Finite.ML

 open Finite;
 
 goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)";
 br lfp_mono 1;
 by (REPEAT (ares_tac basic_monos 1));
-val Fin_mono = result();
+qed "Fin_mono";
 
 goalw Finite.thy Fin.defs "Fin(A) <= Pow(A)";
 by (fast_tac (set_cs addSIs [lfp_lowerbound]) 1);
-val Fin_subset_Pow = result();
+qed "Fin_subset_Pow";
 
 (* A : Fin(B) ==> A <= B *)
 val FinD = Fin_subset_Pow RS subsetD RS PowD;
 
 (*Discharging ~ x:y entails extra work*)

 \    |] ==> P(F)";
 by (rtac (major RS Fin.induct) 1);
 by (excluded_middle_tac "a:b" 2);
 by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3);   (*backtracking!*)
 by (REPEAT (ares_tac prems 1));
-val Fin_induct = result();
+qed "Fin_induct";
 
 (** Simplification for Fin **)
 
 val Fin_ss = set_ss addsimps Fin.intrs;
 
 (*The union of two finite sets is finite*)
 val major::prems = goal Finite.thy
     "[| F: Fin(A);  G: Fin(A) |] ==> F Un G : Fin(A)";
 by (rtac (major RS Fin_induct) 1);
 by (ALLGOALS (asm_simp_tac (Fin_ss addsimps (prems@[Un_insert_left]))));
-val Fin_UnI = result();
+qed "Fin_UnI";
 
 (*Every subset of a finite set is finite*)
 val [subs,fin] = goal Finite.thy "[| A<=B;  B: Fin(M) |] ==> A: Fin(M)";
 by (EVERY1 [subgoal_tac "ALL C. C<=B --> C: Fin(M)",
 	    rtac mp, etac spec,

 by (rtac (fin RS Fin_induct) 1);
 by (simp_tac (Fin_ss addsimps [subset_Un_eq]) 1);
 by (safe_tac (set_cs addSDs [subset_insert_iff RS iffD1]));
 by (eres_inst_tac [("t","C")] (insert_Diff RS subst) 2);
 by (ALLGOALS (asm_simp_tac Fin_ss));
-val Fin_subset = result();
+qed "Fin_subset";
 
 (*The image of a finite set is finite*)
 val major::_ = goal Finite.thy
     "F: Fin(A) ==> h``F : Fin(h``A)";
 by (rtac (major RS Fin_induct) 1);
 by (simp_tac Fin_ss 1);
 by (asm_simp_tac (set_ss addsimps [image_eqI RS Fin.insertI, image_insert]) 1);
-val Fin_imageI = result();
+qed "Fin_imageI";
 
 val major::prems = goal Finite.thy 
     "[| c: Fin(A);  b: Fin(A);  				\
 \       P(b);       						\
 \       !!(x::'a) y. [| x:A; y: Fin(A);  x:y;  P(y) |] ==> P(y-{x}) \
 \    |] ==> c<=b --> P(b-c)";
 by (rtac (major RS Fin_induct) 1);
 by (rtac (Diff_insert RS ssubst) 2);
 by (ALLGOALS (asm_simp_tac
                 (Fin_ss addsimps (prems@[Diff_subset RS Fin_subset]))));
-val Fin_empty_induct_lemma = result();
+qed "Fin_empty_induct_lemma";
 
 val prems = goal Finite.thy 
     "[| b: Fin(A);  						\
 \       P(b);        						\
 \       !!x y. [| x:A; y: Fin(A);  x:y;  P(y) |] ==> P(y-{x}) \
 \    |] ==> P({})";
 by (rtac (Diff_cancel RS subst) 1);
 by (rtac (Fin_empty_induct_lemma RS mp) 1);
 by (REPEAT (ares_tac (subset_refl::prems) 1));
-val Fin_empty_induct = result();
+qed "Fin_empty_induct";