--- a/src/HOL/Old_Number_Theory/Euler.thy Sat Nov 12 21:10:56 2011 +0100
+++ b/src/HOL/Old_Number_Theory/Euler.thy Sun Nov 13 19:26:53 2011 +0100
@@ -85,7 +85,7 @@
apply (auto simp add: MultInvPair_def)
apply (subgoal_tac "~ (StandardRes p j = StandardRes p (a * MultInv p j))")
apply auto
- apply (metis MultInvPair_distinct Pls_def StandardRes_def aux number_of_is_id one_is_num_one)
+ apply (metis MultInvPair_distinct StandardRes_def aux)
done
@@ -227,7 +227,7 @@
lemma Euler_part1: "[| 2 < p; zprime p; ~([x = 0](mod p));
~(QuadRes p x) |] ==>
[x^(nat (((p) - 1) div 2)) = -1](mod p)"
- by (metis Wilson_Russ number_of_is_id zcong_sym zcong_trans zfact_prop)
+ by (metis Wilson_Russ zcong_sym zcong_trans zfact_prop)
text {* \medskip Prove another part of Euler Criterion: *}
@@ -280,13 +280,13 @@
apply (subgoal_tac "p \<in> zOdd")
apply (auto simp add: QuadRes_def)
prefer 2
- apply (metis number_of_is_id numeral_1_eq_1 zprime_zOdd_eq_grt_2)
+ apply (metis zprime_zOdd_eq_grt_2)
apply (frule aux__1, auto)
apply (drule_tac z = "nat ((p - 1) div 2)" in zcong_zpower)
apply (auto simp add: zpower_zpower)
apply (rule zcong_trans)
apply (auto simp add: zcong_sym [of "x ^ nat ((p - 1) div 2)"])
- apply (metis Little_Fermat even_div_2_prop2 mult_Bit0 number_of_is_id odd_minus_one_even one_is_num_one mult_1 aux__2)
+ apply (metis Little_Fermat even_div_2_prop2 odd_minus_one_even mult_1 aux__2)
done