--- a/Quot/QuotOption.thy	Tue Jan 26 00:18:48 2010 +0100
+++ b/Quot/QuotOption.thy	Tue Jan 26 00:47:40 2010 +0100
@@ -17,19 +17,23 @@
 
 declare [[map option = (Option.map, option_rel)]]
 
+text {* should probably be in Option.thy *}
+lemma split_option_all: 
+  shows "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>a. P (Some a))"
+apply(auto)
+apply(case_tac x)
+apply(simp_all)
+done
+
 lemma option_quotient[quot_thm]:
   assumes q: "Quotient R Abs Rep"
   shows "Quotient (option_rel R) (Option.map Abs) (Option.map Rep)"
   unfolding Quotient_def
-  apply(auto)
-  apply(case_tac a, simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
-  apply(case_tac a, simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
-  apply(case_tac [!] r)
-  apply(case_tac [!] s)
-  apply(simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q] )
+  apply(simp add: split_option_all)
+  apply(simp add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
   using q
   unfolding Quotient_def
-  apply(blast)+
+  apply(blast)
   done
   
 lemma option_equivp[quot_equiv]:
@@ -37,16 +41,10 @@
   shows "equivp (option_rel R)"
   apply(rule equivpI)
   unfolding reflp_def symp_def transp_def
-  apply(auto)
-  apply(case_tac [!] x)
-  apply(simp_all add: equivp_reflp[OF a])
-  apply(case_tac [!] y)
-  apply(simp_all add: equivp_symp[OF a])
-  apply(case_tac [!] z)
-  apply(simp_all)
-  apply(clarify)
-  apply(rule equivp_transp[OF a])
-  apply(assumption)+
+  apply(simp_all add: split_option_all)
+  apply(blast intro: equivp_reflp[OF a])
+  apply(blast intro: equivp_symp[OF a])
+  apply(blast intro: equivp_transp[OF a])
   done
 
 lemma option_None_rsp[quot_respect]:
@@ -74,18 +72,11 @@
 lemma option_map_id[id_simps]: 
   shows "Option.map id \<equiv> id"
   apply (rule eq_reflection)
-  apply (auto simp add: expand_fun_eq)
-  apply (case_tac x)
-  apply (auto)
-  done
+  by (simp add: expand_fun_eq split_option_all)
 
 lemma option_rel_eq[id_simps]: 
   shows "option_rel (op =) \<equiv> (op =)"
   apply(rule eq_reflection)
-  apply(auto simp add: expand_fun_eq)
-  apply(case_tac x)
-  apply(case_tac [!] xa)
-  apply(auto)
-  done
+  by (simp add: expand_fun_eq split_option_all)
 
 end