--- a/Quot/QuotProd.thy	Tue Jan 26 07:42:52 2010 +0100
+++ b/Quot/QuotProd.thy	Tue Jan 26 08:09:22 2010 +0100
@@ -14,37 +14,37 @@
   assumes a: "equivp R1"
   assumes b: "equivp R2"
   shows "equivp (prod_rel R1 R2)"
-unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
-apply(auto simp add: equivp_reflp[OF a] equivp_reflp[OF b])
-apply(simp only: equivp_symp[OF a])
-apply(simp only: equivp_symp[OF b])
-using equivp_transp[OF a] apply blast
-using equivp_transp[OF b] apply blast
-done
+  unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
+  apply (auto simp add: equivp_reflp[OF a] equivp_reflp[OF b])
+  apply (simp only: equivp_symp[OF a])
+  apply (simp only: equivp_symp[OF b])
+  using equivp_transp[OF a] apply blast
+  using equivp_transp[OF b] apply blast
+  done
 
 lemma prod_quotient[quot_thm]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"
-unfolding Quotient_def
-using q1 q2
-apply (simp add: Quotient_abs_rep Quotient_abs_rep Quotient_rel_rep Quotient_rel_rep)
-using Quotient_rel[OF q1] Quotient_rel[OF q2] 
-by blast
+  unfolding Quotient_def
+  using q1 q2
+  apply (simp add: Quotient_abs_rep Quotient_rel_rep)
+  using Quotient_rel[OF q1] Quotient_rel[OF q2]
+  by blast
 
 lemma pair_rsp[quot_respect]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"
-by simp
+  by simp
 
 lemma pair_prs[quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(Rep1 ---> Rep2 ---> (prod_fun Abs1 Abs2)) Pair = Pair"
-apply(simp add: expand_fun_eq)
-apply(simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
-done
+  apply (simp add: expand_fun_eq)
+  apply (simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
+  done
 
 lemma fst_rsp[quot_respect]:
   assumes "Quotient R1 Abs1 Rep1"
@@ -56,32 +56,30 @@
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(prod_fun Rep1 Rep2 ---> Abs1) fst = fst"
-apply(simp add: expand_fun_eq)
-apply(simp add: Quotient_abs_rep[OF q1])
-done
+  apply (simp add: expand_fun_eq)
+  apply (simp add: Quotient_abs_rep[OF q1])
+  done
 
 lemma snd_rsp[quot_respect]:
   assumes "Quotient R1 Abs1 Rep1"
   assumes "Quotient R2 Abs2 Rep2"
   shows "(prod_rel R1 R2 ===> R2) snd snd"
   by simp
-  
+
 lemma snd_prs[quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(prod_fun Rep1 Rep2 ---> Abs2) snd = snd"
-apply(simp add: expand_fun_eq)
-apply(simp add: Quotient_abs_rep[OF q2])
-done
+  apply (simp add: expand_fun_eq)
+  apply (simp add: Quotient_abs_rep[OF q2])
+  done
 
-lemma prod_fun_id[id_simps]: 
-  shows "prod_fun id id \<equiv> id"
-  by (rule eq_reflection) 
-     (simp add: prod_fun_def)
+lemma prod_fun_id[id_simps]:
+  shows "prod_fun id id = id"
+  by (simp add: prod_fun_def)
 
-lemma prod_rel_eq[id_simps]: 
-  shows "prod_rel (op =) (op =) \<equiv> (op =)"
-  apply (rule eq_reflection)
+lemma prod_rel_eq[id_simps]:
+  shows "(prod_rel (op =) (op =)) = (op =)"
   apply (rule ext)+
   apply auto
   done