1.1 --- a/src/HOL/Algebra/abstract/NatSum.ML	Thu Nov 11 10:25:17 1999 +0100
     1.2 +++ b/src/HOL/Algebra/abstract/NatSum.ML	Thu Nov 11 10:25:29 1999 +0100
     1.3 @@ -4,8 +4,6 @@
     1.4      Author: Clemens Ballarin, started 12 December 1996
     1.5  *)
     1.6  
     1.7 -open NatSum;
     1.8 -
     1.9  section "Basic Properties";
    1.10  
    1.11  Goalw [SUM_def] "SUM 0 f = (f 0::'a::ring)";
    1.12 @@ -164,12 +162,12 @@
    1.13  \    SUM n (%i. if i = x then f i else <0>) = (if x <= n then f x else <0>)";
    1.14  by (nat_ind_tac "n" 1);
    1.15  (* Base case *)
    1.16 -by (simp_tac (simpset() setloop (split_tac [expand_if])) 1);
    1.17 +by (Simp_tac 1);
    1.18  (* Induction step *)
    1.19 -by (asm_simp_tac (simpset() setloop (split_tac [expand_if])) 1);
    1.20 +by (Asm_simp_tac 1);
    1.21  by (Clarify_tac 1);
    1.22  by (res_inst_tac [("f", "f")] arg_cong 1);
    1.23 -by (fast_arith_tac 1);
    1.24 +by (arith_tac 1);
    1.25  qed "SUM_if_singleton";
    1.26  
    1.27  Goal