InterI RS subsetI) 1; by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1)); -val Inter_greatest = result(); +qed "Inter_greatest"; val prems = goal Set.thy "a:A ==> (INT x:A. B(x)) <= B(a)"; br subsetI 1; by (REPEAT (resolve_tac prems 1 ORELSE etac INT_D 1)); -val INT_lower = result(); +qed "INT_lower"; val [prem] = goal Set.thy "[| !!x. x:A ==> C<=B(x) |] ==> C <= (INT x:A. B(x))"; br (INT_I RS subsetI) 1; by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1)); -val INT_greatest = result(); +qed "INT_greatest"; goal Set.thy "(INT x. B(x)) <= B(a)"; br subsetI 1; by (REPEAT (resolve_tac prems 1 ORELSE etac INT1_D 1)); -val INT1_lower = result(); +qed "INT1_lower"; val [prem] = goal Set.thy "[| !!x. C<=B(x) |] ==> C <= (INT x. B(x))"; br (INT1_I RS subsetI) 1; by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1)); -val INT1_greatest = result(); +qed "INT1_greatest"; (*** Finite Union -- the least upper bound of 2 sets ***) goal Set.thy "A <= A Un B"; by (REPEAT (ares_tac [subsetI,UnI1] 1)); -val Un_upper1 = result(); +qed "Un_upper1"; goal Set.thy "B <= A Un B"; by (REPEAT (ares_tac [subsetI,UnI2] 1)); -val Un_upper2 = result(); +qed "Un_upper2"; val prems = goal Set.thy "[| A<=C; B<=C |] ==> A Un B <= C"; by (cut_facts_tac prems 1); by (DEPTH_SOLVE (ares_tac [subsetI] 1 ORELSE eresolve_tac [UnE,subsetD] 1)); -val Un_least = result(); +qed "Un_least"; (*** Finite Intersection -- the greatest lower bound of 2 sets *) goal Set.thy "A Int B <= A"; by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1)); -val Int_lower1 = result(); +qed "Int_lower1"; goal Set.thy "A Int B <= B"; by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1)); -val Int_lower2 = result(); +qed "Int_lower2"; val prems = goal Set.thy "[| C<=A; C<=B |] ==> C <= A Int B"; by (cut_facts_tac prems 1); by (REPEAT (ares_tac [subsetI,IntI] 1 ORELSE etac subsetD 1)); -val Int_greatest = result(); +qed "Int_greatest"; (*** Set difference ***) @@ -126,10 +126,10 @@ by (rtac Un_least 1); by (rtac (Un_upper1 RS (prem RS monoD)) 1); by (rtac (Un_upper2 RS (prem RS monoD)) 1); -val mono_Un = result(); +qed "mono_Un"; val [prem] = goal Set.thy "mono(f) ==> f(A Int B) <= f(A) Int f(B)"; by (rtac Int_greatest 1); by (rtac (Int_lower1 RS (prem RS monoD)) 1); by (rtac (Int_lower2 RS (prem RS monoD)) 1); -val mono_Int = result(); +qed "mono_Int";