src/HOL/mono.ML

     Copyright   1991  University of Cambridge
 
 Monotonicity of various operations
 *)
 
-goal Set.thy "!!A B. A<=B ==> f``A <= f``B";
+Goal "A<=B ==> f``A <= f``B";
 by (Blast_tac 1);
 qed "image_mono";
 
-goal Set.thy "!!A B. A<=B ==> Pow(A) <= Pow(B)";
+Goal "A<=B ==> Pow(A) <= Pow(B)";
 by (Blast_tac 1);
 qed "Pow_mono";
 
-goal Set.thy "!!A B. A<=B ==> Union(A) <= Union(B)";
+Goal "A<=B ==> Union(A) <= Union(B)";
 by (Blast_tac 1);
 qed "Union_mono";
 
-goal Set.thy "!!A B. B<=A ==> Inter(A) <= Inter(B)";
+Goal "B<=A ==> Inter(A) <= Inter(B)";
 by (Blast_tac 1);
 qed "Inter_anti_mono";
 
-val prems = goal Set.thy
+val prems = Goal
     "[| A<=B;  !!x. x:A ==> f(x)<=g(x) |] ==> \
 \    (UN x:A. f(x)) <= (UN x:B. g(x))";
 by (blast_tac (claset() addIs (prems RL [subsetD])) 1);
 qed "UN_mono";
 
 (*The last inclusion is POSITIVE! *)
-val prems = goal Set.thy
+val prems = Goal
     "[| B<=A;  !!x. x:A ==> f(x)<=g(x) |] ==> \
 \    (INT x:A. f(x)) <= (INT x:A. g(x))";
 by (blast_tac (claset() addIs (prems RL [subsetD])) 1);
 qed "INT_anti_mono";
 
-goal Set.thy "!!C D. C<=D ==> insert a C <= insert a D";
+Goal "C<=D ==> insert a C <= insert a D";
 by (Blast_tac 1);
 qed "insert_mono";
 
-goal Set.thy "!!A B. [| A<=C;  B<=D |] ==> A Un B <= C Un D";
+Goal "[| A<=C;  B<=D |] ==> A Un B <= C Un D";
 by (Blast_tac 1);
 qed "Un_mono";
 
-goal Set.thy "!!A B. [| A<=C;  B<=D |] ==> A Int B <= C Int D";
+Goal "[| A<=C;  B<=D |] ==> A Int B <= C Int D";
 by (Blast_tac 1);
 qed "Int_mono";
 
-goal Set.thy "!!A::'a set. [| A<=C;  D<=B |] ==> A-B <= C-D";
+Goal "!!A::'a set. [| A<=C;  D<=B |] ==> A-B <= C-D";
 by (Blast_tac 1);
 qed "Diff_mono";
 
-goal Set.thy "!!A B. A<=B ==> Compl(B) <= Compl(A)";
+Goal "A<=B ==> Compl(B) <= Compl(A)";
 by (Blast_tac 1);
 qed "Compl_anti_mono";
 
 (** Monotonicity of implications.  For inductive definitions **)
 
-goal Set.thy "!!A B x. A<=B ==> x:A --> x:B";
+Goal "A<=B ==> x:A --> x:B";
 by (rtac impI 1);
 by (etac subsetD 1);
 by (assume_tac 1);
 qed "in_mono";
 
-goal HOL.thy "!!P1 P2 Q1 Q2. [| P1-->Q1; P2-->Q2 |] ==> (P1&P2) --> (Q1&Q2)";
+Goal "[| P1-->Q1; P2-->Q2 |] ==> (P1&P2) --> (Q1&Q2)";
 by (Blast_tac 1);
 qed "conj_mono";
 
-goal HOL.thy "!!P1 P2 Q1 Q2. [| P1-->Q1; P2-->Q2 |] ==> (P1|P2) --> (Q1|Q2)";
+Goal "[| P1-->Q1; P2-->Q2 |] ==> (P1|P2) --> (Q1|Q2)";
 by (Blast_tac 1);
 qed "disj_mono";
 
-goal HOL.thy "!!P1 P2 Q1 Q2.[| Q1-->P1; P2-->Q2 |] ==> (P1-->P2)-->(Q1-->Q2)";
+Goal "[| Q1-->P1; P2-->Q2 |] ==> (P1-->P2)-->(Q1-->Q2)";
 by (Blast_tac 1);
 qed "imp_mono";
 
-goal HOL.thy "P-->P";
+Goal "P-->P";
 by (rtac impI 1);
 by (assume_tac 1);
 qed "imp_refl";
 
-val [PQimp] = goal HOL.thy
+val [PQimp] = Goal
     "[| !!x. P(x) --> Q(x) |] ==> (EX x. P(x)) --> (EX x. Q(x))";
 by (blast_tac (claset() addIs [PQimp RS mp]) 1);
 qed "ex_mono";
 
-val [PQimp] = goal HOL.thy
+val [PQimp] = Goal
     "[| !!x. P(x) --> Q(x) |] ==> (ALL x. P(x)) --> (ALL x. Q(x))";
 by (blast_tac (claset() addIs [PQimp RS mp]) 1);
 qed "all_mono";
 
-val [PQimp] = goal Set.thy
+val [PQimp] = Goal
     "[| !!x. P(x) --> Q(x) |] ==> Collect(P) <= Collect(Q)";
 by (blast_tac (claset() addIs [PQimp RS mp]) 1);
 qed "Collect_mono";
 
-val [subs,PQimp] = goal Set.thy
+val [subs,PQimp] = Goal
     "[| A<=B;  !!x. x:A ==> P(x) --> Q(x) \
 \    |] ==> A Int Collect(P) <= B Int Collect(Q)";
 by (blast_tac (claset() addIs [subs RS subsetD, PQimp RS mp]) 1);
 qed "Int_Collect_mono";